General Relativity and Cosmology with Engineering Applications 9781032001623, 9781003173021

Table of contents :
Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
Part I: The General Theory of Relativity and Some of Its Applications
Chapter 1: The Special Theory of Relativity
1.1 Conflict between Newtonian mechanics and Maxwell’s theory of electromagnetism
1.2 The experiments of Michelson and Morley
1.3 Study Projects
1.4 The notion of proper time, time dilation and length contraction
1.5 The twin paradox
1.6 The equations of mechanics in special relativity
1.7 Mass, velocity, momentum and energy in special relativity, Einstein’s derivation of the energy mass relation E = mc2
1.8 Four vectors and tensors in special relativity and their Lorentz transformation laws
1.9 The general from of the Lorentz group consisting of boosts and rotations
1.10 The Poincare group consisting of Lorentz tranformations with space-time translations
1.11 Irreducible representations of the Poincare group with applications to Wigner’s particle classfication theory
1.12 Lorentz transformations of the electromagnetic field
1.13 Relative velocity in inspecial relativity
1.14 Fluid dynamics in special relativity
1.15 Plasma physics and magnetohydrodynamics in special relativity
1.16 Particle moving in a constant magnetic field in special relativity
Chapter 2: The General Theory of Relativity
2.1 Drawbacks with the special theory of relativity
2.2 The principle of equivalence
2.3 Why gravitational field is not a force ?
2.4 Four vectors and tensors in the general theory of relativity
2.5 Basics of Riemannian geometry
2.6 The energy-momentum tensor of matter in a background curved metric
2.7 Maxwell’s equations in a background curved metric
2.8 The energy-momentum tensor of the electromagnetic field in a background curved metric
2.9 The Einstein field equations of gravitation (i) In the absence of matter and radiation, (ii) In the presence of matter and radiation
2.10 Proof of the consistency of the Einstein field equations with the fluid dynamical equations based on the Bianchi identity for the Einstein tensor
2.11 The weak field limit of Einstein’s field equations is Newton’s inverse square law of gravitation
2.12 The post-Newtonian equations of celestial mechanics, gravitation and hydrodynamics
Chapter 3: Engineering Applications of General Relativity
3.1 Applications of general relativity to global positioning systems
3.2 General relativistic corrections to the Klein-Gordon wave propagation
3.3 Calculating the effect of general relativity on the motion of a plasma with applications to estimation of the metric from the radiation field produced by the plasma in motion
3.4 Problems with hints
3.5 Quantum theory of fields
3.6 Energy-momentum tensor of matter with viscous and thermal corrections
3.7 Energy-momentum tensor of the electromagnetic field in a background curved space-time
3.8 Relativistic Fermi fluid in a gravitational field
3.9 The post-Newtonian approximation
3.10 Energy-Momentum tensor of matter with viscous and thermal corrections
3.11 Energy-momentum tensor of the electromagnetic field in a background curved spacetime
3.12 Relativistic Fermi fluid in a gravitational field. The Dirac equation in a gravitational field has the form
3.13 The post-Newtonian approximation
3.14 The BCS theory of superconductivity
3.15 Quantum scattering theory in the presence of a gravitational field
3.16 Maxwell’s equations in the Schwarzchild space-time
3.17 Some more problems in general relativity
3.18 Neural networks for learning the expansion of our universe
3.19 Quantum stochastic differential equations in general relativity
Chapter 4: Some Basic Problems in Electromagnetics Related to General Relativity (gtr)
4.1 Em waves and quantum communication
4.2 Cavity resonator antennas with current source in a gravitational field
4.3 Cq coding theorem
4.4 Restricted quantum gravity in one spatial dimension and one time dimension
4.5 Quantum theory of fields
4.6 Energy-momentum tensor of matter with viscous and thermal corrections
4.7 Energy-momentum tensor of the electromagnetic field in a background curved spacetime
4.8 Relativistic Fermi fluid in a gravitational field
4.9 The post-Newtonian approximation
4.10 The BCS theory of superconductivity
4.11 Quantum scattering theory in the presence of a gravitational field
4.12 Maxwell’s equations in the Schwarzchild spacetime
4.13 Some more problems in general relativity
Chapter 5: Basic Problems in Algebra, Geometry and Differential Equations
5.1 Algebra, Triangle geometry, Integration and basic probability
5.2 Mechanics
5.3 Brownian motion simulation
5.4 Geometric series
5.5 Surface area
5.6 Hamiltonian mechanics from Lagrangians
5.7 Rate of a chemical reaction
5.8 Linearization of the Navier-Stokes Fluid equations with gravitational self interaction
5.9 Wave equations in mechanics
5.10 Surface of revolution
5.11 1-D Schrodinger equation
5.12 Lagrange’s triangle in mechanics
5.13 Number theory
5.14 Blurring of 3-D objects in random motion
5.15 Commutators of products of matrices
5.16 Path of a light ray in an medium having inhomogeneous refractive index
5.17 Re-ection matrices
5.18 Rotation matrices
5.19 Jacobian formula for multiple integrals
5.20 Existence of only five regular polyhedra in nature
5.21 Definition of the derivative and its properties
5.22 Pattern recognition using group representations
5.23 Using characters of group representations to estimate the group transformation element
5.24 Explicit formulas for the induced representation for semidirect products of finite groups
5.25 Applications of the Extended Kalman filter and the Recursive Least Squares Algorithm to System Identification Problems using Neural Networks
5.26 Application of neural networks to the gravitational metric estimation problem
5.27 Problems in quantum scattering theory
5.28 Compact operators
5.29 Estimating the metric parameters from geodesic measurements
5.30 Perturbations to the band structure of semiconductors
5.31 Scattering into cones for Schrodinger Hamiltonians
5.32 Study projects involving conventional field theory in curved background metrics
5.33 Intuitive explanation of an invariance principle in scattering theory
5.34 Scattering theory for the Dirac Hamiltonian in curved space-time
5.35 Derivation of the approximate Schrodinger Hamiltonian for a particle in curved spacetime with corrections upto fourth order in the space derivatives
5.36 Quantum scattering theory in the presence of time dependent Hamiltonians arising in general relativity
5.37 Band structure of a semiconductor altered by a massive gravitational field
5.38 Design of quantum gates using quantum physical systems in a gravitational field
5.39 Quantum phase estimation
5.40 Noisy Schrodinger equations, pure and mixed states
5.41 Constructions using ruler and compass
5.42 Application of the Jordan canonical form for matrices in general relativity
5.43 Application of the Jordan canonical form in solving fluid dynamical equations when the velocity field is a small perturbation of a constant velocity field
5.44 The Jordan canonical form
5.45 Some topics in scattering theory in L2(Rn)
5.46 MATLAB problems on applications of linear algebra to signal processing
5.47 Applications of the RLS lattice algorithms to general relativity
5.48 Knill-Laflamme theorem on quantum coding theory, a different proof
5.49 Ashtekar’s quantization of gravity
5.50 Example of an error correcting quantum code from quantum mechanics
5.51 An application of the Jordan canonical form to noisy quantum theory
5.52 An algorithm for computing the Jordan canonical form
5.53 Rotating blackhole analysis using the tetrad formalism
5.54 Maxwell’s equations in the rotating blackhole metric
5.55 Some notions on operators in an infinite/finite dimensional Hilbert space
5.56 Some versions of the quantum Boltzmann equation
Part II: Quantum Mechanics
1 The De-Broglie Duality of particle and wave properties of matter
2 Bohr’s correspondence principle
3 Bohr-Sommerfeld’s quantization rules
4 The principle of superposition of wave functions and its application to the Young double slit diffraction experiment
5 Schrodinger’s wave mechanics and Heisenberg’s matrix mechanics
6 Dirac’s replacement of the Poisson bracket by the quantum Lie bracket
7 Duality between the Schrodinger and Heisenberg mechanics based on Dirac’s idea
8 Quantum dynamics in Dirac’s interaction picture
9 The Pauli equation: Incorporating spin in the Schrodinger wave equation in the presence of a magnetic field
10 The Zeeman effect
11a The spectrum of the Hydrogen atom
11b The spectrum of particle in a 3 − D box
11c The spectrum of a quantum harmonic oscillator
12 Time independent perturbation theory
13 Time dependent perturbation theory
14 The full Dyson series for the evolution operator of a quantum system in the presence of a time varying potential
15 The transition probabilities in the presence of a stochastically time varying potential
16 Basics of quantum gates and their realization using perturbed quantum systems
17 Bounded and unbounded linear operators in a Hilbert space
18 The spectral theorem for compact normal and bounded and unbounded self-adjoint operators in a Hilbert space
19 The general theory of Events, states and observables in the quantum theory
20 The evolution of the density operator in the absence of noise
21 The Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation for noisy quantum systems
22 Distinguishable and indistinguishable particles
23 The relationship between spin and statistics
24(a) Tensor products of Hilbert spaces
24(b) Symmetric and antisymmetric tensor products of Hilbert spaces, the Fock spaces
24(c) Coherent/exponential vectors in the Fock spaces
25 Creation, Conservation and Annihilation Operators in the Boson Fock Space
26 The general theory of quantum stochastic processes in the sense of Hudson and Parthasarathy
27 The quantum Ito formula of Hudson and Parthasarathy
28 The general theory of quantum stochastic differential equations
29 The Hudson-Parthasarathy noisy Schrodinger equation and the derivation of the GKSL equation from its partial trace
30 The Feynman path integral for solving the Schrodinger equation
31 Comparison between the Hamiltonian (Schrodinger-Heisenberg) and Lagrangian (path integral) approaches to quantum mechanics
32 The quantum theory of fields
33 Dirac’s wave equation in a gravitational field
34 Canonical quantization of the gravitational field
35 The scattering matrix for the interaction between photons, electrons, positrons and gravitons
36 Atom interacting with a Laser
37 The classical and quantum Boltzmann equations
38 Bands in a semiconductor
39 The Hartree-Fock apporoximate method for computing the wave functions of a many electron atom
40 The Born-Oppenheimer approximate method for computing the wave functions of electrons and nuclei in a lattice
41 The performance of quantum gates in the presence of classical and quantum noise
42 Design of quantum gates by applying a time varying electromagnetic field on atoms and oscillators
43 Solution of Dirac’s equation in the Coulomb potential
44 Dirac’s equation in general radial potentials
45 The Schrodinger equation in an electromangetic field described as a quantum stochastic process
46 Dirac’s equation in an electromagnetic field described as a quantum stochastic process
47 General Scattering theory, the Moller and wave operators, the scattering matrix, the Lippman-Schwinger equation for the scattering matrix, Born scattering
48 Design of quantum gates using time dependent scattering theory
49 Evans-Hudson flows and its application to the quantization of the fluid dynamical equations in noise
50 Classical non-linear filtering
51 Derivation of the extended Kalman filter (EKF) as an approximation to the Kushner filter
52 Belavkin’s theory of non-demolition measurements and quantum filtering in coherent states based on the Hudson- Parthasarathy Boson Fock space theory of quantum noise, The quantum Kallianpur-Striebel formula
53 Classical control of a stochastic dynamical system by error feedback based on a state observer derived from the EKF
54 Quantum control using error feedback based on Belavkin quantum filters for the quantum state observer
55 Lyapunov’s stability theory with application to classical and quantum dynamical systems
56 Imprimitivity systems as a description of covariant observables under a group action
57 Schwinger’s analysis of the interaction between the electron and a quantum electromagnetic field
58 Quantum Control
59 Quantum error correcting codes
60 Quantum hypothesis testing
61 The Sudarshan-Lindblad equation for observables in an open quantum system
62 The Yang-Mills field and its quantization using path integrals
63 A general remark on path integral computations for gauge invariant actions
64 Calculation of the normalized spherical harmonics
65 Volterra systems in quantum mechanics
66a RLS lattice algorithms for quantum observable estimation
66b Quantum scattering theory, the wave operators and the scattering matrix
67 Quantum systems driven by Stroock-Varadhan martingales
Appendix
References
Index

Citation preview

General Relativity and Cosmology with Engineering Applications

General Relativity and Cosmology with Engineering Applications

Harish Parthasarathy Professor Electronics & Communication Engineering Netaji Subhas Institute of Technology (NSIT) New Delhi, Delhi-110078

First published 2021 by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN and by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 © 2021, Manakin Press Pvt. Ltd. CRC Press is an imprint of Informa UK Limited The right of Harish Parthasarathy to be identified as author of this work has been asserted by him in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. For permission to photocopy or use material electronically from this work, access www. copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Print edition not for sale in South Asia (India, Sri Lanka, Nepal, Bangladesh, Pakistan or Bhutan). British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record has been requested ISBN: 978-1-032-00162-3(hbk) ISBN: 978-1-003-17302-1(ebk)

Table of Contents

Part I:

The General Theory of Relativity and Some of Its Applications

Chapter 1: The Special Theory of Relativity 1-18

1.1 Conflict between Newtonian mechanics and Maxwell’s

theory of electromagnetism 1

1.2 The experiments of Michelson and Morley 3

1.3 Study Projects 4

1.4 The notion of proper time, time dilation and length contraction 4

1.5 The twin paradox 5

1.6 The equations of mechanics in special relativity 6

1.7 Mass, velocity, momentum and energy in special relativity,

Einstein’s derivation of the energy mass relation E = mc2 7

1.8 Four vectors and tensors in special relativity and their

Lorentz transformation laws 8

1.9 The general from of the Lorentz group consisting of boosts

and rotations 10

1.10 The Poincare group consisting of Lorentz tranformations

with space-time translations 11

1.11 Irreducible representations of the Poincare group

with applications to Wigner’s particle classfication theory 12

1.12 Lorentz transformations of the electromagnetic field 13

1.13 Relative velocity in inspecial relativity 15

1.14 Fluid dynamics in special relativity 16

1.15 Plasma physics and magnetohydrodynamics

in special relativity 16

1.16 Particle moving in a constant magnetic field in special relativity 17

Chapter 2: 2.1 2.2 2.3 2.4 2.5 2.6

The General Theory of Relativity Drawbacks with the special theory of relativity The principle of equivalence Why gravitational field is not a force ? Four vectors and tensors in the general theory of relativity Basics of Riemannian geometry The energy-momentum tensor of matter in a background

curved metric 2.7 Maxwell’s equations in a background curved metric

19-44

19

19

20

21

22

37

38

VI

General Relativity and Cosmology with Engineering Applications

2.8 The energy-momentum tensor of the electromagnetic

field in a background curved metric 2.9 The Einstein field equations of gravitation (i) In the absence of matter and radiation, (ii) In the presence of matter and radiation 2.10 Proof of the consistency of the Einstein field equations

with the fluid dynamical equations based on the Bianchi

identity for the Einstein tensor 2.11 The weak field limit of Einstein’s field equations is Newton’s

inverse square law of gravitation 2.12 The post-Newtonian equations of celestial mechanics,

gravitation and hydrodynamics

39

40

41

42

42

Chapter 3: Engineering Applications of General Relativity 45-106

3.1 Applications of general relativity to global positioning systems 45

3.2 General relativistic corrections to the Klein-Gordon

wave propagation 48

3.3 Calculating the effect of general relativity on the motion of

a plasma with applications to estimation of the metric from

the radiation field produced by the plasma in motion 49

3.4 Problems with hints 50

3.5 Quantum theory of fields 51

3.6 Energy-momentum tensor of matter with viscous and

thermal corrections 66

3.7 Energy-momentum tensor of the electromagnetic field

in a background curved space-time 69

3.8 Relativistic Fermi fluid in a gravitational field 70

3.9 The post-Newtonian approximation 71

3.10 Energy-Momentum tensor of matter with viscous and

thermal corrections 75

3.11 Energy-momentum tensor of the electromagnetic field

in a background curved spacetime 79

3.12 Relativistic Fermi fluid in a gravitational field. The Dirac

equation in a gravitational field has the form 80

3.13 The post-Newtonian approximation 81

3.14 The BCS theory of superconductivity 85

3.15 Quantum scattering theory in the presence of

a gravitational field 87

3.16 Maxwell’s equations in the Schwarzchild space-time 89

3.17 Some more problems in general relativity 91

General Relativity and Cosmology with Engineering Applications

3.18 3.19

VII

Neural networks for learning the expansion of our universe 101

Quantum stochastic differential equations in general relativity 102

Chapter 4: Some Basic Problems in Electromagnetics Related to General Relativity (gtr) 107-164

4.1 Em waves and quantum communication 107

4.2 Cavity resonator antennas with current source in

a gravitational field 108

4.3 Cq coding theorem 110

4.4 Restricted quantum gravity in one spatial dimension and

one time dimension 112

4.5 Quantum theory of fields 113

4.6 Energy-momentum tensor of matter with viscous and

thermal corrections 126

4.7 Energy-momentum tensor of the electromagnetic field

in a background curved spacetime 129

4.8 Relativistic Fermi fluid in a gravitational field 130

4.9 The post-Newtonian approximation 130

4.10 The BCS theory of superconductivity 135

4.11 Quantum scattering theory in the presence of

a gravitational field 137

4.12 Maxwell’s equations in the Schwarzchild spacetime 138

4.13 Some more problems in general relativity 141

Chapter 5: Basic Problems in Algebra, Geometry and

Differential Equations 165-252

5.1 Algebra, Triangle geometry, Integration and basic probability 165

5.2 Mechanics 169

5.3 Brownian motion simulation 169

5.4 Geometric series 170

5.5 Surface area 170

5.6 Hamiltonian mechanics from Lagrangians 170

5.7 Rate of a chemical reaction 171

5.8 Linearization of the Navier-Stokes Fluid equations

with gravitational self interaction 171

5.9 Wave equations in mechanics 172

5.10 Surface of revolution 172

5.11 1-D Schrodinger equation 172

5.12 Lagrange’s triangle in mechanics 173

5.13 Number theory 174

General Relativity and Cosmology with Engineering Applications

VIII

5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25

5.26 5.27 5.28 5.29 5.30 5.31 5.32 5.33 5.34 5.35

5.36 5.37 5.38

Blurring of 3-D objects in random motion Commutators of products of matrices Path of a light ray in an medium having inhomogeneous

refractive index Re-ection matrices Rotation matrices Jacobian formula for multiple integrals Existence of only five regular polyhedra in nature Definition of the derivative and its properties Pattern recognition using group representations Using characters of group representations to estimate

the group transformation element Explicit formulas for the induced representation for

semidirect products of finite groups Applications of the Extended Kalman filter and the Recursive

Least Squares Algorithm to System Identification Problems

using Neural Networks Application of neural networks to the gravitational

metric estimation problem Problems in quantum scattering theory Compact operators Estimating the metric parameters from geodesic

measurements Perturbations to the band structure of semiconductors Scattering into cones for Schrodinger Hamiltonians Study projects involving conventional field theory in curved

background metrics Intuitive explanation of an invariance principle

in scattering theory Scattering theory for the Dirac Hamiltonian in

curved space-time Derivation of the approximate Schrodinger Hamiltonian for

a particle in curved spacetime with corrections upto fourth

order in the space derivatives Quantum scattering theory in the presence of time dependent

Hamiltonians arising in general relativity Band structure of a semiconductor altered by a massive

gravitational field Design of quantum gates using quantum physical

systems in a gravitational field

175

176

176

176

177

178

179

180

181

187

188

189

215

216

217

217

218

218

219

224

225

226

226

228

229

General Relativity and Cosmology with Engineering Applications

5.39 5.40 5.41 5.42 5.43

5.44 5.45 5.46 5.47 5.48 5.49 5.50 5.51 5.52 5.53 5.54 5.55 5.56

Quantum phase estimation Noisy Schrodinger equations, pure and mixed states Constructions using ruler and compass Application of the Jordan canonical form for matrices

in general relativity Application of the Jordan canonical form in solving fluid

dynamical equations when the velocity field is a small

perturbation of a constant velocity field The Jordan canonical form Some topics in scattering theory in L2(Rn) MATLAB problems on applications of linear algebra

to signal processing Applications of the RLS lattice algorithms to general relativity Knill-Laflamme theorem on quantum coding theory,

a different proof Ashtekar’s quantization of gravity Example of an error correcting quantum code from

quantum mechanics An application of the Jordan canonical form to noisy

quantum theory An algorithm for computing the Jordan canonical form Rotating blackhole analysis using the tetrad formalism Maxwell’s equations in the rotating blackhole metric Some notions on operators in an infinite/finite dimensional

Hilbert space Some versions of the quantum Boltzmann equation

IX

230

231

232

232

233

233

234

236

238

239

241

245

246

246

247

247

248

250

Part II: Quantum Mechanics 1 2 3 4 5 6

The De-Broglie Duality of particle and wave properties of matter Bohr’s correspondence principle Bohr-Sommerfeld’s quantization rules The principle of superposition of wave functions and its application to the Young double slit diffraction experiment Schrodinger’s wave mechanics and Heisenberg’s matrix mechanics Dirac’s replacement of the Poisson bracket by the quantum Lie bracket

273

273

274

275

276

278

General Relativity and Cosmology with Engineering Applications

X

7

Duality between the Schrodinger and Heisenberg

mechanics based on Dirac’s idea 278

8 Quantum dynamics in Dirac’s interaction picture 280

9 The Pauli equation: Incorporating spin in the Schrodinger

wave equation in the presence of a magnetic field 281

10 The Zeeman effect 281

11a The spectrum of the Hydrogen atom 282

11b The spectrum of particle in a 3 − D box 284

11c The spectrum of a quantum harmonic oscillator 285

12 Time independent perturbation theory 285

13 Time dependent perturbation theory 287

14 The full Dyson series for the evolution operator of

a quantum system in the presence of a time varying potential 288

15 The transition probabilities in the presence of a stochastically

time varying potential 288

16 Basics of quantum gates and their realization using

perturbed quantum systems 289

17 Bounded and unbounded linear operators in a Hilbert space 290

18 The spectral theorem for compact normal and bounded

and unbounded self-adjoint operators in a Hilbert space 291

19 The general theory of Events, states and observables in

the quantum theory 293

20 The evolution of the density operator in the absence of noise 296

21 The Gorini-Kossakowski-Sudarshan-Lindblad (GKSL)

equation for noisy quantum systems 296

22 Distinguishable and indistinguishable particles 296

23 The relationship between spin and statistics 296

24(a) Tensor products of Hilbert spaces 296

24(b) Symmetric and antisymmetric tensor products of

Hilbert spaces, the Fock spaces 297

24(c) Coherent/exponential vectors in the Fock spaces 299

25 Creation, Conservation and Annihilation Operators

in the Boson Fock Space 300

26 The general theory of quantum stochastic processes

in the sense of Hudson and Parthasarathy 300

27 The quantum Ito formula of Hudson and Parthasarathy 301

28 The general theory of quantum stochastic differential equations 301

29 The Hudson-Parthasarathy noisy Schrodinger equation

and the derivation of the GKSL equation from its partial trace 301

General Relativity and Cosmology with Engineering Applications

30 31

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

48 49 50 51

The Feynman path integral for solving the Schrodinger equation Comparison between the Hamiltonian

(Schrodinger-Heisenberg) and Lagrangian (path integral) approaches to quantum mechanics The quantum theory of fields Dirac’s wave equation in a gravitational field Canonical quantization of the gravitational field The scattering matrix for the interaction between photons,

electrons, positrons and gravitons Atom interacting with a Laser The classical and quantum Boltzmann equations Bands in a semiconductor The Hartree-Fock apporoximate method for computing

the wave functions of a many electron atom The Born-Oppenheimer approximate method for computing

the wave functions of electrons and nuclei in a lattice The performance of quantum gates in the presence of

classical and quantum noise Design of quantum gates by applying a time varying

electromagnetic field on atoms and oscillators Solution of Dirac’s equation in the Coulomb potential Dirac’s equation in general radial potentials The Schrodinger equation in an electromangetic

field described as a quantum stochastic process Dirac’s equation in an electromagnetic field described

as a quantum stochastic process General Scattering theory, the Moller and wave operators,

the scattering matrix, the Lippman-Schwinger equation for the scattering matrix, Born scattering Design of quantum gates using time dependent

scattering theory Evans-Hudson flows and its application to the quantization

of the fluid dynamical equations in noise Classical non-linear filtering Derivation of the extended Kalman filter (EKF)

as an approximation to the Kushner filter

XI

301

302

303

327

327

329

329

333

335

337

338

340

341

342

342

348

348

350

352

353

355

356

General Relativity and Cosmology with Engineering Applications

XII

52

53

54 55 56 57 58 59 60 61 62 63 64 65 66a 66b 67

Belavkin’s theory of non-demolition measurements

and quantum filtering in coherent states based on

the Hudson- Parthasarathy Boson Fock space theory of

quantum noise, The quantum Kallianpur-Striebel formula 357

Classical control of a stochastic dynamical system

by error feedback based on a state observer derived

from the EKF 361

Quantum control using error feedback based on

Belavkin quantum filters for the quantum state observer 361

Lyapunov’s stability theory with application to classical

and quantum dynamical systems 363

Imprimitivity systems as a description of covariant observables

under a group action 363

Schwinger’s analysis of the interaction between the electron

and a quantum electromagnetic field 365

Quantum Control 366

Quantum error correcting codes 367

Quantum hypothesis testing 375

The Sudarshan-Lindblad equation for observables in an open

quantum system 378

The Yang-Mills field and its quantization using path integrals 379

A general remark on path integral computations for gauge

invariant actions 381

Calculation of the normalized spherical harmonics 383

Volterra systems in quantum mechanics 386

RLS lattice algorithms for quantum observable estimation 391

Quantum scattering theory, the wave operators and

the scattering matrix 392

Quantum systems driven by Stroock-Varadhan martingales 395

Appendix

397-653

References

654-655

Index

656-668

Preface This book introduces the reader to the important and beautiful subject of General Relativity as founded by Albert Einstein in 1915. General relativity and quantum mechanics are the foundations on which modern physics is based and hence a large section on quantum mechanics has also been included here. The hope is that the reader, after reading this book will be in a position to understand the subject of quantum gravity which as it stands today is incom­ plete owing to renormalization problems for the gravitational field. The book is broadly divided into two parts. In the first part of the book, we first discuss spe­ cial relativity, starting with many attempts made to explain the results of the Michelson-Morley experiment culminating with the derivation of the Lorentz transformation of Einstein that treats space and time on the same footing and based on the postulate of the constancy of the speed of light in all inertial frames. Ernst Mach’s definition of an absolute inertial frame as one that its far separated from the non-rotating galaxies in the universe and non-accelerating with respect to them is introduced. Any other frame that moves with a con­ stant velocity relative to this absolute inertial frame is termed as an inertial frame. That the constancy of the speed of light is in perfect agreement with the Maxwell theory of electromagnetism and not in agreement with Galilean rela­ tivity on which Newtonian mechanics is based is explained in detail which led Einstein to conclude that Maxwell’s theory is right and accurate as opposed to Newtonian mechanics which is inaccurate according to the principles of special relativity. The main features of Galilean relativity are that the speed of light is not a constant and is subject to the usual law of velocity addition and further that time is absolute. The main features of the special relativity are that the speed of light in vacuum is a constant and that time is not absolute. This causes the law of velocity addition to get modified nonlinearly in such a way that if one of the velocities is the velocity of light and the other arbitrary, then the resultant velocity will be the velocity of light in vacuum. The second postulate of special relativity is introduced according to which the laws of physics are the same in all mutually inertial frames and hence the correct laws of physics should be ex­ pressed as four vector or more generally tensor equations where each four vector transforms according to a Lorentz transformation and each tensor transforms according to a tensor product of Lorentz transformations. This led Einstein to formulate the correct equations of special relativistic mechanics as a modifica­ tion of Newtonian mechanics, finally leading to the famous Einstein mass energy relation E = mc2 . Various consequences of special relativistic mechanics like time dilation, length contraction and the twin paradox are explained. Finally, using the Lorentz transformation for tensors, the transformation rules for the electric and magnetic fields between two inertial frames are derived by regard­ ing the scalar and vector potential of electromagnetism as the components of a four vector resulting in the electromagnetic field being a 4 × 4 antisymmetric tensor field and the charge and current densities as a four vector field. All this is in perfect agreement with the Maxwell theory which results in wave equations for the four potential with sources being the four current density. It should be noted that the wave operator is Lorentz invariant in contrast to the Laplacian. XIII

XIV

General Relativity and Cosmology with Engineering Applications In the first part, we then introduce Einstein’s general theory of relativity. Why the gravitational field should be regarded as a curvature of the space-time manifold rather than as a force is clarified. This is based on Einstein’s principle of equivalence which states that any gravitational field can be locally cancelled out by choosing our frame to be accelerating over an infinitesimal region of space-time. This leads us to understand that a gravitational field is simply a set of space-time dependent coefficients of a metric. Particles follow geodesics (ie trajectories of shortest Riemannian distance between two fixed points in spacetime) with respect to such a curved space-time metric and these geodesics are therefore curved paths on the space-time manifold which appear to us as accel­ erated motion. Einstein’s general principle of equivalence which states that the laws of physics should remain invariant with respect to all observers in the uni­ verse ie with respect to arbitrary diffeomorphisms of space-time and not merely with respect to Lorentz transformations which transform between two mutually inertial systems. The Einstein field equations for a weak gravitational field re­ duce to Newton’s inverse square law of gravitation provided that we interpret the metric tensor in terms of the Newtonian gravitational potential as in the geodesic equation of motion of a single particle. It should be noted here that the inertial mass of the particle does not appear here, but the gravitational mass of particles that generate the gravitational field implicitly appear in the metric. Einstein’s general relativity also asserts the equivalence of inertial and gravitational mass as in Newton’s theory. This means that if M is the mass of a body to which a force F is applied, then its acceleration will be a = F/M ’ ie inversely proportional to M while if g is the gravitational field produced by the same body, then g will also be proportional to M . Today this is a subject of hot debate and some experimental physicists say that the gravitational and inertial masses of a body are not proportional because the force produced by M also includes a fifth force which is not proportional to M and depends on the quan­ tum mechanical nature of elementary particles constituting M . Plasma physics in a gravitational field taking into account the tensor conductivity is formulated as a tensor equation. How viscous and thermal effects contribute to the energymomentum tensor of a fluid whose vanishing covariant divergence leads tot he correct tensor fluid dynamical equations is mentioned. Perturbation theoretic tools for approximately solving the equations of motion of a particle and fluid in general relativity as first developed by S.Chandrasekhar in his famous papers on post-Newtonian hydrodynamics is discussed. The principle of equivalence leads us to the concept of a tensor equation, ie, the laws of physics should be expressible as tensor equations where the trans­ ¯ is in terms formation law of a tensor with respect to a diffeomorphism X → X of the tensor product of the corresponding Jacobian matrix—specifically con­ travariant indices of the tensor are transformed using the Jacobian matrix while covariant indices are transformed using the inverse Jacobian matrix. The Ja­ cobian matrix of a diffeomorphism in general relativity replaces the Lorentz transformation used in special relativity. For a given metric, we can write down the geodesic equation ie the curve of shortest Riemannian length joining two points on the Riemannian manifold of curved space-time. In the weak field limit,

General Relativity and Cosmology with Engineering Applications ie, when the metric is a small perturbation of the Minkowski metric, geodesics reduce to Newton’s law of motion in a gravitational potential U : d2 r(t) ∂U (t, r) =− dt2 ∂r provided that we make an appropriate identification of the metric with the Newtonian potential. Next, Newton’s inverse square law of gravitation is the same as Poisson’s equation ∇2 U (t, r) = 4πGρ(t, r) This equation represents action at a distance and further, it is not a tensor equation and hence by the principle of equivalence, it does not represent a correct law of physics. We may think of modifying it to the Lorentz invariant wave equation ∂2 (∇2 − c−2 2 )U (t, r) = 4πGρ(t, r) ∂t with U and ρ regarded as scalar fields or as the time components of four vector or tensor fields but although this overcomes the problem of action at a distance, it is not diffeomorphism invariant. Einstein proposed the replacement of Newton’s law of gravitation by the tensor equation 1 Rμν − Rg μν = −8πGT μν 2 where Rμν is the Ricci tensor obtained by contracting two indices of the Rie­ mann curvature tensor Rμνρσ and T μν is the energy-momentum tensor of matter plus electromagnetic radiation. This curvature tensor is defined by the discrep­ ancy involved in parallely translating a vector around a small closed loop on the curved Riemannian manifold of space-time. Parallel transport of a vector on a curved surface is an important concept, perhaps the most fundamental concept in Riemannian geometry because if two vectors, tangent to a curved surface at two neighbouring points are subtracted, the resultant vector will not generally be tangent to the surface. So if our universe is a curved surface, then vector fields tangential to this surface are the only valid vector fields in our universe and their partial derivatives are no longer tensors which live in our universe. The way our of this difficulty is to replace partial derivatives by covariant deriva­ tives which involve subtracting from the vector Aμ (x + dx) at x + dx, the vector Aμ (x) + δAμ (x) which is obtained from Aμ (x) after parallely translating it to x + dx, ie, translating it in the usual Euclidean sense followed by projecting it onto the tangent plane at x + dx. The connection components of the covariant derivative of a vector field can be expressed as a linear combination of first or­ der partial derivatives of the metric tensor, ie, the Christoffel symbols. All the known physical laws like Maxwell’s laws of electromagnetism, Dirac’s relativis­ tic wave equation for the electron, Navier-Stokes fluid dynamical equations, the Klein-Gordon equation with Higgs potentials, the Yang-Mills non-commutative gauge field and matter field equations involve partial derivatives with respect

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General Relativity and Cosmology with Engineering Applications to space-time coordinates. In these equations, when a background gravitational field is present, the field should be replaced by scalar fields, four vector fields or tensor fields and partial derivatives by covariant derivatives in order to get tensor equations valid in all reference frames, ie, diffeomophism invariant field equations. Since covariant derivatives involve partial derivatives of the metric tensor which is related to the ambient gravitational field, it follows that the gravitational field interacts with all the fields of physics like the electromagnetic field, the fluid dynamical velocity field, the Dirac electron-positron field and the Yang-Mills field describing electro-weak and strong nuclear fields. These aspects are discussed in the book. Some material on general connections on a differentiable Riemannian manfiold is presented. These involve notions like torsion and curvature of a connection expressed in coordinate free language. We derive Cartan’s basic equations of structure that express the torsion and curvature tensor in the language of differential forms with the forms being the connection forms. In the special case when the connection is derived from a metric, ie the covariant derivative of the metric is zero and the torsion is also zero, we show that the connection reduces to the standard metrical connection. The notion of a general connection gives us another method to define a geodesic as a natural generalization of a straight line on a flat surface, namely, by the rule that the covariant derivative of the tangent vector to the curve along the curve is zero or equivalently by the rule that if the tangent vector to the curve is parallely displaced along the curve, then we would get the tangent vector to curve at the displaced point. When the connection is the metrical connection, then this definition of the geodesic coincides with the standard minimal length one. The Cartan structure equations enable us to formulate the Einstein field equations in arbitrary coordinate systems nicely, for example, Chandrasekhar uses this method to obtain the components of the curvature tensor in terms of local vector field components (ie tetrads ) not necessarily coordinate compo­ nents. The computations are then simpler and one can quickly arrive at the Schwarzchild spherical blackhole solution, the Kerr rotating blackhole solution, the Reissner-Nordstorm solution involving the presence of a charge at the centre of the spherical distribution of matter and the Kerr-Newman solution, ie charge at the centre of a rotating blackhole. It should be noted that the curvature tensor of a Riemanian manifold can also be neatly described by the difference in taking two successive covariant derivatives of a vector field along two different coordinate directions with the order of the covariant derivatives interchanged. This property enables us to prove the Bianchi identities for the curvature ten­ sor very easily and one of these identities is that the covariant divergence of the Einstein tensor Rμν − 12 Rg μν vanishes thus causing the energy-momentum tensor of the matter plus radiation to vanish which leads at once to the general relativistic fluid dynamical equations for a plasma. The combined equations involving the Einstein field equations in the presence of matter and radiation, the Maxwell equations and the fluid dynamical equations can all be derived from an action principle with the total action being the sum of the gravitational action being the curvature scalar times the invariant four dimensional volume element, the matter action being simply the integral of the matter rest density

General Relativity and Cosmology with Engineering Applications taking the four dimensional invariant volume element √ into consideration and the standard action for the electromagnetic field Fμν F μν −g. Any other field like the Dirac field will also have a diffeomorphic invariant action and that can be added to this action to derive the overall set of field equations. It should also be noted that if the matter consists of discrete point particles, then the sum of their invariant proper times multiplied by their respective masses must be taken as the matter action. Further, it should be mentioned that the curvature tensor of space time in the neighbourhood of a point vanishes iff the space time there is locally flat, ie, it can be transformed into the Minkowskian flat space-time metric in that neighbourhood of that point by a local diffeomorphism. This fundamental theorem is due to Riemann. Einstein’s law of gravitation states that if there is no matter at all in the universe, then the Riemann curvature tensor vanishes and space-time is flat. However, if matter is present then it pro­ duces a gravitational field in its vicinity and hence even if there is no matter in this region, the curvature tensor would not vanish, only its Ricci tensor would vanish. Thus, in the presence of matter/radiation, in a region A, the Ricci tensor would vanish in its complement Ac but the curvature tensor would not vanish, in other words, matter and radiation generate curved space-time man­ ifolds that cannot be transformed away to flat space-time manifolds. Genuine flat space-time manifolds, ie having Minkowski metric in some system of coordi­ nates are characterized by the vanishing of the curvature tensor. We may have a flat space-time but appears curved because the coordinate system is not chosen properly. For example, if the gravitational field g is a constant everywhere, then we can remove it completely by passing over to a freely falling elevator. Thus, Einstein’s theory beautifully relates matter to the geometry of our universe. In this book, a brief description of Boltzmann’s kinetic transport equation for the particle distribution function of a plasma of ions in position-velocity space is given when the background space-time is curved, ie, there is a background gravitational field. Waveguide and cavity resonator electromagnetic fields in a background gravitational field are also described. The precise definition of the energy- momentum tensor of the plasma for particle distribution functions fa (t, r, v), a = 1, 2, ..., N of N species of particles is not easy and some attention has been devoted to this problem. At this stage, it should be remarked that the Einstein field equations in the presence of radiation are a set of ten partial differential equations and the Maxwell field equations are a set of four partial differential equations. The fluid dynamical equations can be derived from the Einstein field equations by taking the covariant divergence and using the fact that divergence of the Einstein tensor vanishes, this leads to the vanishing of the energy-momentum tensor of matter plus radiation, which are precisely the plasma fluid equations for the velocity field and density of the fluid assuming an equation of state that relates the pressure to the density. Since we can choose our coordinate system arbitrarily, there are four degrees of freedom involved in specifying our metric, ie, we can impose four gauge conditions on the metric. Thus in all, we have ten Einstein field equations, four Maxwell equations and four coordinate conditions on our metric, totally in all, eighteen equations for eighteen functions, namely ten metric components, three velocity components,

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General Relativity and Cosmology with Engineering Applications

one density function and four electromagnetic potential components. This ar­ gument proves the consistency of general relativity. Some engineering applications are discussed towards the end of the first part of the book. These involve GPS corrections based on bending of light rays in a gravitational field, the influence of gravitational waves on the Maxwell photon field coming from a laser as demonstrated recently in the Louisiana experiment involving laser detection of gravitational waves coming from the collision of blackholes several billion years ago and estimating the metric of the gravita­ tional field from measurements on particle motion and electromagnetic wave fields. Electromagnetic wave propagation through a plasma with gravitational effects described by tensor equations is important in communication through the ionosphere. We give a brief description of this problem too. Part II of the book discusses some other mathematical aspects of general relativity espe­ cially related to quantum communication and electromagnetics in a background gravitational field. The focus in this part is on waveguide fields in a back­ ground gravitational field, Application of the Hudson-Parthasarathy quantum stochastic calculus to problems involving interaction of the gravitational field with a noisy photon bath field caused by the interaction Lagrangian between the Maxwell field and the gravitational field with the Maxwell four potential modeled as a superposition of the distributional time derivatives of the cre­ ation, annihilation and conservation processes. We also discuss here the BCS theory of superconductivity in a background gravitational field, ie, how a second quantized Fermionic liquid interacts with a gravitational field etc. Some of the omissions in this book which are standard material found in textbooks on the gtr are derivation of the Schwarzchild solution to the Einstein field equations, study of particle and photon orbits in the Schwarzchild metric, bending of light rays in the Schwarzchild metric, proper time and coordinate time taken for particles and photons to arrive at the event horizon of a spherical blackhole, precession of the perihilion of a planet revolving around a spherical body as general relativistic effect, the polarization of gravitational waves con­ firming that if gravitons exist, they must the be spin two bosons, spherically symmetric solution to the Einstein field equations in a region having spherical matter distribution that is static or moving with a radial velocity dependent only on the radial coordinate and the gravitational collapse of a spherical body, the Robertson metric for cosmology as a natural candidate for a homogeneous and isotropic expanding universe having curvatures ±1 or zero, the gravita­ tional redshift in the Schwarzhild solution when a ray of light propagates from a strong to a weak gravitational field, the redshift of galactic radiation in a uni­ formly expanding universe described by the Robertson-Walker metric and its relation to the scale factor of the expanding universe, the Friedman solutions to the scale factor and density of the expanding universe, the Robertson Walker metric solution for the scale factor S(t) , the density ρ(t) and the pressure p(t) from the Einstein field equations given the equation of state of the galactic fluid, derivation of the Friedman models from Newtonian cosmology and finally the propagation of inhomogeneities of matter and radiation in an expanding universe by considering small metric, density and velocity perturbations and linearizing

General Relativity and Cosmology with Engineering Applications the Einstein field equations taking into account viscous and thermal effects in the energy-momentum tensor of the matter field. Excellent books describing all these are (a) Steven Wienberg, ”Gravitation and Cosmology, principles and applications of the general theory of relativity” published by Wiley and Lan­ dau and Lifshitz ”The classical theory of fields”, published by Butterworth and Heinemann. Again, in the first part of the book, we discuss elementary planar geometry as taught in schools. This section has been introduced so that school students can immediately relate planar geometry to curved Riemannian geometry re­ quired in the gtr, especially theorems involving how the parallel line postulate of Euclid fails in curved geometry, and how the sum of angles of a triangle on a curved surface is π radians plus the integral of the curvature scalar over the surface of the triangle. Some elementary algebra has been included here that will motivate school students to take up general relativity and cosmology as a research subject. In this part, we also discuss some group theoretic problems with the hope that somebody may be able to develop a representation theory for the diffeomorphism group required for the quantization of the gravitational field and background independent physics. Since matter is quantized and mat­ ter produces gravitation, which acts back on matter, gravitation should also be quantized and thus we must have a background independent physics, namely quantum gravity (See Thomas Thiemann, ”Modern canonical quantum general relativity”, Cambridge university press). A nice way to write down the EinsteinHilbert action in a way appropriate for quantization of gravity has been given at the end of the book. The portion involving only spatial derivatives of the position fields has been separated from the portion involving time derivatives in the action integral. This allows us to calculate easily the canonical momentum fields and hence obtain a neat expression for the Hamiltonian of the gravita­ tional field to which canonical quantization can be applied. We’ve not discussed the Ashtekar variables which are obtained by applying a canonical nonlinear transformation to the position and momentum fields. The second part of the book deals with basic quantum mechanics and quantum field theory written once again with the hope that some day a renormalizable quantum field theory of gravity will be constructed using which the S-matrix for interaction between photons, electrons, positrons, gravitons and perhaps other particles like quarks, leptons, mesons, pions and gauge bosons appearing in the Yang-Mills theory will be constructed.

XIX

Part I:

The General Theory of Relativity

and Some of Its Applications

Chapter 1

The special theory of relativity 1.1

Conflict between Newtonian mechanics and Maxwell’s theory of electromagnetism

According to Newtonian mechanics, if light has a velocity of c relative to a frame at absolute rest (That the notion of a frame at absolute rest is valid is one of the postulates of Newtonian mechanics, it is not there in Einstein’s postulates of special relativity) and if a person carrying a torch moves with velocity v relative to a frame K at absolute rest, then the speed of light will be c + v relative to K. This is a contradiction if we assume Maxwell’s theory of electromagnetism to be correct in all inertial frames, for Maxwell’s theory then predicts the same speed c of light in all inertial frames. We should note that according to Ernst Mach, there exists a frame at absolute rest and any frame that is in uniform motion relative to this frame should be regarded as an inertial frame an no other. Such a frame at absolute rest is a frame that is non-moving relative to the distant stars and galaxies in the universe and is at infinite distance from all such matter. This notion was used by Einstein in his general relativity theory but not in his special relativity theory. Einstein assumed that Maxwell was right and Newton was not right and this led him to postulate two principles for the special theory of relativity which would modify Newtonian mechanics. These postulates were: (i) The velocity of light is the same in all inertial frames in vacuum. (ii) The laws of physics are the same in two frames that are at uniform motion relative to each other. If one accepts these postulates and regards Maxwell’s theory of electromag­ netism and light as a correct physical theory, then the wave equation for the electric and magnetic fields in vacuum that one derives from the Maxwell equa­ tions should predict the same speed c for these waves in two inertial frames.

1

2

General Relativity and Cosmology with Engineering Applications Thus if K ' moves relative to K with a uniform velocity, then the wave equa­ tions for light in these two frames are (∂ 2 /∂t2 − c2 ∇2 )ψ = 0, '

'

(∂ 2 /∂t 2 − c2 ∇ 2 )ψ ' = 0 and hence the wave operators in the two frames are the same, ie, '

'

∂ 2 /∂t 2 − c2 ∇ 2 = ∂ 2 /∂t2 − c2 ∇2 Moreover, space-time forms a uniform manifold, which means that the trans­ formation law (t, r) → (t' , r' ) of the space-time coordinates from K to K ' must be linear. Thus, we must have x' = Lx '

where x = (t, rT )T , (t' , r T )T ∈ R4 and L is a 4 × 4 matrix. The invariance of the wave operator ' ∂ μ ∂ μ = ∂ μ ∂ μ' where

x0 = ct, (xr )3r=1 = (x, y, z) = r

and likewise for x' with x0 = ct, xr = −r = (−x, −y, −x) and ∂μ =

∂ ∂ , ∂μ = ∂x'μ ∂xμ

implies LT GL = G where G = diag[1, −1, −1, −1] Any real 4 × 4 matrix satisfying the above equation is called a Lorentz trans­ formation. Problem: [a] Show that this condition for L is equivalent to saying that '

'

'

c2 t 2 − r' 2 = x'μ xμ = xμ xμ = c2 t2 − r2 What does this equation mean physically if we consider a photon as a particle and not as a wave? [b] Show that for motion of K ' relative to K along the x direction alone with velocity v, we have t' = a11 t + a12 x, x' = a21 t + a22 x, y ' = y, z ' = z and

'

'

c 2 t 2 − x 2 = c 2 t2 − x 2

General Relativity and Cosmology with Engineering Applications with

x' = 0

when x = vt. Show that these imply the Lorentz transformation for boosts: t' = γ(v)(t − vx/c2 ), x' = γ(v)(x − vt), γ(v) = (1 − v 2 /c2 )−1/2 [c] Show that if L is any Lorentz transformation, we can express it as L = P RL0 where P = diag[1, −1, −1, −1] is a reflection,

( R=

1 0T

0 R1

) , R1 ∈ SO(3)

and L0 is a boost along the x direction, or more generally, along any fixed direction n ˆ. [d] Show that Lorentz transformations form a group G and that all Lorentz transformations having unit determinant form a subgroup G+ of G. Show that the coset space G/G0 has just two elements, namely G0 and P G0 . [e] Show that the Lorentz transformation corresponding to boosts along the direction n ˆ with speed v can be expressed as ' t' = γ(v)(t − vˆ n.r/c2 ), r' = r⊥ + r||' ,

where

' r⊥ = r⊥ = r − (r.n ˆ )ˆ n

r||' = γ(v)(r|| − vˆ nt), r|| = (r.ˆ n)ˆ n

Remark: The invariance of the speed of light in two frames that are moving relative to each other with a uniform velocity can be cast either in the wave format or in the particle format. In the wave format, this means that the wave operator ∂ μ ∂μ is an invariant while in the particle format, this means that the ' metric xμ x'μ is an invariant.

1.2

The experiments of Michelson and Morley

Before Einstein made his postulates of special relativity, many physicists thought that Light needed a medium called the Ether to propagate. To test the presence of Ether, Michelson and Morley performed an experiment involving interference of light rays propagating along different directions, one parallel to the direction of the earth’s motion and another perpendicular to this direction. If Ether

3

4

General Relativity and Cosmology with Engineering Applications existed, then the Newtonian laws of addition of velocities would be valid even for light, which would mean that light propagating parallel to the direction of Earth’s motion would have a speed of either c + v or c − v where v is the speed of the earth while light propagating perpendicular to the direction of Earth’s √ motion would have a speed of c2 − v 2 . Thus there would be a delay between the paths of equal length with one path parallel to the earth’s motion and another perpendicular. This delay will result in a phase difference between the two paths causing a shift in the interference pattern for two such rays. Such a shift was not observed in the experiments of Michelson and Morley. Lorentz tried to explain this absence of shift in the interference pattern by saying that although the speed of light would be different along the different paths, rods with length placed along the direction of the earth’s motion would contract due to molecular forces while those with lengths perpendicular to the earth’s motion would not contract. However, physicists were dissatisfied with this explanation since according to Lorentz, Ether would be present causing Maxwell’s theory of constancy of the velocity of light to fail under inertial motion. It was only after Einstein banished the concept of ether and introduced his postulates of special relativity that the experiments of Michelson and Morley could be successfully explained. Specifically, light according to Einstein would travel with the same speed both parallel to the direction of the Earth’s motion and perpendicular to it.

1.3

Study projects

[1] Explain Lorentz’s interpretation of the results of Michelson and Morely [2] Explain the banishing of the ether by Einstein and successful explanation of the Michelson-Morley experiment [3] Write down Einstein’s postulates of the special principle of relativity with the subsequent derivation of the Lorentz transformation equations [4] Einstein’s derivation of the Lorentz transformation between frames mov­ ing with relative constant velocity along a direction n ˆ [5] What is the meaning of an inertial frame ?

1.4

The notion of proper time, time dilation and length contraction

Let K be an inertial frame in the sense of Ernst Mach, ie, far removed from all the static matter in the universe and at rest or in uniform motion relative to this matter. Let t, r(t) denote the space-time coordinates of a moving particle relative to such a frame. Then, the time t' as measured by a clock attached to the moving particle is given according to the Lorentz transformation equation over a small interval of time dt (during which the velocity of the particle can be assumed to be a constant so that the Lorentz transformation equations of

General Relativity and Cosmology with Engineering Applications special relativity can be applied over this infinitesimal time interval) by where (v(t) = r' (t)) dt' = γ(v)(dt − v.dr/c2 ) = γ(v)(dt − v.(dr/dt)dt/c2 ) = γ(v)(dt − v 2 dt/c2 ) √ = 1 − v 2 (t)/c2 dt = dt/γ(v(t)) This means that a clock attached to the particle will measure a time duration of ∫ t2 √ t'2 − t'1 = 1 − v 2 (t)/c2 dt t1

which is smaller than the time duration t2 − t1 measured by a clock attached to K. This phenomenon is called time dilation: Moving objects measure lesser time.

1.5

The twin paradox

After Einstein derived the time dilation equation, many physicists came up with the twin paradox which apparently seemed to contradict the special principle of relativity. This paradox can be stated in the following way: Suppose A is fixed to the earth’s surface and B is initially located at A' position along with A. Then, B gets into a rocket and travels far away into space directly above A' s location with a velocity of v(t) where t is the time measured by A' s clock. After reaching a sufficiently great height, B reverses the direction of motion of his rocket and comes down back to A. Let t' denote time as measured by B ' s clock during his motion. If t1 and t2 denote respectively the times as measured by A when B departs from A and when B arrives back to A respectively and t'1 , t'2 the corresponding times as measured by B ' s clock, then according to the time dilation principle, we should have ∫ t2 √ 1 − v 2 (t)/c2 dt < t2 − t1 t'2 − t'1 = t1

which implies that B would have aged less than A during his travel. However, if we look at this situation in another way, namely denote by K ' B ' s frame of rest. Then relative to B, A travels with a speed of u(t' ) = −v(t) and hence we should have ∫ t'2 √ ∫ t'2 √ ' 2 ' 2 t 2 − t1 = 1 − u (t )/c dt = 1 − v 2 /c2 dt' < t'2 − t'1 t'1

t'1

ie, A would have aged lesser than B. This contradiction is settled by noting that A' s frame is nearly inertial as compared to B because A is almost at rest ' relative to the distant matter in the universe √ while B s ' frame is non-inertial in 2 2 this sense. So the second formula dt = 1 − u /c dt is incorrect. The two frames are not equivalent if we agree with Ernst Mach’s notion of an inertial frame.

5

6

General Relativity and Cosmology with Engineering Applications

1.6

The equations of mechanics in special rela­ tivity

Let f μ (x) be a four vector field, ie, a vector field which transforms under Lorentz transformations to f ' (x' ) = Lf (x), x' = Lx, or more precisely,

'

f μ (x' ) = Lμν f ν (x)

If an equation of physics in K has the form f μ (x) = 0 then we get

'

f μ (x' ) = Lμν f ν (x) = 0

ie the corresponding equation of physics in K ' is '

f μ (x' ) = 0 μ ...μ

More generally if T = Tν11...νrp (x) is a (p, r) tensor field in K, then this tensor field in K ' is given by T ' (x' ) = L⊗p T (x)L⊗rT or equivalently, in terms of components, '

'

...ρp 1 ...μp Tν1μ...ν (x' ) = Lμρ11 ...Lμρpp Lσν11 ...Lσνrr Tσρ11...σ (x), x μ = Lμν xν r r

Thus, if the law of physics T (x) = 0 holds in K, then, it also holds in K ' , ie, T ' (x' ) = 0 In other words, laws of physics in the special theory of relativity must be ex­ pressed only as tensor equations, where by a tensor, we mean that it transforms according to a Lorentz transformation that link the space-time coordinates in the two frames. √ Now, the proper time defined by dτ = 1 − v 2 /c2 dt for a particle moving with a velocity v(t) in an inertial frame K is obviously a Lorentz invariant since no matter what frame we choose in place of K, dτ will always be the time differential measured by a clock attached to the particle. Hence out of the space-time four-vector differential dxμ and the invariant/scalar dτ , we can form a four vector uμ = dxμ /dτ Thus, if m0 denotes the rest mass of the particle (which is obviously an invari­ ant), we can form the four momentum vector for the particle pμ = m0 uμ

General Relativity and Cosmology with Engineering Applications and Newton’s second law of motion should be replaced by a four vector equation (ie a tensor equation) dpμ = fμ dτ where f μ is another four vector called the four force on the particle. Newton’s law of motion is invariant under Galilean transformations while the above Ein­ stein’s law of motion is invariant under Lorentz transformations which are the correct laws of transformation between inertial frames. It follows from the above that dpr /dt = γf r , r = 1, 2, 3, γ = dt/dτ = (1 − v 2 /c2 )−1/2 , v r = dxr /dt = ur /γ We have pr = m0 ur = γm0 v r This is the three momentum of the particle, and we see from the above discussion that the three force should be defined as √ dpr /dt = m0 d(γ(v)v r )/dt = m0 d(v r / 1 − v 2 /c2 )/dt Problem: Show that in the limit c → ∞, Einstein’s laws of motion reduce to Newton’s laws of motion and also Lorentz transformations reduce to Galilean transformations: r → r − vt, t → t ie, time is absolute and the Galilean law of addition of velocities is valid.

1.7

Mass, velocity, momentum and energy in special relativity, Einstein’s derivation of the energy mass relation E = mc2

Now we are in a position to derive Einstein’s famous mass energy relationship. First note that if we define m0 m= √ = γ(v)m0 1 − v 2 /c2 then the three momentum can be expressed as pr = mv r , r = 1, 2, 3 Hence m = γ(v)m0 should be regarded as the mass of the body relative to K when it moves with a velocity of v relative to K. Thus, the total work done by external forces on the body cause the body’s energy to increase in time [0, t] by an amount ∫ t ∫ v ∫ t r r r r r r (dp /dt)v dt = v .dp = p v − pr dv r ΔE = 0

0

0

7

8

General Relativity and Cosmology with Engineering Applications = γ(v)m0 v 2 − m0

∫

t 0

γ(v)v r dv r = γ(v)m0 v 2 − m0

∫

v

γ(v)vdv 0

= m0 (γ(v(t)) − γ(v(0)))c2 = (m(t) − m(0))c2 Hence we can express the energy of the body as E = mc2 = γ(v)m0 c2 To check that this coincides with the kinetic energy of the body at small particle velocities we observe that by the binomial theorem, γ(v) = (1 − v 2 /c2 )−1/2 = 1 + v 2 /2c2 + O(v 4 /c4 ) and hence,

E = m0 c2 + m0 v 2 /2 + O(v 4 /c2 )

This equation states that m0 c2 should be regarded as the internal energy of the body at rest and m0 v 2 /2 then nearly coincides with the increase in the body’s energy when it is in motion with a velocity of v. We also note the physical significance of f 4 : dpμ /dτ = f μ gives 2m0 f μ uμ = (d/dτ )(pμ pμ ) = 0 since

uμ uμ = c2 γ 2 − γ 2 v 2 = c2

Thus,

f 4 = f r v r /c = P/c

where P is the rate at which the external forces do work on the body, ie, the instantaneous power pumped into the body.

1.8

Four vectors and tensors in special relativity and their Lorentz transformation laws

A Lorentz transformation is a linear transformation on R4 that preserves the light cone ie the quadratic form x02 −

3 ∑

xk2 = xμ xμ = ημν xμ xν

k=1

It is not hard to see that such a 4 × 4 matrix L is defined by the condition LT ηL = η

General Relativity and Cosmology with Engineering Applications where η = diag[1, −1, −1, −1] is the Minkowski metric. Suppose L transforms the event [t, 0, 0, 0]T to an event [t' , x' , y ' , z ' ]T . Then, '

'

'

t2 = t 2 − x 2 − y 2 − z It follows that

'

2

= ((L00 )2 − (L10 )2 − (L20 )2 − (L30 )2 )t2

(L00 )2 = 1 + (L10 )2 + (L20 )2 + (L30 )2

Accordingly, we can partition the group G of all Lorentz transformations into two disjoint subsets, G↑ and G↓ , where G↑ consists of all L ∈ G for which √ 0 L0√= 1 + (L10 )2 + (L20 )2 + (L30 )2 and G↓ consists of all L ∈ G for which L00 = − 1 + (L10 )2 + (L20 )2 + (L30 )2 . Equivalently, G↑ consists of all L ∈ G such that L00 > 0 and G↓ consists of all L ∈ G such that L00 < 0. G↑ is called the set of orthochronous Lorentz transformations and G↓ is called the set of all non­ orthochronous Lorentz transformations. If L, S are orthochronous, then so is T = LS. To prove this, we first show that if L transforms [t, x, y, z]T into [t' , x' , y ' , z ' ]T and if further [t, x, y, z] is time like, ie, t2 > x2 + y 2 + z 2 , then t > 0 implies t' > 0. Indeed, since LT is also a Lorentz transformation, it follows from the above argument that (L00 )2 = 1 + (L01 )2 + (L02 )2 + (L03 )2 and hence, t' = L00 t + L01 x + L02 y + L03 z > L00 t − ≥ L00 t −

√ (L01 )2 + (L02 )2 + (L03 )2 (x2 + y 2 + z 2 )1/2

√ 1 + (L01 )2 + (L02 )2 + (L03 )2 (x2 + y 2 + z 2 )1/2 √ = L00 (t − x2 + y 2 + z 2 ) > 0

Note that L00 ≥ 1 since we are assuming L ∈ G↑ . Conversely, if L ∈ G transforms a time like vector [t, x, y, z]T into [t' , x' , y ' , z ' ]T and t > 0 implies t' > 0 and that this is true for all vectors, then L ∈ G↑ . Indeed, we then have on taking [t, 0, 0, 0] as the first vector with t > 0, t' = L00 t, x' = L10 t, y ' = L20 t, z ' = L30 t, so that in particular t, t' > 0 imply L00 > 0 and so L ∈ G↑ . We next show that G↑ is a group. Indeed, suppose L, M ∈ G↑ . Then, let [t, x, y, z]T be timelike with t > 0. Define M [t, x, y, z]T = [t' , x' , y ' , z ' ]T , L[t' , x' , y ' , z ' ]T = [t'' , x'' , y '' , z '' ]T Then since L, M ∈ G↑ , it follows that first t' > 0 and next t'' > 0 proving that LM ∈ G↑ . Further, let L ∈ G↑ and define M = L−1 . Let M [t, x, y, z]T = [t' , x' , y ' , z]'T , t > 0 and let [t, x, y, z]T be timelike. Then we have to show that t' > 0. Suppose t' < 0. Then, we have 0 < t = L00 t' + L01 x' + L02 y ' + L03 z ' √ ' ' ' ≤ L00 t' + (L01 )2 + (L02 )2 + (L03 )2 (x 2 + y 2 + z 2 )1/2

9

10

General Relativity and Cosmology with Engineering Applications ≤ L00 t' +

√

'

'

'

1 + (L01 )2 + (L02 )2 + (L03 )2 (x 2 + y 2 + z 2 )1/2 √ = L00 (t' + x' 2 + y ' 2 + z ' 2 ) < 0

since t’¡0 by assumption and L00 > 0. We have also used the fact that t' 2 > x2 + y 2 + z 2 for then since t' < 0, it follows that √ t' < − x' 2 + y ' 2 + z ' 2 This contradiction proves that t' > 0 and therefore L−1 = M ∈ G↑ and com­ pletes the proof that G↑ is a group. We further note that the equation LT ηL = η that characterizes a Lorentz transformation L gives (detL)2 = 1 and hence detL = ±1. We can thus par­ tition G, into two disjoint sets G+ and G− where G+ consists of all elements in G having determinant 1 and G− consists of all elements in G having deter­ minant −1. Clearly G+ is a group. G+ is called the group of proper Lorentz transformations. Now, we define four disjoint subsets of G: ∐ ∐ ∐ G = G1 G2 G3 G4 where G1 = G↑ ∩ G+ , G2 = G↓ ∩ G+ ∩, G3 = G↑ ∩ G− , G4 = G↓ ∩ G− It is clear that G1 , G2 , G3 , G4 are pairwise disjoint with union G and that G1 is a group. G1 is called the subgroup of all proper orthochronous Lorentz transfor­ mations. Now, define P = diag[1, −1, −1, −1] and T = diag[−1, 1, 1, 1]. Then, it is clear that G2 = T G1 , G3 = P G1 , G4 = P T G1

1.9

The general from of the Lorentz group con­ sisting of boosts and rotations

The group G1 of proper orthochronous Lorentz transformations, consists of all L ∈ G for which L00 > 0 and detL = 1. Any such L is expressible as the product of a rotation of space with a boost along some direction. Specifically, a boost along the x direction has a transformation law of the form x' = γ(x − bt), t' = γ(t − vx), y ' = y, z ' = z ˆ with a speed of where γ = (1 − v 2 )−1/2 and hence a boost along the direction n v has a transformation law of the form ˆ n − vˆ nt) + r − (r.ˆ n)ˆ n, r' = γ((r, n)ˆ

General Relativity and Cosmology with Engineering Applications

11

t' = γ(t − v(r, n ˆ )) where

γ = (1 − v 2 )−1/2

Here, r = [x, y, z]T , r' = [x' , y ' , z ' ]T . Exercise [a]: Write down explicitly the 4×4 matrix of the above boost matrix B(ˆ n, v). [b] Write down the explicitly the matrix Rm ˆ (φ) of a rotation around the direction m ˆ by an angle φ. hint: If Rm ˆ (φ)r(0) = r(φ), then r' (φ) = m ˆ × r(φ) Express this equation in matrix notation and solve it using matrix exponentials. [c] Prove that any L ∈ G1 can be expressed as ˜m L = B(ˆ n, v)R ˆ (φ) (

where ˜m R ˆ (φ) =

1 0

0 Rm ˆ (φ)

)

Explain how given the matrix L, you would determine n ˆ , v, m, ˆ φ. [d] Prove that the set G of all Lorentz transformations forms a group. hint: LT ηL = η, M T ηM = η together imply (LM )T ηLM = η and L−T ηL = η.

1.10

The Poincare group consisting of Lorentz tranformations with space-time translations

The Poincare group GP is defined as the group of all ordered pairs (ξ, L) where L is a Lorentz transformation and ξ ∈ R4 with its composition law determined by its action on R4 as (ξ, L).x = Lx + ξ Thus, if (ξk , Lk ) ∈ GP , then (ξ2 , L2 ).(ξ1 , L1 ) = (L2 ξ1 + ξ2 , L1 L2 ) In other words, GP is the semidirect product of R4 with the Lorentz group G. The subgroup R4 of GP ie (ξ, I4 ), ξ ∈ R4 consists of all space-time translations while the subgroup G of GP ie (0, L), L ∈ G consists of all spatial rotations, spatial reflections, time reversal and boosts. When we consider unitary repre­ sentations of GP , then the generators of spatial translations will go over into momentum operators, the generator of time translations will go over to the energy operator and the generators of rotations will go over into angular mo­ mentum operators.

12

General Relativity and Cosmology with Engineering Applications

Exercise: Let H denote the set of all 2 × 2 Hermitian matrices. Show that any such matrix X can be expressed as ( ) t + z x − iy X = tI2 + xσ1 + yσ2 + zσ3 = x + iy t − z Denote X by Φ(ξ) where ξ = [t, x, y, z]T ∈ R4 . Prove that the map Φ : R4 → H is a vector space isomorphism and that detΦ(ξ) = q(ξ) = ξ T ηξ = t2 − x2 − y 2 − z 2 Let A ∈ SL(2, C). Then, AΦ(ξ)A∗ ∈ H and hence we can define ζ ∈ R4 by the equation Φ(ζ) = AΦ(ξ)A∗ or equivalently,

ζ = Φ−1 (AΦ(ξ)A∗ )

We get on taking determinants and using detA = detA∗ = 1 that detΦ(ζ) = detΦ(ξ) which implies on setting ξ = [t, x, y, z]T , ζ = [t' , x' , y ' , z ' ]T that

'

'

'

t2 − x 2 − y 2 − z 2 = t 2 − x 2 − y 2 − z

'

2

and hence LA = Φ−1 AΦ is a Lorentz transformation. We wish to show further that all the Lorentz transformations LA belong to G1 , ie, they have determinant 1 and are orthochronous. Exercise: Prove the above statement using the fact that Φ−1 AΦ as A varies over SL(2, C) is connected and contains the identity element. Use also the fact that the G1 , G2 , G3 , G4 are mutually disjoint and all are topologically isomorphic to G1 which is a closed subset of G.

1.11

Irreducible representations of the Poincare group with applications to Wigner’s parti­ cle classification theory

Let G be the Lorentz group and G1 the subgroup of proper orthochronous Lorentz transformations. The Poincare group GP = R4 ⊗s G1 We know that SL(2, C) is the double cover of G1 . Hence, constructing rep­ resentations of G1 is equivalent to constructing representations of SL(2, C).

General Relativity and Cosmology with Engineering Applications In fact, it is easy to see that G1 is isomorphic as a Lie group to the group SL(2, C)/{I, −I}. Hence, we may equivalently express the Poincare group GP as the semidirect product GP = R4 ⊗s SL(2, C) Let V be a unitary representation of G1 and let χ0 be a Character of R4 . Then under the adjoint action of G1 or equivalently, SL(2, C) χ0 varies over an orbit Oχ0 of the character group of R4 . Wigner proved that there are exactly four types disjoint orbits and the representative elements of these orbits are [m, 0, 0, 0]T , [−m, 0, 0, 0]T , [im, 0, 0, 0]T where m > 0 and [1, 1, 0, 0]T . The first orbit corresponds to positive mass, the second to negative mass, the third to imaginary mass, ie, particles traveling faster than the speed of light and finally, the last one corresponds to zero mass. To this end, let H denote the stability subgroup of χ0 and L an irreducible representation of H. Then χ0 ⊗ L is 1 an irreducible representation of G10 = R4 ⊗s H and if U = indG G10 V , then U is an irreducible representation of G1 . The representation U of G1 is that induced by the representation V of G10 . There are many methods to express this induced representation of a semidirect product of an Abelian group and another subgroup H that normalizes N . Some of these methods are discussed in detail in the following books: [1] K.R.Parthasarathy, ”Mathematical foundations of quantum mechanics”, Hindustan Book Agency. [2] Barry Simon, ”Representations of finite and compact groups”, American Mathematical Society. [3] V.S.Varadarajan, ”Supersymmetry for mathematicians”, Courant insti­ tute lecture notes”.

1.12

Lorentz transformations of the electromag­ netic field

We have seen that the Maxwell equations in flat space-time can be expressed in tensor form as F,νμν = −μμJ This equation is invariant under Lorentz transformations provided that J μ is a (1, 0) four vector and F μν is a (2, 0) tensor. Note that ∂/∂xμ is a (0, 1) vector. The transformation law of a general (p, q) tensor ...μp Tνμ11...ν (x) q

under a Lorentz transformation L (ie a transformation that connects two sys­ tems moving at uniform velocity relative to each other after an appropriate rotation of the frame) is given by ...μp (¯ x) = T¯νμ11...ν q

13

14

General Relativity and Cosmology with Engineering Applications ρ1 ...ρp 1 ...αq Lμρ11 ...Lμρpp Lα ν1 ...νq Tα1 ...αq (x)

This transformation law is the definition of a (p, q)-tensor. Thus, F μν transforms as x) = F αβ (x)Lμα Lνβ F¯ μν (¯ Equivalently, we can first Lorentz transform the four potential x) = Lμν Aν (x0 A¯μ (¯ and then transform the partial derivatives ∂/∂xμ as ∂/∂x ¯μ = Lνμ ∂/∂xν and then evaluate the electric and magnetic fields in the barred frame using ¯ (¯ ¯ A0 (¯ x) − ∂A¯(¯ x)/∂x ¯0 E x) = −∇ and ¯ x) = ∇ ¯ × A¯(¯ B(¯ x) ¯ moves Exercise: Show using the fact that if K is an inertial frame and K relative to K with a velocity v along the x-axis, then the space-time coordinates ¯ are given in terms of those with respect to K by the equations in K x ¯ = γ(x − vt), t¯ = γ(t − vx/c2 ), y¯ = y, z¯ = z that the electric and magnetic fields in K ' are related to those in K by ¯x (ξ¯) = Ex (ξ), E ¯y (¯ E xi) = γ(E2 (ξ) − vB3 (ξ)), ¯z (ξ¯) = γ(Ez (ξ) + vBy (ξ)) E and using duality, ie, the fact that the Maxwell curl equations are invariant under the transformations E → B, B → −E (assume c = 1) ¯y (ξ¯) = γ(By (ξ) + vEz (ξ)), ¯x (ξ¯) = Bx (ξ), B B ¯z (ξ¯) = γ(Bz (ξ) − vEy (ξ)) B Here, ¯ x, ξ = [t, x, y, z], ξ¯ = [t, ¯ y, ¯ z] ¯ hint: Aμ is a four vector and hence transforms just like xμ under Lorentz transformations, x) = γ(A0 (x) − γA1 (x)), A¯0 (¯ x) = γ(A1 (x) − vA1 (x)), A¯1 (¯ x) = A2 (x), A¯3 (¯ x) = A3 (x) A¯2 (¯ Also use the inverse Lorentz transformation in the form ¯), y = y¯, z = z¯ x = γ(¯ x + vt¯), t = γ(t¯ + vx

General Relativity and Cosmology with Engineering Applications

15

so that ∂/∂ x ¯ = γ∂/∂x + γv∂/∂t, ∂/∂ y¯ = ∂/∂y, ∂/∂ z¯ = ∂/∂z = ∂/∂z ∂/∂t¯ = γv∂/∂x + γ∂/∂t

¯ moves relative to K along the direction n ˆ with a speed v, Exercise: If K then show that the em fields transform as ¯|| (¯ E x) = E|| (x), ¯⊥ (¯ x) = γ(E(x) + v × B(x))⊥ = γ(E⊥ (x) + v × B(x)), E ¯|| (¯ x) = B|| (x), B ¯⊥ (¯ x) = γ(B(x) − v × E(x))⊥ = γ(B⊥ (x) − v × E(x)) B

1.13

Relative velocity inspecial relativity

¯ move relative to K with a speed of v along Let K be an inertial frame and let K ¯ with a speed of w ¯ along the x ¯ axis. the x axis and a particle move relative to K Then, we wish to find the velocity w of the particle relative to K. We have if x(t) denotes the x-position of the particle relative to K at time t and x ¯(t¯) the ¯ that x-position of the particle relative to K x ¯ = γ(v)(x − vt), t¯ = γ(v) = (t − vx/c2 ) and hence w ¯ = d¯ x/dt¯ =

dx − vdt dx/dt − v = 1 − (v/c2 )dx/dt dt − vdx/c2

or equivalently, w ¯=

w−v 1 − vw/c2

Equivalently, using the inverse Lorentz transformation ¯ t = γ(v)(t¯ + v x/c x = γ(v)(¯ x + v t), ¯ 2) that w = dx/dt = =

d¯ x/dt¯ + v 1 + (v/c2 )d¯ x/dt¯

w ¯+v 1 + v w/c ¯ 2

16

General Relativity and Cosmology with Engineering Applications

1.14

Fluid dynamics in special relativity

The energy momentum tensor of the fluid in the absence of viscous and thermal effects can be expressed as T μν = (ρ + p)v μ v ν − pη μν ρ = ρ(x), p = p(x), v μ = v μ (x), ημν v μ v ν = 1 for then, if f μ (x) denotes the four force density field, we get the Euler equations of the fluid as T,νμν = f μ or μ − p,μ = f μ ((ρ + p)v ν ),ν v μ + (ρ + p)v ν v,ν

This can be simplified further by multiplying both sides with vμ giving ((ρ + p)v ν ),ν − p,μ v μ − fμ v μ = 0 This is the mass conservation equation in special relativity and substituting this into the previous equation gives us the special-relativistic version of the Euler equation (ie the Navier-Stokes equation without the viscous term): μ (ρ + p)v ν v,ν + (fα + p,α )v α v μ = f μ + p,μ

Exercise: equation by taking μ = r = 1, 2, 3 and substituting √ write down this∑ 3 v 0 = 1 + v 2 where v 2 = r=1 v r2 .

1.15

Plasma physics and magnetohydrodynam­ ics in special relativity

The energy-momentum tensor of the matter fluid plus electromagnetic radiation is given by T μν + S μν where T μν = (ρ + p)v μ v ν − pη μν is the energy-momentum of the matter fluid and S μν = (−1/4)Fαβ F αβ η μν + Fαμ Fαν Using the Maxwell equations F,νμν = −μ0 J μ , Fμν,α + Fνα,μ = Fαμ,ν = 0 it is easy to show that S μν , ν = F μν Jμ

General Relativity and Cosmology with Engineering Applications

17

and hence the total energy-momentum conservation equation of matter plus radiation given by the tensor equation (T μν + S μν ),ν = 0 becomes T,νμν = F μν Jν For a conducting fluid, we usually write J = σ(E + v × B) Its tensor generalization should be of the form J μ = σF μν vν where σ is the conductivity scalar, or more generally, if the conductivity is a tensor, then μν F αβ v ρ J μ = σαβρ Assuming scalar conductivity, we obtain the following special relativistic gener­ alization of the MHD equation: T,νμν = σF μν Fνα v α This equation should be jointly solved with the Maxwell equations F,νμν = −μ0 J μ = −μ0 σF μν vν

1.16

Particle moving in a constant magnetic field in special relativity

The equations of motion are derived from the Lagrangian √ L(r, v, t) = −m0 c2 1 − v 2 /c2 − q(Φ(t, r) − (v, A(t, r))) The Euler-Lagrange equations read d m v √ 0 = q(E(t, r) + v × B(t, r)) dt 1 − v 2 /c2 where q is the charge of the particle and m0 is its rest mass. Here v = dr/dt is the usual three velocity. For a zero electric field and constant magnetic field B0 zˆ, the equations of motion are (γ(v)vx )' = (qB0 /m0 )vy , (γ(v)vy )' = (−qB0 /m0 )vx ,

18

General Relativity and Cosmology with Engineering Applications (γ(v)vz )' = 0 We seek a solution with vz = 0 so that γ(v) = (1 − (vx2 + vy2 )/c2 )−1/2 From now onwards, we shall write vx x ˆ + vy yˆ for v, so that v 2 = vx2 + vy2 . One of the integrals is obtained using γ(v)vx (γ(v)vx )' + γ(v)vy (γ(v)vy )' = 0 so that

γ(v)2 v 2 = K

where K is a constant. Thus v 2 , γ(v) are also constants and v 2 /(1 − v 2 /c2 ) = K so that say. This gives This gives

v 2 = K/(1 + K/c2 ), γ(v) = (1 + K/c2 )1/2 = γ0 vx' = ωvy , vy' = −ωvx , ω = qB0 /m0 γ0 vx'' = ωvy' = −ω 2 vx

and hence vx (t) = A.cos(ωt + φ), vy (t) = −A.sin(ωt + φ) Thus, and

v 2 = A2 = K/(1 + K/c2 ) γ0 = (1 − A2 /c2 )−1/2

Chapter 2

The General theory of relativity 2.1

Drawbacks with the special theory of rela­ tivity

The principal drawback with the STR is that it is not covariant for all observers in the universe, it is covariant only for the equivalence class of all relatively inertial observers, ie, observers who are moving relative to each other with constant relative velocity.

2.2

The principle of equivalence

Einstein first postulated that the gravitational field should not be treated as a force, it should only be treated as a curvature of the space-time manifold on which particles execute motion. This stems from the following thought exper­ iment. Suppose we have an inertial frame K, ie a frame that is infinitely far from all the stars and galaxies in our universe and moves with uniform relative velocity with respect to these distant stars and galaxies. Herein, we are assum­ ing that the distant stars and galaxies in our universe are at rest relative to each other or at most, they are moving relative to each other with constant rel­ ative velocities. The metric of space-time in K is then the standard Minkowski metric: dτ 2 = dt2 − (dx2 + dy 2 + dz 2 )/c2 This metric guarantees in accordance with Einstein’s time dilation principle of STR that the proper time of a particle moving with a velocity u(t) relative to K is given by dτ = dt(1 − u2 /c2 )1/2

19

20

General Relativity and Cosmology with Engineering Applications Now, suppose that there is a constant gravitational field in K defined by the vector gn where n is a constant unit vector (usually n = −zˆ). Let K ' denote the frame that is freely falling in K in this gravitational field. Then, K ' is an inertial frame and K is no longer inertial since particles experience a gravitational force in K but not in K ' . Then, if (x' , y ' , z ' , t' ) denote space-time coordinates relative to K ' , it follows that relative to K the coordinates relative to K and K ' are related by r = r' + gnt2 /2, t' = t assuming Galilean relativity. Thus, dr' = dr − gndt' , dt' = dt and hence, the metric relative to K ' and K are respectively given by '

dτ 2 = dt 2 − |dr' |2 /c2 , dτ 2 = dt2 − |dr − gndt|2 /c2 = (1 − g 2 /c2 )dt2 + 2g(n, dr)dt/c2 − |dr|2 which shows that the metric in K is no longer flat Minkowskian but rather curved, with its metric coefficients dependent upon the gravitational field gn. This means that if the gravitational field in K is not a constant in space-time, then we can still cancel it over an infinitesimal region of space-time by moving to a locally freely falling frame. This raises the important question as to given a metric in K which is non-Minkowskian, then when does there exist a global coordinate transformation that brings it to Minkowskian and when does there not exist any such global transformation of coordinates ? It is a classic theorem in differential geometry due to Riemann that such a global transformation exists if and only if the four space-time index curvature tensor (to be defined latter) is identically zero in K. Then, this tensor will vanish in all coordinate systems. Einstein conjectured that this tensor will vanish, ie, that there will exist a global coordinate transformation (a diffeomorphism) that brings it to flat Minkowskian iff there is no matter that generates the gravitational field. When matter is present, the curvature tensor can never vanish and hence there can never exist and global coordinate transformation that reduces the metric to Minkowskian.

2.3

Why gravitational field is not a force ?

As we just saw, it is possible to cancel out a constant gravitational field by ap­ plying a transformation of space-time coordinates and if the gravitational field is non-constant, we can cancel it out only locally. Nevertheless, this leads us to believe that the gravitational field is not a force but rather a distortion of the space-time manifold and that there exists a reference frame in which there is no gravitational field iff the metric can be transformed by a global change of coor­ dinates into the flat Minkowskian metric iff the Riemann Christoffel curvature

General Relativity and Cosmology with Engineering Applications

21

tensor vanishes in any one system iff this tensor vanishes in all systems (owing to the tensor transformation law). Study project: Gravitational field as a curvature of space-time.

2.4

Four vectors and tensors in the general the­ ory of relativity

We shall be discussing in this section, the laws of tensor transformation under diffeomorphisms of space-time. Let S be a p-dimensional manifold immersed in an N dimensional Euclidean system. The equations of the surface are defined by y n = y n (x1 , ..., xp ), n = 1, 2, ..., N Let A be a vector on this surface at x. Then, by definition, its cartesian com­ ponents are given by n An = Aμ y,μ where Aμ are the curvilinear components of this vector. Now, suppose, we make a change of coordinates of the curvilinear system to x ¯μ = f μ (x), μ = 1, 2, ..., p or equivalently, by its inverse x), μ = 1, 2, ..., p xμ = g μ (¯ Let A¯μ denote the curvilinear components of the vector A relative to the barred system x ¯. Then, by definition, n

∂y An = A¯μ μ ∂x ¯ ν

n ∂x = A¯μ y,ν ∂x ¯μ ν n = A y,ν

and hence, we infer that

ν

∂x Aν = A¯μ μ ∂x ¯ or equivalently,

μ

∂x ¯ A¯μ = Aν ν ∂x More precisely, x and x ¯ = f (x) refer to the same point P on the curvilinear manifold respectively relative to the unbarred and barred coordinate systems. With this understanding, we write the above relations as Aν (x) = A¯μ (¯ x)

∂xν ∂x ¯μ

22

General Relativity and Cosmology with Engineering Applications ν

x) ∂g (¯ = A¯μ (¯ x) μ ∂x ¯ or equivalently, μ

∂f (x) x) = Aν (x) A¯μ (¯ ∂xν

2.5 2.5.1

Basics of Riemannian geometry Parallel displacement

Even without referring to a cartesian system in which the Riemannian manifold is imbedded, we can develop Riemannian geometry directly from the metric gμν (x). First, we assume that parallel displacement is given by a bilinear form δAμ = −Γμαβ Aα δxβ Then, we assume that if S is a scalar field, its parallel displacement is zero, ie δS(x) = 0 Now, take two vector fields Aμ , B μ and construct the scalar field S = gμν Aμ B ν We also assume that the covariant derivative of the metric tensor is zero relative to the connection Γμαβ . In other words, we are assuming that the connection is the metric connection, ie, derived from the metric. Since the covariant differen­ tial of the metric is zero, we must have dgμν − δgμν = 0 or equivalently, δgμν = dgμν = gμν,α δxα In all these formulae, δ refers to parallel displacement of scalars, vectors and tensors from x to x + δx and d refers to the ordinary differential, ie, d = δxα

∂ ∂xα

Then, we get 0 = δ(gμν Aμ B ν ) = = (δgμν )Aμ B ν + gμν δAμ B ν + gμν Aμ δB ν = (gμν,β Aμ B ν − gμν B ν Aα Γμαβ − gμν Aμ B α Γναβ )δxβ

General Relativity and Cosmology with Engineering Applications

23

and hence, we infer that (gμν,β Aμ B ν − gμν B ν Aα Γμαβ − gμν Aμ B α Γναβ ) = 0 from which we get on using the arbitrariness of A, B α gμν,β − gαν Γα μβ − gμα Γνβ = 0

or equivalently, gμν,β = Γνμβ + Γμνβ where Γμνα = gμβ Γβνα We further assume that the connection is symmetric, ie, torsionless: Γμαβ = Γμβα It then follows from the above that Γμνα = (1/2)(gμν,α + gμα,ν − gνα,μ )

2.5.2

Riemannian metric on a curved manifold of dimen­ sion p immersed in an N > p dimensional Euclidean space

The distance dτ between two neighbouring points x and x + dx on a surface is defined by the quadratic form dτ 2 = gμν (x)dxμ dxν The total distance along a curve Γ : λ → x(λ) with λ1 ≤ λ ≤ λ2 is then given by the integral ∫ λ2 s(1, 2) = dτ = ∫

λ1 λ2

(gμν (x(λ))(dxμ (λ)/dλ)(dxν (λ)/dλ))1/2 dλ

λ1

We denote this distance by ∫

2

1,Γ

(gμν (x)dxμ dxν )1/2

24

General Relativity and Cosmology with Engineering Applications

2.5.3

Parallel displacement of a vector on a curved sur­ face. An approach based on immersing the curvi­ linear manifold in a higher dimensional Euclidean manifold

Given a vector with curvilinear components Aμ (x) at x on the curved surface of dimension p immersed in N dimensional Euclidean space with Cartesian coordinates y n , the Cartesian components An (x) are by definition given by ∑ n Aμ y,μ An = μ

We parallely displace this vector to x + dx and project it onto the tangent space to the surface at x + dx. We denote the curvilinear components of the resulting vector by Aμ (x) + δAμ (x). Thus, we can write An = An |tan + An |nor where An |tan is tangential to the surface at x + dx and An |nor is normal to the surface at x + dx. We then have n An |nor y,μ (x + dx) = 0

summation over the Cartesian indices n = 1, 2, ..., N being understood. Also, by definition, the Cartesian components of the displaced and projected vector are given by n (x + dx) An |tan = (Aμ + δAμ )y,μ Thus, n n n n n (x)y,ν (x + dx) = An y,ν (x + dx) = (Aμ + δAμ )y,μ (x + dx)y,ν (x + dx) Aν (x)y,μ

or equivalently, n n y,νρ (x)dxρ ) Aν (x)(gμν (x) + y,μ

= (Aμ (x) + δAμ (x))gμν (x + dx) or equivalently, Aμ (x) + Aν (x)Γμνρ (x)dxρ = = Aν (x) + Aμ (x)gμν,ρ dxρ + gμν (x)δAμ (x) from which we easily deduce that δAμ (x) = −Γμνρ (x)Aν (x)dxρ where Γμνρ = g μα Γανρ with Γανρ = (1/2)(gαν,ρ + gαρ,ν − gνρ,α )

General Relativity and Cosmology with Engineering Applications

25

n n = y,α y,νρ

Exercise: Deduce from the relation n n y,ν gμν = y,μ

that n n y,νρ (1/2)(gμν,ρ + gμρ,ν − gνρ,μ ) = y,μ

Study project: Notion of covariant derivative as a natural generalization of ordinary derivative to curved manifolds

2.5.4

The geodesic equations

Geodesics on a Riemannian manifold are natural generalizations of the notion of a straight line in Euclidean∑space. In Euclidean space, (ie a space of p dimensions p having the metric ds2 = i=1 (dxi )2 ), a straight line is uniquely characterized by any one of the following two properties: (i) Given any two points, it is the shortest path joining the two points, (ii) given a tangent vector to a curve at a point, if this tangent vector is parallely displaced along the curve to any other point on the curve, then the resulting vector is once again tangent to the curve the the new point, if this is the case, then the curve must necessarily be a straight line. We shall now prove that for a Riemannian manifold, these two characterizations give the same curve, or more precisely they lead to the same differential equations for the curve, namely the geodesic equations. First, we observe that if λ → xμ (λ) is a curve of shortest distance between two points xμ (λ1 ) = xμ1 and xμ (λ2 ) = xμ2 , then by Lagrange’s variational principle ∫ 2√ δ gμν (x(λ))dxμ (λ)dxν (λ) = 0 1

This leads to the Euler-Lagrange equations for the optimal trajectory: d ∂L ∂L ' = μ dλ ∂x ∂xμ where '

L(xμ , xμ ) = and

'

√

gμν (x)xμ' xν '

'

xμ = xμ (λ) = dxμ (λ)/dλ Problem: Show that the above Euler Lagrange equations can be expressed as

where

dxα (τ ) dxβ (τ ) d2 xμ (τ ) μ + Γ (x(τ )) =0 αβ dτ 2 dτ dτ dτ 2 = gμν dxμ dxν

26

General Relativity and Cosmology with Engineering Applications or equivalently, dτ /dλ = L Now, we derive the same equations using the second method. First note that vμ =

dxμ dτ

is a unit tangent vector to the curve at x(τ ). Tangency is by the definition and unit length property follows from gμν v μ v ν = gμν dxμ dxν /dτ 2 = dτ 2 /dτ 2 = 1 Now, if this vector v μ is displaced parallely along the curve from x(τ ) to x(τ + dτ ), the new vector will be v μ + δv μ = v μ − Γμαβ (x(τ ))v α δxβ where δxα = v α dτ If this displaced vector is to be tangent to the curve at x(τ +dτ ), then we require that v μ + δv μ be proportional to v μ (τ + dτ ) = v μ + dv μ , ie, v μ − Γμαβ v α v β dτ = (1 + φ(τ )dτ )(v μ + dv μ ) where φ(τ ) is a scalar function along the curve. Using dv μ = in the equation dv μ + Γμαβ v α v β + φv μ = 0 dτ

dv μ dτ dτ ,

this results

If we put the normalization condition vμ v μ = gμν v μ v ν = 1, then it is easy to see that along the curve 0=

d d (vμ v μ ) = (gμν v μ v ν ) = dτ dτ

gμν,α v α v μ v ν + 2gμν v ν

dv μ dτ

= gμν,α v α v μ v ν + 2vμ (−Γμαβ v α v β − φv μ ) = −φ ie φ = 0 and hence, we end up with the same differential equations, ie, the above Euler-Lagrange equations.

General Relativity and Cosmology with Engineering Applications

2.5.5

The general theory of connections and definition of the Riemann-Christoffel curvature tensor and the torsion of a connection

Let M be any differentiable manifold. For any two vector fields X = X μ (x)∂μ and Y = Y μ (x)∂μ , their Lie bracket [X, Y ]is defined b [X, Y ]f = (XY − Y X)(f ) = X(Y f ) − Y (Xf ) = X μ ∂μ (Y ν ∂ν f ) − Y μ ∂μ (X ν ∂ν f ) ν ν = (X μ Y,μ − Y μ X,μ )∂ν f

Thus although XY and Y X being second order linear differential operators are not vector fields, their difference [X, Y ] is a first order differential operator and hence a vector field with components ν ν dxν ([X, Y ]) = [X, Y ]ν = X μ Y,μ − Y μ X,μ

A connection is a map ∇ on M which maps an ordered pair (X, Y ) of vector fields on M to another vector field ∇X Y such that ∇f X Y = f ∇X Y, ∇X (f Y ) = X(f )Y + f ∇X Y for all smooth functions f on M. The torsion and curvature of the connection ∇ are respectively defined by the equations T (X, Y ) = ∇X Y − ∇Y X − [X, Y ] and R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z where X, Y, Z are vector fields. Clearly, both T (X, Y ) and R(X, Y )Z are vector fields on M. Let us now look at the components of a connection relative to a basis {eμ } for the space of vector fields on M, not necessarily a coordinate basis {∂μ }. Note that eμ can be expressed as eμ (x) = Xμν (x)∂ν We have for X = X μ eμ , X μ = eμ (X), where {eμ } is a basis of one forms on M that is dual to the basis {eμ } for T M. Note that the space of one forms on M is denoted by T M∗ and is called the cotangent bundle as opposed to the space of vector fields on M which is denoted by T M and also called the tangent bundle. We have ∇X Y = ∇eμ (X)eμ (eν (Y )eν ) = eμ (X)∇eμ (eν (Y )eν ) = eμ (X)(eμ (eν (Y ))eν + eν (Y )∇eμ eν ) Writing α ∇eμ eν = ωμν eα

27

28

General Relativity and Cosmology with Engineering Applications we get α eα ∇X Y = X(eν (Y ))eν + eμ (X)eν (Y )ωμν

We write α ν e = ωμα ωμν

and then obtain ∇X Y = X(eν (Y ))eν + ωμα (Y )eμ (X)eα = (X(eα (Y )) + ωμα (Y )eμ (X))eα Equivalently, eα (∇X Y ) = X(eα (Y )) + ωμα (Y )eμ (X) Hence, eα (T (X, Y )) = X(eα (Y )) − Y (eα (X)) + ωμα (Y )eμ (X) − ωμα (X)eμ (Y ) −eα ([X, Y ]) Now let ω be any one form on M. Then, we have relative to a coordinate basis, on writing ω = ωa dxa dω(X, Y ) = ωa,b dxb ∧ dxa (X, Y ) = ωa,b (X a Y b − X b Y a ) Therefore, X(ω(Y )) − Y (ω(X)) − ω([X, Y ]) = X(ωa Y a ) + Y (ωa X a ) − ωa [X, Y ]a a = X b (ωa Y a ),b − Y b (ωa X a ),b − ωa (X b Y,ba − Y b X,b )

= ωa,b (X b Y a − X a Y b ) = −dω(X, Y ) This result is valid for any basis, coordinate or not since it is a tensor equation, although we have used local coordinates for proving it. Going back now to the general basis eα , we get on using the above result, T α (X, Y ) = eα (T (X, Y )) = ωμα (Y )eμ (X) − ωμα (X)eμ (Y ) − deα (X, Y ) = (eμ ∧ ωμα − deα )(X, Y ) or equivalently, T α = eμ ∧ ωμα − deα This is called Cartan’s first equation of structure. This equation is important since it tells us how to compute the components of the torsion in any local basis, not necessarily a coordinate basis. Likewise, we shall now derive Cartan’s second equation of structure which tells us how to compute the components of the curvature tensor in any local basis. This result is important in general

General Relativity and Cosmology with Engineering Applications relativity since there exist many metrics like the Kerr metric which can be brought to diagonal form relative to only a local basis that is not a coordinate basis and hence if we write down the Einstein field equations using the Ricci tensor (which is obtained from the curvature tensor by contraction) in such a local non-coordinate basis, the equations will have a much simpler structure. Many examples of such situations in general relativity have been dealt with in a masterly way in the book ”The Mathematical theory of blackholes”, by S.Chandrasekhar, Oxford University Press. In the particular case when eα = ∂α and hence eα = dxα , ie, the coordinate basis, we get since deα = d2 dxα = 0 T α (X, Y ) = dxα (T (X, Y )) = dxμ ∧ ωμα (X, Y ) α Further, we use the notation Γα μν = ωμν for this special case, ie

∇∂μ ∂ν = Γα μν ∂α Then, it follows that T α (X, Y ) = dxμ ∧ Γα μ (X, Y ) α ν = X μ Γμν Y ν − Y μ Γα μν X α α = (Γμν − Γνμ )X μ Y ν

It follows that the torsion vanishes iff α Γα μν = Γνμ

ie the connection components are symmetric in the last two indices in a coor­ dinate frame. We now look at a further special case when the connection is derived from a metric (X, Y ) → g(X, Y ) which in components means that g(X, Y ) = gμν (x)X μ (x)Y ν (x) where we are taking components w.r.t. a coordinate basis dxμ . The metric is assumed to be symmetric, ie, gμν (x) = gνμ (x), or equivalently, g(X, Y ) = g(Y, X) for all vector fields X, Y . We say that the connection ∇ is derived from the metric g if ∇g = 0 which means that ∇X g = 0 for all vector fields X, which further means that X(g(Y, Z)) − g(∇X Y, Z) − g(Y, ∇X Z) = 0

29

30

General Relativity and Cosmology with Engineering Applications for all vector fields X, Y, Z. Looking at this tensor equation in a coordinate basis ∂μ gives us ∂μ g(∂α , ∂β ) − g(∇∂μ ∂α , ∂β ) −g(∂α , ∇∂μ ∂β ) = 0 or equivalently, using ∇∂μ ∂β = Γρμβ ∂ρ , we get ρ gαβ,μ − Γρμα gρβ − Γμβ gαρ = 0

or equivalently, using the standard method of lowering tensor indices, Γβμα + Γαμβ = gαβ,μ which on using the symmetry of the Γ-symbols in view of the assumption that the torsion vanishes, gives us Γαβμ = (1/2)(gαβ,μ + gαμ,β − gμβ,α ) Exercise: Prove the above formula. Cartan’s second equation of structure: R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z Now using a local basis eα (not necessarily a coordinate basis) and its dual basis eα , we define μ ∇eα eβ = ωαβ eμ or equivalently, μ ∇X eβ = ∇X α eα eβ = X α ωαβ eμ

= ωβμ (X)eμ μ where ωβμ is a one form with components ωαβ . We have

∇Y Z = ∇Y (Z α eα ) = Y (Z α )eα + Z α ωαμ (Y )eμ ∇X ∇Y Z = XY (Z α )eα + Y (Z α )ωαμ (X)eμ + X(Z α )ωαμ (Y )eμ +Z α X(ωαμ (Y ))eμ + Z α ωαμ (Y )ωμν (X)eν Interchanging X and Y and forming the difference gives [∇X , ∇Y ]Z = [X, Y ](Z α )eα + +Z α (X(ωαμ (Y )) − Y (ωαμ (X))eμ + Z α (ωαμ (Y )ωμν (X) − ωαμ (X)ωμν (Y ))eν

General Relativity and Cosmology with Engineering Applications

31

Also, ∇[X,Y ] Z = ∇[X,Y ] (Z α eα ) = [X, Y ](Z α )eα + Z α ωαμ ([X, Y ])eμ Taking the difference of the above two expressions gives us R(X, Y )Z = −Z α ωαμ ∧ ωμν (X, Y )eν +Z α dωαν (X, Y )eν = Z α (dωαν + ωμν ∧ ωαμ )(X, Y )eν This is Cartan’s second equation of structure.

2.5.6

The Riemann-Christoffel curvature tensor associated with a Riemannian metric using parallel displace­ ment of a vector around an infinitesimal loop, and as the difference between two covariant derivatives with exchanged order

Let Aμ (x) be a covariant vector field. Then we consider the (0, 3)-tensor Tμνα = Aμ:ν:α − Aμ:α:ν where : denotes covariant derivative. In flat space time covariant derivatives reduce to ordinary partial derivatives and hence Tμνα = 0. In curved space-time with a metric gμν (x), we get using the expression for the covariant derivative of a (0, 2) tensor field, ρ Aρ:ν − Γρνα Aμ:ρ Aμ:ν:α = Aμ:ν,α − Γμα

= Aμ,να − (Γρμν Aρ ),α β −Γρμα (Aρ,ν − Γρν Aβ ) β Aβ ) −Γρνα (Aμ,ρ − Γμρ

Interchanging ν and α in this expression and subtracting, gives us ρ + Γρμα,ν )Aρ Tμνα = (−Γμν,α ρ ρ β +(−Γμα Γβρν + Γνα Γμρ )Aβ β = Rμνα Aβ β where by the quotient theorem, the Rμνα is a (1, 3) tensor given by β Rμνα =

−Γβμν,α + Γβμα,ν )

32

General Relativity and Cosmology with Engineering Applications ρ ρ β −Γμα Γβρν + Γνα Γμρ

It is clear from it definition that this tensor, called the Riemann-Christoffel curvature tensor measures the degree of non-commutativity between the co­ variant derivatives of a vector along two different coordinate directions. This non-commutativity arises because parallel displacement of a vector along two different paths from one given point to another given point gives two results or equivalently, the parallel displacement around a closed loop does not bring back the original vector to itself. This provides yet another equivalent way to determine the curvature tensor: Let C be an infinitesimal loop around the point x. Take a vector Aμ (x) at x and displace it parallely to a point x + ξ on C. ξ varies over C. Then we get the vector Aμ (x + ξ) = Aμ (x) + Γμαβ (x)Aα (x)ξ β assuming that ξ is infinitesimal since C is infinitesimal. Now displace this vector Aμ (x + ξ) from the point x + ξ on C to the neighbouring point x + ξ + dξ on C. The change in the resulting vector is δAμ (x + ξ) = −Γμαβ (x + ξ)Aα (x + ξ)dξ β ν β = −(Γμαβ (x) + Γμαβ,ρ (x)ξ ρ )(Aα (x) + Γα νσ (x)ξ )dξ μ ρ β = −Γμαβ (x)Aα (x)dξ β − Γαβ (x)Γα νσ (x)ξ dξ

−Γμαβ,ρ (x)Aα (x)ξ ρ dξ β where we have neglected third order terms like ξ∫ρ ξ σ dξ α . Now integrate this expression once around the loop C. Noting that C dξ β = 0, we get that the change in the vector on displacing it once around this loop is given upto second order terms in the loop dimension by ∫ μ δAμ (x + ξ) = ΔC A (x) = C

∫

μ α −Γαβ (x)Γνσ (x)

ξ ρ dξ β C

∫ −Γμαβ,ρ (x)Aα (x) Now noting that

∫

ξ ρ dξ β C

∫ ξ dξ = − ρ

C

σ

ξ σ dξ ρ C

∫ 1 = (ξ ρ dξ σ − ξ σ ξ ρ ) = aρσ 2 C the area tensor, it follows from the above, by anti-symmetrization that we can write μ aβσ Aα ΔC Aμ (x) = Rαβσ μ where Rαβσ is the tensor defined earlier. Note that aρσ = −aσρ .

General Relativity and Cosmology with Engineering Applications

2.5.7

33

The Ricci tensor and the Bianchi identities

The (0, 4) Riemann curvature tensor is α Rμνρσ = gμα Rνρσ

It is antisymmetric in the first two arguments and also in the last two arguments. Further, it is antisymmetric under interchange of the ordered pair of its first two arguments with the ordered pair of the last two arguments (Prove all these facts). Note that the definition β Aβ Aμ:ρ:σ − Aμ:σ:α = Rμρσ

implies that β Rβ μρσ = −Rμσρ

and hence Rβμρσ = −Rβμσρ ie the Riemann curvature tensor is antisymmetric in its last two arguments. We also note that for any two (0, 1) vector fields Aμ and Bμ , we have (Aμ Bν ):ρ:σ − (Aμ Bν ):σ:ρ = (Aμ:ρ:σ − Aμ:σ:ρ )Bν + Aμ (Bν:ρ:σ − Bν:σ:ρ ) +(Aμ:ρ Bν:σ − Aμ:σ Bν:ρ ) β β = Rμρσ Aβ Bν + Rνρσ Aμ Bβ

+(Aμ:ρ Bν:σ − Aμ:σ Bν:ρ Putting B = A in this equation, it follows that if Tμν is any symmetric (0, 2)­ tensor, ie, Tμν = Tνμ , then Tμν:ρ:σ − Tμν:σ:ρ = β β Tβν + Rνρσ Tμβ Rμρσ

We’ve already noted the following symmetries of the Riemann curvature tensor: Rμναβ = −Rνμαβ = −Rμνβα = Rαβμν We now prove the Bianchi identity: Rμναβ:ρ + Rμνβρ:α + Rμνρα:β = 0

34

General Relativity and Cosmology with Engineering Applications This can be expressed as

∑

Rμναβ:ρ = 0

[αβρ]

where [αβρ] runs over the three cyclic permutations of the same indices. This is proved by calculating (Aν:α:β − Aν:β:α ):ρ +(Aν:β:ρ − Aν:ρ:β ):α +(Aν:ρ:α − Aν:α:ρ ):β To calculate this use the formula for Tμν:ρ:σ − Tμν:σ:ρ for a (0, 2)-tensor Tμν and apply it to Tμν = Aμ:ν . Exercise: Evaluate the above quantity and hence prove the Bianchi identi­ ties. Solution: Bianchi identities for the Riemann curvature tensor. First we show that ∑ β =0 Rμρσ (μρσ)

where (μρσ) runs over the three cyclic permutations of this ordered triplet of indices. To prove this, consider β Aβ = Aμ:ρ:σ − Aμ:σ:ρ Rμρσ

Using this, we get

∑

β Rμρσ Aβ =

(μρσ)

(Aμ:ρ:σ − Aμ:σ:ρ ) + (Aρ:σ:μ − Aρ:μ:σ ) + (Aσ:μ:ρ − Aσ:ρ:μ ) = (Aμ:ρ − Aρ:μ ):σ + (Aρ:σ − Aσ:ρ ):μ +(Aσ:μ − Aμ:σ ):ρ = Tμρ:σ + Tρσ:μ + Tσμ:ρ where Tμρ = Aμ:ρ − Aρ:μ = Aμ,ρ − Aρ,μ is a skew symmetric tensor. We get using the standard formula for the covariant derivative of a tensor, α Tμρ:σ = Tμρ,σ − Γα μσ Tαρ − Γρσ Tμα

Noting that

∑

Tμρ,σ = 0

μρσ)

gives us finally −

∑ (μρσ)

β Rμρσ Aβ =

General Relativity and Cosmology with Engineering Applications ∑

α Γα μσ Tαρ + Γρσ Tμα

(μρσ) α = Γα μσ Tαρ + Γρσ Tμα α α +Γρμ Tασ + Γσμ Tρα α +Γα σρ Tαμ + Γμρ Tσα = 0

on using α Tμν = −Tνμ , Γα μν = Γνμ

This proves the desired Bianchi identity. Now we prove ∑ β Rμνρ:σ =0 (νρσ)

For this, we compute β Aβ ):σ = Aμ:ν:ρ:σ − Aμ:ρ:ν:σ = (Rμνρ β β Aβ + Rμνρ Aβ:σ Rμνρ:σ

So the desired claim will be proved if we can show that ∑ (Aμ:ν:ρ:σ − Aμ:ρ:ν:σ ) (νρσ)

=

∑

β Rμνρ Aβ:σ

(νρσ)

Now,

∑

(Aμ:ν:ρ:σ − Aμ:ρ:ν:σ )

(νρσ)

= (Aμ:ν:ρ:σ + Aμ:ρ:σ:ν + Aμ:σ:ν:ρ ) −(Aμ:ρ:ν:σ + Aμ:σ:ρ:ν + Aμ:ν:σ:ρ ) = Tμν:ρ:σ − Tμν:σ:ρ +Tμρ:σ:ν − Tμρ:ν:σ +Tμσ:ν:ρ − Tμσ:ρ:ν where Tμν = Aμ:ν It follows that

∑

(Aμ:ν:ρ:σ − Aμ:ρ:ν:σ )

(νρσ) β β = Rμρσ Tβν + Rνρσ Tμβ β β +Rμσν Tβρ + Rρσν Tμβ

35

36

General Relativity and Cosmology with Engineering Applications β β Rμνρ Tβσ + Rσνρ Tμβ β β β = Rμρσ Tβν + Rμσν Tβρ + Rμνρ Tβσ

by the previous Bianchi identity. But this is precisely equal to ∑ β Aβ:σ Rμνρ (νρσ)

and this completes the proof of the second Bianchi identity. We now define the Ricci tensor by α Rμν = Rμνα

and the curvature scalar by R = Rμμ and note that by the Bianchi identity, (Rμν − Rg μν /2):ν = 0 (Take this as an exercise).

2.5.8

The approximate relationship between the Rieman­ nian metric and the Newtonian gravitational poten­ tial

Given the metric in the form dτ 2 = (1 + 2U (t, r))dt2 − dx2 − dy 2 − dz 2 The geodesic equations give d2 xr /dτ 2 ≈ −Γr00 (dt/dτ )2 where x1 = x, x2 = y, x3 = z, c = 1 and we assume that |dxr /dτ | = T r(ρJ μ (x)) Here, ρ calculated by taking the ψa' s at time zero in H while J μ (x) requires the ψa' at time t. This is obtained using the equations of motion ψa,t (t, r) = i[H, ψa (t, r)] with the commutator evaluated using the anticommutation rules for the ψa' s ' and ψa∗ s . The anticommutation rules are valid at every time t provided that all the observables are evaluated at the same time. H is conserved ie H(t) = H(0) = H since A, V are assumed to be time independent. Hence the ψa' s and ' ψa∗ s in the integral expression for H can be taken at any time t. As a first order approximation, we can take ρ = exp(−βH0 )/T r(exp(−βH0 ) where H0 is obtained by setting A = 0, V = 0 (ie equilibrium density in the absence of external forces). With this preliminary discussion about quantum electrodynamics and quan­ tum field theory, we would now like to generalize this theory when the back­ ground space- time is curved with a metric of gμν . Assume first that the metric is time independent. Then, the scalar field φ(x) satisfies the curved space-time KG wave equation √ (g μν φ,μ −g),ν + m2 φ = 0 This equation can be derived from a variational principle ∫ √ δ L −gd4 x = 0 where

L = (1/2)g μν φ,ν φ,ν − m2 φ2 /2

is the Lagrangian density. The propagator for this scalar field can be derived from the Feynman path integral ∫ ∫ √ Dφ (x, y) = φ(x)φ(y)exp(i L(z) −g(z)d4 z)Πz∈R4 dφ(z) This path integral can be evaluated formally using the formula for the covariance of a Gaussian random vector. It evaluates to Dφ (x, y) = iK −1 (x, y), √ ∂2 (g μν (x)δ 4 (x − y) −g(x)) − m2 δ 4 (x − y) μ ν ∂x ∂x Calculating this inverse kernel can be quite complicated. Hence, we indicate approximate methods for calculating it. K(x, y) = (1/2)

65

66

General Relativity and Cosmology with Engineering Applications

3.6

Energy-momentum tensor of matter with vis­ cous and thermal corrections

Assume first the special relativistic case. The energy-momentum tensor without the corrections is T μν = (ρ + p)v μ v ν − pg μν Let ΔT αβ be the correction to the tensor due to viscosity and heat conduction. We have the conservation law (T μν + ΔT μν ),ν = 0 and hence μ ((ρ + p)v ν ),ν v μ + (ρ + p)v ν v,ν − p,μ + ΔT,νμν = 0

from which, we deduce by multiplying by vμ , the mass conservation equation ((ρ + p)v ν ),ν − p,μ v μ + vμ ΔT,νμν = 0 − − − (1) Let n denote the particle number density field. Then its conservation law can be expressed as (nv μ ),μ = 0 − − − (2) The basic energy conservation equation in thermodynamics with σ denoting the entropy per particle is given by T dσ = d(ρ/n) + pd(1/n) = d((ρ + p)/n) − dp/n so that T σ,μ v μ = ((ρ + p)/n),μ v μ − p,μ v μ /n = (ρ + p),μ v μ /n − (ρ + p)n,μ v μ /n2 − p,μ v μ /n μ = (ρ + p),μ v μ /n + (ρ + p)v,μ /n − p,μ v μ /n

= ((ρ + p)v μ ),μ /n − p,μ v μ /n where in the last but one equality, we have made use of (2). Thus, T σ,μ v μ = = −(ΔT μν ),ν vμ /n or equivalently, nσ,μ v μ = −(ΔT,νμν vμ /T or using again (2), ((nσ + ΔT μν vμ /T ),ν = ΔT μν (vμ /T ),ν = ΔT μν vμ,ν /T − ΔT μν vμ T,ν /T 2

General Relativity and Cosmology with Engineering Applications The lhs can be regarded as the rate of change of entropy per unit volume of the fluid which according to the second law of thermodynamics, should be non­ negative. Now choose a space-time point P and a frame that is comoving at P, ie, the three velocity v i at P is zero. Therefore at P , v 0 = 1. Then the equation 0 = 0 and also v0,ν = 0. v μ vμ = 1 implies on differentiating w.r.t. xν that at P v,ν Evaluating the rhs of the above equation at P thus gives us ΔT i0 vi,0 /T + ΔT ij vi,j /T − ΔT 0i T,i /T 2 ≥ 0 Noting the symmetry of ΔT μν , this can be guaranteed provided we choose k ΔT ij = χ1 (T )(vi,j + vj,i ) − χ2 (T )v,k δij ,

ΔT i0 = χ3 (T )(T vi,0 − T,i ) where χj (T ), j = 1, 2, 3 are positive scalar functions of the temperature T . We now define the four tensor α μν ΔT˜ μν = χ1 (T )(v μ,ν + v ν,μ ) + χ2 (T )v,α g

the four vector Qμ = χ3 (T )T ,μ and the four tensor S μν = Qμ v ν + Qν v μ We then have at the space-time point P k ΔT˜ij = χ1 (vi,j + vj,i ) − χ2 v,k δij ,

ΔT˜i0 = −χ1 vi,0 , S ij = 0, S i0 = Qi = −χ3 T,i Now, define the four tensor ΔT μν = H μα H νβ ΔT˜αβ − Hρμ Qρ v ν − Hρν Qρ v μ where H μν = −g μν + v μ v ν Then at P , we have H i0 = 0, H ij = δij , H 00 = 0, H0i = 0, Hji = −δij , Hi0 = 0, H00 = 0, and hence at P , ΔT ij = ΔT˜ij , ΔT i0 = Qi Thus we get from the tensor character of ΔT μν that in any frame, ΔT μν = H μα H νβ ΔT˜αβ − Hρμ Qρ v ν − Hρν Qρ v μ

67

68

General Relativity and Cosmology with Engineering Applications where α μν ΔT˜μν = χ1 (T )(v μ,ν + v ν,μ ) + χ2 (T )v,α g

or equivalently, ρ αβ ΔT μν = H μα H νβ (χ1 (T )(v α,β + v β,α ) + χ2 (T )v:ρ g )

−χ3 (T )T ,ρ (Hρμ v ν + Hρν v μ ) The general relativistic equations of hydrodynamics taking viscous and thermal effects into account are T:νμν + ΔT:νμν = 0 To start with, we neglect viscous and thermal effects. Then, the momentum equation and equation of continuity of the fluid are given by ((ρ + p)v μ v ν ):ν − p,μ = 0 which give ((ρ + p)v ν ):ν v μ + (ρ + p)v ν v:μν − p,μ = 0 − − − (1) and so we get the equation of continuity as ((ρ + p)v ν ):ν − p,μ v μ = 0 − − − (2) This is the same as ((ρ + p)v ν a),ν − ap,μ v μ = 0 − − − (3) where a=

√

−g

The first few terms in the perturbation expansion of this equation are based on ρ = ρ(2) + ρ(4) + ..., p = p(4) + p(6) + ..., v r = v r(1) + v r(3) + v (r(5) + ..., v 0 = 1 + v 0(2) + v 0(4) + ... a = 1 + a(2) + a(4) + ... We get as the third order perturbative contribution to the equation of continuity, (2)

(ρ(2) v r(1) ),r + ρ,0 = 0 and as the fifth order contribution, (4)

r(3) r(1) (ρ(2) v,r + (ρ(2) a(2) ),0 + (ρ(2) v r(1) a(2) ),r − p(4) − p,0 = 0 ,r v

General Relativity and Cosmology with Engineering Applications

69

We now compute the first few approximations for the Navier-Stokes equation. These approximations are obtained by first setting up the exact general rela­ tivistic Navier-Stokes equation by substituting (2) into (1). p,ν v ν v μ + (ρ + p)v ν v:μν − p,μ = 0 − − − (1) The first few approximants to this equation taking μ = r are: O(v 4 ) ρ(2) (v k(1) (v,k

r(1)

r(1)

+ v,0

+ Γ00 ) + p(4) ,r = 0 r(2)

O(v 6 ) ρ(2) (v k(3) v,k

r(1)

r(3)

+ v k(1) v,k

r (1)

+v 0(2) (v,0

r (3)

+ Γ00 ) + v,0

+ρ(4) (v k(1) v,k

r(1)

3.7

r(2)

r(2)

+ v k(1) Γks v s(1) r(4)

+ Γ00 ))

4) + v,0 ) − g rs(2) p,s =0 r(1)

Energy-momentum tensor of the electromag­ netic field in a background curved spacetime

The action functional of the em field is given by ∫ √ SEM [A, g] = (−1/4) Fμν F μν −gd4 x One way to determine its energy-momentum tensor, is to compute it as the √ coefficient of δgμν −g when the metric is allowed to vary slightly. The reason for this is that if the action functional of the gravitational field is given by ∫ √ SG [g] = L −gd4 x then, the variational principle δg (SG [g] + SEM [A, g]) = 0 would give rise to the Einstein field equations EM Gμν = K.Tμν

√ where Gμν is the coefficient of gδgμν in δg SG [g]. Gμν is the Einstein tensor Rμν − (1/2)Rgμν and it satisfies the Bianchi identity Gμν :ν = 0. Hence, we would get conservation of the energy-momentum of the EM field (assuming absence

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General Relativity and Cosmology with Engineering Applications

√ of matter), ie, T:νEM μν = 0. Here, T EM μν is the coefficient of −gδgμν in the variation δSEM [g]. Now, we compute √ √ δg SEM [A, g] = (−1/4)Fμν F μν δ −g − ( −g/4)Fμν Fαβ δ(g μα g νβ ) √ = (−1/8)Fμν F μν −gg αβ δgαβ √ +( −g/2)Fμν Fαβ (g μα g νρ g βσ δgρσ √ = (−1/8)Fμν F μν −gg αβ δgαβ √ +( −g/2)Fμρ F μσ δgρσ Thus after removing a proportionality constant, T EM αβ = (−1/4)Fμν F μν g αβ + Fμα F μβ We now check this result for the special relativistic case, ie, when the background space-time is flat Minkowskian.

3.8

Relativistic Fermi fluid in a gravitational field

The Dirac equation in a gravitational field has the form [γ a Vaμ (x)i(∂μ + Γμ (x)) − m]ψ(x) = 0 where Γμ (x), μ = 0, 1, 2, 3 are 4 × 4 matrix valued functions of the space-time coordinates and Vaμ (x) is a tetrad basis for the metric gμν (x): g μν (x) = η ab Vaμ (x)Vbν (x) with η ab being the Minkowski metric. If Λ(x) is a local Lorentz transformation and D the spinor representation of the Lorentz group, then we get D(Λ)γ a D(Λ)−1 Vaμ i(D(Λ)∂μ D(Λ)−1 + D(Λ)Γμ D(Λ)−1 )D(Λ)ψ = 0 or equivalently, Λab γ b Vaμ i(∂μ + (D(Λ)Γμ D(Λ)−1 + D(Λ)(∂μ D(Λ)−1 ))D(Λ)ψ = 0 which is same as the Dirac equation in a gravitational field with locally Lorentz transformed tetrad V˜bμ = Λab Vaμ , and locally Lorentz transformed gravitational connection ˜ = D(Λ)Γμ D(Λ)−1 + D(Λ)(∂μ D(Λ)−1 ) Γ

General Relativity and Cosmology with Engineering Applications

71

Now we can identify the Dirac Hamiltonian in a gravitational field as HD =

Dirac Fermionic liquid in a static electromagnetic field: ∫ H0 = ψa∗ (x)((α, −i∇ + eA(x)) + βm − eV (x))ψa (x)d3 x ∫ +

3.9

V (x, y)ψa (x)∗ ψa (x)ψb (y)∗ ψb (y)d3 xd3 y

The post-Newtonian approximation

Quantities are expanded in powers of the three velocity magnitude or equiva­ lently velocity relationship in Newtonian mechanics, √in view of the mass-orbital √ v = GM/r in powers of M . Thus for any physical quantity X, we have the perturbative expansion X = X (0) + X (1) + ... where X (r) is O(v r ) = O(M r/2 ). By analogy with the Schwarzchild metric for which g00 = 1 − 2GM/rc2 and g11 = (1 − 2GM/rc2 )−1 , we expand (2)

(4)

(2m)

g00 = 1 + g00 + g00 + ... + g00

+ ...

(2) (4) (2m) + grs + ... + grs + ... grs = −δrs + grs (1)

(3)

(2m+1)

gr0 = gr0 + gr0 + ... + gr0

+ ...

The odd powers of velocity expansion of gr0 can be seen as follows. Suppose K is a one dimensional Minkowski frame with metric dτ 2 = dt2 − dx2 /c2 . If another frame K ' moves relative to K with a uniform velocity v along the x axis, then a simple Galilean transformation gives x' = x − vt, t' = t so that ' ' ' dτ 2 = dt 2 − (dx' + vdt' )2 /c2 = dt 2 (1 − v 2 /c2 ) − dx 2 − 2vdt' dx' /c2 which ' ' shows that g00 contains only zeroth and second powers of the velocity, and g10 contains only the first power of the velocity. Now substituting this perturbative expansion of the metric into the geodesic equation gives the following: d2 xr /dτ 2 = d/dτ ((dt/dτ )dxr /dt) = (d2 t/dτ 2 )dxr /dt + (dt/dτ )2 d2 xr /dt2 The exact geodesic equation can be expressed as with v k = dxk /dt: r (dt/dτ )2 dv r /dt + (d2 t/dτ 2 )v r + (dt/dτ )2 Γkm v k v m + 2(dt/dτ )2 Γr0m v m

72

General Relativity and Cosmology with Engineering Applications +(dt/dτ )2 Γr00 = 0 or equivalently, r dv r /dt + Γrkm v k v m + (t,τ τ /t2,τ )v r + 2Γr0m v m + Γ00 =0

We have Γrkm = g r0 Γ0km + g rs Γskm Now, g μν gνρ = δρμ implies the following:

g 00 g0r + g 0s gsr = 0, g 00 g00 + g 0r g0r = 1, g r0 g0s + g rk gks = δsr = δrs

Thus, writing

g 0r = g 0r(1) + g (0r)(3) + ..., g 00 = 1 + g (00)(2) + g (00)4 + ..., g rs = −δrs + g (rs)(2) + g (rs(4) + ...

we get

(1)

(1 + g (00)2 + g (00)4 + ...)(g0r + g (0r(3) + ...)+ (2) + ...) = 0 (g 0s(1) + g (0s(3) + ...)(−δsr + grs

so that

g (r0)(1) = gr0(1) , (1)

(2) g (0s)(1) grs − g (r0(3) + g (00)(2) g0r + g (r0(3) = 0 (1)

(1)

In particular, if we assume that gr0 = 0, then it follows that gr0 = 0 and (3) g (r0(3) = gr0 . We shall henceforth make such an assumption which is in agree­ ment with the Schwarzchild metric, according to which g00 and g11 contain only upto O(v 2 ) = O(M ) terms while the non-diagonal metric components identi­ cally vanish. We then get from g 00 g00 + g 0r g0r = 1, g r0 g0s + g rk gks = δrs the following: (2)

(4)

(1 + g (00(2) + g (00(4) + ...)(1 + g00 + g00 + ..) (3)

+(g (0r(3) + g (0r(5) + ...)(g0r + ...) = 1 and hence

(2)

g (00(2) = −g00 ,

General Relativity and Cosmology with Engineering Applications (4)

(2)

g (00(4) + g00 + g (00(2) g00 = 0 implying thereby that

(4)

(2)

g (00(4) = −g00 + (g00 )2 , and further, (2)

(4)

(−δrk + g rk((2) + g (rk(4) + ...)(−δks + gks + gks + ...) = δrs which yields

(2) (2)

(2) (4) , −g (rs(4) − grs − grk gks = 0 g (rs(2) = −grs

so that

(2) (2)

(4) − grk gks g (rs(4) = −grs

We have the following perturbation expansions for the Christoffel symbols: (2)

(4)

(3)

(5)

Γrkm = Γrkm + Γrkm + .. Γ0km = Γ0km + Γ0km + ... because

(2)

(4)

gkm = −δkm + gkm + gkm + ... (2)

(2)

gkm,0 = (gkm ),0 + (gkm ),0 + ... and since one time derivative increases the perturbation order by one, we have (2)

(gkm,0 )(3) = (gkm ),0 etc. Also,

(3)

(5)

g0k,m = (g0k ),m + (g0k ),m + ... and since spatial derivatives do not change the perturbation order, we have (3)

(g0k ),m = (g0k,m )(3) etc. Remark: Roman indices like i, j, k, m, r, s, l denote spatial components ie they assume the values 1, 2, 3 only while Greek indices like μ, ν, α, β, ρ, σ, γ, δ denote space-time components, ie, they assume all four values 0, 1, 2, 3. We have further, (3) (5) Γrm0 = Γrm0 + Γrm0 + ... and

(2)

(4)

Γr00 = Γr00 + Γr00 + ... For example, 1 (2) (2) (2) (2) ),k − (gkm ),r ) Γrkm = ( )((grk ),m + (grm 2 (2)

(2)

(2)

= (1/2)(grk,m + grm,k − gkm,r )

73

74

General Relativity and Cosmology with Engineering Applications (3)

(3)

(3)

(2) Γrm0 = (1/2)((grm ),0 + (gr0 ),m − (gm0 ),r ) (2)

(2)

Γr00 = (1/2)(−(g00 ),r ) (4)

(3)

(4)

Γr00 = (1/2)(2(gr0 ),0 − (g00 ),r ) (3)

(5)

Γ000 = Γ000 + Γ000 + ... where

(3)

(2)

Γ000 = (1/2)(g00 ),0 etc.

(2)

(4)

Γ00r = Γ0r0 = Γ00r + Γ00r + ... where

(2)

(2)

(4)

(4)

Γ00r = (1/2)(g00 ),r Γ00r = (1/2)(g00 ),r For the Christoffel symbols of the second kind, we likewise have the perturbation expansions Γrkm = g r0 Γ0km + g rs Γskm r(2)

r(4)

= Γkm + Γkm + ... where

(2)

r(2)

Γkm = −Γrkm , (4)

r(4)

(2)

Γkm = −Γrkm + g rs(2) Γskm The equations of motion of a particle assume the perturbative form r dv r /dt + Γrkm v k v m + (t,τ τ /t2,τ )v r + 2Γ0m v m + Γr00 = 0

Now,

t,τ τ /t2,τ = −d/dt(log(τ,t ) = −τ,tt /τ,t

Now,

τ,t = (g00 + 2g0r v r + grs v r v s )1/2 (2)

(4)

(3)

(2) r s 1/2 = (1 + g00 + g00 + 2g0r v r − v 2 + grs v v )

with neglect of O(v 6 ) and higher terms. So, (2)

(4)

(3)

(2) log(τ,t ) = 1 + g00 /2 + g00 /2 + g0r v r − v 2 /2 + grs (2)

+(−1/8)(g00 )2 + (−1/8)v 4 + O(v 5 ) Thus, in particular, (log(τ,t )),t = (2)

(2)

k (1/2)((g00 ),0 + (g00 ),r v r − v k v,t + O(v 4 )

75

General Relativity and Cosmology with Engineering Applications With neglect of O(v 7 ) terms, our equations of motion are r(2)

r(4)

r(3)

r(5)

r(2)

(r(4)

(r(6)

dv r /dt+Γkm +Γkm )v k v m −(log(τ,t )),t v r +2(Γ0m +Γkm )v m +Γ00 +Γ00 +Γ00

=0

while with neglect of O(v 5 ) terms, the equations of motion are r(2)

r(2)

r + Γ00 + Γkm v k v m v,0 (2)

(2)

k r +[(1/2)((g00 ),0 + (g00 ),k v k − v k v,0 ]v r(3)

r(2)

+2Γ0m v m + Γ00 = 0 These constitute the post-Newtonian equations of celestial mechanics. We now derive the post-Newtonian equations of hydrodynamics. The energy momentum tensor of matter taking into account viscous and thermal effects is given by T μν + ΔT μν where T μν = (ρ + p)v μ v ν − pg μν and ΔT μν = H μα H νβ ΔT˜αβ − Hρμ Qρ v ν − Hρν Qρ v μ where H μν = −g μν + v μ v ν , Qμ = χ3 (T )T ,μ = χ3 (T )g μν T,ν and ˜ T αβ Δ is as computed in the next paragraph.

3.10

Energy-momentum tensor of matter with viscous and thermal corrections

The main idea here is to start with the energy-momentum tensor of the matter fluid taking into account an unknown correction to this tensor due to viscous and thermal effects, the conservation law of the number of particles (Baryon number conservation) and the first law of thermodynamics using the entropy per particle as a measure to calculate the heat energy input to that particle, to arrive at a differential equation for the rate of entropy increase per unit volume of the fluid in terms of the unknown energy-momentum tensor correction. Then, we make use of the second law of thermodynamics that for an adiabatic fluid, the entropy in a given volume can only increase with time to arrive at a general form for the energy-momentum tensor correction due to viscous and thermal effects. This final corrected energy-momentum tensor is used in calculating the

76

General Relativity and Cosmology with Engineering Applications evolution of inhomogeneities like galaxies in our universe by using a linearized form of the Einstein field equations for the perturbations in the metric, the velocity field and the density field. Assume first the special relativistic case. The energy-momentum tensor without the corrections is T μν = (ρ + p)v μ v ν − pg μν Let ΔT αβ be the correction to the tensor due to viscosity and heat conduction. We have the conservation law (T μν + ΔT μν ),ν = 0 and hence μ − p,μ + ΔT,νμν = 0 ((ρ + p)v ν ),ν v μ + (ρ + p)v ν v,ν

from which, we deduce by multiplying by vμ , the mass conservation equation ((ρ + p)v ν ),ν − p,μ v μ + vμ ΔT,νμν = 0 − − − (1) Let n denote the particle number density field. Then its conservation law can be expressed as (nv μ ),μ = 0 − − − (2) The basic energy conservation equation in thermodynamics with σ denoting the entropy per particle is given by T dσ = d(ρ/n) + pd(1/n) = d((ρ + p)/n) − dp/n so that T σ,μ v μ = ((ρ + p)/n),μ v μ − p,μ v μ /n = (ρ + p),μ v μ /n − (ρ + p)n,μ v μ /n2 − p,μ v μ /n μ = (ρ + p),μ v μ /n + (ρ + p)v,μ /n − p,μ v μ /n

= ((ρ + p)v μ ),μ /n − p,μ v μ /n where in the last but one equality, we have made use of (2). Thus, T σ,μ v μ = = −(ΔT μν ),ν vμ /n or equivalently, nσ,μ v μ = −(ΔT,νμν vμ /T or using again (2), ((nσ + ΔT μν vμ /T ),ν = ΔT μν (vμ /T ),ν = ΔT μν vμ,ν /T − ΔT μν vμ T,ν /T 2

General Relativity and Cosmology with Engineering Applications

77

The lhs can be regarded as the rate of change of entropy per unit volume of the fluid which according to the second law of thermodynamics, should be non­ negative. Now choose a space-time point P and a frame that is comoving at P, ie, the three velocity v i at P is zero. Therefore at P , v 0 = 1. Then the equation 0 = 0 and also v0,ν = 0. v μ vμ = 1 implies on differentiating w.r.t. xν that at P v,ν Evaluating the rhs of the above equation at P thus gives us ΔT i0 vi,0 /T + ΔT ij vi,j /T − ΔT 0i T,i /T 2 ≥ 0 Noting the symmetry of ΔT μν , this can be guaranteed provided we choose k ΔT ij = χ1 (T )(vi,j + vj,i ) − χ2 (T )v,k δij ,

ΔT i0 = χ3 (T )(T vi,0 − T,i ) where χj (T ), j = 1, 2, 3 are positive scalar functions of the temperature T . We now define the four tensor α μν ΔT˜ μν = χ1 (T )(v μ,ν + v ν,μ ) + χ2 (T )v,α g

the four vector Qμ = χ3 (T )T ,μ and the four tensor S μν = Qμ v ν + Qν v μ We then have at the space-time point P k ΔT˜ij = χ1 (vi,j + vj,i ) − χ2 v,k δij ,

ΔT˜i0 = −χ1 vi,0 , S ij = 0, S i0 = Qi = −χ3 T,i Now, define the four tensor ΔT μν = H μα H νβ ΔT˜αβ − Hρμ Qρ v ν − Hρν Qρ v μ where H μν = −g μν + v μ v ν Then at P , we have H i0 = 0, H ij = δij , H 00 = 0, H0i = 0, Hji = −δij , Hi0 = 0, H00 = 0, and hence at P , ΔT ij = ΔT˜ij , ΔT i0 = Qi Thus we get from the tensor character of ΔT μν that in any frame, ΔT μν = H μα H νβ ΔT˜αβ − Hρμ Qρ v ν − Hρν Qρ v μ

78

General Relativity and Cosmology with Engineering Applications where α μν ΔT˜μν = χ1 (T )(v μ,ν + v ν,μ ) + χ2 (T )v,α g

or equivalently, ρ αβ ΔT μν = H μα H νβ (χ1 (T )(v α,β + v β,α ) + χ2 (T )v:ρ g )

−χ3 (T )T ,ρ (Hρμ v ν + Hρν v μ ) The general relativistic equations of hydrodynamics taking viscous and thermal effects into account are T:νμν + ΔT:νμν = 0 To start with, we neglect viscous and thermal effects. Then, the momentum equation and equation of continuity of the fluid are given by ((ρ + p)v μ v ν ):ν − p,μ = 0 which give ((ρ + p)v ν ):ν v μ + (ρ + p)v ν v:μν − p,μ = 0 − − − (1) and so we get the equation of continuity as ((ρ + p)v ν ):ν − p,μ v μ = 0 − − − (2) This is the same as ((ρ + p)v ν a),ν − ap,μ v μ = 0 − − − (3) where a=

√

−g

The first few terms in the perturbation expansion of this equation are based on ρ = ρ(2) + ρ(4) + ..., p = p(4) + p(6) + ..., v r = v r(1) + v r(3) + v (r(5) + ..., v 0 = 1 + v 0(2) + v 0(4) + ... a = 1 + a(2) + a(4) + ... We get as the third order perturbative contribution to the equation of continuity, (2)

(ρ(2) v r(1) ),r + ρ,0 = 0 and as the fifth order contribution, (4)

r(3) r(1) (ρ(2) v,r + (ρ(2) a(2) ),0 + (ρ(2) v r(1) a(2) ),r − p(4) − p,0 = 0 ,r v

General Relativity and Cosmology with Engineering Applications

79

We now compute the first few approximations for the Navier-Stokes equation. These approximations are obtained by first setting up the exact general rela­ tivistic Navier-Stokes equation by substituting (2) into (1). p,ν v ν v μ + (ρ + p)v ν v:μν − p,μ = 0 − − − (1) The first few approximants to this equation taking μ = r are: O(v 4 ) ρ(2) (v k(1) (v,k

r(1)

r(1)

+ v,0

+ Γ00 ) + p(4) ,r = 0 r(2)

O(v 6 ) ρ(2) (v k(3) v,k

r(1)

r(3)

+ v k(1) v,k

r (1)

+v 0(2) (v,0

r (3)

+ Γ00 ) + v,0

+ρ(4) (v k(1) v,k

r(1)

3.11

r(2)

r(2)

+ v k(1) Γks v s(1) r(4)

+ Γ00 ))

4) + v,0 ) − g rs(2) p,s =0 r(1)

Energy-momentum tensor of the electro­ magnetic field in a background curved spacetime

The action functional of the em field is given by ∫ √ SEM [A, g] = (−1/4) Fμν F μν −gd4 x One way to determine its energy-momentum tensor, is to compute it as the √ coefficient of δgμν −g when the metric is allowed to vary slightly. The reason for this is that if the action functional of the gravitational field is given by ∫ √ SG [g] = L −gd4 x then, the variational principle δg (SG [g] + SEM [A, g]) = 0 would give rise to the Einstein field equations EM Gμν = K.Tμν

√ where Gμν is the coefficient of gδgμν in δg SG [g]. Gμν is the Einstein tensor Rμν − (1/2)Rgμν and it satisfies the Bianchi identity Gμν :ν = 0. Hence, we would get conservation of the energy-momentum of the EM field (assuming absence

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General Relativity and Cosmology with Engineering Applications

√ of matter), ie, T:νEM μν = 0. Here, T EM μν is the coefficient of −gδgμν in the variation δSEM [g]. Now, we compute √ √ δg SEM [A, g] = (−1/4)Fμν F μν δ −g − ( −g/4)Fμν Fαβ δ(g μα g νβ ) √ = (−1/8)Fμν F μν −gg αβ δgαβ √ +( −g/2)Fμν Fαβ (g μα g νρ g βσ δgρσ √ = (−1/8)Fμν F μν −gg αβ δgαβ √ +( −g/2)Fμρ F μσ δgρσ Thus after removing a proportionality constant, T EM αβ = (−1/4)Fμν F μν g αβ + Fμα F μβ We now check this result for the special relativistic case, ie, when the background space-time is flat Minkowskian.

3.12

Relativistic Fermi fluid in a gravitational field. The Dirac equation in a gravitational field has the form [γ a Vaμ (x)i(∂μ + Γμ (x)) − m]ψ(x) = 0

where Γμ (x), μ = 0, 1, 2, 3 are 4 × 4 matrix valued functions of the space-time coordinates and Vaμ (x) is a tetrad basis for the metric gμν (x): g μν (x) = η ab Vaμ (x)Vbν (x) with η ab being the Minkowski metric. If Λ(x) is a local Lorentz transformation and D the spinor representation of the Lorentz group, then we get D(Λ)γ a D(Λ)−1 Vaμ i(D(Λ)∂μ D(Λ)−1 + D(Λ)Γμ D(Λ)−1 )D(Λ)ψ = 0 or equivalently, Λab γ b Vaμ i(∂μ + (D(Λ)Γμ D(Λ)−1 + D(Λ)(∂μ D(Λ)−1 ))D(Λ)ψ = 0 which is same as the Dirac equation in a gravitational field with locally Lorentz transformed tetrad V˜bμ = Λab Vaμ , and locally Lorentz transformed gravitational connection ˜ = D(Λ)Γμ D(Λ)−1 + D(Λ)(∂μ D(Λ)−1 ) Γ

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Now we can identify the Dirac Hamiltonian in a gravitational field as HD =

Dirac Fermionic liquid in a static electromagnetic field: ∫ H0 = ψa∗ (x)((α, −i∇ + eA(x)) + βm − eV (x))ψa (x)d3 x ∫ +

3.13

V (x, y)ψa (x)∗ ψa (x)ψb (y)∗ ψb (y)d3 xd3 y

The post-Newtonian approximation

Quantities are expanded in powers of the three velocity magnitude or equiva­ lently velocity relationship in Newtonian mechanics, √in view of the mass-orbital √ v = GM/r in powers of M . Thus for any physical quantity X, we have the perturbative expansion X = X (0) + X (1) + ... where X (r) is O(v r ) = O(M r/2 ). By analogy with the Schwarzchild metric for which g00 = 1 − 2GM/rc2 and g11 = (1 − 2GM/rc2 )−1 , we expand (2)

(4)

(2m)

g00 = 1 + g00 + g00 + ... + g00

+ ...

(2) (4) (2m) + grs + ... + grs + ... grs = −δrs + grs (1)

(3)

(2m+1)

gr0 = gr0 + gr0 + ... + gr0

+ ...

The odd powers of velocity expansion of gr0 can be seen as follows. Suppose K is a one dimensional Minkowski frame with metric dτ 2 = dt2 − dx2 /c2 . If another frame K ' moves relative to K with a uniform velocity v along the x axis, then a simple Galilean transformation gives x' = x − vt, t' = t so that ' ' ' dτ 2 = dt 2 − (dx' + vdt' )2 /c2 = dt 2 (1 − v 2 /c2 ) − dx 2 − 2vdt' dx' /c2 which ' ' contains only zeroth and second powers of the velocity, and g10 shows that g00 contains only the first power of the velocity. Now substituting this perturbative expansion of the metric into the geodesic equation gives the following: d2 xr /dτ 2 = d/dτ ((dt/dτ )dxr /dt) = (d2 t/dτ 2 )dxr /dt + (dt/dτ )2 d2 xr /dt2 The exact geodesic equation can be expressed as with v k = dxk /dt: r (dt/dτ )2 dv r /dt + (d2 t/dτ 2 )v r + (dt/dτ )2 Γkm v k v m + 2(dt/dτ )2 Γr0m v m

82

General Relativity and Cosmology with Engineering Applications +(dt/dτ )2 Γr00 = 0 or equivalently, r dv r /dt + Γrkm v k v m + (t,τ τ /t2,τ )v r + 2Γr0m v m + Γ00 =0

We have Γrkm = g r0 Γ0km + g rs Γskm Now, g μν gνρ = δρμ implies the following:

g 00 g0r + g 0s gsr = 0, g 00 g00 + g 0r g0r = 1, g r0 g0s + g rk gks = δsr = δrs

Thus, writing

g 0r = g 0r(1) + g (0r)(3) + ..., g 00 = 1 + g (00)(2) + g (00)4 + ..., g rs = −δrs + g (rs)(2) + g (rs(4) + ...

we get

(1)

(1 + g (00)2 + g (00)4 + ...)(g0r + g (0r(3) + ...)+ (2) + ...) = 0 (g 0s(1) + g (0s(3) + ...)(−δsr + grs

so that

g (r0)(1) = gr0(1) , (1)

(2) g (0s)(1) grs − g (r0(3) + g (00)(2) g0r + g (r0(3) = 0 (1)

(1)

In particular, if we assume that gr0 = 0, then it follows that gr0 = 0 and (3) g (r0(3) = gr0 . We shall henceforth make such an assumption which is in agree­ ment with the Schwarzchild metric, according to which g00 and g11 contain only upto O(v 2 ) = O(M ) terms while the non-diagonal metric components identi­ cally vanish. We then get from g 00 g00 + g 0r g0r = 1, g r0 g0s + g rk gks = δrs the following: (2)

(4)

(1 + g (00(2) + g (00(4) + ...)(1 + g00 + g00 + ..) (3)

+(g (0r(3) + g (0r(5) + ...)(g0r + ...) = 1 and hence

(2)

g (00(2) = −g00 ,

General Relativity and Cosmology with Engineering Applications (4)

83

(2)

g (00(4) + g00 + g (00(2) g00 = 0 implying thereby that

(4)

(2)

g (00(4) = −g00 + (g00 )2 , and further, (2)

(4)

(−δrk + g rk((2) + g (rk(4) + ...)(−δks + gks + gks + ...) = δrs which yields

(2) (2)

(2) (4) , −g (rs(4) − grs − grk gks = 0 g (rs(2) = −grs

so that

(2) (2)

(4) − grk gks g (rs(4) = −grs

We have the following perturbation expansions for the Christoffel symbols: (2)

(4)

(3)

(5)

Γrkm = Γrkm + Γrkm + .. Γ0km = Γ0km + Γ0km + ... because

(2)

(4)

gkm = −δkm + gkm + gkm + ... (2)

(2)

gkm,0 = (gkm ),0 + (gkm ),0 + ... and since one time derivative increases the perturbation order by one, we have (2)

(gkm,0 )(3) = (gkm ),0 etc. Also,

(3)

(5)

g0k,m = (g0k ),m + (g0k ),m + ... and since spatial derivatives do not change the perturbation order, we have (3)

(g0k ),m = (g0k,m )(3) etc. Remark: Roman indices like i, j, k, m, r, s, l denote spatial components ie they assume the values 1, 2, 3 only while Greek indices like μ, ν, α, β, ρ, σ, γ, δ denote space-time components, ie, they assume all four values 0, 1, 2, 3. We have further, (3) (5) Γrm0 = Γrm0 + Γrm0 + ... and

(2)

(4)

Γr00 = Γr00 + Γr00 + ... For example, 1 (2) (2) (2) (2) ),k − (gkm ),r ) Γrkm = ( )((grk ),m + (grm 2 (2)

(2)

(2)

= (1/2)(grk,m + grm,k − gkm,r )

84

General Relativity and Cosmology with Engineering Applications (3)

(3)

(3)

(2) Γrm0 = (1/2)((grm ),0 + (gr0 ),m − (gm0 ),r ) (2)

(2)

Γr00 = (1/2)(−(g00 ),r ) (4)

(3)

(4)

Γr00 = (1/2)(2(gr0 ),0 − (g00 ),r ) (3)

(5)

Γ000 = Γ000 + Γ000 + ... where

(3)

(2)

Γ000 = (1/2)(g00 ),0 etc.

(2)

(4)

Γ00r = Γ0r0 = Γ00r + Γ00r + ... where

(2)

(2)

(4)

(4)

Γ00r = (1/2)(g00 ),r Γ00r = (1/2)(g00 ),r For the Christoffel symbols of the second kind, we likewise have the perturbation expansions Γrkm = g r0 Γ0km + g rs Γskm r(2)

r(4)

= Γkm + Γkm + ... where

(2)

r(2)

Γkm = −Γrkm , (4)

r(4)

(2)

Γkm = −Γrkm + g rs(2) Γskm The equations of motion of a particle assume the perturbative form r dv r /dt + Γrkm v k v m + (t,τ τ /t2,τ )v r + 2Γ0m v m + Γr00 = 0

Now,

t,τ τ /t2,τ = −d/dt(log(τ,t ) = −τ,tt /τ,t

Now,

τ,t = (g00 + 2g0r v r + grs v r v s )1/2 (2)

(4)

(3)

(2) r s 1/2 = (1 + g00 + g00 + 2g0r v r − v 2 + grs v v )

with neglect of O(v 6 ) and higher terms. So, (2)

(4)

(3)

(2) log(τ,t ) = 1 + g00 /2 + g00 /2 + g0r v r − v 2 /2 + grs (2)

+(−1/8)(g00 )2 + (−1/8)v 4 + O(v 5 ) Thus, in particular, (log(τ,t )),t = (2) (1/2)((g00 ),0

(2)

k + (g00 ),r v r − v k v,t + O(v 4 )

85

General Relativity and Cosmology with Engineering Applications With neglect of O(v 7 ) terms, our equations of motion are r(2)

r(4)

r(3)

r(5)

r(2)

(r(4)

(r(6)

dv r /dt+Γkm +Γkm )v k v m −(log(τ,t )),t v r +2(Γ0m +Γkm )v m +Γ00 +Γ00 +Γ00

=0

while with neglect of O(v 5 ) terms, the equations of motion are r(2)

r(2)

r v,0 + Γ00 + Γkm v k v m (2)

(2)

k r +[(1/2)((g00 ),0 + (g00 ),k v k − v k v,0 ]v r(3)

r(2)

+2Γ0m v m + Γ00 = 0 These constitute the post-Newtonian equations of celestial mechanics. We now derive the post-Newtonian equations of hydrodynamics. The energy momentum tensor of matter taking into account viscous and thermal effects is given by T μν + ΔT μν where T μν = (ρ + p)v μ v ν − pg μν and ΔT μν = H μα H νβ ΔT˜αβ − Hρμ Qρ v ν − Hρν Qρ v μ where H μν = −g μν + v μ v ν , Qμ = χ3 (T )T ,μ = χ3 (T )g μν T,ν and T˜αβ =

3.14

The BCS theory of superconductivity

ψ1 (t, x), ψ2 (t, x) are the two Fermionic fields corresponding respectively to up and down spin states of the electron. They satisfy the canonical anticommuta­ tion relations {ψa (t, x), ψb (t, x' )∗ } = δab δ 3 (x − x' ) We use the notation ψa (x) for ψa (0, x) and likewise for ψa (x)∗ . The BCS Hamil­ tonian is then defined as H=

∑ ∫

ψa (x)∗ (−∇2 /2m+V (x))ψa (x)d3 x+

∫

f0 (x) < ψ1 (x)ψ2 (x) > ψ1 (x)∗ ψ2 (x)∗ d3 x

a=1,2

∫ +

f¯0 (x) < ψ2 (x)∗ ψ1 (x)∗ > ψ2 (x)ψ1 (x)d3 x

86

General Relativity and Cosmology with Engineering Applications ∫ +

(f1 (x) < ψ1 (x)∗ ψ1 (x) > ψ2 (x)∗ ψ2 (x)+f2 (x) < ψ2 (x)∗ ψ2 (x) > ψ1 (x)∗ ψ1 (x))d3 x

We write for convenience of notation Δ(x) =< ψ1 (x)ψ2 (x) > so that

Δ(x)∗ =< ψ2 (x)∗ ψ2 (x) >

Here, the quantum expectation < . > is taken w.r.t. the Gibbs density ρG = exp(−βH)/Z(β), Z(β) = T r(exp(−βH)) we note that H is a constant of the motion since it is by definition, time inde­ pendent, ie, the coefficients functions V, f0 , f1 , f2 do not explicitly depend on time. We get using the Fermionic anticommutation relations [H, ψ1 (x)] = (∇2 /2m − V (x))ψ1 (x) − f0 (x)Δ(x)ψ2 (x)∗ − f2 (x)n2 (x)ψ1 (x) where

na (x) =< ψa (x)∗ ψa (x) >, a = 1, 2

and likewise, [H, ψ2 (x)] = (∇2 /2m − V (x))ψ2 (x) + f0 (x)Δ(x)ψ1 (x)∗ − f1 (x)n1 (x)ψ2 (x) Define the following Green’s functions: G(t, x|t' , x' ) =< T (ψ1 (t, x)ψ1 (t' , x' )∗ ) >, F (t, x|t' , x' ) =< T (ψ1 (t, x)ψ2 (t' , x' )) > where T is the time ordering operator. Remark: If the Fermions are subject to a gravitational field described by a static metric tensor gμν (x), then we can approximate the energy of a such a par­ ticle due to motion and gravitational effects by considering first the Lagrangian of the particle: L = −mτ,t = −m(g00 + 2g0r v r + grs v r v s )1/2 √ 2 ≈ −m g00 (1 + g0r v r /g00 + (grs g00 − g0r g0s )v r v s /2g00 ) To express the corresponding Hamiltonian in terms of canonical coordinates and momenta, we first compute the momenta as pr = −L,vr = m(g0r + grs v s )/τ,t , pr = −pr and then the Hamiltonian using the Legendre transformation as H = pr v r − L =

General Relativity and Cosmology with Engineering Applications

87

= −m(g0r v r + grs v r v s )/τ,t + mτ,t = [mτ,t2 − m(g0r v r + grs v r v s )]/τ,t = (mg00 + mg0r v r )/τ,t = mg0μ uμ = mu0 Now, pr = mgrμ v μ /τ,t = mgrμ uμ and writing p0 = mg0μ v μ /τ,t = mg0μ uμ = mu0 so that pν = mgνμ v μ /τ,t = mgνμ uμ = muν we get pμ = g μν pν = muμ and hence pμ = muμ and in particular, H = p0 We note that the equation g μν uμ uν = 1 implies

g μν pμ pν = m2

Thus the energy p0 = H satisfies the quadratic equation g 0 p20 − 2g 0r p0 pr + g rs pr ps − m2 = 0 Solving this for p0 and replacing pr by −i∂r gives us the Hamiltonian operator H = p0 of the particle in terms of ∂r and it is this operator p0 that must be used to compute the free particle energy of the Fermi liquid: ∑ ∫ ψa (x)∗ p0 ψa (x)d3 x H0 = a=1,2

3.15

Quantum scattering theory in the presence of a gravitational field

The Dirac equation for an electron in the presence of an electromagnetic field and a gravitational field described by a tetrad Vμa (x) and a corresponding connection Γμ (x) which is a 4 × 4 matrix valued function of the space-time coordinates x is given by (Steven Weinberg, Gravitation and Cosmology) [γ a Vaμ (i∂μ + eAμ + iΓμ ) − m]ψ = 0

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General Relativity and Cosmology with Engineering Applications

This equation can be derived from a variational principle with Lagrangian den­ sity √ L = ψ ∗ αa Vaμ (i∂μ + eAμ + iΓμ )ψ −g where αa = γ 0 γ a , a = 0, 1, 2, 3 Note that α0 = 1. Unfortunately, this Lagrangian density is not real and hence we replace it by its real part: √ L = Re[(ψ ∗ αa Vaμ (i∂μ + eAμ + iΓμ )ψ) −g] Let us compute the Hamiltonian density corresponding to this Lagrangian den­ sity. The canonical momentum densities are π=

∂L = ∂ψ,0

√ (i/2) −gVa0 αaT ψ¯ π ¯=

∂L = ∂ψ¯,0

√ (−i/2) −galphaa Va0 ψ Note that the αa matrices are Hermitian. So the Hamiltonian density is ¯ T ψ¯ − L = H = πT ψ + π √ √ Var Re[psi∗ αa (i∂r ψ)] −g − Vaμ Re[ψ ∗ αa (eAμ + iΓμ )ψ] −g The first term represents the kinetic energy of the Dirac particle in curved spacetime and the second terms represents the interaction energy between the Dirac particle and the electromagnetic and gravitational field. This is the second quantized picture and can be used in the BCS theory of superconductivity. In quantum scattering theory, we are concerned with first quantized Hamiltonians. Thus, we write √ H0 = − −g(x)Var (x)αa P r , P r = −i∂r for the unperturbed energy of the incoming projectile in a background gravita­ tional field and √ V = − −gVaμ (x)αa (eAμ + iΓμ (x)) More precisely, V should be defined as the Hermitian part of the above matrix valued function of position. When we assume that the gravitational field is time independent and so is the electromagnetic field, then V becomes a matrix valued function of the spatial coordinates only while H0 becomes a vector field whose coefficients are time independent. The scattering matrix in this case is defined by S = Ω∗+ Ω− where Ω+ = limt→∞ exp(it(H0 + V )).exp(−itH0 ), Ω− = limt→−∞ exp(it(H0 + V )).exp(−itH0 )

General Relativity and Cosmology with Engineering Applications

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More generally, if H0 is time independent but V is time dependent, then one could ask the question how one defines the scattering matrix. The answer is as follows. Write H1 (t) = H0 + V (t). Then if φi is the input free particle state that gets scattered to the input scattered state ψi while ψf is the final scattered state that evolves into the free particle state ψf , then we have U (0, −T )−1 ψi − U0 (0, −T )−1 )φi → 0, T → ∞, U (T, 0)ψf − U0 (T, 0)φf → 0, T → ∞ Thus, Ω= limT →∞ U (0, −T )U0 (−T ), Ω+ = limT →∞ U (T, 0)−1 U0 (T ) and hence, the scattering matrix is defined by S = Ω∗+ Ω− = limT →∞ U0 (−T )U (T, 0)U (0, −T )U0 (−T ) ∫ T = limT →∞ exp(iT H0 ).T {exp(i V (t)dt)}.exp(−iT H0 ) −T

3.16 Maxwell’s equations in the Schwarzchild spacetime dτ 2 = α(r)dt2 − α(r)−1 dr2 − r2 (dθ2 + sin2 (θ)dφ2 ) α(r) = 1 − 2m/r, m = GM, c = 1 This is the metric of space-time. g00 = α(r), g11 = −α(r)−1 , g22 = −r2 , g33 = −r2 sin2 (θ) The contravariant electromagnetic four potential is A1 = Ar , A2 = Aθ , A3 = Aφ , A0 = V The covariant electromagnetic four potential is A0 = g00 A0 = α(r)A0 , A1 = g11 A1 = −α(r)−1 A1 , A2 = g22 A2 = −r2 A2 , A3 = g33 A3 = −r2 sin2 (θ)A3 F01 = A1,0 − A0,1 = −α−1 A1,0 − αA0,1 , F02 = A2,0 − A0,2 = −r2 A2,0 − αA0,2 ,

90

General Relativity and Cosmology with Engineering Applications F03 = A3,0 − A0,3 = −r2 sin2 (θ)A3,0 − αA0,2 F 01 = g 00 g 11 F01 = −F01 , F 02 = g 00 g 22 F02 = −αr−2 F02 , F 03 = g 00 g 33 F03 = −α(r.sin(θ))−2 F03

The Maxwell equations in the absence of current sources but in the presence of the Schwarzchild gravitationl field are √ (F μν −g),ν = 0 We list these equations below: √ √ √ √ (F 0r −g),r = (F 01 −g),1 + (F 02 −g),2 + (F 03 −g),3 = 0 and

√ √ F,r00 −g + (F rs −g),s = 0

or equivalently, r2 sin(θ)F,010 + (F 12 r2 sin(θ)),2 + F,313 r2 sin(θ) = 0 r2 sin(θ)F,020 + (F 21 r2 sin(θ)),1 + F,323 r2 sin(θ) = 0 r2 sin(θ)F,030 + (F 31 r2 sin(θ)),1 + (F 32 r2 sin(θ)),2 = 0 Remark: We wish to give meaning to F μν in terms of electric and magnetic fields. For that purpose, we consider the Minkowskian flat space-time metric and evaluate F μν using this metric. The Minkowskian metric is dτ 2 = dt2 − dr2 − r2 (dθ2 + sin2 (θ)dφ2 ) for which

g00 = 1, g11 = −1, g22 = −r2 , g3 = −r2 sin2 (θ)

Then the Cartesian components of the electromagnetic four potential Ax , Ay , Az , V and the polar components Ar , Aθ , Aφ , At are related by Ar = Ax r,x + Ay r,y + Az r,z = = Ax cos(φ)sin(θ) + Ay sin(φ)sin(θ) + Az cos(θ) which is the usual definition for the radial component of the magnetic vector potential. Aθ = Ax θ,x + Ay θ,y + Az θ,z = (A, ∇θ) which is the usual θ component of the magnetic vector potential multiplied by |∇θ| = 1/r. Finally, Aφ = (A, ∇φ)

General Relativity and Cosmology with Engineering Applications

91

which is the usual φ component of the magnetic vector potential divided by r.sin(θ). We write these relations as Ar = Ar , Aθ = r−1 Aθ , Aφ = (r.sinθ)−1 Aφ , At = V Thus, we have F01 = A1,0 − A0,1 = −Ar,0 − V,1 = −Ar,0 − V,r which is Er , the radial component of the electric field. F02 = A2,0 − A0,2 = −r2 Aθ,0 − V,θ = −rAθ,0 − V,θ = rEθ F03 = −r.sin(θ)Eφ Further,

F 01 = g 00 g 11 F01 = −Er , F 02 = g 00 g 22 F02 = −r−1 Eθ F 03 = g 00 g 33 F03 = −(r.sin(θ))−1 Eφ F12 = A2,1 − A1,2 = −(r2 Aθ ),r + Ar,θ = = −(rAθ ),r + Ar,θ = −rBφ

(using the formula for the curl in spherical polar coordinates). Thus, F 12 = g 11 g 22 F12 = (−1/r2 )F12 = Bφ /r F23 = A3,2 − A2,3 = −r2 (sin2 (θ)Aφ ),θ + (r2 Aθ ),φ = −r(sin(θ)Aφ ),θ + rAθ,φ = −r2 sin(θ)Br F 23 = g 22 g 33 F23 = (−1/r2 sin(θ))Br and Exercise: Evaluate F31 .

3.17

Some more problems in general relativity

3.17.1

Gauss and Riemann curvatures of a 2-D surface

Consider a two dimensional surface parametrized by u, v so that a general point on the surface can be expressed as r = r(u, v) = (x(u, v), y(u, v), z(u, v)) Calculate the metric on the surface in the form ds2 = |dr|2 = g11 (u, v)du2 + g22 (u, v)dv 2 + 2g12 (u, v)dudv

92

General Relativity and Cosmology with Engineering Applications Now choose a curve on the surface parametrized by t → (u(t), v(t)) or more precisely as t → r(u(t), v(t)). Calculate its curvature at t: K(t) = |d2 r/ds2 | Now consider a point say (u0 , v0 ) on the surface. Draw the unit normal n(u0 , v0 ) at this point to the surface at this point. Now consider the set of all planes con­ taining this normal and let the maximum and minimum curvatures of the curves at (u0 , v0 ) in which this plane intersects the surface be K1 and K2 respectively. Determine the Gauss Curvature of the surface at (u0 , v0 ) defined by K1 K2 . Also determine the components of the Riemann-Christoffel curvature tensor of the surface at (u0 , v0 ). Remark: Consider the curve (u(s), v(s)) on the surface or equivalently, r(s) = r(u(s), v(s)) parametrized by the curve length parameter s, ie ds2 = |dr|2 . Assume that this curve is the intersection of the surface and a plane passing through the normal n to the surface at (u(s), v(s)). Then '

'

dr/ds = ru u' + rv v ' , d2 r/ds2 = ruu u 2 + rvv v 2 + 2ruv u' v ' + ru u'' + rv v '' where u' = du/ds, u'' = d2 u/ds2 etc. Show that (dr/ds, d2 r/ds2 ) = 0 and hence d2 r/ds2 is a normal to the curve at r(s). Equivalently, u' (ru , d2 r/ds2 ) + v ' (rv , d2 r/ds2 ) = 0 Since (n, ru ) = (n, rv ) = 0, we get '

'

K(u' , v ' ) = (n, d2 r/ds2 ) = (n, ruu )u 2 + (n, rvv )v 2 + 2(n, ruv )u' v ' Now determine the maximum and minimum values of K(u' , v ' ) as u' , v ' vary in ' ' such a way that g11 u 2 + g22 v 2 + 2g12 u' v ' = 1.

3.17.2

Parallel displacement on a 2-D surface

Compute the formulas for parallel displacement on a two dimensional surface specified by (u, v) → r(u, v) ∈ R3 . Specifically, for a vector on the surface (Au , Av ) defined by A(u, v) = Au (u, v)ru + Av (u, v)rv Compute u u δAu = Γuuu Au du + Γuuv Av du + Γvv Av dv + Γuv Au dv, v v δAv = Γvuu Au du + Γvuv Av du + Γvv Av dv + Γuv Au dv,

General Relativity and Cosmology with Engineering Applications

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3.17.3 Einstein field equations in a homogeneous and isotropic universe Calculate the components R00 , R11 , R22 , R33 of the Ricci tensor for the RobertsonWalker metric dτ 2 = dt2 −

S 2 (t) dr2 − S 2 (t)r2 (dθ2 + sin2 (θ)dφ2 ) 1 − kr2

ie g00 = 1, g11 = −S 2 (t)/(1 − kr2 ), g22 = −S 2 (t)r2 , g33 = −S 2 (t)r2 sin2 (θ) / ν. Calculate the equations resulting and show further that Rμν = 0 for μ = from the Einstein field equations Rμν = K(Tμν − T gμν /2) where K = −8πG and Tμν = (ρ(t) + p(t))Vμ Vν − p(t)gμν where ρ(t), p(t) are only functions of t and V0 = 1, Vk = 0, k = 1, 2, 3 (Comoving system) and prove that the Einstein field equations yield only two independent equations for the three functions of time S(t), ρ(t), p(t). The third independent equation is obtained by specifying the equation of state of the fluid p = f (ρ(t)).

3.17.4

Klein-Gordon Equation in a Robertson-Walker Uni­ verse

For the Robertson-Walker metric specified in the previous problem, write down the Klein-Gordon wave equation for the scalar wave field ψ(x) = ψ(t, r) in an electromagnetic field Aμ (x): √ √ [(g μν (x)(ψ,μ (x) + ieAμ (x)ψ(x))) −g(x)],ν + m2 ψ(x) −g(x) = 0 and solve it approximately assuming that k = 0, S(t) = 1+δS(t), δ being a small perturbation parameter and Aμ (x) is the same as δ.Aμ (x), ie, the background radiation field is weak, and of the same order as the rate of expansion of the universe.

3.17.5

Shift in the atomic energy levels in the presence of a blackhole gravitational field

Solve the Klein-Gordon equation for a quantum mechanical particle in the Schwarzchild metric using perturbation theory. The equation can be formu­ lated as in the previous exercise. Also solve it in when there is an external

94

General Relativity and Cosmology with Engineering Applications electrostatic potential field V (x) = V (r, θ, φ). This problem enables us to deter­ mine the shift in the energy levels of an atom when it interacts with the strong gravitational field produced by a blackhole.

3.17.6

Random perturbations of metric and the energy momentum tensor of matter and radiation in the Einstein field equations

Consider a solution to the Einstein field equations in the presence of the energy momentum tensor Tμν : Rμν = K(Tμν − T gμν /2) Note that this automatically implies that T:νμν = 0 Now, suppose T μν suffers a small random perturbation δT μν such that δT:νμν = 0. Then, give an algorithm for calculating the change in the metric tensor δgμν upto first order and also its autocorrelation function E(δgμν (x)δgαβ (x' )) in terms of the autocorrelation function E(δTμν (x)δTαβ (x' )) of the energy momentum ten­ sor perturbations δTμν (x). Explain how you would apply this idea to simulta­ neous matter and electromagnetic radiation perturbations, ie, perturbations in the energy momentum tensor of matter plus radiation. Recall that the energymomentum tensor of matter is T μν = (ρ + p)V μ V ν − pg μν and the energy-momentum tensor of radiation is S μν = (−1/4)Fαβ F αβ g μν + F μα Fαν where Fμν = Aν,μ − Aμ,ν

3.17.7

Discretized fluid dynamical and MHD equations in curved background metric; Formulation of the fil­ tering equations for velocity field estimation from noisy sparse pixel set measurements

Write down the fluid dynamical equations of a fluid in a curved space-time background metric gμν (x) in the form ((ρ + p)v μ v ν ):ν − g μν p,ν = f μ (x)

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General Relativity and Cosmology with Engineering Applications

where f μ is a random external four force. Assuming an equation of state p = ψ(ρ) and the equation gμν v μ v ν = 1 cast this equation in the form r k (x) = f0r (x, v k (x), v,m (x), ρ(x), ρ,m (x)), r = 1, 2, 3, v,0 k ρ,0 (x) = χ(x, v k (x), v,m (x), ρ(x), ρ,m (x)),

in the noiseless case. Now discretize space over a finite region [−N δ, N δ]3 into a grid of size (2N + 1) × (2N + 1) × (2N + 1) and denote by v r (t), the (2N + 1)3 × 1 vector N ∑ v r (t) = v r (t, r1 δ, r2 δ, r3 δ)e(r1 ) ⊗ e(r2 ) ⊗ e(r3 ) r1 ,r2 ,r3 =−N

and by ρ(t) the (2N + 1)3 × 1 vector ρ(t) =

N ∑

ρ(t, r1 δ, r2 δ, r3 δ)e(r1 ) ⊗ e(r2 ) ⊗ e(r3 )

r1 ,r2 ,r3 =−N

where e(r) is the 2N + 1 × 1 vector having a one at its N + r + 1th position and a zero at all its other positions. Now replace spatial partial derivatives by finite differences, for example v,r1 will become ∑ δ −1 (v r (t, (r1 + 1)δ, r2 δ, r3 δ) − v r (t, r1 δ, r2 δ, r3 δ))e(r1 ) ⊗ e(r2 ) ⊗ e(r3 ) r1 ,r2 ,r3

∑

=

δ −1 e(r1 , r2 , r3 )(e(r1 + 1, r2 , r3 ) − e(r1 , r2 , r3 ))T v r (t) = A1 v r (t)

r1 ,r2 ,r3

where e(r1 , r2 , r3 ) = e(r1 ) ⊗ e(r2 ) ⊗ e(r3 ) and A1 =

∑

δ −1 e(r1 , r2 , r3 )(e(r1 + 1, r2 , r3 ) − e(r1 , r2 , r3 ))T ∈ R(2N +1)

3

×(2N +1)3

r1 ,r2 ,r3

Using this idea, cast the equations of motion in the form '

v r (t) = F r (t, v m (t), m = 1, 2, 3, ρ(t)) + Gkl (t, v m (t), m = 1, 2, 3, ρ(t))dBl (t)/dt ρ' (t) = Ψ(t, v m (t), m = 1, 2, 3, ρ(t)) + H kl (t, v m (t), m = 1, 2, 3, ρ(t))dBl (t)/dt in the noisy case. Now assume that velocity and pressure (or equivalently, density) are measured at only certain spatial point, say at (r1 δ, r2 δ, r3 δ) where (r1 , r2 , r3 ) ∈ E where E ⊂ {−N, −N + 1, ..., N − 1, N }3 . These measurements can be represented as dZ(t) = Kv(t)dt + σV dV (t), r = 1, 2, 3

96

General Relativity and Cosmology with Engineering Applications where K r is a μ(E) × 3(2N + 1)3 matrix consisting of only ones and zeroes and ( 1 ) v (t) 3 v(t) = ( v 2 (t) ) ∈ R3(2N +1) ×1 v 3 (t) and μ(E) denotes the number of points in E. V (t) is a Rμ(E) -valued Brownian motion. Write down the EKF filtering equations for estimating ρ(t), v(t) based on measurements Z(s), s ≤ t. Remark: Suppose T μν and S μν denote respectively the energy-momentum tensors of matter and radiation and f μν is the energy-momentum tensor of the random external field. Then, in accordance with Bianchi’s identity for the Einstein field equations, we should have (T μν + S μν + f μν ):ν = 0 and hence the external random force field should have the form f μ = −(S μν + f μν ):ν Now if there are external charges and currents in the picture described by a four current density J μ , then the Maxwell equations F:νμν = J μ should be satisfies. This would imply that μν = F μν Jν S:ν

and hence μν f μ = −F μν Jν − f:ν

If the fluid is a conducting fluid with conductivity σ or more generally a con­ μ , then the classical equation ductivity tensor σναβ J = σ(E + v × B) should be replaced by μ F αβ v ρ J μ = σαβρ

and hence the general relativistic MHD equations become μ F αβ v ρ , F:νμν = σαβρ μν T:νμν = −F μν σναβρ F αβ v ρ − f:ν

where T μν = (ρ + p)v μ v ν − pg μν Now write down these MHD equations taking into account viscous and thermal conduction effects.

General Relativity and Cosmology with Engineering Applications

3.17.8

97

Joint Einstein-Maxwell-Dirac equations

Write down the joint equations satisfied by the Dirac wave function ψ(x) in a gravitational field and an electromagnetic field taking into account the spinor connection for curved space-time Γμ (x) in terms of a tetrad basis eaμ (x), the Maxwell equations satisfied by the electromagnetic four potential Aμ (x) in the presence of the Dirac four current and the Einstein field equations in the pres­ ence of matter with an energy-momentum tensor dictated by the Dirac field. Specifically, derive all these equations from the total Lagrangian density L = L1 + L2 + L3 , √ β α β L1 = K1 .g μν −g(Γα μν Γαβ − Γμβ Γνα ), √ L2 = K2 .Re(ψ ∗ γ 0 γ a eμa (i∂μ + eAμ + Γμ )ψ −g), √ L3 = K3 .Fμν F μν −g, Fμν = Aν,μ − Aμ,ν

3.17.9

Learning the metric from geodesic trajectory mea­ surements in coordinate time domain

This problem deals with trying to learn about the metric of space-time using a multilayered neural network. Assume that the metric of space-time is gμν (x). This metric affects both the propagation of electromagnetic waves as well as the motion of particles. Suppose we choose a set of test functions ψn (r), n = 1, 2, ..., N where r are spatial coordinates. Assume to start with that the metric is time independent, ie gμν (x) = gμν (r). The equation of motion of a particle in this metric can be expressed as d2 xr (t)/dt2 = f r (xk (t), dxk (t)/dt, k = 1, 2, 3) To see how to derive the form of the function f r , we start with the spatial components of the geodesic equation d2 xr /dτ 2 + Γr00 (dt/dτ )2 + 2Γr0k (dt/dτ )(dxk /dτ ) + Γrkm (dxk /dτ )(dxm /dτ ) = 0 Now, dxr /dτ = γdxr /dt, d2 xr /dτ 2 = γ 2 d2 xr /dt2 + γ.(dγ/dt)dxr /dt where γ = dt/dτ = (dτ /dt)−1 = (g00 + 2g0k dxk /dt + gkm (dxk /dt)(dxm /dt))−1/2 so that dγ/dt = (1/2γ)(g00,m dxm /dt + 2g0k,m (dxk /dt)(dxm /dt)+

98

General Relativity and Cosmology with Engineering Applications 2g0m d2 xm /dt2 + gkm,s (dxk /dt)(dxm /dt)(dxs /dt) + 2gkm (dxk /dt)(d2 xm /dt2 )) = (1/2γ)(g00,m v m + 2g0k,m v k v m + 2g0m dv m /dt + gkm,s v k v m v s + 2gkm v k dv m /dt) = (1/2γ)(A(r, v) + Bm (r, v)dv m /dt) where v = ((v m ))3m=1 , A(r, v) = g00,m (r)v m + 2g0k,m (r)v k v m + gkm,s v k v m v s and Bm (r, v) = 2g0m (r) + 2gmk (r)v k and v k = dxk /dt We then get our geodesic equations in the form dv k /dt + (1/2γ 2 )(A(r, v) + Bm (r, v)dv m /dt)v k k +2Γk0r v r + Γrm vr vm = 0

We note that γ = dt/dτ = (dτ /dt)−1 = (g00 (r) + 2g0k (r)v k + gkm (r)v k v m )−1/2 = γ(r, v) We can rearrange the geodesic equations by defining C(r, v) = A(r/v)/2γ(r, v)2 , Dm (r, v) = Bm (r, v)/2γ(r, v)2 so that (δkm + Dm (r, v)v k )dv m /dt + (C(r, v)δkm + 2Γk0m )v m + Γkrm v r v m = 0 Denoting F (r, v) = (I + vD(r, v)T )−1 we get our geodesic equations in the form s m r s dv k /dt + Fkm (r, v)C(r, v)v m + 2Fkm (r, v)Γm 0s v + Fkm (r, v)Γrs v v = 0

which is of the required form. Exercise: Explain how you could generalize this idea to the case when the metric depends on time.

General Relativity and Cosmology with Engineering Applications

3.17.10

99

General relativistic corrections to the motion of a system of N point particles moving under mutual gravitational interaction

This problem deals with determining the approximate equations of motion of point masses interacting with each other via the gravitational field produced by them calculated using Einstein’s general theory of relativity. The approximate metric of space-time is given by dτ 2 = (1 + 2φ)dt2 − (1 − 2φ)(dx2 + dy 2 + dz 2 ) where φ(t, x, y, z) = φ(t, r) is the gravitational field produced by the system of particles. By analogy with the Schwarzchild metric, this form is justified in the far field zone where the system of point particles appears as a smooth sphere. Here we are taking c = 1. The three velocity of the ath particle is defined by vai = dxia /dt, i = 1, 2, 3, a = 1, 2, ..., N We have

va2 = O(φ) = O(ma )

where ma , a = 1, 2, ..., N are the masses of the particles. This order of magnitude formula follows from Newton’s formula for the orbital velocity v 2 = GM/r = −φ(r) for a particle moving in the gravitational field φ produced by a spherical mass M . We can write the approximate metric as g00 = 1 + 2φ, grs = −(1 − 2φ)δrs , g0r = 0 The energy-momentum tensor of this system of masses is T μν =

∑ dxμ (−g(t, ra ))−1/2 ma δ 3 (r − ra ) a dxνa /dt dτa a

= (−g(t, r))−1/2

∑

ma δ 3 (r − ra )vaμ vaν dt/dτa

a

where

va0

=

dx0a /dt

= 1. Noting that dτa2 = gμν (t, ra )dxμa dxνa

and hence suppressing the subscript a, dt/dτ = (dτ /dt)−1 = (g00 + 2g0r v r + grs v r v s )−1/2 = (1 + 2φ − v 2 )−1/2 = 1 − φ + v 2 /2 where we have neglected O(v 4 ) = O(φ2 ) = O(v 2 .φ) terms. Remark: ∫ ∫ ∑ √ ma dxμa uνa − − − (1) T μν −gd4 x =

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General Relativity and Cosmology with Engineering Applications

where uμa = dxμa /dτ is the four velocity of the ath particle. Since the four volume √ element −gd4 x is an invariant and the rhs of (1) is also a tensor, it follows μν that T is a tensor. Note that both uμ and dxμ are four vectors and ma is a scalar, ie, the rest mass of the ath particle in an inertial frame. Now, since −g = (1 + 2φ)(1 − 2φ)3 = (1 − 4φ2 )(1 − 2φ)2 = (1 − 4φ2 )(1 − 4φ + 4φ2 ) = 1 − 4φ + O(φ2 ) = 1 − 4φ + O(v 4 ) it follows that (−g)−1/2 = 1 + 2φ + O(φ2 ) and hence, 2 T 00 = (1 + 2φ)3 T00 = g00

∑

ma δ 3 (r − ra )(1 − φ + va2 /2)

a

=

∑ a

=

ma (1 + 6φ)(1 − φ + va2 /2)δ 3 (r − ra ) ∑

ma (1 + 5φ + va2 /2)δ 3 (r − ra )

a

T0r = g00 grr T 0r = −(1 + 2φ)(1 − 2φ) =

∑

∑

ma δ 3 (r − ra )(dt/dτa )var

a

ma δ 3 (r − ra )var (1 + φ − va2 /2) =

a

∑

ma δ 3 (r − ra )var

a

where we have neglected O(v 5 ) terms. Note that ma = O(φ) = O(v 2 ). Further, by the same procedure, ∑ ma δ 3 (r − ra )var vas Trs = a

again with neglect of O(v 5 ) terms. To obtain the equations of motion, we must first calculate the O(v 4 ) = O(φ2 ) general relativistic corrected value of g00 , g0r , grs in terms of ma , va using the Einstein field equations Rμν = −8πG(Tμν − T gμν /2) and then write down the corrected value of the Lagrangian of the system of particles ∑ ∑ dτa /dt = (1 + g00 (t, ra ) + 2g0r (t, ra )var + grs (t, ra )var vas )1/2 L= a

a

The equations Rrs = −8πG(Trs −T grs /2), R00 = −8πG(T00 −T g00 /2), R0r = −8πG(T0r −T g0r /2)

General Relativity and Cosmology with Engineering Applications

101

are to be set up first. We note that T = gμν T μν = gμν (−g)−1/2

∑

ma δ 3 (r − ra )uμa uνa dτa /dt

a

= (−g)−1/2 =

∑

∑

ma δ 3 (r − ra )dτa /dt

a

ma δ 3 (r − ra )(g00 (t, ra ) + 2g0r (t, ra )var + grs (t, ra )var vas )1/2

a

3.18

Neural Networks for learning the expan­ sion of our universe

This problem deals with developing a deep neural network (DNN) architecture for modeling the dynamics and galaxy formation in the expanding universe. The unperturbed metric, ie, metric in the absence of inhomogeneous matter and radiation is given by dτ 2 = dt2 − S 2 (t)dt2 /(1 − kr2 ) − S 2 (t)r2 (dθ2 + sin2 (θ)dφ2 ) Let ρ(t), p(t) denote the unperturbed density and radiation pressure. Then, the unperturbed Einstein field equations give us only two independent equations for the three functions S(t), ρ(t), p(t) which can be expressed as fm (S '' (t), S ' (t), S(t), ρ(t), p(t), k) = 0, m = 1, 2 − − − (1) Here, the curvature k appears as a parameter. In the DNN, assuming that we do not know anything about the equation of state of the radiation, ie, relationship between p(t) and ρ(t). Thus, we can regard the pressure p(t) as the input signal and ρ(t), S(t) as output signals. We wish to learn about this dynamics using a DNN. Thus, we model (1) in discrete time as Z[n + 1] = ψ0 (W [n], Z[n], p[n]) + εZ [n], S[n] = ψ1 (Z[n]) + εS [n], ρ[n] = ψ2 (Z[n]) + ερ [n] '

where the ε s are white noise processes and Z[n] is a state vector. W [n] is the evolving weight vector of the recurrent neural network (DNN-RNN). By taking the measurements on S[n] (via red shift measurements), on p[n] (via sensors sen­ sitive to very weak electromagnetic fields) and on ρ[n], we can use the EKF to estimate Z[n], W [n] dynamically, ie, [Z[n], W [n]] form the extended state vector. This problem can be further generalized as follows: Let gμν (x, θ) be a metric of space-time dependent on an unknown parameter vector θ which we wish to es­ timate. By taking measurements on the motion of material particles (following geodesic trajectories) or on the velocity-density-pressure of a fluid moving in this

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General Relativity and Cosmology with Engineering Applications

curved space-time or on electromagnetic fields satisfying the Maxwell equations in this curved space-time, we wish to estimate θ. Alternatively, we can learn about the parameter θ even when it varies slowly with time using a DNN-RNN as follows: First compute the theoretical trajectory of a set of material particles as a function of θ using perturbation theory for ordinary differential equations. Alternately, for a given current density field, calculate the radiated electromag­ netic field produced by solving Maxwell’s equations in the background metric gμν (x, θ). This field will be a function of θ. Now take measurements (noisy) on the particle trajectories or on the electromagnetic fields and build a neural net­ work that takes in initial values of the particle trajectories or the initial values of the electromagnetic field as input and predicts the measured values. This is done by estimating the weights of the DNN-RNN either by direct matching or using the EKF.

3.19

Quantum stochastic differential equations in general relativity

Suppose we have matter in a finite region D of spatial volume. The Einstein field equations within D are derived from the Lagrangian density √ β α β L = g μν −g(Γα μν Γαβ − Γμβ Γνα ) √ +KT μν gμν −g where K is a constant and T μν is the energy momentum tensor of matter within the region D. Assuming that the matter consists of discrete point particles, we have ∑ √ ma δ 3 (r − ra )(dxμa /dτ )(dxνa /dt) T μν −g = a

∫

so that

∑ √ T μν −gd4 x = ma

∫ (dxμa /dτ )dxνa

a

which is clearly a tensor. We write vaμ = dxμa /dt so that

va0 = 1

and then

∑ √ ma δ 3 (r − ra (t))vaμ vaν (dτa /dt)−1 T μν −g(t, r) = a

where (dτa /dt)−1 = (g00 (t, ra ) + 2g0k (t, ra )vak + gkm (t, ra )vak vam )−1/2

General Relativity and Cosmology with Engineering Applications Note that

103

dra /dt = va , va = ((vak ))3k=1

The Lagrangian of the system is ∫ L(ra (t), va (t), gμν (t, .), gμν,0 (t, .)) = ∫

Ld3 r =

√ β α 3 β g μν −g(Γα μν Γαβ − Γμβ Γνα )d r ∫ +K

√ T μν −gd3 r

The Euler-Lagrange equations give the Einstein field equations as well as the equations of motion of the matter particles inside the region D. These are d ∂L ∂L = , k = 1, 2, 3 k ∂xak dt ∂va which reduce to the geodesic equations and Rμν = K0 (T μν − (T /2)g μν ) Note that T = gμν T μν =

∑

√ ma (δ 3 (r − ra )/ −g(t, r))gμν (dxμa /dτ )(dxνa /dτ )(dτa /dt)

a

= =

∑

∑

√ ma (δ 3 (r − ra )/ −g)dτa /dt

a

√ ma (δ 3 (r − ra )/ −g(t, ra ))(g00 (t, ra ) + 2g0k (t, ra )vak + gkm (t, ra )vak vam )1/2

a

By applying the Legendre transformation to this total Lagrangian L, taking as our canonical position variables ra (t) = (xka (t))3k=1 , gμν (t, .) and canonical velocities vak = dxka /dt.k = 1, 2, 3 and gμν,0 (t, .), we obtain as our canonical momenta ∂L ∂L = pa = ∂va ∂va and π μν =

∂L ∂L = ∂gμν,0 ∂gμν,0

The Hamiltonian of the matter plus gravitational field within D is then ∫ ∑ H= (pa , va ) + π μν gμν,0 (t, r)d3 r − L a

= H(ra (t), pa (t), gμν (t, .), gμν,0 (t, .))

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General Relativity and Cosmology with Engineering Applications

We now quantize this Hamiltonian and allow the gravitational field and mat­ ter within D to interact with the electromagnetic field photon bath outside D described by the usual creation and annihilation processes Ak (t), Ak (t)∗ , k = 1, 2, ... of Hudson and Parthasarathy satisfying the quantum Ito formula dAj (t)dAk (t)∗ = δjk dt, dAk (t)dAm (t) = 0, dAk (t)∗ dAm (t)∗ = 0 If we assume that only second degree contributions of the position and momen­ tum fields in D after also including a system electromagnetic field within D, then the Hamiltonian of the system in D (consisting of matter, gravitational fields and electromagnetic fields) can be brought to the standard Harmonic oscillator form: ∑ ωk a∗k ak HS = k

and the qsde satisfied by the unitary evolution operator of the system and bath is of the general form ∑ ∑ ∗ ¯ k a∗ +¯ dU (t) = −(i ωk a∗k ak +P )dt+ (λk ak +μk a∗k )dAk (t)−(λ k μk ak )dAk (t) )U (t) k

k

where P is the quantum Ito correction term required to make U (t) unitary: P =

1∑ ¯ k a∗ + μ ¯ k ak ) (λk ak + μk a∗k )(λ k 2 k

Note that [ak , a∗m ] = δkm , [ak , am ] = 0, [ak , Am (t)] = 0, [ak , Am (t)∗ ] = 0 The GKSL equation for system state ρS (t) associated with the above HP equa­ tion is obtained as ∑ dρS (t)/dt = −i[HS , ρS (t)] − (1/2) (Lk L∗k ρS (t) + ρS (t)Lk L∗k − 2L∗k ρS (t)Lk ) k

where

Lk = λk ak + μk a∗k

The corresponding GKSL-Heisenberg equation for system observables X is ob­ tained by duality as ∑ Lk L∗k X(t) + X(t)Lk L∗k − 2Lk X(t)L∗k ) X ' (t) = i[HS , X(t)] − (1/2) k

The GKSL generator is therefore θ where ∑ θ(X) = (−1/2) (Lk [L∗k , X] + [X, Lk ]L∗k ) k

Now, let W (z) = W (z, I) denote the Weyl operator acting on the system Hilbert space, ie, the Hilbert space for translations alone, no rotations on which the

General Relativity and Cosmology with Engineering Applications

105

ak , a∗k act. We know that the creation and annihilation processes for the sys­ tem can be obtained from the generators of one parameter unitary subgroups t → W (tz) (Ref:K.R.Parthasarathy,”An introduction to quantum stochastic calculus, Birkhauser, 1992). We have ∑ θ(W (z)) = (−1/2) [Lk [L∗k , W (z)] + [W (z), Lk ]L∗k k

Now, < e(v), [ak , W (z)]e(u) >=< e(v)|ak W (z)|e(u) > − < e(v)|W (z)ak |e(u) > < e(v)|ak W (z)|e(u) >= exp(−|z|2 /2− < z, u >)(uk + zk ) < e(v), e(u + z) > so writing ¯ = a(λ)

∑

λ k ak

l

we get ¯ u+z >< e(v), e(u+z) > ¯ )W (z)|e(u) >= exp(−|z|2 /2− < z, u >) < λ, < e(v), a(λ ¯ )|e(u) >= < e(v), W (z)a(λ ¯ u > W (z)|e(u) >=< λ, ¯ u >< e(v), W (z)|e(u) > < e(v), < λ, ¯ u > exp(−|z|2 /2− < z, u >) < e(v)|e(u + z) > =< λ, Thus, ¯ ), W (z)]|e(u) >= < e(v), [a(λ ¯ z >< e(v)|e(u + z) > exp(−|z|2 /2− < z, u >) < λ, In other words, we have proved that ¯ ), W (z)] =< λ, ¯ z > W (z) [a(λ Taking the adjoint of this equation and using W (z)∗ = W (z)−1 = W (−z) we get or equivalently,

¯ > W (−z) ¯ )∗ , W (−z)] = − < z, λ [a(λ ¯ > W (z) ¯ )∗ , W (z)] =< z, λ [a(λ

Chapter 4

Some basic problems in electromagnetics related to the gtr 4.1

EM waves and quantum communication

[a] Consider a cavity resonator with interior region D ⊂ R3 and boundary surface ∂D. The wave field within the cavity ψ(ω, r) satisfies the Helmholtz equation (∇2 + k 2 )ψ(ω, r) = 0, r ∈ D with boundary condition ψ(r) = ψ0 (r), r ∈ ∂D Explain how you would solve this problem using the Green’s function, ie, in terms of a function G(r|r' ), r, r' ∈ D satisfying (∇2r + k 2 )G(r|r' ) = δ 3 (r − r' ), r, r' ∈ D and

G(r|r' ) = 0, r ∈ ∂D, r' ∈ D

How would the solution be modified if the above Dirichlet boundary condition on ψ is replaced by the Neumann boundary condition ∂ψ(r)/∂n ˆ = ψ0 (r), r ∈ ∂D hint: Use Green’s identity ∫ ∫ [G(r|r' )∇2 ψ(r)−ψ(r)∇2 G(r|r' )] = D

[G(r|r' )∂ψ(r)/∂n ˆ −ψ(r)∂G(r|r' )/∂n ˆ ]dS(r) ∂D

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Explain how you would obtain corrections to ψ if the gravitational field in the form of a time independent gμν (r) is taken into account and ψ satisfies the Laplace-Beltrami-Helmholtz equation: √ √ √ (g km −gψ,m ),k + jω(g k0 −gψ,k ) + jω(g k0 −gψ),k √ −ω 2 g 00 −gψ = 0

4.2

Cavity resonator antennas with current source in a gravitational field

Consider an RDRA (Rectangular dielectric resonator antenna) of side lengths a, b, d respectively along the x, y, z axes. There is a current density J(ω, r) = J(r) inside the antenna box coming from a probe source. The relevant Maxwell equations are ∇ × E(r) = −jωμH(r), ∇ × H(r) = J(r) + jωεE(r) Assuming the walls to be perfect electric conductors, the boundary conditions are that the normal components of H and the tangential components of E vanish. The source current density is zero in a neighbourhood of the walls. Thus, in this region the different components of E and H will have expansions in terms of the eight combinations of the basis functions {cos(mπx/a), sin(mπx/a)}⊗{cos(nπy/b), sin(nπy/b)}⊗{cos(pπz/d), sin(pπz/d)} where m, , n, p assume non-negative integer values. Using divJ(r) + jωρ(r) = 0 and divE(r) = ρ(r)/ε, divH(r) = 0 We get by taking the curl of the above Maxwell curl equations ∇2 E(r) + k 2 E(r) = jωμJ(r) + ∇ρ(r)/ε − − − (1) ∇2 H(r) + k 2 H(r) = −∇ × J(r) − − − (2) where k 2 = ω 2 εμ. For different components of E, H different sin cosine bases are used. For example, for Hz , we must choose the expansion sin(mπx/a)sin(nπy/b)sin(pπz/d)

so that Hz vanishes when x = 0, a, y = 0, b, z = 0, d. Using the standard ex­ pressions for the tangential components of the electric and magnetic fields in a guide, we have ˆ E⊥ = (−γ/h2 )∇⊥ Ez − (jωμ/h2 )∇⊥ Hz × z,

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H⊥ = (−γ/h2 )∇⊥ Hz + (jωε/h2 )∇⊥ Ez × zˆ where γ stands for the operator −∂/∂z. Thus applying the boundary condi­ tion that Ez vanishes at x = 0, a, y = 0, b and Ex , Ey vanish when z = 0, d, it follows that Ez should have the expansion in terms of the basis functions sin(mπx/a)sin(nπy/b)cos(pπz/d) and using further that Ex vanishes when y = 0, b, z = 0, d while Ey vanishes when x = 0, a, z = 0, d, and the above formulas for E⊥ , we get that Ex should have the expansion in terms of cos(mπx/a)sin(nπy/b)sin(pπz/d

Likewise, Ey should have the expansion in terms of sin(mπx/a)cos(nπy/b)sin(pπz/d Finally, from the above formulas, and the boundary conditions on Hx , Hy , it fol­ lows that Hx should have the expansion in terms of sin(mπx/a)cos(nπy/b)cos(pπz/d and Hy should have the expansion in terms of sin(mπx/a)cos(nπy/b)cos(pπz/d). If all the walls are perfect magnetic conductors, then the above boundary con­ ditions hold with E and H interchanged. So finally, if ψ denotes any one com­ ponent of the six components of E, H, we must solve (1) or (2) in the form (∇2 + k 2 )ψ(r) = s(r) where s(r) is one of the six components of the source fields appearing on the rhs of (1) and (2). While solving this equation, we must represent s(r) as a three dimensional half wave Fourier series using the same basis functions as used for ψ. For example, if ψ = Hz , then we must expand s(r) as ∑ s(r) = s(x, y, z) = s[mnp]sin(mπx/a)sin(nπy/b)sin(pπz/d) mnp

and this gives ψ(r) = ψ(x, y, z) =

∑

(k 2 −π 2 (m2 /a2 +n2 /b2 +p2 /d2 ))−1 sin(mπx/a)sin(nπy/b)sin(pπz/d)

mnp

Exercise: Solve explicitly for the six components of E, H in terms of the appropriate basis function expansions of the sources jωμJ(r) + ∇ρ(r)/ε − − − (1' ) −∇ × J(r) − − − (2' ) appearing on the rhs of (1) and (2). Remark: The general solution for the fields is a superposition of the particu­ lar solution given above and the general solution to the homogeneous equations, ie, without sources. Exercise: Derive the general relativistic Helmholtz equation at frequency ω for the electromagnetic four potential Aμ in terms of the source field J μ (x) in a background gravitational field specified a time independent metric gμν (x, y, z) √ assuming the general relativistic Lorentz gauge condition (Aμ −g),μ = 0. Note that the time dependence everywhere is exp(jωt), so that this gauge condition assumes the form √ √ jωA0 −g + (Ak −g),k = 0

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The general relativistic Helmholtz equation with source is obtained from the Maxwell equations √ √ jωF μ0 −g + (F μk −g),k = 0 where F0k = −Fk0 = jωAk − ∂k A0 Frs = As,r − Ar,s with

A0 = g00 A0 + g0k Ak , Ak = gk0 A0 + gkm Am

Formulate the boundary conditions on the em fields in terms of the em four potential. Reference: R.S.Yaduvanshi and H.Parthasarathy, ”Polarization of electro­ magnetic fields in a RDRA in terms of the the polarization of the current density source”, Technical report, NSIT, 2017.

4.3

Cq-coding theorem

A is the source alphabet and corresponding to each x ∈ A we have a density ρ(x). If u ∈ An , we write N (x|u) for the number of times x appears in u and Pu (x) = N (x|u)/n is the relative frequency with which x appears in u. If p is any probability distribution on A, we define√T (n, p, δ) to be the set of all sequences u ∈ An such that |N (x|u) − np(x)| < np(x)(1 − p(x)) for all x ∈ A. It follows that for all u ∈ T (n, p, δ), p(x)np(x)−O(

√

n

√

) ≤ p(x)N (x|u) ≤ p(x)np(x)−O(

n)

,x ∈ A

and hence taking the product over all x ∈ A, we get 2−nH(p)−O(

√

n

) ≤ p(u) ≤ 2−nH(p)+O(

√

n)

Now let ρ be any state with spectral representation ρ=

N ∑

|i > Pρ (i) < i|

i=1

Then H(ρ) = −T r(ρ.log(ρ)) = −

N ∑

Pρ (i)log(Pρ (i))

i=1

and so if we define E(ρ⊗n , δ) =

∑ (i1 ,...,in )∈T (n,Pρ ,δ)

|i1 ...in >< i1 ...in |

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General Relativity and Cosmology with Engineering Applications then it follows that 2−nH(ρ)− Also,

√

n

E(ρ⊗n , δ) ≤ ρ⊗n E(ρ⊗n , δ) ≤ 2−nH(ρ)+O(

√

n)

E(ρ⊗n , δ)

T r(ρ⊗n E(ρ⊗n , δ)) = Pρ⊗n (T (n, Pρ , δ) ≥ 1 − 1/δ 2

by Chebyshev’s inequality. Now, define for u ∈ An , ρ(u) = ⊗x∈A ρ(x)⊗N (x|u) and correspondingly E(n, u, δ) = ⊗x∈A E(ρ(x)⊗N (x|u) , δ) Then, from the above, it easily follows that √

2−N (x|u)H(ρ(x))−O(

n)

E(ρ(x)⊗N (x|u) , δ) ≤ ρ(x)⊗N (x|u) E(ρ(x)⊗N (x|u) , δ) ≤

2−N (x|u)H(ρ(x))+O(

√

n)

E(ρ(x)⊗N (x|u) , δ)

and hence taking the tensor product over all x ∈ A (after choosing an order in A), we get 2−n

∑ x∈A

√ Pu (x)H(ρ(x))−O( n)

E(n, u, δ) ≤ ρ(u)E(n, u, δ) ≤ 2−n

∑ x∈A

√ Pu (x)H(ρ(x))+ n

E(n, u, δ)

Also by Chebyshev’s inequality, for all u ∈ An , T r(ρ(u)E(n, u, δ)) ≥ (1 − 1/δ 2 )a ≥ 1 − a/δ 2 where a is the number of elements in A. The greedy algorithm: Let u1 , ..., uM be sequences in T (n, p, δ) and D1 , ..., DM ∑M operators in the Hilbert space H⊗n such that 0 ≤ Dk ≤ I∀k, k=1 Dk ≤ I, Dk ≤ E(n, uk , δ)∀k and T r(ρ(uk )Ek ) > 1 − ε∀k. Let M be maximal subject to ∑M these constraints. Then define D = k=1 Dk . We have 0 ≤ D ≤ I. Suppose T r(ρ(u)D) < γ for some u ∈ T (n, p, δ). Then, define √ √ D' = 1 − DE(n, u, δ) 1 − D We have since T r(ρ(u)(1 − D)) > 1 − γ, √ √ |T r[ρ(u)(E(n, u, δ) − 1 − DE(n, u, δ) 1 − D)]| √ √ = |T r[(ρ(u) − 1 − Dρ(u) 1 − D)E(n, u, δ)]| √ √ ≤|| ρ(u) − 1 − Dρ(u) 1 − D ||1 < 1 − ε1

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and hence T r(ρ(u)E(n, u, δ)) − T r[ρ(u)(E(n, u, δ) −

√

√ 1 − DE(n, u, δ) 1 − D)]

≥ 1 − ε2 ie,

T r(ρ(u)D' ) ≥ 1 − ε2

and further, D' +

M ∑

Dk ≤ 1 − D +

k=1

M ∑

Dk ≤ 1

k=1

contradicting maximality of M . Lemma: Let 0 ≤ T, Z ≤ I. Suppose for some density ρ, we have ρT ≤ θT . Then, T r(Z) ≥ θ−1 (T r(ρZ) − T r(ρ(1 − T )))

Define ρ¯ =

∑

p(x)ρ(x)

x∈A

where p is a given probability distribution on A. Problem: Using the above lemma, derive a lower bound on T r(Z) in terms of the probability T r(¯ ρ⊗n Z) and the probability ρ⊗n , δ)) T r(¯ ρ⊗n E(¯ and hence by applying the lower bound for this latter probability in terms of typical projection theory, derive a lower bound on T r(Z) in terms of the entropy of ρ¯

4.4

Restricted quantum gravity in one spatial dimension and one time dimension

The metric is dτ 2 = (1 + 2U (t, x))dt2 − (1 + 2V (t, x))dx2 The position fields are U (t, x) and V (t, x) and to find the momentum fields, we must first evaluate the Lagrangian density √ β β α L = g μν −g(Γα μν Γαβ − Γμβ Γαβ ) in terms of U, V and then compute the path integral for the corresponding action by integrating over the paths of U and V to compute their propagators.

General Relativity and Cosmology with Engineering Applications

4.5

113

Quantum theory of fields

The Klein-Gordon field: The Lagrangian density for this field is L(φ(x), φ,μ (x) = (1/2)(∂μ φ)(∂ μ φ) − m2 φ2 /2 = (1/2)φ2,0 − (1/2)(∇φ)2 − m2 φ2 /2 We are assuming here that the field is real. The canonical classical field equa­ tions are ∂L ∂L − =0 ∂μ ∂φ ∂φ,μ which give us the Klein-Gordon equation ∂μ ∂ μ φ + m2 φ = 0 or equivalently,

φ,00 − ∇2 φ + m2 φ = 0

This has the following interpretation in quantum mechanics (not yet quantum field theory): In the special theory of relativity, the energy momentum relation is E 2 − p2 c2 − m2 c4 = 0, p2 = p2x + p2y + p2z According to the standard rules of quantum mechanics, the momentum three vector p is replaced by −ih∇/2π and the energy E by (ih/2π)∂/∂t yielding thereby the KG equn. (E 2 − p2 c2 − m2 c4 )φ = 0 or or

[−h2 ∂t2 /4π 2 + (h2 c2 /4π 2 )∇2 − m2 c4 ]φ = 0 [∂t2 − c2 ∇2 + 4π 2 m2 c4 /h2 ]φ = 0

which is the KG equation once normalized units are used: h/2π = 1, c = 1. To canonically quantize this field, we first construct the Hamiltonian density corre­ sponding to the Lagrangian density by performing a Legendre transformation: π(x) =

∂L = φ,0 ∂φ,0

The Hamiltonian density is then H(φ, ∇φ, pi) = πφ,0 − L = (1/2)(π 2 + (∇φ)2 + m2 φ2 ) We can check that classical Hamiltonian equations of motion of the field obtained from the Hamiltonian ∫ H = Hd3 x

114

General Relativity and Cosmology with Engineering Applications as π,0 = −δH/δφ, φ,0 = δH/δπ with the variational derivative δH/δφ =

∂H ∂H − (∇, ), ∂∇φ ∂φ

∂H ∂π yield the correct classical KG field equations. Remark: The variational derivative of H w.r.t φ is obtained by considering the change in H under a small change δφ in φ: ∫ ∂H H δH = δφ + ( , ∇δφ)d3 x ∂φ ∂∇φ ∫ ∂H ∂H = [ − (∇, )]δφd3 x ∂φ ∂∇φ where we have used integration by parts, δ∇φ = ∇δφ and the assumption that δφ(x) vanishes when the spatial coordinates go to ∞. The Schrodinger equation for this Hamiltonian reads: δH/δπ =

∫

[(1/2)

(−δ 2 /δφ(r)2 +(∇φ(r))2 +m2 φ(r)2 )d3 r]ψt (φ(r) : r ∈ R3 ) = (ih/2π)

∂ ψt (φ(r) : r ∈ R3 ) ∂t

In other words, the second quantized Klein Gordon field is determined by a continuous infinity of quantum Harmonic oscillators. This does not make much sense so we work in the spatial frequency domain wherein we express the solution the KG equation as a superposition of plane waves: ∫ φ(t, r) = [f (k)a(k)exp(−i(ω(k)t − k.r)) + f¯(k)a(k)∗ exp(i(ω(k)t − k.r))]d3 k ω(k) = (k 2 + m2 )1/2 so that

∫

π(t, r) = φ,0 = −i

[ω(k)f (k)a(k)exp(−i(ω(k)t−k.r))−f¯(k)a(k)∗ exp(i(ω(k)t−k.r))]d3 k

In the second quantized picture, a(k), a(k)∗ are operators and φ(t, r) is also (field) operator. The canonical equal time commutation rules (we are assuming φ to be a bosonic field) are [φ(t, r), π(t, r' )] = iδ(r − r' ) φ is called the canonical position field and π the canonical momentum field. In accordance with the commutation rules for creation and annihilation operators of a harmonic oscillator, we assume that [a(k), a(k ' )∗ ] = δ(k − k ' ), [a(k), a(k ' )] = [a(k)∗ , a(k ' )∗ ] = 0

General Relativity and Cosmology with Engineering Applications

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Then we get ∫

'

iδ(r−r ) = i

[ω(k ' )f (k)f¯(k ' )[a(k), a(k ' )∗ ]exp(−i(ω(k)−ω(k ' ))t+i(k.r−k ' .r' ))+

ω(k ' )f (k)f¯(k ' )[a(k ' ), a(k)∗ ]exp(i(ω(k) − ω(k ' ))t − i(k.r − k ' .r' ))]d3 kd3 k ' or equivalently, δ(r − r' ) =

∫

[ω(k ' )f (k)f¯(k ' )exp(−i(ω(k) − ω(k ' ))t + i(k − k ' ).r)+

ω(k ' )f (k)f¯(k ' )exp(i(ω(k) − ω(k ' ))t − i(k − k ' ).r)]δ(k − k ' )d3 kd3 k ' ∫ = |f (k)|2 ω(k)(exp(ik.(r − r' )) + exp(−ik.(r − r' )))d3 k so we require that |f (k)|2 ω(k) = (1/2)(1/(2π)3 ) = 1/16π 3 We may thus take

f (k) = (1/4π 3/2 )ω(k)−1/2

a(k) is to be interpreted as the annihilation operator of a KG boson having momentum k (spinless boson) and a(k)∗ is to be interpreted as teh creation operator of a KG boson. We now compute the second quantized Hamiltonian in terms of the creation and annihilation fields: ∫ H = (1/2) [φ2,0 + (∇φ)2 + m2 ]d3 r ∫ =

ω(k)a(k)∗ a(k)

Exercise:Verify this formula by performing the integrations w.r.t d3 k, d3 k ' d3 r. Now to check this, we calculate the Heisenberg equation of motion of the creation and annihilation operators: ∫ da(k, t)/dt = i[H, a(k, t)] = i ω(k ' )[a(k ' , t)∗ , a(k, t)]a(k, t)d3 k ' ∫ = −i

ω(k ' )δ(k − k ' )d3 k ' a(k, t) = iω(k)a(k, t)

and hence a(k, t) = a(k)exp(−iω(k)t) Likewise,

a(k, t)∗ = a(k)∗ exp(iω(k)t)

Thus, we get ∫ φ(t, r) =

(f (k)a(k, t)exp(ik.r) + f¯(k)a(k, t)∗ exp(−ik.r))d3 k

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and hence 2

∫

2

∇ φ−m φ=−

ω(k)2 (f (k)a(k, t)exp(ik.r) + f¯(k)a(k, t)∗ exp(−ik.r))d3 k = φ,00

in agreement with the KG equation. Now suppose that we apply an external field f (x) to the KG system. The resulting Lagrangian density is then given by L = L0 + f (x)V (φ(x)) where V (φ(x)) is some linear/nonlinear function of the KG field. We wish to approximately compute the transition probability of the KG field from the initial state |φi > at time t1 in which the field is exactly a given function φi (r), r ∈ R3 of the spatial variables to the final state |φf > at time t2 in which the field is exactly another given function φf (r), r ∈ R3 of the spatial variables. The corresponding Hamiltonian will then have the form ∫ ∫ H(t) = H0 (φ(x), ∇φ(x), π(x))d3 x − f (x)V (φ(x))d3 x (Note: x = (t, r), d3 x = d3 r, d4 x = d3 xdt = d3 rdt) where H0 is the Hamiltonian density corresponding to the Lagrangian density L0 (ie obtained by applying the Legendre transform to L0 ): ∫ H0 = (1/2) (π 2 + (∇φ)2 + m2 φ2 )d3 x The transition probability amplitude from |φi >→ |φf > in the time duration [t1 , t2 ] can be calculated using the Feynman path integral formula: ∫ C ∫ =C

φ(t1 ,.)=φi ,φ(t2 ,.)=φf

∫

exp(iS0 )(1+i

S0 =

[t1 ,t2 ]×R3

f (x)V (φ(x))dtd3 x+(i2 /2!)

∫

where

< φf |S[t2 , t1 ]|φi >= ∫ exp(−( Ldtd3 x))Πr∈R3 ,t∈(t1 ,t2 ) dφ(x)

L0 dtd3 x = (1/2)

∫

∫

f (x)f (x)' φ(x)φ(x' )dtdt' d3 xd3 x' +..)Πdφ(x)

(∂μ φ∂ μ φ − m2 φ2 )dt d3x

By expanding V (φ(x)) as a power series in φ(x), the computation of the above path integral reduces to computing the moments of a complex infinite dimen­ sional zero mean Gaussian distribution sinc S0 is a quadratic functional of φ. In particular, we note that the odd moments of a symmetric Gaussian distribution are zero and the even moments can be computed by summing the products of the second moments taken over all partitions of the product fields into pairs. Thus,

General Relativity and Cosmology with Engineering Applications

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computation of the second moments of such a Gaussian distribution beomes significant, ie, ∫ D(x, y) = C exp(iS0 )φ(x)φ(y)Πz∈R4 dφ(z) if we are interested in transitions from t = −∞ to t = +∞. From standard methods in quantum mechanics, it is easily seen that D(x, y) =< 0|T {φ(x)φ(y)}|0 > provided that we use the interaction representation which removes the effect of the unperturbed Hamiltonian H0 . If we use the Schrodinger representation, then we would have to compute D as D(x, y) =< 0|T {U (∞, −∞)φ(x)φ(y)}|0 >= < 0|U (∞, tx )φ(x)U (tx , ty )φ(y)U (ty , −∞)|0 > assuming tx ≥ ty and where U is the unperturbed Schrodinger evolution oper­ ator. Here |0 > is the vacuum state of the field. The function D(x, y) is called the propagator. The complete propagator taking into account interactions is defined as ∫ Dc (x, y) = C ∫

where S = S0 +

exp(iS)φ(x)φ(y)Πz dφ(z) f (x)V (φ(x))d4 x =

∫

Ld4 x

We can write a perturbative expansion for Dc as ∫ Dc (x, y) = exp(iS0 )(1 + iS1 + i2 S12 /2! + ..)φ(x)φ(y)Πz dφ(z) ∫

where S1 =

f (x)V (φ(x))d4 x

is the perturbation to the action caused by external field coupling. Even if there is no external field, but there is a small perturbation to the Lagrangian density/Hamiltonian density, the above series expansion can be used to deter­ mine the complete propagator It was Feynman’s genius to recognize that the various perturbation terms in Dc can be calculated easily using a diagrammatic method which could be applied to more complex situations like quantum elec­ trodynamics wherein the quantum fields are the electromagnetic four potential Aμ (x) and the Dirac four component spinor wave function ψ(x). Let us now formally compute the propagator of the unperturbed KG field: ∫ S0 = φ(x)[(1/2)∂μ ∂ μ − m2 /2)δ 4 (x − y)]φ(y)d4 xd4 y

118

General Relativity and Cosmology with Engineering Applications ∫ =

φ(x)K(x, y)φ(y)d4 xd4 y

and hence a simple Gaussian second moment evaluation gives ∫ D(x, y) = exp(iS0 [φ])φ(x)φ(y)Πz dφ(z) = C1 (det(iK))−1/2 .K −1 (x, y) In other words D(x, y) is proportional to K −1 (x, y) where K −1 is the inverse

∫

Kernel of K:

K −1 (x, y)K(y, z)d4 y = δ 4 (x − z)

We can write K(x, y) = K(x − y) and then defining its four dimensional Fourier transform: ∫ ˆ (p) = K

we get Clearly, and hence

K(x)exp(−ip.x)d4 x, p.x = pμ xμ = p0 x0 − p1 x1 − p2 x2 0 − p3 x3

K −1 (p) = 1/K(p) K(x) = (1/2)∂μ ∂ μ − m2 /2)δ 4 (x) ˆ (p) = (pμ pμ − m2 )/2 K

Thus, ˆ (p) = D where

p2

C0 − m2

p2 = pμ pμ = p02 − p12 − p22 − p32

Finally, D(x, y) = D(x − y) = C0 /(2π)4

∫

exp(ip.x) 4 d p p2 − m2

The corrected (complete) propagator: ∫ ∫ Dc (x, y) = exp(iS0 [φ])φ(x)φ(y)(1 + i f (x)V (φ(z))d4 z + ...)Πu dφ(u) Clearly, we can write this in operator kernel notation as Dc = D + DΣD + DΣDΣD + ... using the property of moments of a Gaussian distribution. For ∫example, if 4 4

V (φ) = φ4 and f = c0 , then in the Gaussian average of the product φ(x)φ(y)

f (z)φ(z) d z,

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we get the coupling terms 4 < φ(x)φ(z) >< φ(z)2 >< φ(z)φ(y) > so if we define Σ(z) = 4f (z)0 < φ(z)2 >, we can write ∫ ∫ < φ(x)φ(y)( f (z)φ(z)4 d4 z) >= D(x − z)Σ(z)D(z − y)d4 z Likewise, for the next perturbation term ∫ < φ(x)φ(y)( f (z)φ(z)4 d4 z)2 >= ∫

f (z1 )f (z2 ) < φ(x)φ(y)φ4 (z1 )φ4 (z2 ) > d4 z1 d4 z2

Again, this can be expressed using the Gaussian moments formula as a sum of terms of the form ∫ f (z1 )f (z2 ) < φ(x)φ(z1 ) >< φ(z1 )3 φ(z2 )3 >< φ(z2 )φ(y) > d4 z1 d4 z2 and

∫

f (z1 )2 < φ(x)φ(z1 ) >< φ(z1 )2 φ(z2 )4 >< φ(z1 )φ(y) > d4 z1 d4 z2

etc. Now, each term < φ(z1 )m φ(z2 )m > is a product of propagators D(z1 − z2 ) and D(0) so the above general form is valid. Dirac brackets for constraints: Suppose Q1 , ..., Qn , P1 , ..., Pn are the uncon­ strained positions and momenta of a system. The constraints are Qj = Pj = 0, j = n+1, ..., n+p. Without loss of generality, we are choosing our constrained variables as new positions and momenta. The Poisson bracket relations are {f, g} =

n+p ∑

f,Qi g,Pi − f,Pi g,Qi )

i=1

In particular, we get the contradiction {f, Qi } = −f,Pi , {f, Pi } = f,Qi , i > n since Qi = Pi = 0, i > n. In order to rectify this problem, Dirac introduced a new kind of bracket defined as follows: Let χij = {ηi , ηj } = Jij J is the standard symplectic matrix of size 2p × 2p. where η = [Qn+1 , ..., Qn+p , Pn+1 , ..., Pn+p ]T Qn+i , Pn+i , i = 1, 2, ..., p, ie ηi are functions of Qi , Pi , i = 1, 2, ..., n and the bracket {., .}ef f is calculated using Qi , Pi , i = 1, 2, ..., n and regarding Qn+i , Pn+i

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as functions of Qj , Pj , j ≤ n. The bracket {f, g}P is computed using Qi , Pi , i ≤ n and taking Qn+i = 0, Pn+i = 0: {f, g}P =

n ∑

(f,Qi g,Pi − f,Pi g,Qi )

i=1

We have

C = χ−1 = −J

as 2p × 2p matrices. Then, the Dirac bracket is defined as ∑ {f, g}D = {f, g} + {f, ηi }Jij {ηj , g} i,j

We see that for k ≤ n, {f, Qk }D = {f, Qk } = −f,Pk since {ηj , Qk } = 0, k ≤ n Note that {., .} is the unconstrained Poisson bracket. Again, we note that {f, Pk }D = {f, Pk } = f,Qk since {Pk , ηj } = 0, k ≤ n. Further, for i, j ≥ 1, we have {f, ηi }D = {f, ηi } +

∑

{f, ηk }Jkl {ηl , ηi }

k,l

= {f, ηi } −

∑

{f, ηk }Ckl χli = 0

k,l

since

∑

Ckl χli = δki

l

We note that {f, g}D = {f, g} −

∑ {f, ηi }Jij {ηj , g} i,j

∑ ∑ ∑ (f,ηj ηj,Qi ))(g,Pi + g,ηj ηj,Pi ) = (f,Qi + j

i≤n

+ =

∑ i≤n

∑

j

−interchangeof f andg ∑ f,ηi Jij g,ηj + {f, ηi }Jij {ηj , g}

i

(f,Qi +

∑ ∑ (f,ηj ηj,Qi ))(g,Pi + g,ηj ηj,Pi ) j

j

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This formula tells us that the Dirac bracket between two observables is cal­ culated using the Poisson bracket w.r.t. the unconstrained variables and by regarding the constrained variables as functions of the unconstrained variables.

Quantum electrodynamics using creation and annihilation operators for pho­ tons, electrons and positrons: We work in the Coulomb gauge so that divA = 0 and this implies ∇2 A0 = −J 0 , ie, A0 is a matter field. The Maxwell wave equation for A in the absence of matter, ie charge and current densities is given by ∇2 A − A,00 = 0 and the general solution to this is ∫ Ak (t, r) =

er (K, σ)[a(K, σ)exp(−i(|K|t−K.r))+¯ er (K, σ)a(K, sigma)∗ exp(i(|K|t−K.r))]d3 K

Here, the summation is over σ = 1, 2 corresponding to only ∑3 two linearly inde­ pendent polarizations of the photon, ie, divA = 0 implies r=1 K r er (K, σ) = 0. The energy of the electromagnetic field in the Coulomb gauge is ∫ ∫ HF = (1/2) (E 2 + B 2 )d3 x = (1/2) [(A2,t + (∇ × A)2 ]d3 x ∫ =

2|K|2 |e(K, σ)|2 a(K, σ)∗ a(K, σ)d3 K

once we make use of the fact that |K × e(K, σ)| = |K||e(K, σ)|. For this to be interpretable as the sum of energies of harmonic oscillators, each oscillator in the spatial frequency domain having energy |K|, ie, the frequency of the wave. This means that we must have |e(K, σ)| = (2|K|)−1/2 in order to ensure that ∫ HF =

|K|a(K, σ)∗ a(K, σ)d3 K

We can cross check this result as follows. Assuming that the a(K, σ)' s satisfy the canonical commutation relations: [a(K, σ), a(K ' , σ ' )∗ ] = δ 3 (K − K ' )δσ,σ' it follows from the Heisenberg equations of motion that a(t, K, σ),t = i[HF , a(t, K, σ)] = −i|K|a(t, K, σ), a∗ (t, K, σ),t = i[HF , a(t, K, σ)∗ ] = i|K|a∗ (t, K, σ)

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These equations imply a(t, K, σ),t = −|K|2 a(t, K, σ) a∗ (t, K, σ),tt = −|K|2 a∗ (t, K, σ) which are the correct equations for the spatial Fourier transform of the vector potential arrived from the wave equation. Another way to check these commu­ tation relations which we leave as an exercise, is to start with the Lagrangian density LF = (1/2)(A,t )2 − (1/2)(∇ × A)2 so that the momentum density is πk (t, r) =

∂LF = Ak,t ∂Ak,t

then apply the canonical commutation relations k 3 δ (r − r' ) [Ak (t, r), πm (t, r' )] = iδm

and verify that these relations are satisfied by the above Fourier integral repre­ sentation of A assuming the canonical commutation relations between a(K, σ) and a(K ' , σ ' ). We leave this verification as an exercise to the reader. Now consider the second quantized Dirac field described by the four compo­ nent field operators ψ(x), ψ(x)∗ where x = (t, r), t ∈ R, r ∈ R3 . In the absence of any classical or quantum electromagnetic field ,ψ satisfies the Dirac equation [iγ μ ∂μ − m]ψ(x) = 0 or equivalently, [γ μ pμ − m]ψ = 0, pμ = i∂μ The solutions to ψ are plane waves: ∫ ψ(x) = (u(P, σ)a(P, σ)exp(−ip.x) + v(P, σ)b(P, σ)∗ exp(ip.x))d3 P where p.x = pμ xμ = E(P )t−P.r, E(P ) =

√ m2 + P 2 , p = (π μ ) = (E, P ), u(P, σ), v(P, σ) ∈ C4

Here, the summation is over σ = ±1/2 corresponding to the fact that Dirac’s equation can be expressed as [i∂0 − (α, P ) − βm]ψ(x) = 0, P = −i∇ and hence if P denotes an ordinary 3-vector (not an operator), then u(P )exp(−ip.x) satisfies the Dirac equation iff [p0 − (α, P ) − βm]u(P ) = 0

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123

and likewise, v(P )exp(ip.x) satisfies the Dirac equation iff (−p0 + (α, P ) − βm)v(P ) = 0 Thus, u(P ) is an eigenvector of the matrix HD (P ) = (α, P )+βm with eigenvalue p0 and v(−P ) is an eigenvector of HD (P ) with eigenvalue p0 . Now since HD (P ) is a 4 × 4 Hermitian matrix, it has four real eigenvalues √ taking all multiplicities into account. These eigenvalues are ±E(P ), E(P ) = m2 + P 2 with each one have a multiplicity of two. We denote the corresponding mutually orthogonal eigenvectors by u(P, σ), v(−P, σ), σ = ±1/2. On applying second quantization, the free Dirac Hamiltonian becomes ∫ HDQ = ψ(x)∗ ((α, −i∇) + βm)ψ(x)d3 x and it is easy to verify that the normalizations of u(P, σ) and v(P, σ) are chosen so that ∫ HDQ = E(P )(a(P, σ)∗ a(P, σ) + b(P, σ)b(P, σ)∗ )d3 P and if we postulate the anticommutation relations {a(P, σ), a(P ' , σ ' )∗ } = {b(P, σ), b(P ' , σ ' )∗ } = δσ,σ' δ 3 (P − P ' ) then and only then we can ensure the canonical anticommutation relations (CAR) {ψl (t, r), πm (t, r' )} = iδlm δ 3 (r − r' ) where πm is the canonical momentum associated with the canonical position field ψm . From the free Dirac Lagrangian density LD = ψ(x)∗ (i∂0 − (α, −i∇) − βm)ψ(x) = ψ(x)∗ γ 0 (iγ μ ∂μ − m)ψ(x) we infer that πm (x) =

∂LD = iψl (x)∗ ∂ψl,0

so that the CAR gives {ψl (t, r), ψm (t, r' )∗ } = δlm δ 3 (r − r' ) Thus in particular, we can subtract an infinite constant from the second quan­ tized Dirac Hamiltonian to get ∫ HDQ = E(P )(a(P, σ)∗ a(P, σ) − b(P, σ)∗ b(P, σ)) − − − (1) This equation has the following nice interpretation: a(P, σ)∗ creates an electron with momentum P and spin σ, a(P, σ) annihilates an electron with momentum P and spin σ. b(P, σ)∗ creates positron with momentum P and spin σ while

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b(P, σ) annihilates a positron with momentum P and spin σ. a(P, σ)∗ a(P, σ) is the number operator density for electrons and b(P, σ)∗ b(P, σ) is the number operator for positrons. Since the presence of an additional electron increases the energy of the Dirac sea of electrons by E(P ) while the presence of an additional positron decreases the energy of the Dirac sea by E(P ), equn (1) has the correct physical interpretation for the energy of the second quantized Dirac field. Now suppose we have a collection of photons, electrons and positrons. The total Lagrangian density is then L = LEM + LD + Lint = (−1/4)Fμν F μν + ψ ∗ γ 0 (γ μ (i∂μ + eAμ ) − m)ψ so that LEM = (−1/4)Fμν F μν , LD = ψ ∗ γ 0 (iγ μ ∂μ − m)ψ, Lint = −J μ Aμ , J μ = −eψ ∗ γ 0 γ μ ψ J μ is the Dirac four current density. It is easily verified to be conserved even when an electromagnetic field is present. In other words, we can verify using the Dirac equation [γ μ (i∂μ + eAμ ) − m]ψ = 0 that ∂μ J μ = 0 ie, the current is conserved. We can further show that the matrices K μν = (−1/4)[γ μ , γ ν ] satisfy the same commutation relations as do the standard skew-symmetric gen­ erators of the Lorentz group do. Hence these matrices furnish a representation of the Lie algebra of the Lorentz group. Let D denote the corresponding repre­ sentation of the Lorentz group. D is called the Dirac spinor representation of the Lorentz group and if Λ is any Lorentz transformation, we write D(Λ) = exp(ωμν K μν ) where Λ = exp(ωμν Lμν ) with ω a skew symmetric matrix and Lμν the standard generators of the Lorentz group: (Lμν )αβ = η μα η νβ − η μβ η να Further, we note the following: D(Λ)γ μ D(Λ)−1 = Λμν γ ν

General Relativity and Cosmology with Engineering Applications

125

and hence, the Dirac equation is invariant under Lorentz transformations ie if xμ → Λμν xν and ψ(x) → D(Λ)ψ(x), Aμ → Λμν Aν , then the Dirac equation remains invariant. Further, the existence of the positron follows from the fact that if we start with the Dirac equation, conjugate it and multiply by the unitary matrix iγ 2 , then we get γ μ )(iγ 2−1 )(−i∂μ + eAμ ) − m]iγ 2 ψ¯ = 0 [(iγ 2 )(¯ It is easily verified that this equation is the same as [γ μ (i∂μ − eAμ ) − m]ψ˜ = 0 where

ψ˜ = iγ 2 ]ψ¯

In other words ψ˜ satisfies the Dirac equation in an electromagnetic field but with the charge e replaced by −e or equivalently, −e replaced by e. This observation led Dirac to conclude the existence of the positron, namely the antiparticle of the electron, having the same mass but opposite charge as that of the electron. The positron was discovered in an accelerator later by Anderson. Current in the BCS theory of superconductivity: The BCS Hamiltonian is ∫ H = ψa∗ (x)(E(−i∇ + eA(x)) + V (x))ψa (x)d3 x+ ∫

Va1 a2 a3 a4 (x1 , x2 , x3 , x4 )ψa1 (x)∗ ψa2 (x)∗ ψa3 (x3 )ψa4 (x4 )d3 x1 d3 x2 d3 x3 d4 x4

where all the x' s have the same time component. Here, E(P ) = P 2 /2m in the case of non-relativistic particles and E(P ) = (α, P ) + βm in the case of relativistic particles. V (x) = V (t, r) is the external potential and A(x) is the external magnetic vector potential. The ψa' s satisfy the CAR {ψa (t, r), ψb (t, r' )∗ } = δab δ 3 (r − r' ) The current density operator is given by J μ (x) = −eψa∗ (t, r)αμ ψa (x) in the relativstic case and in the non-relativistic case, J 0 = −eψa (x)∗ ψa (x), J r = (e/2m)Im(ψa (x)∗ (∂r + ieAr (x))ψa (x))

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summation over repeated indices is implied. If the state is the Gibbs state with A(x), V (x) both being time independent, then the perturbed Gibbs state of the Fermi fluid is given by ρ = exp(−βH)/T r(exp(−βH)) and the measured current as a function of A, V is given by < J μ (x) >= T r(ρJ μ (x)) Here, ρ calculated by taking the ψa' s at time zero in H while J μ (x) requires the ψa' at time t. This is obtained using the equations of motion ψa,t (t, r) = i[H, ψa (t, r)] with the commutator evaluated using the anticommutation rules for the ψa' s ' and ψa∗ s . The anticommutation rules are valid at every time t provided that all the observables are evaluated at the same time. H is conserved ie H(t) = H(0) = H since A, V are assumed to be time independent. Hence the ψa' s and ' ψa∗ s in the integral expression for H can be taken at any time t. As a first order approximation, we can take ρ = exp(−βH0 )/T r(exp(−βH0 ) where H0 is obtained by setting A = 0, V = 0 (ie equilibrium density in the absence of external forces).

4.6 Energy-momentum tensor of matter with vis­ cous and thermal corrections Assume first the special relativistic case. The energy-momentum tensor without the corrections is T μν = (ρ + p)v μ v ν − pg μν Let ΔT αβ be the correction to the tensor due to viscosity and heat conduction. We have the conservation law (T μν + ΔT μν ),ν = 0 and hence μ − p,μ + ΔT,νμν = 0 ((ρ + p)v ν ),ν v μ + (ρ + p)v ν v,ν

from which, we deduce by multiplying by vμ , the mass conservation equation ((ρ + p)v ν ),ν − p,μ v μ + vμ ΔT,νμν = 0 − − − (1) Let n denote the particle number density field. Then its conservation law can be expressed as (nv μ ),μ = 0 − − − (2)

General Relativity and Cosmology with Engineering Applications

127

The basic energy conservation equation in thermodynamics with σ denoting the entropy per particle is given by T dσ = d(ρ/n) + pd(1/n) = d((ρ + p)/n) − dp/n so that T σ,μ v μ = ((ρ + p)/n),μ v μ − p,μ v μ /n = (ρ + p),μ v μ /n − (ρ + p)n,μ v μ /n2 − p,μ v μ /n μ = (ρ + p),μ v μ /n + (ρ + p)v,μ /n − p,μ v μ /n

= ((ρ + p)v μ ),μ /n − p,μ v μ /n where in the last but one equality, we have made use of (2). Thus, T σ,μ v μ = = −(ΔT μν ),ν vμ /n or equivalently, nσ,μ v μ = −(ΔT,νμν vμ /T or using again (2), ((nσ + ΔT μν vμ /T ),ν = ΔT μν (vμ /T ),ν = ΔT μν vμ,ν /T − ΔT μν vμ T,ν /T 2 The lhs can be regarded as the rate of change of entropy per unit volume of the fluid which according to the second law of thermodynamics, should be non­ negative. Now choose a space-time point P and a frame that is comoving at P, ie, the three velocity v i at P is zero. Therefore at P , v 0 = 1. Then the equation 0 = 0 and also v0,ν = 0. v μ vμ = 1 implies on differentiating w.r.t. xν that at P v,ν Evaluating the rhs of the above equation at P thus gives us ΔT i0 vi,0 /T + ΔT ij vi,j /T − ΔT 0i T,i /T 2 ≥ 0 Noting the symmetry of ΔT μν , this can be guaranteed provided we choose k ΔT ij = χ1 (T )(vi,j + vj,i ) − χ2 (T )v,k δij ,

ΔT i0 = χ3 (T )(T vi,0 − T,i ) where χj (T ), j = 1, 2, 3 are positive scalar functions of the temperature T . We now define the four tensor α μν ΔT˜ μν = χ1 (T )(v μ,ν + v ν,μ ) + χ2 (T )v,α g

the four vector Qμ = χ3 (T )T ,μ

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General Relativity and Cosmology with Engineering Applications

and the four tensor S μν = Qμ v ν + Qν v μ We then have at the space-time point P k δij , ΔT˜ij = χ1 (vi,j + vj,i ) − χ2 v,k

ΔT˜i0 = −χ1 vi,0 , S ij = 0, S i0 = Qi = −χ3 T,i Now, define the four tensor ΔT μν = H μα H νβ ΔT˜αβ − Hρμ Qρ v ν − Hρν Qρ v μ where H μν = −g μν + v μ v ν Then at P , we have H i0 = 0, H ij = δij , H 00 = 0, H0i = 0, Hji = −δij , Hi0 = 0, H00 = 0, and hence at P , ΔT ij = ΔT˜ij , ΔT i0 = Qi Thus we get from the tensor character of ΔT μν that in any frame, ΔT μν = H μα H νβ ΔT˜αβ − Hρμ Qρ v ν − Hρν Qρ v μ

Exercise: Consider the Robertson-Walker metric of homogeneous and isotropic space-time. the zero three velocity vector is known to satisfy the geodesic equa­ tions, ie, the metric is a comoving metric. We consider small (inhomogeneous in space-time) perturbations of the metric, the three velocity and the density field and linearize the Einstein field equations around the homogeneous and isotropic values of these taking into account the above formula for the corrections in the energy-momentum tensor. Then, determine the pde’s satisfied by the above per­ turbations and derive appropriate dispersion relations in terms of the expansion factor S(t) of the Robertson-Walker universe.

General Relativity and Cosmology with Engineering Applications

4.7

129

Energy-momentum tensor of the electromag­ netic field in a background curved spacetime

The action functional of the em field is given by ∫ √ SEM [A, g] = (−1/4) Fμν F μν −gd4 x One way to determine its energy-momentum tensor, is to compute it as the √ coefficient of δgμν −g when the metric is allowed to vary slightly. The reason for this is that if the action functional of the gravitational field is given by ∫ √ SG [g] = L −gd4 x then, the variational principle δg (SG [g] + SEM [A, g]) = 0 would give rise to the Einstein field equations EM Gμν = K.Tμν

√ where Gμν is the coefficient of gδgμν in δg SG [g]. Gμν is the Einstein tensor Rμν − (1/2)Rgμν and it satisfies the Bianchi identity Gμν :ν = 0. Hence, we would get conservation of the energy-momentum of the EM field (assuming absence √ μν EM μν = 0. Here, T is the coefficient of −gδg of matter), ie, T:EM μν in the ν variation δSEM [g]. Now, we compute √ √ δg SEM [A, g] = (−1/4)Fμν F μν δ −g − ( −g/4)Fμν Fαβ δ(g μα g νβ ) √ = (−1/8)Fμν F μν −gg αβ δgαβ √ +( −g/2)Fμν Fαβ (g μα g νρ g βσ δgρσ √ = (−1/8)Fμν F μν −gg αβ δgαβ √ +( −g/2)Fμρ F μσ δgρσ Thus after removing a proportionality constant, T EM αβ = (−1/4)Fμν F μν g αβ + Fμα F μβ We now check this result for the special relativistic case, ie, when the background space-time is flat Minkowskian.

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4.8 Relativistic Fermi fluid in a gravitational field The Dirac equation in a gravitational field has the form [γ a Vaμ (x)i(∂μ + Γμ (x)) − m]ψ(x) = 0 where Γμ (x), μ = 0, 1, 2, 3 are 4 × 4 matrix valued functions of the space-time coordinates and Vaμ (x) is a tetrad basis for the metric gμν (x): g μν (x) = η ab Vaμ (x)Vbν (x) with η ab being the Minkowski metric. If Λ(x) is a local Lorentz transformation and D the spinor representation of the Lorentz group, then we get D(Λ)γ a D(Λ)−1 Vaμ i(D(Λ)∂μ D(Λ)−1 + D(Λ)Γμ D(Λ)−1 )D(Λ)ψ = 0 or equivalently, Λab γ b Vaμ i(∂μ + (D(Λ)Γμ D(Λ)−1 + D(Λ)(∂μ D(Λ)−1 ))D(Λ)ψ = 0 which is same as the Dirac equation in a gravitational field with locally Lorentz transformed tetrad V˜ μ = Λa V μ , b

b

a

and locally Lorentz transformed gravitational connection ˜ = D(Λ)Γμ D(Λ)−1 + D(Λ)(∂μ D(Λ)−1 ) Γ Now we can identify the Dirac Hamiltonian in a gravitational field as HD =

Dirac Fermionic liquid in a static electromagnetic field: ∫ H0 = ψa∗ (x)((α, −i∇ + eA(x)) + βm − eV (x))ψa (x)d3 x ∫ +

4.9

V (x, y)ψa (x)∗ ψa (x)ψb (y)∗ ψb (y)d3 xd3 y

The post-Newtonian approximation

Quantities are expanded in powers of the three velocity magnitude or equiva­ lently in view of the mass-orbital velocity relationship in Newtonian mechanics,

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√ √ v = GM/r in powers of M . Thus for any physical quantity X, we have the perturbative expansion X = X (0) + X (1) + ... where X (r) is O(v r ) = O(M r/2 ). By analogy with the Schwarzchild metric for which g00 = 1 − 2GM/rc2 and g11 = (1 − 2GM/rc2 )−1 , we expand (2)

(4)

(2m)

g00 = 1 + g00 + g00 + ... + g00

+ ...

(2) (4) (2m) grs = −δrs + grs + grs + ... + grs + ... (1)

(3)

(2m+1)

gr0 = gr0 + gr0 + ... + gr0

+ ...

The odd powers of velocity expansion of gr0 can be seen as follows. Suppose K is a one dimensional Minkowski frame with metric dτ 2 = dt2 − dx2 /c2 . If another frame K ' moves relative to K with a uniform velocity v along the x axis, then a simple Galilean transformation gives x' = x − vt, t' = t so that ' ' ' dτ 2 = dt 2 − (dx' + vdt' )2 /c2 = dt 2 (1 − v 2 /c2 ) − dx 2 − 2vdt' dx' /c2 which ' ' contains only zeroth and second powers of the velocity, and g10 shows that g00 contains only the first power of the velocity. Now substituting this perturbative expansion of the metric into the geodesic equation gives the following: d2 xr /dτ 2 = d/dτ ((dt/dτ )dxr /dt) = (d2 t/dτ 2 )dxr /dt + (dt/dτ )2 d2 xr /dt2 The exact geodesic equation can be expressed as with v k = dxk /dt: r v k v m + 2(dt/dτ )2 Γr0m v m (dt/dτ )2 dv r /dt + (d2 t/dτ 2 )v r + (dt/dτ )2 Γkm

+(dt/dτ )2 Γr00 = 0 or equivalently, r =0 dv r /dt + Γrkm v k v m + (t,τ τ /t2,τ )v r + 2Γr0m v m + Γ00

We have Γrkm = g r0 Γ0km + g rs Γskm Now, g μν gνρ = δρμ implies the following:

g 00 g0r + g 0s gsr = 0, g 00 g00 + g 0r g0r = 1, g r0 g0s + g rk gks = δsr = δrs

Thus, writing

g 0r = g 0r(1) + g (0r)(3) + ..., g 00 = 1 + g (00)(2) + g (00)4 + ...,

132

General Relativity and Cosmology with Engineering Applications g rs = −δrs + g (rs)(2) + g (rs(4) + ... we get

(1)

(1 + g (00)2 + g (00)4 + ...)(g0r + g (0r(3) + ...)+ (2) + ...) = 0 (g 0s(1) + g (0s(3) + ...)(−δsr + grs

so that

g (r0)(1) = gr0(1) , (1)

(2) − g (r0(3) + g (00)(2) g0r + g (r0(3) = 0 g (0s)(1) grs (1)

(1)

In particular, if we assume that gr0 = 0, then it follows that gr0 = 0 and (3) g (r0(3) = gr0 . We shall henceforth make such an assumption which is in agree­ ment with the Schwarzchild metric, according to which g00 and g11 contain only upto O(v 2 ) = O(M ) terms while the non-diagonal metric components identi­ cally vanish. We then get from g 00 g00 + g 0r g0r = 1, g r0 g0s + g rk gks = δrs the following: (2)

(4)

(1 + g (00(2) + g (00(4) + ...)(1 + g00 + g00 + ..) (3)

+(g (0r(3) + g (0r(5) + ...)(g0r + ...) = 1 and hence

(2)

g (00(2) = −g00 , (4)

(2)

g (00(4) + g00 + g (00(2) g00 = 0 implying thereby that

(4)

(2)

g (00(4) = −g00 + (g00 )2 , and further, (2)

(4)

(−δrk + g rk((2) + g (rk(4) + ...)(−δks + gks + gks + ...) = δrs which yields

(2) (2)

(2) (4) , −g (rs(4) − grs − grk gks = 0 g (rs(2) = −grs

so that

(2) (2)

(4) g (rs(4) = −grs − grk gks

We have the following perturbation expansions for the Christoffel symbols: (2)

(4)

(3)

(5)

Γrkm = Γrkm + Γrkm + .. Γ0km = Γ0km + Γ0km + ...

General Relativity and Cosmology with Engineering Applications because

(2)

133

(4)

gkm = −δkm + gkm + gkm + ... (2)

(2)

gkm,0 = (gkm ),0 + (gkm ),0 + ... and since one time derivative increases the perturbation order by one, we have (2)

(gkm,0 )(3) = (gkm ),0 etc. Also,

(3)

(5)

g0k,m = (g0k ),m + (g0k ),m + ... and since spatial derivatives do not change the perturbation order, we have (3)

(g0k ),m = (g0k,m )(3) etc. Remark: Roman indices like i, j, k, m, r, s, l denote spatial components ie they assume the values 1, 2, 3 only while Greek indices like μ, ν, α, β, ρ, σ, γ, δ denote space-time components, ie, they assume all four values 0, 1, 2, 3. We have further, (3) (5) Γrm0 = Γrm0 + Γrm0 + ... and

(2)

(4)

Γr00 = Γr00 + Γr00 + ... For example, 1 (2) (2) (2) (2) Γrkm = ( )((grk ),m + (grm ),k − (gkm ),r ) 2 (2)

(2)

(2)

= (1/2)(grk,m + grm,k − gkm,r ) (3)

(3)

(3)

(2) Γrm0 = (1/2)((grm ),0 + (gr0 ),m − (gm0 ),r ) (2)

(2)

Γr00 = (1/2)(−(g00 ),r ) (4)

(3)

(4)

Γr00 = (1/2)(2(gr0 ),0 − (g00 ),r ) (3)

(5)

Γ000 = Γ000 + Γ000 + ... where

(3)

(2)

Γ000 = (1/2)(g00 ),0 etc.

(2)

(4)

Γ00r = Γ0r0 = Γ00r + Γ00r + ... where

(2)

(2)

(4)

(4)

Γ00r = (1/2)(g00 ),r Γ00r = (1/2)(g00 ),r For the Christoffel symbols of the second kind, we likewise have the perturbation expansions Γrkm = g r0 Γ0km + g rs Γskm

134

General Relativity and Cosmology with Engineering Applications r(2)

r(4)

= Γkm + Γkm + ... where

(2)

r(2)

Γkm = −Γrkm , (4)

r(4)

(2)

Γkm = −Γrkm + g rs(2) Γskm The equations of motion of a particle assume the perturbative form r =0 dv r /dt + Γrkm v k v m + (t,τ τ /t2,τ )v r + 2Γr0m v m + Γ00

Now, t,τ τ /t2,τ = −d/dt(log(τ,t ) = −τ,tt /τ,t Now, τ,t = (g00 + 2g0r v r + grs v r v s )1/2 (2)

(4)

(3)

(2) r s 1/2 = (1 + g00 + g00 + 2g0r v r − v 2 + grs v v )

with neglect of O(v 6 ) and higher terms. So, (2)

(4)

(3)

(2) log(τ,t ) = 1 + g00 /2 + g00 /2 + g0r v r − v 2 /2 + grs (2)

+(−1/8)(g00 )2 + (−1/8)v 4 + O(v 5 ) Thus, in particular, (log(τ,t )),t = (2)

(2)

k + O(v 4 ) (1/2)((g00 ),0 + (g00 ),r v r − v k v,t

With neglect of O(v 7 ) terms, our equations of motion are r(2)

r(4)

r(3)

dv r /dt + Γkm + Γkm )v k v m − (log(τ,t )),t v r + 2(Γ0m + r(5)

r(2)

(r(4)

Γkm )v m + Γ00 + Γ00

(r(6)

+ Γ00

=0

while with neglect of O(v 5 ) terms, the equations of motion are r(2)

r(2)

r v,0 + Γ00 + Γkm v k v m (2)

(2)

k r +[(1/2)((g00 ),0 + (g00 ),k v k − v k v,0 ]v r(3)

r(2)

+2Γ0m v m + Γ00 = 0 These constitute the post-Newtonian equations of celestial mechanics. We now derive the post-Newtonian equations of hydrodynamics.

General Relativity and Cosmology with Engineering Applications

4.10

135

The BCS theory of superconductivity

ψ1 (t, x), ψ2 (t, x) are the two Fermionic fields corresponding respectively to up and down spin states of the electron. They satisfy the canonical anticommuta­ tion relations {ψa (t, x), ψb (t, x' )∗ } = δab δ 3 (x − x' ) We use the notation ψa (x) for ψa (0, x) and likewise for ψa (x)∗ . The BCS Hamil­ tonian is then defined as H=

∑ ∫

ψa (x)∗ (−∇2 /2m+V (x))ψa (x)d3 x+

∫

f0 (x) < ψ1 (x)ψ2 (x) > ψ1 (x)∗ ψ2 (x)∗ d3 x

a=1,2

∫ + ∫ +

f¯0 (x) < ψ2 (x)∗ ψ1 (x)∗ > ψ2 (x)ψ1 (x)d3 x

(f1 (x) < ψ1 (x)∗ ψ1 (x) > ψ2 (x)∗ ψ2 (x)+f2 (x) < ψ2 (x)∗ ψ2 (x) > ψ1 (x)∗ ψ1 (x))d3 x

We write for convenience of notation Δ(x) =< ψ1 (x)ψ2 (x) > so that Δ(x)∗ =< ψ2 (x)∗ ψ2 (x) > Here, the quantum expectation < . > is taken w.r.t. the Gibbs density ρG = exp(−βH)/Z(β), Z(β) = T r(exp(−βH)) we note that H is a constant of the motion since it is by definition, time inde­ pendent, ie, the coefficients functions V, f0 , f1 , f2 do not explicitly depend on time. We get using the Fermionic anticommutation relations [H, ψ1 (x)] = (∇2 /2m − V (x))ψ1 (x) − f0 (x)Δ(x)ψ2 (x)∗ − f2 (x)n2 (x)ψ1 (x) where na (x) =< ψa (x)∗ ψa (x) >, a = 1, 2 and likewise, [H, ψ2 (x)] = (∇2 /2m − V (x))ψ2 (x) + f0 (x)Δ(x)ψ1 (x)∗ − f1 (x)n1 (x)ψ2 (x) Define the following Green’s functions: G(t, x|t' , x' ) =< T (ψ1 (t, x)ψ1 (t' , x' )∗ ) >, F (t, x|t' , x' ) =< T (ψ1 (t, x)ψ2 (t' , x' )) > where T is the time ordering operator.

136

General Relativity and Cosmology with Engineering Applications Remark: If the Fermions are subject to a gravitational field described by a static metric tensor gμν (x), then we can approximate the energy of a such a par­ ticle due to motion and gravitational effects by considering first the Lagrangian of the particle: L = −mτ,t = −m(g00 + 2g0r v r + grs v r v s )1/2 √ 2 ≈ −m g00 (1 + g0r v r /g00 + (grs g00 − g0r g0s )v r v s /2g00 ) To express the corresponding Hamiltonian in terms of canonical coordinates and momenta, we first compute the momenta as pr = −L,vr = m(g0r + grs v s )/τ,t , pr = −pr and then the Hamiltonian using the Legendre transformation as H = pr v r − L = = −m(g0r v r + grs v r v s )/τ,t + mτ,t = [mτ,t2 − m(g0r v r + grs v r v s )]/τ,t = (mg00 + mg0r v r )/τ,t = mg0μ uμ = mu0 Now, pr = mgrμ v μ /τ,t = mgrμ uμ and writing p0 = mg0μ v μ /τ,t = mg0μ uμ = mu0 so that pν = mgνμ v μ /τ,t = mgνμ uμ = muν we get pμ = g μν pν = muμ and hence pμ = muμ and in particular, H = p0 We note that the equation g μν uμ uν = 1 implies

g μν pμ pν = m2

Thus the energy p0 = H satisfies the quadratic equation g 0 p20 − 2g 0r p0 pr + g rs pr ps − m2 = 0 Solving this for p0 and replacing pr by −i∂r gives us the Hamiltonian operator H = p0 of the particle in terms of ∂r and it is this operator p0 that must be used to compute the free particle energy of the Fermi liquid: ∑ ∫ H0 = ψa (x)∗ p0 ψa (x)d3 x a=1,2

General Relativity and Cosmology with Engineering Applications

4.11

137

Quantum scattering theory in the presence of a gravitational field

The Dirac equation for an electron in the presence of an electromagnetic field and a gravitational field described by a tetrad Vμa (x) and a corresponding connection Γμ (x) which is a 4 × 4 matrix valued function of the space-time coordinates x is given by (Steven Weinberg, Gravitation and Cosmology) [γ a Vaμ (i∂μ + eAμ + iΓμ ) − m]ψ = 0 This equation can be derived from a variational principle with Lagrangian den­ sity √ L = ψ ∗ αa Vaμ (i∂μ + eAμ + iΓμ )ψ −g where

αa = γ 0 γ a , a = 0, 1, 2, 3

Note that α0 = 1. Unfortunately, this Lagrangian density is not real and hence we replace it by its real part: √ L = Re[(ψ ∗ αa Vaμ (i∂μ + eAμ + iΓμ )ψ) −g] Let us compute the Hamiltonian density corresponding to this Lagrangian den­ sity. The canonical momentum densities are π=

∂L = ∂ψ,0

√ (i/2) −gVa0 αaT ψ¯ π ¯=

∂L = ∂ψ¯,0

√ (−i/2) −galphaa Va0 ψ Note that the αa matrices are Hermitian. So the Hamiltonian density is H = πT ψ + π ¯ T ψ¯ − L = √ √ Var Re[psi∗ αa (i∂r ψ)] −g − Vaμ Re[ψ ∗ αa (eAμ + iΓμ )ψ] −g The first term represents the kinetic energy of the Dirac particle in curved spacetime and the second terms represents the interaction energy between the Dirac particle and the electromagnetic and gravitational field. This is the second quantized picture and can be used in the BCS theory of superconductivity. In quantum scattering theory, we are concerned with first quantized Hamiltonians. Thus, we write √ H0 = − −g(x)Var (x)αa P r , P r = −i∂r for the unperturbed energy of the incoming projectile in a background gravita­ tional field and √ V = − −gVaμ (x)αa (eAμ + iΓμ (x))

138

General Relativity and Cosmology with Engineering Applications

More precisely, V should be defined as the Hermitian part of the above matrix valued function of position. When we assume that the gravitational field is time independent and so is the electromagnetic field, then V becomes a matrix valued function of the spatial coordinates only while H0 becomes a vector field whose coefficients are time independent. The scattering matrix in this case is defined by S = Ω∗+ Ω− where Ω+ = limt→∞ exp(it(H0 + V )).exp(−itH0 ), Ω− = limt→−∞ exp(it(H0 + V )).exp(−itH0 ) More generally, if H0 is time independent but V is time dependent, then one could ask the question how one defines the scattering matrix. The answer is as follows. Write H1 (t) = H0 + V (t). Then if φi is the input free particle state that gets scattered to the input scattered state ψi while ψf is the final scattered state that evolves into the free particle state ψf , then we have U (0, −T )−1 ψi − U0 (0, −T )−1 )φi → 0, T → ∞, U (T, 0)ψf − U0 (T, 0)φf → 0, T → ∞ Thus, Ω= limT →∞ U (0, −T )U0 (−T ), Ω+ = limT →∞ U (T, 0)−1 U0 (T ) and hence, the scattering matrix is defined by S = Ω∗+ Ω− = limT →∞ U0 (−T )U (T, 0)U (0, −T )U0 (−T ) ∫ T V (t)dt)}.exp(−iT H0 ) = limT →∞ exp(iT H0 ).T {exp(i −T

4.12 Maxwell’s equations in the Schwarzchild spacetime dτ 2 = α(r)dt2 − α(r)−1 dr2 − r2 (dθ2 + sin2 (θ)dφ2 ) α(r) = 1 − 2m/r, m = GM, c = 1 This is the metric of space-time. g00 = α(r), g11 = −α(r)−1 , g22 = −r2 , g33 = −r2 sin2 (θ)

General Relativity and Cosmology with Engineering Applications

139

The contravariant electromagnetic four potential is A1 = Ar , A2 = Aθ , A3 = Aφ , A0 = V The covariant electromagnetic four potential is A0 = g00 A0 = α(r)A0 , A1 = g11 A1 = −α(r)−1 A1 , A2 = g22 A2 = −r2 A2 , A3 = g33 A3 = −r2 sin2 (θ)A3 F01 = A1,0 − A0,1 = −α−1 A1,0 − αA0,1 , F02 = A2,0 − A0,2 = −r2 A2,0 − αA0,2 , F03 = A3,0 − A0,3 = −r2 sin2 (θ)A3,0 − αA0,2 F 01 = g 00 g 11 F01 = −F01 , F 02 = g 00 g 22 F02 = −αr−2 F02 , F 03 = g 00 g 33 F03 = −α(r.sin(θ))−2 F03 The Maxwell equations in the absence of current sources but in the presence of the Schwarzchild gravitationl field are √ (F μν −g),ν = 0 We list these equations below: √ √ √ √ (F 0r −g),r = (F 01 −g),1 + (F 02 −g),2 + (F 03 −g),3 = 0 and

√ √ F,r00 −g + (F rs −g),s = 0

or equivalently, r2 sin(θ)F,010 + (F 12 r2 sin(θ)),2 + F,313 r2 sin(θ) = 0 r2 sin(θ)F,020 + (F 21 r2 sin(θ)),1 + F,323 r2 sin(θ) = 0 r2 sin(θ)F,030 + (F 31 r2 sin(θ)),1 + (F 32 r2 sin(θ)),2 = 0 Remark: We wish to give meaning to F μν in terms of electric and magnetic fields. For that purpose, we consider the Minkowskian flat space-time metric and evaluate F μν using this metric. The Minkowskian metric is dτ 2 = dt2 − dr2 − r2 (dθ2 + sin2 (θ)dφ2 ) for which

g00 = 1, g11 = −1, g22 = −r2 , g3 = −r2 sin2 (θ)

Then the Cartesian components of the electromagnetic four potential Ax , Ay , Az , V and the polar components Ar , Aθ , Aφ , At are related by Ar = Ax r,x + Ay r,y + Az r,z =

140

General Relativity and Cosmology with Engineering Applications = Ax cos(φ)sin(θ) + Ay sin(φ)sin(θ) + Az cos(θ)

which is the usual definition for the radial component of the magnetic vector potential. Aθ = Ax θ,x + Ay θ,y + Az θ,z = (A, ∇θ) which is the usual θ component of the magnetic vector potential multiplied by |∇θ| = 1/. Finally, Aφ = (A, ∇φ) which is the usual φ component of the magnetic vector potential divided by r.sin(θ). We write these relations as Ar = Ar , Aθ = r−1 Aθ , Aφ = (r.sinθ)−1 Aφ , At = V Thus, we have F01 = A1,0 − A0,1 = −Ar,0 − V,1 = −Ar,0 − V,r which is Er , the radial component of the eletric field. F02 = A2,0 − A0,2 = −r2 Aθ,0 − V,θ = −rAθ,0 − V,θ = rEθ F03 = −r.sin(θ)Eφ Further,

F 01 = g 00 g 11 F01 = −Er , F 02 = g 00 g 22 F02 = −r−1 Eθ F 03 = g 00 g 33 F03 = −(r.sin(θ))−1 Eφ F12 = A2,1 − A1,2 = −(r2 Aθ ),r + Ar,θ = = −(rAθ ),r + Ar,θ = −rBφ

(using the formula for the curl in spherical polar coordinates). Thus, F 12 = g 11 g 22 F12 = (−1/r2 )F12 = Bφ /r F23 = A3,2 − A2,3 = −r2 (sin2 (θ)Aφ ),θ + (r2 Aθ ),φ = −r(sin(θ)Aφ ),θ + rAθ,φ = −r2 sin(θ)Br F 23 = g 22 g 33 F23 = (−1/r2 sin(θ))Br and finally, F31 =

General Relativity and Cosmology with Engineering Applications

141

4.13

Some more problems in general relativity

4.13.1

Gaussian curvature of a two dimensional surface

Consider a two dimensional surface parametrized by u, v so that a general point on the surface can be expressed as r = r(u, v) = (x(u, v), y(u, v), z(u, v)) Calculate the metric on the surface in the form ds2 = |dr|2 = g11 (u, v)du2 + g22 (u, v)dv 2 + 2g12 (u, v)dudv Now choose a curve on the surface parametrized by t → (u(t), v(t)) or more precisely as t → r(u(t), v(t)). Calculate its curvature at t: K(t) = |d2 r/ds2 | Now consider a point say (u0 , v0 ) on the surface. Draw the unit normal n(u0 , v0 ) at this point to the surface at this point. Now consider the set of all planes con­ taining this normal and let the maximum and minimum curvatures of the curves at (u0 , v0 ) in which this plane intersects the surface be K1 and K2 respectively. Determine the Gauss Curvature of the surface at (u0 , v0 ) defined by K1 K2 . Also determine the components of the Riemann-Christoffel curvature tensor of the surface at (u0 , v0 ). Remark: Consider the curve (u(s), v(s)) on the surface or equivalently, r(s) = r(u(s), v(s)) parametrized by the curve length parameter s, ie ds2 = |dr|2 . Assume that this curve is the intersection of the surface and a plane passing through the normal n to the surface at (u(s), v(s). Then '

'

dr/ds = ru u' + rv v ' , d2 r/ds2 = ruu u 2 + rvv v 2 + 2ruv u' v ' + ru u'' + rv v '' where u' = du/ds, u'' = d2 u/ds2 etc. Show that (dr/ds, d2 r/ds2 ) = 0 and hence d2 r/ds2 is a normal to the curve at r(s). Equivalently, u' (ru , d2 r/ds2 ) + v ' (rv , d2 r/ds2 ) = 0 Since (n, ru ) = (n, rv ) = 0, we get '

'

K(u' , v ' ) = (n, d2 r/ds2 ) = (n, ruu )u 2 + (n, rvv )v 2 + 2(n, ruv )u' v ' Now determine the maximum and minimum values of K(u' , v ' ) as u' , v ' vary in ' ' such a way that g11 u 2 + g22 v 2 + 2g12 u' v ' = 1.

142

4.13.2

General Relativity and Cosmology with Engineering Applications

Parallel displacement on a two dimensional surface

Compute the formulas for parallel displacement on a two dimensional surface specified by (u, v) → r(u, v) ∈ R3 . Specifically, for a vector on the surface (Au , Av ) defined by A(u, v) = Au (u, v)ru + Av (u, v)rv Compute u u δAu = Γuuu Au du + Γuuv Av du + Γvv Av dv + Γuv Au dv, v v δAv = Γvuu Au du + Γvuv Av du + Γvv Av dv + Γuv Au dv,

4.13.3

Linearized dynamics in general relativity

If there is a small random fluctuation in the density, velocity and pressure of a Newtonian fluid, then determine the first order perturbation equations for the same. Assume an appropriate equation of state p = p(ρ) and derive your formulae. Now consider a small perturbation in the metric, fluid velocity field, density and pressure assuming an appropriate equation of state. Choose coordinates so that the metric perturbations satisfy δg0μ = 0 and calculate the linear pde’s satisfied by δgrs (x), δv r (x) and δρ(x).

4.13.4

MHD equations in general relativity

Consider the Einstein-Maxwell equations in the presence of a conducting fluid. These equations are of the form Rμν − (1/2)Rgμν = K(Tμν + Sμν ) where Tμν = (ρ + p)vμ vν − pgμν is the energy-momentum tensor of the matter field and Sμν is the energy-momentum tensor of the electromagnetic field. These equations imply the general relativistic generalizations of the MHD equations: (T μν + S μν ):ν = 0 and are to be combined with the Maxwell equations for the conducting fluid expressed in the form F:νμν = J μ = σF μν vν where Fμν = Aν,μ − Aμ,ν Derive the first order perturbation equations satisfied by the metric pertur­ bations, the four potential perturbations, the density, pressure and velocity perturbations.

General Relativity and Cosmology with Engineering Applications

4.13.5

143

Energy momentum tensor of a system of particles T μν (x) = (−g(x))−1/2

∑

Mn δ 3 (x − xn )(dxμn /dτ )(dxνn /dt)

n

In special relativity, ie, flat space-time, this equals ∑ T μν (x) = γ(xn )Mn δ 3 (x − xn )(dxμn /dt)(dxνn /dt) n

where

γ(xn ) = dt/dτ = (1 − vn2 )−1/2 , vnr = dxrn /dt

Mn is the rest mass of the nth particle and hence γ(xn )Mn is the mass of the nth particle as measured in the laboratory frame. Another way to express this tensor is by noting that En = γ(xn )Mn is the energy of the nth mass as measured in the laboratory frame and then ∑ T μν = (Pnμ Pnν /En )δ 3 (x − xn ) n

where Pnμ = γ(xn )Mn dxμn /dt is the four momentum of the nth particle as measured in the laboratory frame.

4.13.6

EKF for estimating the fluid velocity field in gen­ eral relativity

Discretize the equations of motion of an Eulerian incompressible fluid in a curved background metric with respect to the spatial indices and show how it can be simulated. hint: ((ρ + p(x))v μ (x)v ν (x)):ν − g μν p,ν (x) = 0 This gives ((ρ + p)v ν ):ν v μ + (ρ + p)v ν v:μν − g μν p,ν = 0 This implies the equation of continuity ((ρ + p)v ν ):ν − p,ν v ν = 0 Assuming the fluid to be incompressible means that ρ is a constant and then the equation of continuity reduces to ρv:νν + (pv ν ):ν − p,ν v ν = 0 − − − (1) Substituting the incompressibility condition into the above equation for energymomentum conservation then gives (ρ + p)v ν v:μν + p,α v α v μ − p,μ = 0 − − − (2)

144

General Relativity and Cosmology with Engineering Applications (1) and (2) are our basic set of four equations for the four fields v r , r = 1, 2, 3, p. Taking μ = r in (2) results in r r r + Γr00 v 0 + Γr0k v k ) + v m (v,m + Γmk v k )) (ρ + p)(v 0 (v,0

+(p,0 v 0 + p,k v k )v r − g r0 p,0 − g rs p,s = 0 where This gives

g00 v 02 + 2g0r v r v 0 + grs v r v s = 0 v 0 = (−g0r v r /g00 ) +

√ (g0r g0s − g00 g rs )v r v s /g00

The plus sign for the discriminant has been chosen so that in the limit of flat space-time, v 0 reduces to (1 − v 2 )−1/2 . We define the ”spatial metric” 2 γrs = (g0r g0s − g00 grs )/g00

Also define hr = −g0r /g00 Then, we can write

v 0 = hr v r +

√ γrs v r v s

Now consider the problem of dynamically estimating the velocity and pressure field when there is noise in the system. The above equations of motion are then of the form r = f r (t, v k , k = 1, 2, 3), v,0 without noise and with noise, dv r (t) = f r (t, v k (t), k = 1, 2, 3, p(t))dt + σ(r, k)dBk (t) dp(t) = f 0 (t, v k (t), k = 1, 2, 3, p(t))dt + σ0 dB0 (t) where summation over the repeated index k = 1, 2, 3 is implied. Measurements are taken on the velocity field at the spatial points (r1 δ, r2 δ, r3 δ) for (r1 , r2 , r3 ) ∈ E where E is a set of integers. Note that the above notation means that if the spatial discretization grid size is (2N + 1) × (2N + 1) × (2N + 1), then v r (t) =

N ∑

v r (t, r1 δ, r2 δ, r3 δ)e(r1 ) ⊗ e(r2 ) ⊗ e(r3 )

r1 ,r2 ,r3 =−N

and e(r) is the 2N + 1 × 1 vector with a 1 at the (r + N + 1)th position and zeros at all the other positions. The measurement model is clearly dz(t) = Hr v r (t)dt + σv dV (t) where Hr is a sparse matrix consisting of only ones and zeroes, and is defined by the fact that its column indices corresponding to (r1 , r2 , r3 ) ∈ E and serially arranged row indices with the number of rows equal to the number of elements in E is precisely one and the other entries of H r are zeroes. We leave it as an exercise to formulate the EKF equations for obtaining estimates of the velocity field at all the (2N + 1)3 pixels in a real time manner.

General Relativity and Cosmology with Engineering Applications

4.13.7

145

Some aspects of group representation theory

The group under which the equations of general relativity are invariant is the group of all space-time diffeomorphism. This is an infinite dimensional group and we do not know at present, how to construct all of its irreducible representa­ tions. However, we can by approximations reduce this symmetry group to some finite dimensional subgroups and keeping this in mind we discuss some aspects of representation theory of SL(2, C) which in the adjoint representation, is the Lorentz group of special relativity. [1] Explain how you would construct all the finite dimensional irreducible representations of SL(2, C) using the Lie algebra commutation relations [H, X] = 2X, [H, Y ] = −2Y, [X, Y ] = H Hence calculate the character of these representations. [2] Let G be a compact Lie group with Lie algebra g and let h be a Cartan subalgebra of g, ie, h is a maximal Abelian subalgebra of g. Show that if h' is any other Cartan subalgebra of g, then there exists an x ∈ G such that mathf rakh' = Ad(x)(h). Show that H = exp(h) is a maximal torus in G and hence that any two maximal torii in G are conjugate to each other. For X, Y ∈ g, define B(X, Y ) = T r(ad(X).ad(Y )) Show that B([X, Y ], Z) = B(X, [Y, Z]), X, Y, Z ∈ g Also show that for g ∈ G, B(Ad(g)(X), Ad(g)(Y )) = B(X, Y ), X, Y ∈ g Tale any inner product < ., . >0 on g. Show that ∫ < X, Y >= < Ad(g)(X), Ad(g)(Y ) >0 dg G

defines an inner product on g that is Ad(G) invariant, ie, < Ad(g)(X), Ad(g)(Y ) >=< X, Y >, g ∈ G Hence, deduce that Ad(g) is unitary with respect to some basis for g and there­ fore has all eigenvalues on the unit circle. Deduce by differentiating the expres­ sion f (t) =< Ad(exp(tX)(Y ), Ad(exp(tX)(Z) > with respect to t at t = 0 that < ad(X)(Y ), Z >= − < Y, ad(X)(Z) >

146

General Relativity and Cosmology with Engineering Applications

and therefore ad(X) is a skew symmetric real matrix with respect to some basis for g regarded as a real vector space. Deduce that the eigenvalues of ad(H) for H ∈ h are all pure imaginary and hence we can diagonalize g as ⊕ gC = hC ⊕ gα α∈P

Show that ad(H) for each H ∈ h has purely imaginary eigenvalues α(H), α ∈ P , so that α ∈ P is a linear functional on h with values in iR. Note that gα = {X ∈ g : [H, X] = α(H)X∀H ∈ h} Prove that dimgα = 1, α ∈ P .

4.13.8

Waveguide equations in a curved background static metric

Consider the metric dτ 2 = dt2 − a(x, y)dx2 − b(x, y)dy 2 − c(x, y)dz 2 Formulate the Maxwell equations in this metric with dependence of the em ∂ occurs, replace it fields on z being proportional to exp(−γz), ie, wherever ∂z by multiplication with −γ. Further, work in the frequency domain, ie, assume ∂ with multiplication a time dependence of exp(iωt), or equivalently, replace ∂t by iω. Derive the generalized waveguide equations in such a metric. hint: We have √ √ −g = abc, −g = abc, The Maxwell equations are √ √ (F 0r )sqrt−g),r = 0, (F r0 −g),0 + (F rs −g),s = 0 − −(1) where Fmuν = Aν,μ − Aμ,ν or equivalently, Fμν,α + Fνα,μ + Fαμ,ν = 0 − − − (2) The second of (1) and the special case of (3) obtained by taking one of the indices as zero form the general relativistic form of the Maxwell curl equations. Using these equations, express the x, y components of the electric and magnetic fields in terms of the partial derivatives of the z components of these fields with respect to x and y.

General Relativity and Cosmology with Engineering Applications

4.13.9

147

Waveguide of arbitrary cross section in a gravita­ tional field w = q1 + iq2 = f (z) = f (x + iy)

is an analytic function of a complex variable with inverse x + iy = z = g(w) = g(q1 + iq2 ) By the Cauchy-Riemann equations, (q1 , q2 ) form an orthogonal curvilinear co­ ordinate system in the xy plane. We have ∇2⊥ = ∂x2 + ∂y2 = 4 = 4|g ' (w)|−2

∂2 ∂z∂z¯

∂2 ∂w∂ w ¯

= |g ' (q1 + iq2 )|−2 (∂q21 + ∂q22 ) So the Helmholtz equation satisfied by Ez (q1 , q2 ), Hz (q1 , q2 ) inside the guide is [∂q21 + ∂q22 + h2 G0 (q1 , q2 )]ψ(q1 , q2 ) = 0 with boundary conditions Ez |∂D = 0, ∂Hz /∂q1 |∂D = 0 for pec sidewalls and ∂Ez /∂q1 |∂D = 0, Hz |∂D = 0 for pmc walls. All this is in the absence of a gravitational field. In the presence of a gravitational field with metric of the form g00 = 1, g11 = −a(q1 , q2 ), g22 = −b(q1 , q2 ), g33 = −c(q1 , q2 ) so that

dτ 2 = dt2 − a(q1 , q2 )dq12 − b(q1 , q2 )dq22 − c(q1 , q2 )dz 2

under weak curvature, ie, the metric is approximately dτ 2 = dt2 − |g ' (q1 + iq2 )|2 (dq12 + dq22 ) − dz 2 the Helmholtz equation for ψ = Ez , Hz gets modified to [(∂q21 + ∂q22 + h2 G0 (q)) + δF1 (h2 , q)∂q1 + δF2 (h2 , q)∂q2 ][Ez , Hz ] = 0 where F1 (h2 , q), F2 (h2 , q) are 2 × 2 matrices. Suppose the metric has the form dτ 2 = dt2 − a(x, y)dx2 − b(x, y)dy 2 − (x, y)dz 2

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Assume time dependence of the electromagnetic fields as exp(iωt) and z depen­ dence of the em fields as exp(−γz). Then, F01 = A1,0 − A0,1 = iωA1 − A0,1 , F02 = iωA2 − A0,2 , F03 = iωA3 + γA0 F12 = A2,1 − A1,2 , F23 = A3,2 + γA2 , F31 = −γA1 − A3,1 F 01 = −F01 /a, F 02 = −F02 /b, F 03 = −F03 /c, F 12 = F12 /ab, F 23 = F23 /bc, F 31 = F31 /ca, √ √ −g = abc = f (x, y) say. The Maxwell equations are (F μν f ),ν = 0 In terms of components, iωf F μ0 + (f F μ1 ),1 + (f F μ2 ),2 − γf F μ3 = 0, or iωf F01 /a + (f F12 /ab),2 − γf F13 /ca = 0, iωf F02 /b − (f F12 /ab),1 − γf F23 /bc = 0, iωf F03 /c + (f F31 /ca),1 − (f F23 /bc),2 = 0, −(f F01 /a),1 − (f F02 /b),2 + γf F03 = 0 Note that β Fμβ Fμν:α = Fμν,α − Γβμα Fβν − Γνα

so by the Maxwell equation 0 = Fμν:α + Fνα:μ + Fαμ:ν = = Fμν,α + Fνα,μ + Fαμ,ν − β Fμβ (Γβμα Fβν + Γνα β +Γβνμ Fβα + Γαμ Fνβ β +Γβαν Fβμ + Γμν Fαβ )

= Fμν,alpha + Fνα,μ + Fαμ,ν This Maxwell equation is equivalent to the existence of a four potential Aμ whose four curl equals the field tensor Fμν . The relevant curl equations from these are F01,2 + iωF12 − F02,1 = 0

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−γF01 − iωF31 − F03,1 = 0 −γF02 + iωF23 − F03,2 = 0 To summarize, we have first the following four linear algebraic equations to solve for F01 , F02 , F23 , F31 in terms of F03 , F12 and their the partial derivatives w.r.t x1 = x and x2 = y: iωf F01 /a + (f F12 /ab),2 + γf F31 /ca = 0, iωf F02 /b − (f F12 /ab),1 − γf F23 /bc = 0, −γF01 − iωF31 − F03,1 = 0 −γF02 + iωF23 − F03,2 = 0 After solving these linear equations for F01 , F02 , F23 , F31 in terms of F12 , F03 and their partial derivatives w.r.t x and y, we substitute these expressions into the z components of the curl equations, namely, iωf F03 /c + (f F31 /ca),1 − (f F23 /bc),2 = 0, F01,2 + iωF12 − F02,1 = 0 to get generalized coupled Helmholtz equations for F03 and F12 . This whole process is the general relativistic generalization of the flat space-time case for a waveguide in which, we solve for Ex (F01 ), Ey (F02 ), Hx (F23 ) and Hy (F31 ) in terms of the partial derivatives of Ez (F03 ) and Hz (F12 ) w.r.t x and y. We leave it as an exercise to the reader to work out the details to determine the modes of wave propagation in a waveguide when the metric coefficients are functions of only x and y so that propagation along the z direction takes place by a factor exp(−γz). Exercise: Derive the coupled two dimensional Helmholtz equations for F 03 , F 12 when the t-z dependence of the em fields in the guide is exp(iωt − γz) and the metric has the form dτ 2 = dt2 − (a11 (x, y)dx2 + a22 (x, y)dy 2 + 2a12 (x, y)dxdy) − a33 (x, y)dz 2 hint: Using the six Maxwell curl equations √ √ √ √ iω(F r0 −g) + (F r1 −g),1 + (F r2 −g),2 − γF r3 −g = 0, r = 1, 2, 3, Frs,0 + Fs0,r + F0r,s = 0, 1 ≤ r < s ≤ 3 obtain expressions for F 0k , k = 1, 2, F 23 , F 31 in term of F 12 , F 03 , F,r12 , F,r03 , r = 1, 2 and consequently the generalized two dimensional Helmholtz equations for F 12 , F 30 .

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4.13.10 Post-Newtonian magneto-hydrodynamics in Gen­ eral relativity ((ρ + p)v μ v ν ):ν − p,μ = σF μα Fαν v ν = f μ F:νμν = J μ = σF μν vν Approximation: ((ρ + p)v ν ):ν v μ + (ρ + p)v ν v:μν − p,μ = f μ ((ρ + p)v ν ):ν − p,μ v μ = f μ vμ f μ vμ = F μα Fαν v ν vμ = = Fμα F αν v μ vν = Fνα F αμ v ν vμ Let

√

−g = a = 1 + a(2) + a(4) + ... = 1 + a ρ = ρ(2) + ρ(4) + ..., p = p(4) + p(6) + ...

Then,

v 0 = 1 + v 0(2) + v (0(4) + ...

Equation of mass conservation can be expressed as ((ρ + p)v ν (1 + a)),ν − p,μ v μ = f μ vμ The third order contribution of this equation is (2)

ρ,0 + (ρ(2) v r(1) ),r = f r(2) vr(1) + f 0(3) The fifth order contribution of the same equation is (4)

(ρ(2) v 0(2) ),0 + (ρ(2) a(2) ),0 + ρ,0 (4)

(2)

+(ρ(2) v r(1) a(2) ),r − p,0 = f r(4) vr(1) + f 0(3) v0 + f 0(5) Note: f μ = F μα Fαν v ν so if we assume that the electromagnetic fields have perturbation expansions of the form

F r0 = F r0(1) + F r0(2) + F r0(3) + ... F rs = F (rs(1) + F (rs)(2) + F rs(3) + ...

General Relativity and Cosmology with Engineering Applications then

151

(1)

(1) s(1) v f 0(3) = F 0r(2) Fr0 + F 0r(1) Frs (1)

(3) s(1) (1) (1) (1) s(3) f 0(5) = F 0r(1) Frs v + F 0r(3) Frs vs + F0r Frs v (2)

(2) s(1) +F 0r(2) Frs v + F 0r(3) Fr0 (1)

f r(2) = F rs(1) Fs0

etc. Note: Unlike the velocity, density, pressure and metric tensor, the electro­ ex­ magnetic fields must contain all powers of the velocity in their perturbation √ pansions, not only either even or odd. This is because although v = O( M ), ρ = O(M ), p = O(M v 2 ) = O(M 2 ) = O(v 4 ), there is no definite order of magnitude of charge in terms of mass. It is clear that from the term v = p + eA in the Hamiltonian that eA = O(v) and hence, A = O(v/e) and since e can be of any order of magnitude, we must assume that the perturbation series for A contains all the powers of the velocity.

4.13.11

Supergravity

First we discuss the notion of supercurrent. Suppose that S[x, θ] is a superfield and that L is a Lagrangian density formed from the component superfields of S. For this to be a valid supersymmetric Lagrangian density, it should under a supersymmetry generator vector field αa La , a = 0, 1, 2, 3 acting on the superfield S transform to a total divergence, ie, writing the component superfields of S as Sa [x], a = 1, 2, ..., N and L = L(Sa , Sa,μ ), we let with αa La S[x, θ] = δS[x, θ] with component superfields δSa [x] so that the change in the Lagrangian density under a supersymmetric transformation is given by δL = L(Sa + δSa , Sa,μ + S˜a,μ ) − L(Sa , Sa,μ ) = ∂μ K μ Now we introduce the Noether current N μ associated with the supersymmetry transformation and the Lagrangian density L. Under the supersymmetry trans­ formation δ = αa La (α is an inifinitesimal Majorana Fermion), the Noether current is given by ∂L δSa Nμ = ∂Sa,μ We note that when the Euler-Lagrange equations are satisfied by the superfield for the Lagrangian density L, ∂μ N μ = ∂L δSa,μ ∂Sa,μ

152

General Relativity and Cosmology with Engineering Applications +∂μ (

∂L )δSa ∂Sa,μ

∂L δSa,μ ∂Sa,μ +

∂L δSa ∂Sa = δL

As a result even if the Noether current is not conserved, ie, the Lagrangian density is not invariant under supersymmetry yet, ∂μ (K μ − N μ ) = 0 ie the supersymmetry current S μ = K μ − N μ is conserved.

4.13.12

Quantum stochastic differential equations in gen­ eral relativity

The metric field of gravitation couples to both the electromagnetic field and to the electron-positron field of matter via the gravitational connection for the Dirac field in curved space-time. By moving to an appropriate coordinate sys­ tem, we may assume that g00 = 1, g0r = 0, r = 1, 2, 3. Then, write down the Einstein field equations for the metric driven by a quantum noisy electromag­ netic field and likewise write down the Maxwell equations in the curved metric in the presence of a quantum noisy current source.

4.13.13

Geodesic equation for a charged particle in an electromagnetic field

Two species plasma in a gravitational field described by a metric gμν (x). The equations of motion of a charge q in this background metric and with an external electromagnetic field present can be expressed as dv μ /dτ + Γμαβ v α v β = (q/m)F μν vν the spatial components of this equation can be expressed as (dt/dτ )(d/dt)((dt/dτ )ur ) + Γr00 (dt/dτ )2 + 2Γr0s (dt/dτ )2 us + Γrsk us uk (dt/dτ )2 = (q/m)(F r0 (g00 u0 + g0k uk ) + F rm (gm0 u0 + gmk uk ))(dt/dτ ) where

ur = dxr /dt, u0 = 1

(ur ) is the three velocity. Equivalently, r r s dur /dt + ((d2 t/dτ 2 )/(dt/dτ )2 )ur + Γ00 + 2Γ0s u + Γrsk us uk =

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153

(q/m)(F 0r + F rm gmk uk )(dt/dτ )−1 ) provided that we choose our coordinate system so that g00 = 1, g0r = 0. We have dt/dτ = (1 + grs ur us )−1/2 Now,

d2 t/dτ 2 = (dt/dτ )d/dt(dt/dτ ), (d2 t/dτ 2 )/(dt/dτ )2 = d/dt(log(dt/dτ )) = (−1/2)(d/dt)log(1 + grs ur us ) = fr (u)dur /dt

Thus the equation of motion of the charge in the mixture of the gravitational field and the electromagnetic field can be expressed as r r + 2Γr0s us + Γsk u s uk = (δrs + fs (u)ur )dus /dt + au)2 )ur + Γ00

(q/m)(1 + gkl uk ul )(F 0r + F rm gmk uk )

4.13.14

Scattering of classical particles after elastic colli­ sion

Suppose two particles of masses m1 and m moving initially with velocities u1 , u respectively collide or interact via a time independent potential and finally get scattered with m1 moving with a final velocity of u'1 and m moving with a final ˆ relative velocity of u' so that the direction of the final relative velocity u'1 −u' is n to the initial relative velocity u1 − u. Momentum and energy conservation give m1 u1 + mu = m1 u'1 + mu' = (m1 + m)U '

'

m1 u21 + mu2 = m1 u12 + mu 2 U is the velocity of the centre of mass of the system. Our aim is to calculate u'1 and u' in terms of u1 , u, n ˆ . That would enable us to formulate the Boltzmann kinetic transport equation for two species of particles having different masses and charges under the influence of an external electromagnetic field. We have from the above, m1 (u1 − u'1 , u1 + u'1 ) = m(u' − u, u' + u) We assume without loss of generality that x, u'1 − u' = |u'1 − u' |ˆ n u1 − u = |u1 − u|ˆ The scattering angle θ is given by cos(θ) = (ˆ n, x ˆ)

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Let

ur = |u1 − u|, u'r = |u'1 − u' |

the initial and final relative speeds of m1 w.r.t m. Then, u1 = u + ur x, ˆ u'1 = u' + u'r n ˆ 2T = m1 u21 + mu2 = m1 (u' + u'r n ˆ )2 + mu' 2, M U0 = m1 u1 + mu = m1 (u' + u'r n ˆ ) + mu' , M = m1 + m Thus,

u' = U0 − m1 u'r n/M ˆ

and

'

'

2T = mu 2 + m1 u12 = m(U0 − m1 u'r n/M ˆ )2 + m1 (U0 − m1 u'r n/M ˆ + u'r n ˆ )2

= M U02 +(mm21 /M 2 +m1 (1−m1 /M )2 )u2r +u'r (−2mm1 (U0 , n ˆ )/M +2m1 (U0 , n ˆ )) = M U02 + m1 (mm1 /M 2 + (1 + m21 /M 2 − 2m1 /M ))u2r + 2m1 u'r (U0 , n ˆ )(1 − m/M ) = M U02 + (m1 /M 2 )(mm1 + M 2 + m21 − 2M m1 )u2r + 2m1 u'r (1 − m/M )(U0 , n ˆ) = M U02 + (m1 /M 2 )(mm1 + m2 )u2r + (2m21 u'r /M )(U0 , n ˆ) = M U02 + (mm1 /M )u2r + (2m21 /M )(U0 , n ˆ )u'r This is a quadratic equation for u'r and can be solved in terms of U0 , n ˆ.

4.13.15

Broken symmetries and Goldstone Bosons in gen­ eral relativity

Let ψ(x) be the wave function on which the symmetry group G ⊂ U (n) acts. Let Aμ (x) be a connection gauge ield with values in }, the Lie algebra of G. The gauge covariant derivative is ∇μ = ∂μ + Aμ (x) Under local G-transformations, ψ(x) transforms into ψ ' (x) = g(x)ψ(x) and correspondingly, the gauge field transforms to A'μ (x) in such a way that (∂μ + A'μ (x))ψ ' (x) = g(x)(∂μ + Aμ (x))ψ(x) This will ensure that if L is a matter field Lagrangian density is G-invariant, ie, L(ψ, ∇μ ψ) satisfies L(gψ, g∇μ ψ) = L(ψ, ∇μ ψ)∀g ∈ G then the Lagrangian L(ψ(x), ∇μ ψ(x)) is invariant even under local G-transformations. In fact, we have

L(ψ ' (x), ∇'μ ψ ' (x) = L(g(x)ψ(x), g(x)∇μ ψ(x)) = L(ψ(x), ∇μ ψ(x))

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Now let H be a subgroup of G (the broken subgroup). We can write ψ(x) = γ(x)ψ˜(x) where γ(x) is a representative element of a coset in G/H and ψ˜(x) transforms according to H. Then, we can write

ψ ' (x) = g(x)ψ(x) = g(x)γ(x)ψ˜(x) = γ ' (x)h(x)ψ˜(x) where h(x) ∈ H. In other words, g(x)γ(x) = γ ' (x)h(x) ie, γ ' (x) is a representative element of the coset g(x)γ(x)H.

4.13.16

Supersymmetric theories of gravity

Let θ be a set of four anticommuting variables, ie, Majorana Fermions: θa θb + θb θa = 0, a, b = 0, 1, 2, 3 We define a superfield as a function of these four anticommuting variables whose coefficients are functions of the space-time variables xμ , μ = 0, 1, 2, 3. Such a superfield can be expressed as S[x, θ] = S0 [x]+θT εS1 [x]+θT εθS2 [x]+θT εγ5 θS3 [x]+θT εγ μ θS4μ [x]+(θT εθ)2 S5 [x] Here, γ μ , μ = 0, 1, 2, 3 are the Dirac Gamma matrices defined by ( ) 0 σμ μ γ = σμ 0 where σ 0 = I2 , σ 1 , σ 2 , σ 3 are the usual Pauli spin matrices and σr = −σ r , r = 1, 2, 3. We have γ μ γ ν + γ ν γ μ = 2η μν and We have

γ5 = −iγ 0 γ 1 γ 2 γ 3 γ 0 γ5 = γ 1 γ 2 γ 3 , γ5 γ 0 = −γ 1 γ 2 γ 3 ,

ie, Also

{γ5 , γ 0 } = 0 γ 1 γ5 = γ 0 γ 2 γ 3 , γ5 γ 1 = −γ 0 γ 2 γ 3

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so that

{γ 1 , γ 5 } = 0

In general, we get {γ5 , γ μ } = 0, μ = 0, 1, 2, 3 {γ 0 , γ 1 , γ 2 , γ 3 , γ 5 } generate a Clifford algebra. Following Salam and Strathdee, we define the supersymmetry generators as the following vector fields on the supermanifold specified by θ, x:

L = (γ μ θ)∂/∂xμ + γ5 ε∂θ or more specifically, the components of L are defined by La = (γ μ θ)a ∂/∂xμ + (γ5 ε)ab ∂/∂θb summation over the repeated variable b being implied. Here, ( 2 ) iσ 0 ε= 0 iσ 2 (

Note that 2

iσ =

0 −1

1 0

)

so iσ 2 is a real skewsymmetric 2 × 2 matrix and so is ε. The square of iσ 2 is −I2 . We have {La , Lb } = {(γ μ θ)a , (γ5 ε)bc ∂/∂θc }∂/∂xμ +{(γ5 ε)ac ∂/∂θc , (γ μ θ)b }∂/∂xμ Using the anticommutation relation {∂/∂θa , θb } = δab we derive {La , Lb } = [−(γ μ )ad (γ5 ε)bc δdc +(γ5 ε)ac (γ μ )bd δcd ]∂/∂xμ = [−(γ μ )(γ5 ε)T + γ5 εγμT ]ab ∂/∂xμ It is easily computed that γ5 = −iγ 0 γ 1 γ 2 γ 3 =

(

I2 0

0 −I2

)

γ5 is a symmetric matrix that commutes with the skew-symmetric matrix ε and hence γ5 ε is a skew-symmetric matrix whose square is −I. Also, γ5 εγ μ γ5 ε = −γ μT (verify this). Thus, {La , Lb } = −2γ μ γ5 ε∂/∂xμ

General Relativity and Cosmology with Engineering Applications

4.13.17

157

Quantum Belavkin filtering and control for gen­ eral quantum Levy output measurements

In this work, we consider the Hudson-Parthasarathy noisy Schrodinger equation with single annihilation, creation and conservation processes and non-demolition Belavkin measurements as a linear combination of annihilation, creation and conservation noise processes. If dY o (t) denotes the differential of the output of such measurements, then Y o forms a Levy process in coherent states and further all positive integer powers (dY o (t))k , k = 1, 2, ... of this measurements process are non-zero and distinct, unlike the cases of pure quadrature or pure photon counting measurements. In pure quadrature measurements, dY o is built out of only creation and annihilation process differentials and hence by quantum Ito’s formula, (dY o (t)k = 0, k ≥ 3 while, in pure photon counting measure­ ments, dY o (t) is built only of conservation process differentials dΛ(t) and all integer powers of dΛ are the same: (dΛ)k = dΛ, k = 1, 2, .... The input mea­ surement process considered in our paper here is Y i (t) = At + A∗t + cΛt and owing to the quantum Ito formula dAt .dΛt = dAt , dΛt .dA∗t = dA∗t , all integer powers of dY i (t) are distinct but they can be expressed as linear combinations of dY i (t), (dY i (t))2 and (dY i (t))3 . Our filtering algorithm is based on this fact. We assume that for the HP unitary evolution equation dU (t) = ((−iH0 + P (t))dt + L1 dAt + L2 dA∗t + SdΛt )U (t) with corresponding system observable evolution jt (X) = U (t)∗ XU (t) the measurement process is Y o (t) = U (t)∗ (At + A∗t + cΛt )U (t) Then, it is well known [Gough et.al] that Y o (t), t ≥ 0 generate an Abelian Von-Neumann algebra and further satisfy the non-demolition propery: [Y o (t), js (X)] = 0, t ≤ s Hence, joint probability distributions of the observables jt (X), Y o (s), s ≤ t (t fixed) exist in any state and in particular in the coherent states |f ⊗ e(u) > exp(− || u ||2 /2). We denote the corresponding conditional expectation by πt (X): πt (X) = E(jt (X)|Y o (s), s ≤ t) and following the reference probability method [Gough et.al], we can assume that the optimal filtering equations are dπt (X) = Ft (X)dt + G1t (X)dY o (t) + G2t (X)(dY o (t))2 + G3t (X)(dY o (t))3 where the operators Ft (X), Gkt (X), k = 1, 2, 3 are measurable w.r.t the Abelian algebra ηt = σ{Y o (s), s ≤ t}. These operators are calculated by applying the

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quantum Ito formula to the equation (the orthogonality principle in estimation theory) E[(jt (X) − πt (X))Ct ] = 0 where Ct satisfies the qsde dCt = (f1 (t)dY o(t) + f2 (t)(dY o (t))2 + f3 (t)(dY o (t))3 )Ct , t ≥ 0, C0 = 1 with f1 , f2 , f3 being arbitrary real valued functions of time. Our presentation details the precise computation of the operators Ft (X), Gkt (X), k = 1, 2, 3 by solving linear matrix equations. We also discuss the implementation of this Belavkin filter in the state domain, ie, writing πt (X) = T r(ρB (t)X) where ρB (t) is now a classical random operator with values in the space of system space density operators (ρB (t) is a system matrix valued function of the Abelian family (Y o (s), s ≤ t) and since owing to the Abelian property, we can write down the joint probability distribution of (Y o (s), s ≤ t) in any bath state, it follows that ρB (t) can indeed be regarded as a classical random system matrix), we can translate the filter equation to the density domain: Ft (X) = T r(ψ0t (ρB (t))X), Gkt (X) = T r(ψkt (ρB (t))X), k = 1, 2, 3 so that the filter equation now reads dρB (t) = ψ0t (ρB (t))dt +

3 ∑

ψkt (ρB (t))(dY o (t))k

k=1

with ψmt , m = 0, 1, 2, 3 being a mapping from the space of system operators to itself. The paper presents explicit determination of the mappings ψmt (.), m = 0, 1, 2, 3. For the quantum control algorithm, we follow [2]. The idea is to choose a system operator Z such that if we apply the control unitary operator Uc (t, t + dt) = exp(iZdY o (t)) = I +

∞ ∑

(iZ)n (dY o (t))n /n!

n=1

at time t = 0 to the Belavkin filtered state at time t + dt = 0 + dt = dt giving the final state giving ρc (t + dt) = Uc (t + dt)ρB (t + dt)Uc (t + dt)∗ where ρB (t + dt) evolves from ρc (t) in accord to Belavkin’s filter for our general noise case: ρB (t + dt) = ρc (t) + ψ0t (ρc (t))dt +

3 ∑ k=1

ψkt (ρc (t))(dY o (t))k

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159

We then design the system observable Z so that the GKSL component of the evolution of ρc (t) in time dt is minimized, ie, noise in the HP equation is min­ imized. This leads us to an optimization criterion involving optimal matching of the GKSL component in the Belavkin filter to that produced by the control Uc (t, t + dt). We solve this optimization problem by an iterative technique and calculate the noise to signal ratio defined by the ratio of the norm of the differ­ ence between the GKSL component in Belavkin’s filter and the corresponding component coming from the control algorithm to the norm of the evolution component of the state in the Belavkin filter coming from the Hamiltonian part and the measurement part. Excellent N SR' s are obtained here, ie, N SR 1, a contradiction.

5.14

Blurring of 3-D object fields in random mo­ tion

Suppose an object intensity field g : R3 → R is subject to a random rotation R(λ) ∈ SO(3) followed by a random translation a(λ) ∈ R3 . Here, λ is a random vector parameter having probability distribution F and density f . Show that the blurred object field is given by ∫ Bg(r) = f (λ)g(R(λ)−1 (r − a(λ))dλ Express this in the form ∫ Bg(r) =

h(r, r' )g(r' )d3 r'

using the Dirac delta function. If the random parameter λ has mean vector μ and covariance matrix C, then determine approximately the blurred object field Bg(r) upto O(|| C ||). hint: If K(λ) is a function of the random parameter vector λ, then we have ∫ EK(λ) = f (λ)K(λ)dλ = K(μ) + E

∑

(

n1 +...+np >1

∂ n1 +...np f (μ) p [(λk − μk )nk /nk !] n Π ∂λn1 1 ...∂λp p k=1

1 = K(μ) + T r(C∇∇T K(μ)) + O(E || λ ||3 ) 2 If the mean and covariance change from (μ, C) to (μ' , C ' ), then the correspond­ ing change in the blurred object field is given approximately by 1 K(μ) − K(μ' ) + T r(C∇∇T K(μ) − C ' ∇∇T K(μ' )) 2 where

K(λ) = K(λ, r) = g(R(λ)−1 (r − a(λ)))

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Exercise: Assuming λ = (φ, θ, ψ), the three Euler angles and a(λ) = 0, ie, zero translation, so that the rotation is given by R(λ) = Rz (φ)Rx (θ)Rz (ψ) calculate the blurred image EK(λ) in terms of the mean and covariance matrix of the three Euler angles upto linear orders in the covariance. This is purely rotational blurring. Likewise, assume that R(λ) = I3 (no rotation) and the translation is a(λ) = (λ1 , λ2 , λ3 ) with a specified mean and covariance matrix. Then, calculate the blurred image upto linear orders in the covariance matrix. Now consider both random rotations and random translations assuming that the translation and rotation parameters are statistically independent and then calculate the blurred image field to linear orders in the covariance matrices.

5.15

Commutators of Products of Matrices

If A1 , ..., An , B1 , ..., Bm are all square matrices of the same size and if [X, Y ] = XY − Y X, then show that [A1 ...An , B1 ...Bm ] = ∑

A1 ...Aj−1 B1 ...Bk−1 [Aj , Bk ]Aj+1 ...An Bk+1 ...Bn

j,k

where A1 ...Aj−1 is taken as I if j = 1 and Aj+1 ...An is taken as I

5.16

Path of a light ray in an medium having inhomogeneous refractive index

Light travels through a medium with refractive index n(x, y, z) = n(r), r = (x, y, z) dependent upon the spatial location. Show that the time taken for the light ∫ ray to travel along a curved path r = r(s), ds = |dr| is given by T = n(r(s))ds/c0 where c0 is the speed of light in vacuum. By applying the calculus of variations to Fermat’s principle of minimum time, deduce the differential equations for the curved path along which light moves.

5.17

Reflection Matrices

Show that the transformation of reflection in the plane (n, r) = d where n is a unit vector is given by T : r → r − 2sn

General Relativity and Cosmology with Engineering Applications

177

where s is determined by the equations r − r0 = sn, (r0 , n) = d Show that (r − sn, n) = d or equivalently, s = (r, n) − d so that T : r → r − 2(r, n)n + 2dn In the special case that the plane passes through the origin, show that d = 0 and then the reflection in such a plane is given by T : r − 2(r, n)n ie, T is the matrix I − 2nnT . Show that det(T ) = −1 and conversely, if T is any 3 × 3 real matrix with determinant −1, then T is the reflection w.r.t. a plane passing through the origin.

5.18

Rotation Matrices

Compute the matrix of a rotation about the unit vector n passing through the origin by an angle φ using two methods. hint: Under an infinitesimal rotation by angle δφ, the vector r goes over to r + δr where δr = n × rδφ (Draw the diagram and convince yourself about it. Thus, if r(φ) is the vector obtained from r(0) after rotating it by a finite angle φ about the unit vector n, then r(φ) satisfies the differential equation r' (φ) = n × r(φ) or equivalently, in matrix notation, r' (φ) = A(n)r(φ) where

(

0 A(n) = ( n3 −n2

−n3 0 n1

) n2 −n1 ) 0

we can write A(n) = n1 A1 + n2 A1 + n3 A3 = n.A

178

General Relativity and Cosmology with Engineering Applications where A1 , A2 , A3 are skew symmetric real matrices defined by ( ) 0 0 0 A1 = ( 0 0 −1 ) 0 1 0 ( ) 0 0 1 A2 = ( 0 0 0 ) −1 0 0 ( ) 0 −1 0 ( 1 0 0 ) 0 0 0 Now solve the differential equation as r(φ) = exp(φn.A) = exp(φ1 A1 + φ2 A2 + φ3 A3 ) where φk = nk φ, k = 1, 2, 3 Finally, the exponential of a matrix A can be evaluated using the Laplace trans­ form: exp(tA) = L−1 ((sI − A)−1 ) by expressing each entry of the matrix (sI −A)−1 as the ratio of two polynomials in s and Laplace inverting each entry by the method of partial fractions. The other way of calculating Rn (φ) = exp(φA.n) is to draw the picture and deduce that r(φ) = (r, n)n + |n × r|cos(φ)(r − (n, r)n)/|r − (n, r)n| + (n × r)sin(φ) = (r, n)n + (n × r)sin(φ) + n × (r × n)cos(φ)

5.19

Jacobian formula for multiple integrals

Prove the Jacobian determinant formula for multiple integrals: Let D ⊂ Rn and F : D → D' be a diffeomorphism, ie, a one-one onto differentiable map with a differentiable inverse. Then, show that if f : D → R, we have ∫ ∫ f (x)dx = f (F (x))|F ' (x)|dx D'

D

where

∂Fi (x) ))| ∂xj hint: Take an infinitesimal cuboid around x of sidelengths dx1 , dx2 , ..., dxn . Under the action of the mapping F this cuboid goes over to the parallelopiped around F (x) with sidelength vectors F,i (x)dxi , i = 1, 2, ..., n an hence the volume of this parallelopiped is |F ' (x)|dx1 ...dxn . |F ' (x)| = |det((

General Relativity and Cosmology with Engineering Applications

5.20

179

Existence of only five regular polyhedra in nature

We wish to show that there are only five regular polyhedra, namely closed convex solids with each face congruent to another and also each face is a regular polygon. These regular solids are (1) Tetrahedron, ie, a four faces with each face an equilateral triangle. (2) A cube, ie, eight faces with each face a square. (3) An octahedron, ie, eight faces with each face an equilateral triangle. (4) A dodecahedron, ie, twelve faces with each face a regular hexagon. (5) An icosahedron, ie, twenty faces with each face a regular hexagon. Proof: Let the solid have v vertices, f faces and e edges with k edges incident at each vertex. Then, we have Euler’s relation v−e+f =2 and secondly, vk/2 = e v, e, f, k are positive integers Further, if n denotes the number of sides on each face, then nf /2 = e So v = 2e/k, f = 2e/n and further, Euler’s relation gives 2e/k − e + 2e/n = 2 so that e=

2 2/k − 1 + 2/n

Since e is a positive integer, we must have that 2/k−1+2/n > 0, or equivalently, 2/k + 2/n > 1 with k and n being positive integers. Thus, 2(n + k) > nk If n, k ≥ 4, then it would follow that 2/k ≤ 1/2, 2/n ≤ 1/2 and this would imply that 2/k + 2/n ≤ 1, so the above inequality can never be satisfied. So the only possible choices for n, k are (n, k, = 1, 2, 3), (n = 4, k = 3)and(n = 3, k = 4). We try all these choices to arrive at the desired result.

180

General Relativity and Cosmology with Engineering Applications

5.21

Definition of the derivative and its proper­ ties

Prove the following using the ε − δ definitions of limit: (a) d df (x) dg(x) (cf (x) + g(x)) = c + dx dx dx assuming that df (x)/dx and dg(x)/dx exist at x. hint: |(cf (x + h) + g(x + h) − cf (x) − g(x))/h − cf ' (x) − g ' (x)| ≤ |c||(f (x + h) − f (x))/h − f ' (x)| + |(g(x + h) − g(x))/h − g ' (x)| (b) d(f (x)g(x)) = f (x)dg(x)/dx + g(x)df (x)/dx dx provided that both dg(x)/dx and df (x)/dx exist at x. hint: |(f (x + h)g(x + h) − f (x)g(x))/h − f (x)g ' (x) − g(x)f ' (x)| = |(f (x + h) − f (x))g(x + h)/h + f (x)(g(x + h) − g(x))/h − f (x)g ' (x) − g(x)f ' (x)| = |g(x)((f (x + h) − f (x))/h − f ' (x)) + (f (x + h) − f (x))(g(x + h) − g(x))/h +f (x)((g(x + h) − g(x))/h − g ' (x))| ≤ |g(x)|(f (x + h) − f (x))/h − f ' (x)| + |f (x)||(g(x + h) − g(x))/h − g ' (x)| +|(f (x + h) − f (x))/h||(g(x + h) − g(x))/h|.h Now take limh → 0. (c) d g(x)f ' (x) − f (x)g ' (x) (f (x)/g(x)) = dx g(x)2 where we use the notation f ' (x) = df (x)/dx etc and assume that g(x) /= 0. (at a given x). hint: |f (x + h)/g(x + h) − f (x)/g(x)|/h =

General Relativity and Cosmology with Engineering Applications

5.22

181

Pattern Recognition using Group Repre­ sentations

Let χλ (x) be the character of an irreducible representation of the permutation group Sn defined by the Young frame λ. Take another irreducible character χμ (x). Here, x ∈ Sn . Now take two object fields f (r1 , ..., rn ) and g(r1 , ..., rn ). Here, r1 , ..., rn are the locations of the n objects in R3 . Now, suppose that these two object fields are subject to permutations amongst the objects and rotations and translations of the overall system of objects. Then f transforms to f1 (r1 , ..., rn ) = f (R−1 (rσ1 − a), ..., R−1 (rσn − a)) and g transforms to g1 (r1 , ..., rn ) = g(R−1 (rσ1 − a), ..., R−1 (rσn − a)) where

σ ∈ Sn , R ∈ SO(3), a ∈ R3

We wish to construct invariants from these two object fields under (R, a, σ) using irreducible representations of the rotation-translation-permutation group. r) denote the spherical harmonic polynomials. We have with First, let Ylm (ˆ τ = σ −1 , ∫ fˆ1 (k1 , ..., kn ) = f1 (r1 , ..., rn )exp(−i(k1 .r1 + ... + kn .rn ))d3 r1 ...d3 rn ∫ =

f (R−1 (r1 − a), ..., R−1 (rn − a))exp(−i(kτ 1 .r1 + ... + kτ n .rn ))d3 r1 ...d3 rn ∫

=

f (r1 , ..., rn )exp(−i(kτ 1 .(Rr1 + a) + ... + kτ n .(Rrn + a)))d3 r1 ...d3 rn = exp(−i(k1 + ... + kn ).a)fˆ(R−1 kτ 1 , ..., R−1 kτ n )

Likewise, gˆ1 (k1 , .., kn ) = exp(−i(k1 + ... + kn ).a)ˆ g (R−1 kτ 1 , ..., R−1 kτ n ) Let χ be a character of the permutation group Sn . Then, we get ψf1 ,g1 (k1 , ..., kn ) = (

∑

|fˆ1 (kσ1 , ..., kσn )||g1 (kρ1 , ..., kρn )|χ(σ −1 ρ)

σ,ρ∈Sn

=

∑

[|fˆ(R−1 kστ 1 , ..., R−1 kστ n )|

σ,ρ

×|gˆ(R−1 kρτ 1 , ..., R−1 kρτ n )|χ(σ −1 ρ)]

182

General Relativity and Cosmology with Engineering Applications =

∑

[|fˆ(R−1 kσ1 , ..., R−1 kσn )|

σ,ρ

kρ1 , ..., R−1 kρn )|χ(τ σ −1 ρτ −1 ) ∑ = [fˆ(R−1 kσ1 , ..., R−1 kσn )|

×|gˆ(R

−1

σ,ρ

×|gˆ(R−1 kρ1 , ..., R−1 kρn )|χ(σ −1 ρ) = ψf,g (R−1 k1 , ..., R−1 kn ) say. This quantity is independent of the permutation τ applied to the objects f, g. We then have ∫ [ψf1 ,g1 (k1 , ..., kn )Y¯l1 ,m1 (kˆ1 )...Y¯ln ,mn (kˆn ) ×dΩ(kˆ1 )...dΩ(kˆn )]

∫ =

ψf,g (k1 , ..., kn )Πnj =1 Y¯lj ,mj (Rkˆj )dΩ(kˆj )

∫ =

[ψf,g (k1 , ..., kn )Πj =

∑ m'1 ,...,m'n

∑

[¯ πlj (R−1 )]m'j ,mj Ylj ,m'j (kˆj )dΩ(kˆj )]

m'j

[[πl1 (R)]m1 ,m'1 ...[πln (R)]mn ,m'n

×ψf,g,l1 ,m1' ,...,ln ,mn' (|k1 |, ..., |kn |)] = [[πl1 (R) ⊗ ... ⊗ πln (R)]ψf,g,l1 ,...,ln (|k1 |, ..., |kn |)]m1 ,...,mn or equivalently, in matrix notation, ψf1 ,g1 ,l1 ,...,ln (|k1 |, ..., |kn |) = [πl1 (R) ⊗ ... ⊗ πln (R)]ψf,g,l1 ,...ln (|k1 |, ..., |kn |) from which it follows that || ψf1 ,g1 ,l1 ,...,ln (|k1 |, ..., |kn |) ||2 = || ψf,g,l1 ,...,ln (|k1 |, ..., |kn |) ||2 ie, we have constructed an invariant for object field pairs under the joint action of the rotation group SO(3), the translation group R3 and the permutation group Sn . Each object field is a function on R3 × ... × R3 n times. Note that we have made use of the fact that the character of a representation is a class function, or more specifically, χ(τ στ −1 ) = χ(σ), σ, τ ∈ Sn

General Relativity and Cosmology with Engineering Applications

183

The effect of noise on the pattern classification algorithm. The object fields f (r1 , ..., rn ) and g(r1 , ..., rn ) undergo transformations with additive noise: f1 (r1 , ..., rn ) = f (R−1 (rσ1 − a), ..., R−1 (rσn − a)) + wf (r1 , ..., rn ), g1 (r1 , ..., rn ) = g(R−1 (rσ1 − a), ..., R−1 (rσn − a)) + wg (r1 , ..., rn ) where ∫ fˆ1 (k1 , ..., kn ) = ∫ =

f1 (r1 , ..., rn )exp(−i(k1 .r1 + ... + kn .rn ))d3 r1 ...d3 rn

f (R−1 (r1 − a), ..., R−1 (rn − a))exp(−i(kτ 1 .r1 + ... + kτ n .rn ))d3 r1 ...d3 rn ∫

=

σ ∈ Sn , R ∈ SO(3), a ∈ R3

f (r1 , ..., rn )exp(−i(kτ 1 .(Rr1 + a) + ... + kτ n .(Rrn + a)))d3 r1 ...d3 rn = exp(−i(k1 + ... + kn ).a)fˆ(R−1 kτ 1 , ..., R−1 kτ n ) + w ˆf (k1 , ..., kn )

Likewise, g (R−1 kτ 1 , ..., R−1 kτ n ) + w(k ˆ 1 , ..., kn ) gˆ1 (k1 , .., kn ) = exp(−i(k1 + ... + kn ).a)ˆ Let χ be a character of the permutation group Sn . Then, we get ψf1 ,g1 (k1 , ..., kn ) = ∑

(

|fˆ1 (kσ1 , ..., kσn )||g1 (kρ1 , ..., kρn )|χ(σ −1 ρ)

σ,ρ∈Sn

=

∑

|exp(−i(k1 + ... + kn ).a)fˆ(R−1 kστ 1 , ..., R−1 kστ n )+

σ,ρ

w ˆf (kσ1 , ..., kσn )||exp(−i(k1 +...+kn ).a)ˆ g (R−1 kρτ 1 , ..., R−1 kρτ n )+w ˆg (kρ1 , ..., kρn )|χ(σ −1 ρ)

=

∑

|exp(−i(k1 + ... + kn ).a)fˆ(R−1 kσ1 , ..., R−1 kσn )+

σ,ρ

w ˆf (kστ −1 1 , ...kστ −1 n )||exp(−i(k1 + ... + kn ).a)ˆ g (R−1 kρ1 , ..., R−1 kρn )+

=

∑

w ˆg (kρτ −1 1 , ..., kρτ −1 n )|χ(τ σ −1 ρτ −1 ) |exp(−i(k1 + ... + kn ).a)fˆ(R−1 kσ1 , ..., R−1 kσn )+

σ,ρ

w ˆf (kστ −1 1 , ..., kστ −1 n )||exp(−i(k1 + ... + kn ).a)ˆ g (R−1 kρ1 , ..., R−1 kρn )|χ(σ −1 ρ) Now,

|exp(−i(k1 + ... + kn ).a)fˆ(R−1 kσ1 , ..., R−1 kσn )+ w ˆf (kστ −1 1 , ..., kστ −1 n )| ≈ |fˆ(R−1 kσ1 , ..., R−1 kσn |)+

184

General Relativity and Cosmology with Engineering Applications

ˆf (kστ −1 1 , ..., kστ −1 n ))/|fˆ(R−1 kσ1 , ..., R−1 kσn | Re(exp(i(k1 +...+kn ).a)ˆ¯f (R−1 kσ1 , ..., R−1 kσn )w

where quadratic and higher order terms in the noise process have been neglected. Likewise for the g term. We write Wf (σ, τ, R, a, k1 , ..., kn ) = Re(exp(i(k1 +...+kn ).a)ˆ¯f (R−1 kσ1 , ..., R−1 kσn )w ˆf (kστ −1 1 , ..., kστ −1 n ))/|fˆ(R−1 kσ1 , ..., R−1 kσn |

and likewise, Wg (ρ, τ, R, a, k1 , ..., kn ) = Re(exp(i(k1 +...+kn ).a)¯ ˆg(R−1 kρ1 , ..., R−1 kρn )w ˆg (kρτ −1 1 , ..., kρτ −1 n ))/|gˆ(R−1 kρ1 , ..., R−1 kρn |

Thus, with neglect of quadratic and higher terms in the noise processes wf and wg , we have ψf1 ,g1 (k1 , ..., kn ) ∑ = [|fˆ(R−1 kσ1 , ..., R−1 kσn )||gˆ(R−1 kρ1 , ..., R−1 kρn )|χ(σ −1 ρ)] σ,ρ

+

∑

(|fˆ(R−1 kσ1 , ...R−1 kσn )|.

σ,ρ

×Re(exp(i(k1 +...+kn ).a)¯ ˆg(R−1 kρ1 , ..., R−1 kρn )w ˆg (kρτ −1 1 , ..., kρτ −1 n ))/|gˆ(R−1 kρ1 , ..., R−1 kρn |

+ +|gˆ(R

−1

kρ1 , ...R−1 kρn )|.

×Re(exp(i(k1 +...+kn ).a)¯ˆf (R−1 kσ1 , ..., R−1 kσn )w ˆf (kστ −1 1 , ..., kστ −1 n ))/|fˆ(R−1 kσ1 , ..., R−1 kσn |)χ(σ −1 ρ)

∑

= ψf,g (R−1 k1 , ..., R−1 kn )+ (|fˆ(R−1 kσ1 , ..., R−1 kσn )|Wg (ρ, τ, R, a, k1 , ..., kn )+

σ,ρ

|gˆ(R−1 kρ1 , ..., R−1 kρn )|Wf (σ, τ, R, a, k1 , ..., kn ))χ(σ −1 ρ) We write this equation as ψf1 ,g1 (k1 , ..., kn ) = ψf,g (R−1 k1 , ..., R−1 kn ) + Wf,g (R, a, τ, k1 , ..., kn ) We then have

∫ [ψf1 ,g1 (k1 , ..., kn )Y¯l1 ,m1 (kˆ1 )...Y¯ln ,mn (kˆn ) ∫ =

×dΩ(kˆ1 )...dΩ(kˆn )] ψf,g (k1 , ..., kn )Πnj =1 Y¯lj ,mj (Rkˆj )dΩ(kˆj )+

∫ Wf,g (R, a, τ, k1 , ..., kn )Y¯l1 m1 (kˆ1 )...Yln ,mn (kˆn )dΩ(kˆ1 )...dΩ(kˆn ) +[Wf,g (R, a, τ, |k1 |, ..., |kn |)]l1 m1 ...ln mn

General Relativity and Cosmology with Engineering Applications ∫ =

[ψf,g (k1 , ..., kn )Πj

185

∑ [¯ πlj (R−1 )]m'j ,mj Ylj ,m'j (kˆj )dΩ(kˆj )] m'j

+[Wf,g (R, a, τ, |k1 |, ..., |kn |)]l1 m1 ...ln mn ∑ = [πl1 (R)]m1 ,m1' ...[πln (R)]mn ,mn' m'1 ,...,m'n

×ψf,g,l1 ,m1' ,...,ln ,mn' (|k1 |, ..., |kn |) +Wf,g (R, a, τ, |k1 |, ..., |kn |)l1 m1 ,...,ln ,mn = [[πl1 (R) ⊗ ... ⊗ πln (R)]ψf,g,l1 ,...,ln (|k1 |, ..., |kn |)]m1 ,...,mn +Wf,g (R, a, τ, |k1 |, ..., |kn |)l1 m1 ,...,ln ,mn or equivalently, in matrix notation, ψf1 ,g1 ,l1 ,...,ln (|k1 |, ..., |kn |) = [πl1 (R) ⊗ ... ⊗ πln (R)]ψf,g,l1 ,...ln (|k1 |, ..., |kn |) +Wf,g,l1 ,...,ln (R, a, τ, |k1 |, ..., |kn |) from which it follows that || ψf1 ,g1 ,l1 ,...,ln (|k1 |, ..., |kn |) ||2 = || ψf,g,l1 ,...,ln (|k1 |, ..., |kn |) ||2 +2Re(([πl1 (R) ⊗ ... ⊗ πln (R)]ψf,g,l1 ,...ln (|k1 |, ..., |kn |))∗ ×Wf,g,l1 ,...,ln (R, a, τ, |k1 |, ..., |kn |)) with neglect of quadratic terms in the noise field. Taking the square root and again neglecting quadratic and higher order terms in the noise fields gives us || ψf1 ,g1 ,l1 ,...,ln (|k1 |, ..., |kn |) || || ψf,g,l1 ,...,ln (|k1 |, ..., |kn |) || +Re(([πl1 (R) ⊗ ... ⊗ πln (R)]ψf,g,l1 ,...ln (|k1 |, ..., |kn |))∗ ×Wf,g,l1 ,...,ln (R, a, τ, |k1 |, ..., |kn |)/| || ψf,g,l1 ,...,ln (|k1 |, ..., |kn |) ||) ˜ f,g,l ,...,l (R, τ, a, |k1 |, ..., |kn |) =|| ψf,g,l1 ,...,ln (|k1 |, ..., |kn |) || +W 1 n from which, the noise to signal ratio for pattern pair matching is easily calcu­ lated: nsr = E[(|| ψf1 ,g1 ,l1 ,...,ln (|k1 |, ..., |kn |) || − || ψf,g,l1 ,...,ln (|k1 |, ..., |kn |) ||)2 ]/[|| ψf,g,l1 ,...,ln (|k1 |, ..., |kn |) ||)2 ]

= E[(Re(([πl1 (R)⊗...⊗πln (R)]ψf,g,l1 ,...ln (|k1 |, ..., |kn |))∗ Wf,g,l1 ,...,ln (R, a, τ, |k1 |, ..., |kn |))2 ]/

| || ψf,g,l1 ,...,ln (|k1 |, ..., |kn |) ||2

186

General Relativity and Cosmology with Engineering Applications

There is another way to construct invariants of the permutation group based on Frobenius’ formula for the generating function of the irreducible characters. Broadly speaking, the procedure is as follows: Let f (1, 2, ..., n) be the origi­ nal image and f1 (1, 2, ..., n) = f (σ1, ..., σn) the transformed image. Likewise, g(1, 2, ..., n) is another image field and g1 (1, 2, ..., n) = g(σ1, ..., σn) the trans­ formed image field. If χλ , λ ∈ I are the irreducible characters of the permutation group, then Frobenius’ formula for their generating function has the from ∑ Pλ (x)χλ (σ) = Q(x, σ) λ∈I

where Pλ (x) are certain polynomials indexed by λ ∈ I and Q(x, σ) is a polyno­ mial in x = (x1 , ..., xn ). THen we have ∑ ψλ (f1 , g1 ) = f1 (τ 1, ..., τ n)g1 (ρ1, ..., ρn)χλ (τ −1 ρ) τ,ρ∈Sn

=

∑

f (τ σ1, ..., τ σn)g(ρσ1, ..., ρσn)χλ (τ −1 ρ)

τ,ρ

=

∑

f (τ 1, ..., τ n)g(ρ1, ..., ρn)χλ (στ −1 ρσ −1 )

τ,ρ

=

∑

f (τ 1, ..., τ n)g(ρ1, ..., ρn)χλ (τ −1 ρ)

τ,ρ

= ψλ (f, g) and hence ψ(x, f1 , g1 ) =

∑

Pλ (x)ψλ (f1 , g1 ) =

λ

ψ(x, f, g) =

∑

Pλ (x)ψλ (f, g)

λ

Note that we can write ψ(x, f, g) =

∑

f (τ 1, ..., τ n)g(ρ1, ..., ρn)Q(x, [τ −1 ρ])

τ,ρ

where [τ ] denotes the class to which τ belongs. We can write [τ ] = (k1 , ..., kn ) where kj is the number of cycles of length j in τ , j=1,2,.., n. Thus, we have ∑ n j=1 jkj = n.

General Relativity and Cosmology with Engineering Applications

5.23

187

Using characters of group representations to estimate the group transformation ele­ ment

ˆ the set of its irreducible characters. Suppose Let G be a semi-simple group and G G acts on a manifold M and : M → C is a known function. We assume that σ ∈ G transforms f to f1 after adding noise, ie, f1 (x) = f (σ −1 x) + w(x), x ∈ M The aim is to estimate σ from measurements of f, f1 . We have for any irreducible representation πα of G with character χα (As α varies over an index set I, χα ˆ . We have varies in a one-one way over G ∫ ∫ ∫ f1 (ρx)πα (ρ)dρ = f (σ −1 ρx)πα (ρ)dρ + w(ρx)πα (ρ)dρ G

∫ = πα (σ)

∫ f (ρx)πα (ρ)dρ +

w(ρx)πα (ρ)dρ

or taking trace, we get ∫ fˆ1 (α) =

f1 (ρx)χα (ρ)dρ = G

fˆ(ρx)χα (σρ)dρ + w(α) ˆ ˆ , we get linear equations for By allowing α to vary over a large subset of G [σρ], ρ ∈ G using which σ can be determined. Example: G = SO(3). The characters are χl (θ) =

l ∑

exp(imθ) = exp(−ilθ)(exp(i(2l + 1)θ) − 1)/(exp(iθ) − 1)

m=−l

= sin((l + 1/2)θ)/sin(θ/2) For any g ∈ SO(3), θ = θ(g) is the angle of rotation corresponding to g. Thus, by the above procedure, we can determine θ(gh), h ∈ G and since T r(gh) = 2cos(θ(gh)) + 1 we can determine from nine linearly indepndent choices of h ∈ G, the element g ∈ SO(3) by solving nine linear equations.

188

5.24

General Relativity and Cosmology with Engineering Applications

Explicit formulas for the induced represen­ tation for semidirect products of finite groups

G = N ⊗s H is semidirect product of an Abelian group N and a subgroup H ˆ denotes the set of characters of that normalizes N , ie, hN h−1 = N, h ∈ H. N N . Choose a character χ0 and let H0 denote its stability subgroup in H. There is a natural correspondence between H/H0 and O(χ0 ) = {h.χ0 : h ∈ H}, ie, the orbit of χ0 . Note that hχ0 (n) = χ0 (h−1 nh). Let L be an irrep of H0 in a finite dimensional Hilbert space Ve and choose an onb {φn,e : n = 1, 2, ..., N } for Ve . Denote < φn,e |L(h)|φm,e >= L(h)nm , h ∈ H0 Let hx be a representative element of the coset x in H/H0 , ie, one representative for each coset. For a given h ∈ H, let y be the (unique) coset in which hhx falls. ˜ ∈ H0 are uniquely determined from the equation Thus, y and h ˜ hhx = hy h We thus get ˜ = hy H = y hx = hhx H = hy hH Define the character χx by χx = hx .χ0 Thus, χy = hy .χ0 Note that χx , χy ∈ O(χ0 ). For each x ∈ H/H0 , define a vector space Vx of same dimension as Ve and write (orthogonal direct sum) V =

⊕

Vx , VH0 = Ve

x∈H/H0

We can view Vx as the vector space lying over the coset x or equivalently above the character χx in the orbit O(χ0 ). Let {φn,x , n = 1, 2, ..., N } be an onb for Vx . Then write U (nh)φk,x = χy (n)

∑

˜ h ˜ )]lk φl,y , hhx = hy h, ˜ ∈ H0 [L(h

l

We claim that U is an irrep of G = N ⊗s H and that if with each orbit O(χ) ˆ under H and an irrep L of the stability group of any fixed element χ in in N this orbit, we define the irrep U of G in this way, then we would exhaust all the irreps of G.

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5.25

189

Applications of the Extended Kalman fil­ ter and the Recursive Least Squares Al­ gorithm to System Identification Problems using Neural Networks

In this work, we discuss some applications of the dual Extended Kalman fil­ ter(DEKF) using EKF to update state estimates when more measurements are taken and RLS with forgetting factor to update weights when more measure­ ments are taken. The current methods for estimating the weights given the state estimates are based either on linear models for the output dependence on the state process or on approximate iterative schemes like the gradient search method [Haykin, Kalman filtering and neural networks]. In this paper, we pro­ pose a method which is close to optimal in the sense that the weight estimation process is based on the optimal least squares method applied to the output equation combined with a linearized approximation of the output equation with the linearization being taken around the previous weight estimate. The forget­ ting factor is also introduced while formulating the weight estimation as a least squares problem. This method will be very close to optimal if the weights do not vary too rapidly as is true for almost all neural network problems (in which the weights are constants). Linearization of the output equation around the previ­ ous weight estimate can be regarded as a first order perturbation approximation method. We can extend this to higher order perturbation approximations by Taylor expanding the output equation around the previous weight estimate upto any given power in the difference between the current weight estimate and the previous weight estimate. This process would then get closer and closer to the optimal estimate (ie the maximum likelihood estimate in the case when the measurement noise is white Gaussian) but the optimization would then involve finding the minimum of a multivariate polynomial which would again involve another iteration loop. We discuss this scheme also here but our simulations are based on only the linearized approximation. We also discuss optimal maximum likelihood estimation when the measurement noise is white but non-Gaussian. In this case, the Edgeworth series expansion for non-Gaussian probability densities that deviate only slightly from a Gaussian density is employed. Such expan­ sions are modeled as Gaussian densities modulated by a polynomial expanded in terms of the Hermite polynomials which are orthonormal with respect to the Gaussian density. Finally, we have considered a DEKF algorithm for continuous time state and measurement models. This involves replacing the discrete time state and measurement models in noise by stochastic differential equations based on the Ito calculus. We show that the analogue of the least squares method in discrete time fails to work since it does not lead to an sde for the weight update but the stochastic gradient method based on instantaneous minimization of the output error works and as literature points out [T.K.Rawat et.al.] continuous time sde approximations for discrete time stochastic gradient algorithms is very effective in proving convergence of the weights and obtaining asymptotic formu­

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las for the limiting expected error energy. We conclude by writing down a joint sde for the state, weight, state estimate, weight estimate and state estimation error covariance matrix which can be analyzed for convergence using standard mean and variance propagation equations. It should be mentioned that most of the DEKF papers involve estimating the state of the neural network by running an EKF and the weights by running another decoupled EKF parallely with the EKF for the state estimate. Such a DEKF is suboptimal, ie, it does not perform as well as a global coupled EKF for the simultaneous estimation of the state and the weights because in a general neural network model, the state dynamics is inherently coupled with the weight dynamics and the measurement process depends both on the state and on the weights: x[k + 1] = fk (x[k], w[k]) + εx [k + 1], w[k + 1] = w[k] + εw [k + 1] y[k] = hk (x[k], w[k]) + εy [k], The use of an EKF for the state estimate and a decoupled RLS for the weight estimate in our paper has not been used extensively in the existing literature. It is computationally more efficient than running two decoupled EKF’s but its performance may not be as good as the latter. Another advanatage with the former is that it is easy to do weight pruning, ie, determine which weights affect the estimation error energy more than the others and accordingly decrease the size of the weight vector by setting those weights to zero which do not have much effect on the error energy. Further, the RLS algorithm that is run parallely for the weight updates, can easily be extended to a lattice RLS algorithm, ie, increase the number of weights (ie the order of the system) and update the new weights in time accordingly. The algorithm thus becomes order and time recursive. This is not possible with two decoupled EKF’s. The final advantage of the RLS algorithm used in our paper for weight updates is that the forgetting factor introduced takes into account the fact that the weight of the original neural network may vary slowly with time and hence the current estimate of the weights should give more importance to the recent signal samples rather than the samples in the remote past. In what follows, we present a literature survey of some existing related work with relation to our work stating how these existing works can be adapted to our work or in what way our work may outperform their work: Eric A.Wan and Alex T.Nelson in [1] have proposed the use of the DEKF for the removal of non-stationary and coloured noise from speech. They model the speech signal using a nonlinear difference equation with some unknown param­ eters called weights and additive noise taken into account both in the speech process dynamics and in the measurement process. At at any given time k, the speech state estimates are updated using the EKF with the weight estimates held fixed. Then by holding the state estimate fixed, another EKF is used to update the weights. In short, the EKF’s run for the state and weight estimates

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191

in a decoupled way. This reduces the computational burden involved in running joint estimates of the state and weights since the cross covariance between the states and weights does not enter into the picture during the computation of the Kalman gain matrix. The advantages and disadvantages of using an RLS in place of an EKF for the weight updates has been mentioned above. Roberto Togneri in [2], has proposed a simplified version of the EM algorithm for learning about the model parameters of a speech sequence based on observed date whose statistics are state dependent. In the EM algorithm, we first evaluate the approximate conditional expectation of the squared error in the dynamic speech model and that in the measured observation model given the output data and the model and output equation parameters. This constitutes the ”Estep”. In the ”M-step”, the above conditional expectation is minimized w.r.t the model and output parameters. The approximate conditional expectations are calculated using the EKF. This algorithm is very complex. The authors simplify this by approximating the conditional expectations using the results of the ESPS formant tracker which essentially involves retaining the time averages but by removing the conditional expectation operation in the expressions for the conditional expectation of the sum of process model and measurement model errors squared. These expressions involve the state process and the EKF is used to replace the state process samples here by their EKF estimates for we do not have direct access to the states, we have access only to the output. Our paper bypasses the need for evaluating the conditional mean square error because we assume that the processes generated are ergodic, so that time averages naturally replace ensemble averages. This time average error energy idea is at the heart of the RLS algorithm. [3] Isabelle Rivals Two methods are proposed here for neural weight training of a feed forward network. First, an output equation is formed involving expressing the output as a function of the states and unknown parameters or weights. Then the sum of errors squared in the output equation is minimized w.r.t these weights using a gradient descent method. The author also considers adding a quadratic function of the weights to this energy function to avoid the optimal weights from getting too large. In the second method, the author uses the EKF to train the weights. Here, the state process has no dynamics but the weight vector has trivial dynamics. [4] J.Sum, Chi-sing Leung, Gilbert H.Young and Wing-Kay Kan. The same model as in [3], ie, the output is given as a function of the state/input and unknown weights/parameters which follow trivial dynamics. The EKF is applied to calculate the weight estimate on a real time basis from the output sequence. However, the matrices appearing in the EKF depend on the input/state vector at certain times and this state forms a random process. Hence, in the computation of the EKF matrices like the Kalman gain matrix and the estimation error covariance matrix, the authors use asymptotic values

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of the matrices which can be expressed as time averages of certain state depen­ dent matrices. These time averages, assuming ergodicity are then expressed in terms of ensemble average w.r.t. the input/state probability distribution and using this, the authors derive a formula for the asymptotic Kalman error co­ variance matrix diagonal elements which they relate to weight pruning, ie, the output estimate error energy when one of the weights is set equal to zero. This formula determines the order of importance of the different weights, ie, given an error threshold, we can decide which weights to reject and which not to. Rejecting some of the weights in this way reduces the algorithmic complexity and is therefore called ”pruning the network”. The method discussed in this paper is applicable only to feedforward networks, ie, when the output at time n is completely determined by the input at time n. In the other case, ie when the output at time n depends on the input at time n as well as on the output at time n − 1 we have a recurrent network and a different approach is required. The method discussed in our paper is applicable to all kinds of networks and we do not run different EKF’s for the state and weight but instead run one EKF for the state and another decoupled EKF for the weights. [5] Eric A.Wan and Alex T.Nelson. This paper reviews both the methods for DEKF estimation of the state and weights for nonlinear, prediction and smoothing namely the least squares method for weight estimation and EKF for state estimation and secondly run­ ning two decoupled EKF’s for state and weight estimation. The authors point out an important advantage of using decoupled EKF’s rather than a single EKF for both the state and weight estimation, namely when the output equation is bilinear in the weights and states: y[n] = w[n − 1]T x[n] as happens in the LPC case, then running two decoupled EKF’s gives linear algorithms for calculating the state and weight estimates while on the other hand, running a big joint EKF for states and weights leads to nonlinear recursions. State and weight model: x[k + 1] = fk (x[k], w[k]) + εx [k + 1], w[k + 1] = w[k] + εw [k + 1], output model: y[k] = hk (x[k], w[k]) + v[k] First let us just consider the derivation of the EKF for a state model with output measurements. x[k + 1] = fk (x[k]) + εx [k + 1], y[k] = hk (x[k]) + εy [k] The noise processes εx and εy are iid random vectors respectively in Rn and in Rd since x[k] ∈ Rn y[k] ∈ Rd . Let Yk = {y[r] : r ≤ k}. Then, x ˆ[k+1|k] = E[x[k+1]|Yk ] = E(fk (x[k])|Yk ) ≈ fk (ˆ x[k|k])+(1/2)fk'' (ˆ x[k|k])V ec(Px [k|k])

General Relativity and Cosmology with Engineering Applications

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where ˆ[k|k])(x[k] − x ˆ[k|k])T |Yk ] Px [k|k] = Cov(x[k]|Yk ) = E[(x[k] − x Let x[k + 1|k])) x ˆ[k + 1|k + 1] = x ˆ[k + 1|k] + K[k](y[k + 1] − hk (ˆ approximately. Then, we choose the n × d matrix K[k] so that E[(|| x[k + 1] − x ˆ[k + 1|k + 1] ||2 |Yk ] is a minimum. This means that we must minimize ψ(K[k]) = x[k + 1|k))) T r(Cov(x[k + 1] − x ˆ[k + 1|k] − K[k](hk (x[k]) + εy [k] − hk (ˆ ≈ T r(Cov(ex [k + 1|k] − K[k](h'k (ˆ x[k + 1|k])ex [k + 1|k] + εy [k + 1]))) = T r[(I − K[k]Hk )Px [k + 1|k](I − K[k]Hk )T ] + T r(K[k]Pεy [k + 1])K[k]T ) Setting the variational derivative of this expression w.r.t K[k] to zero gives us the optimal equation Hk Px [k + 1|k](I − K[k]Hk )T + Pεy [k + 1]K[k]T = 0 where

x[k + 1|k]) ∈ Rd×n Hk = h'k (ˆ

Equivalently, K[k] = Px [k + 1|k]HkT (Hk Px [k + 1|k]HkT + Pε,y [k + 1])−1 Application of the matrix inversion lemma gives K[k] = Px [k + 1|k]HkT (Pε,y [k + 1]−1 − Pε,y [k + 1]−1 Hk (Px [k + 1|k]−1 + HkT Pε,y [k + 1]−1 Hk )−1 HkT Pε,y [k + 1]−1 ) = Px [k + 1|k]HkT Pε,y [k + 1]−1 − Px [k + 1|k]HkT Pε,y [k + 1]−1 Hk (Px [k + 1|k]−1 + HkT Pε,y [k + 1]−1 Hk )−1 HkT Pε,y [k + 1]−1 = Px [k + 1|k]HkT Pε,y [k + 1]−1 − Px [k + 1|k](In − Px [k + 1|k]−1 (Px [k + 1|k]−1 + HkT Pε,y [k + 1]−1 Hk )−1 HkT Pε,y [k + 1]−1 = (Px [k + 1|k]−1 + HkT Pε,y [k + 1]−1 Hk )−1 HkT Pε,y [k + 1]−1 To close the loop of the algorithm, we need to express Px [k + 1|k + 1] in terms of Px [k + 1|k] and Px [k + 1|k] in terms of Px [k|k]. To do this, we observe that ˆ[k + 1|k] = ex [k + 1|k] = x[k + 1] − x x[k|k]) − (1/2)fk'' (ˆ x[k|k])V ec(Px [k|k]) = fk (x[k]) + εx [k + 1] − fk (ˆ

194

General Relativity and Cosmology with Engineering Applications ≈ fk' (ˆ x[k|k])ex [k|k] + εx [k + 1] − (1/2)fk'' (ˆ x[k|k])V ec(Px [k|k])

and hence forming the conditional covariance on both sides given Yk gives us Px [k + 1|k] = Fk Px [k|k]FkT + Pε,x [k + 1] where

Fk = f ' (ˆ x[k|k]) ∈ Rn×n

Finally, we observe that ex [k + 1|k + 1] = x[k + 1] − x ˆ[k + 1|k + 1] = x[k + 1|k])) x[k + 1] − x ˆ[k + 1|k] + K[k](y[k + 1] − hk (ˆ x[k + 1|k])) = ex [k + 1|k] − K[k](hk (x[k + 1]) + εy [k + 1] − hk (ˆ x[k + 1|k])ex [k + 1|k] + εy [k + 1]) ≈ ex [k + 1|k] − K[k](h'k (ˆ and hence (this calculation followed by tracing has already been computed above) Px [k +1|k +1] = (In −K[k]Hk )Px [k +1|k](In −K[k]Hk )T +K[k]Pε,y [k +1]K[k]T This formula can be simplified considerably. We now look at the DEKF. Here, the state process, the weight vector and the output have the model x[k + 1] = fk (x[k], w[k]) + εx [k + 1], w[k + 1] = w[k] + εw [k + 1], output model: y[k] = hk (x[k], w[k]) + εy [k] We assume that w[k] has been estimated based on Yk as w ˆ[k]. Then, x ˆ[k + 1|k] = fk (ˆ x[k|k], w ˆ[k]) x[k + 1|k], w ˆ[k]) x ˆ[k + 1|k + 1] = x ˆ[k + 1|k] + K[k](y[k + 1] − hk (ˆ where K[k] = Px [k + 1|k]−1 + HkT Pε,y [k + 1]−1 Hk )−1 HkT Pε,y [k + 1]−1 where Px [k + 1|k] = Fk Px [k|k]FkT + Pε,x [k + 1] and Hk = hk,x (ˆ x[k + 1|k], w ˆ[k]), Fk = fk,x (ˆ x[k|k], w ˆ[k]) Px [k +1|k +1] = (In −K[k]Hk )Px [k +1|k](In −K[k]Hk )T +K[k]Pε,y [k +1]K[k]T

General Relativity and Cosmology with Engineering Applications

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Now we look at the update of the weight w, ie, we wish to compute w ˆ[k + 1] in terms of the new data y[k + 1]. This is done by linearization of the output equation w.r.t. the weight vector followed by an RlS with a forgetting factor. We minimize k+1 ∑

λk+1−m || y[m] − hm (ˆ x[m|m − 1], w ˆ[m − 1] + δw) ||2

m=0

w.r.t δw and define w ˆ[k + 1] = w ˆ[k] + δw Linearization of hm w.r.t. the weight gives us the objective function to be minimized as ψ(δw) = k+1 ∑

λk+1−m || y[m] − hm (ˆ x[m|m − 1], w ˆ[m − 1]) − Hm,w δw ||2

m=0

Here, x[m|m − 1], w ˆ[m − 1]) Hm,w = hm,w (ˆ The optimal equation ∂ψ(δw)/∂δw = 0 gives us k+1 ∑

T λk+1−m Hm,w (ey [m|m − 1] − Hm,w δw) = 0

m=0

or equivalently, δw = [

k+1 ∑

T Hm,w ]−1 [ λk+1−m Hm,w

m=0

k+1 ∑

T Hm,w ey [m|m − 1]]

m=0

where x[m|m − 1], w ˆ[m − 1]) ey [m|m − 1] = y[m] − hm (ˆ is the output estimation error at time m. In order to cast this determination of w ˆ[k + 1] in recursive form, we define P [k + 1] = [

k+1 ∑

T λk+1−m Hm,w Hm,w ],

m=0

b[k + 1] = [

k+1 ∑

T Hm,w ey [m|m − 1]

m=0

Thus,

w ˆ[k + 1] = w ˆ[k] + P [k + 1]−1 b[k + 1]

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General Relativity and Cosmology with Engineering Applications

Now, P [k + 1] = λP [k] + HkT+1,w Hk+1,w , T b[k + 1] = λb[k] + Hk+1,w ey [k + 1|k]

Then application of the matrix inversion lemma gives P [k+1]−1 = λ−1 P [k]−1 −λ−1 P [k]−1 HkT+1,w (λI+Hk+1,w P [k]−1 Hk+1,w )−1 Hk+1,w P [k]−1

and so w ˆ[k + 1] = w ˆ[k] − P [k]

−1

HkT+1,w (λI

+ Hk+1,w P [k]−1 HkT+1,w )−1 Hk+1,w δw ˆ[k]

T +λ−1 P [k]−1 Hk+1,w ey [k + 1|k] T −λ−1 P [k]−1 Hk+1,w (λI+Hk+1,w P [k]−1 Hk+1,w )−1 Hk+1,w P [k]−1 Hk+1,w ey [k+1|k] T T ˆ = w[k]+P ˆ [k]−1 Hk+1,w (λI+Hk+1,w P [k]−1 Hk+1,w )−1 )(ey [k+1|k]−Hk+1,w δw[k])

Note that δw[k + 1] = w ˆ[k + 1] − w ˆ[k] Example: The state model is x[k + 1] = f (Ax[k] + Bu[k] + C) + εx [k + 1], and the output measurement model is y[k] = h(Dx[k] + Eu[k] + F ) + εy [k] where the matrices A, B, C, D, E, F are functions of a neural weight vector w. This general situation includes the special case when the matrix elements of some or all of these matrices are themselves some or all of the weights. Applying the DEKF derived above to this model gives the following state observer and weight estimate update equations as ˆ [k]u[k]+Cˆ [k])+Aˆ[k]2 f '' (Aˆ[k]ˆ ˆ [k]u[k]+Cˆ [k])V ec(Px [k|k]) x ˆ[k+1|k] = f (Aˆ[k]ˆ x[k|k]+B x[k|k]+B

ˆ [k]ˆ ˆ [k]u[k] + Fˆ [k])) x ˆ[k + 1|k + 1] = x ˆ[k + 1|k] + K[k](y[k + 1] − h(D x[k + 1|k] + E ˆ [k]h' (D ˆ [k]ˆ ˆ [k]u[k] + Fˆ [k]) x[k + 1|k] + E Hk = D ˆ [k]u[k] + Cˆ [k]) x[k|k] + B Fk = Aˆ[k]f ' (Aˆ[k]ˆ K[k] = Px [k + 1|k]−1 + HkT Pε,y [k + 1]−1 Hk )−1 HkT Pε,y [k + 1]−1 where Px [k + 1|k] = Fk Px [k|k]FkT + Pε,x [k + 1] Note that f ' (ξ) is an n×n matrix while h' (x) is a d×n matrix. We have further, Px [k + 1|k + 1] = (In − K[k]Hk )Px [k + 1|k](In − K[k]Hk )T + Pε,y [k + 1]

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The weight vector update equation is w ˆ[k + 1] = = w[k]+P ˆ [k]−1 HkT+1,w (λI+Hk+1,w P [k]−1 HkT+1,w )−1 )(ey [k+1|k]−Hk+1,w δw ˆ[k]) where P [k+1]−1 = λ−1 P [k]−1 −λ−1 P [k]−1 HkT+1,w (λI+Hk+1,w P [k]−1 Hk+1,w )−1 Hk+1,w P [k]−1

ˆ [k]ˆ ˆ [k]u[k + 1] + Fˆ [k]) ey [k + 1|k] = y[k + 1] − hk+1 (D x[k + 1|k] + E ∂ hk+1 (Dˆ x[k + 1|k] + Eu[k + 1] + F )|w=w[k] ˆ ∂w Let p denote the number of weights. Hk+1,w is then a matrix of size d × p whose j th column is given by Hk+1,w =

[Hk+1,w ]j = h'k+1 (D(w)ˆ x[k + 1|k] + E(w)u[k] + F (w))( +

∂D(w) x ˆ[k + 1|k] ∂wj

∂E(w) ∂F (w) u[k] + )|w=w[k] ˆ ∂wj ∂wj

Note: We assume that u[k] is a q × 1 input vector so that A = A(w) ∈ Rn×n , B = B(w) ∈ Rn×q , C = C(w) ∈ Rn×1 , D = D(w) ∈ Rn×n , E = E(w) ∈ Rn×q , F = F (w) ∈ Rn×1 ˆ [k] for B(w[k]) Also we are using the shorthand notations Aˆ[k] for A(w[k]) ˆ B ˆ etc. A special case of one state variable, two weight variables and one observation variable per time sample: x[k + 1] = a[k]x[k] + bu[k] + c + α(a[k]x[k] + bu[k] + c)2 + εx [k + 1], y[k] = d[k]x[k] + eu[k] + f + β(d[k]x[k] + eu[k] + f )2 + εy [k] The EKF equations for state update and RLS equations for weight update are then x ˆ[k + 1|k] = a ˆ[k]ˆ x[k|k] + bu[k] + c + α(ˆ a[k]ˆ x[k|k] + bu[k] + c)2 , Hk = a ˆ[k] + 2αa ˆ[k](ˆ a[k]ˆ x[k + 1|k] + bu[k + 1] + c) K[k] = (1/Px [k + 1|k] + Hk2 /Pε,y )−1 Hk /Pε,y = Hk Px [k + 1|k]/(Px [k + 1|k]Hk2 + Pε,y ) x ˆ[k+1|k+1] = x ˆ[k+1|k]+K[k](y[k+1]−dˆ[k]ˆ x[k+1|k]−eu[k]−f −β(dˆ[k]ˆ x[k+1|k]+eu[k]+f )2 )

Px [k + 1|k] = Fk2 Px [k|k] + Pε,x

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General Relativity and Cosmology with Engineering Applications

where ˆ[k] + 2βa ˆ[k](ˆ a[k]ˆ x[k|k] + bu[k] + c) Fk = a a ˆ[k](1 + 2β(ˆ a[k]ˆ x[k|k] + bu[k] + c)) Px [k + 1|k + 1] = (1 − K[k]Hk )2 Px [k + 1|k] + K[k]2 Pε,y Note that K[k], Hk , Px [k + 1|k], Px [k|k] are all scalars, ie, 1 × 1 matrices. Also 1 − K[k]Hk = Pε,y /(Px [k + 1|k]Hk2 + Pε,y )

Now we look at the update of the weight w, ie, we wish to compute w[k ˆ + 1] in terms of the new data y[k + 1]. P [k+1]−1 = λ−1 P [k]−1 −λ−1 P [k]−1 HkT+1,w (λI+Hk+1,w P [k]−1 HkT+1,w )−1 Hk+1,w P [k]−1

and so w ˆ[k + 1] = T T = w[k]+P ˆ [k]−1 Hk+1,w (λI +Hk+1,w P [k]−1 Hk+1,w )−1 (ey [k+1|k]−Hk+1,w δw ˆ[k])

Here, x[k+1|k]+eu[k+1]+f )−β(dˆ[k]ˆ x[k+1|k]+eu[k+1]+f )2 , ey [k+1|k] = y[k+1]−(dˆ[k]ˆ ˆ T w[k] ˆ = [ˆ a[k], d[k]] ( ) Paa [k] Pad [k] P [k] = Pad [k] Pdd [k] Let P [k]−1 = Q[k] =

(

qaa [k] qad [k]

qad [k] qdd [k]

)

We have Hk+1,w = [0, x ˆ[k + 1|k] + 2βx ˆ[k + 1|k](dˆ[k]ˆ x[k + 1|k] + eu ˆ[k] + f )] Now consider Kw [k] = P [k]−1 HkT+1,w (λI + Hk+1,w P [k]−1 HkT+1,w )−1 We can write the weight update equation as ˆ[k]) w ˆ[k + 1] = w ˆ[k] + Kw [k](ey [k + 1|k] − Hk+1,w δw Now, Kw [k] = Q[k]HkT+1,w /(λ + Hk+1,w Q[k]HkT+1,w ) Note that Hk+1,w Q[k]HkT+1,w = qdd [k](ˆ x[k+1|k]+2βx ˆ[k+1|k](dˆ[k]ˆ x[k+1|k]+eu ˆ[k+1]+f ))2

199

General Relativity and Cosmology with Engineering Applications = qdd [k]ˆ x[k + 1|k]2 (1 + 2β(dˆ[k]ˆ x[k + 1|k] + eu[k] + f )2 ) T Q[k]Hk+1,w = (ˆ x[k+1|k]+2βx ˆ[k+1|k](dˆ[k]ˆ x[k+1|k]+eu[k+1]+f ))[qad [k], qdd [k]]T

=x ˆ[k + 1|k](1 + 2β(dˆ[k]ˆ x[k + 1|k] + eu[k + 1] + f ))[qad [k], qdd [k]]T

A Remark: We have assumed that in the output equation, the output y[k] does not directly depend upon a[k]. This would imply that it would be difficult to obtain an accurate estimate of the weight a[k] using the RLS since the RLS is based only upon minimizing the output error. To rectify this situation, we note that the output y[k + 1] depends on x[k + 1] which depends upon x[k] and a[k]. So y[k + 1] implicitly depends upon a[k]. This suggests that we should replace Hk+1,w by [(∂y[k + 1]/∂x[k + 1])(∂x[k + 1]/∂a[k]), ∂y[k + 1]/∂d[k + 1]] evaluated at a ˆ[k], dˆ[k], x ˆ[k + 1|k]. Doing so, we get Hk+1,w = [(dˆ[k]+2βdˆ[k](dˆ[k]ˆ x[k+1|k]+eu[k+1]+f ))ˆ x[k](1+2α(ˆ a[k]ˆ x[k|k]+bu[k+1]+c),

x ˆ[k + 1|k] + 2βx ˆ[k + 1|k](dˆ[k]ˆ x[k + 1|k] + eu ˆ[k] + f )]

Comparison with the with the global coupled EKF run for both the state and weight estimate updates: The statistical model is x[k + 1] = a[k]x[k] + bu[k] + c + α(a[k]x[k] + bu[k] + c)2 + εx [k + 1], y[k] = d[k]x[k] + eu[k] + f + β(d[k]x[k] + eu[k] + f )2 + εy [k] a[k + 1] = a[k], d[k + 1] = d[k] The EKF run for the extended state ξ[k] = [x[k], a[k], d[k]]T is given by x ˆ[k + 1|k] = a ˆ[k|k]ˆ x[k|k] + bu[k] + c + α(ˆ a[k|k]ˆ x[k|k] + bu[k] + c)2 , a ˆ[k + 1|k] = a ˆ[k|k], dˆ[k + 1|k] = dˆ[k|k] Pξ [k + 1|k] = Fk Pξ [k|k]FkT + Pε,ξ [k + 1] where

Fk = fk' (ξˆ[k|k]) fk (ξ) = fk (x, a, d) = [ax + bu[k] + c + α(ax + bu[k] + c)2 , a, d]T

so that (

a + 2αa(ax + bu[k] + c) 0 fk' (ξ) = ( 0

c + 2αx(ax + bu[k] + c) 1 0

) 0 0 ) 1

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General Relativity and Cosmology with Engineering Applications

so that (

Fk = (

a ˆ[k|k] + 2αa ˆ[k|k](ˆ a[k|k]ˆ x[k|k] + bu[k] + c) 0 0

Note that

(

Pxx [k + 1|k] Pξ [k|k] = ( Pxa [k + 1|k] Pxd [k + 1|k]

) c + 2αx ˆ[k|k](ˆ a[k|k]ˆ x[k|k] + bu[k] + c) 0 1 0 ) 0 1

Pxa [k + 1|k] Paa [k + 1|k] Pad [k + 1|k]

) Pxd [k + 1|k] Pad [k + 1|k] ) Pdd [k + 1|k]

Pε,ξ [k] = diag[Pε,x [k], 0, 0] ξ[k + 1|k + 1] = [ˆ x[k + 1|k + 1], a ˆ[k + 1|k + 1], dˆ[k + 1|k + 1]]T = [ˆ x[k|k], a ˆ[k|k], dˆ[k|k]]T +K[k](y[k+1]−(dˆ[k+1|k]ˆ x[k+1|k]+eu[k+1]+f +β(dˆ[k+1|k]ˆ x[k+1|k]+eu[k+1]+f )2 )) Hk = h'k (ξˆ[k + 1|k]) = [dˆ[k + 1|k] + 2βdˆ[k + 1|k](dˆ[k + 1|k]ˆ x[k + 1|k] + eu[k] + f ), 0, x ˆ[k + 1|k] + 2βx ˆ[k + 1|k](dˆ[k + 1|k]ˆ x[k + 1|k] + eu[k] + f )] ∈ R1×3 K[k] = Pξ [k + 1|k]HkT (Hk Pξ [k + 1|k]HkT + Pε,y [k + 1])−1 ∈ R3×1 Pξ [k+1|k+1] = Pξ [k+1|k+1] = (In −K[k]Hk )Pξ [k+1|k](In −K[k]Hk )T +K[k]Pε,y [k+1]K[k]T ∈ R3×3 Comparison with two decoupled EKF’s run for separately the state and weight updates. x ˆ[k + 1|k] = a ˆ[k|k]ˆ x[k|k] + bu[k] + c + α(ˆ a[k|k]ˆ x[k|k] + bu[k] + c)2 2 Px [k + 1|k] = Fxk Px [k|k] + Pε,x [k + 1]

where Fxk = a ˆ[k|k] + 2αa ˆ[k|k](ˆ a[k|k]ˆ x[k|k] + bu[k] + c) 2 Kx [k] = Px [k + 1|k]Hxk (Hxk Px [k + 1|k] + Pε,y [k + 1])−1

Hxk = dˆ[k|k] + 2βdˆ[k|k](dˆ[k|k]ˆ x[k|k] + eu[k] + f ) x ˆ[k+1|k+1] = x ˆ[k+1|k]+Kx [k](y[k+1]−(dˆ[k|k]ˆ x[k+1|k]+eu[k+1]+f +β(dˆ[k|k]ˆ x[k+1|k]+eu[k+1]+f )2 )

Px [k + 1|k + 1] = (1 − Kx [k]Hxk )2 Px [k + 1|k] + Kx [k]2 Pε,y [k + 1] a ˆ[k + 1|k] = a ˆ[k|k], dˆ[k + 1|k] = dˆ[k|k], or equivalently, w ˆ[k + 1|k] = w[k|k] ˆ

General Relativity and Cosmology with Engineering Applications

201

since w[k] = [a[k], d[k]]T w ˆ[k + 1|k + 1] = w ˆ[k + 1|k] + Kw [k](y[k + 1] − (dˆ[k + 1|k]ˆ x[k + 1|k] + eu[k + 1] + f ) −β(dˆ[k + 1|k]ˆ x[k + 1|k] + eu[k + 1]) where T T (Hwk Pw [k + 1|k]Hwk + Pε,y [k + 1])−1 Kw [k] = Pw [k + 1|k]Hwk

Hwk = [0, x ˆ[k + 1|k] + 2βx ˆ[k + 1|k](dˆ[k + 1|k]ˆ x[k + 1|k] + eu[k + 1] + f )]T and finally, Pw [k+1|k+1] = (I2 −Kw [k]Hwk )Pw [k+1|k](I2 −Kw [k]Hwk )T +Kw [k]Pε,y [k]Kw [k]T

Note: The standard EKF for a state model with output measurements given by ξ[k + 1] = fk (ξ[k]) + εξ [k + 1], y[k] = hk (ξ[k]) + εy [k] Let Yk = {y[r] : r ≤ k}. Then, ξˆ[k + 1|k] = E[ξ[k + 1]|Yk ] = E(fk (ξ[k])|Yk ) ≈ fk (ξˆ[k|k]) Pξ [k|k] = Cov(ξ[k]|Yk ) = E[(ξ[k] − ξˆ[k|k])(x[k] − ξˆ[k|k])T |Yk ] We have ξˆ[k + 1|k + 1] = ξˆ[k + 1|k] + K[k](y[k + 1] − hk+1 (ξˆ[k + 1|k])) approximately. Let

Hk = h'k+1 (ξˆ[k + 1|k]) ∈ Rd×n

Then, from previous analysis K[k] = Pξ [k + 1|k]HkT (Hk Pξ [k + 1|k]HkT + Pε,y [k + 1])−1 = (Pξ [k + 1|k]−1 + HkT Pε,y [k + 1]−1 Hk )−1 HkT Pε,y [k + 1]−1 Pξ [k + 1|k] = Fk Pξ [k|k]FkT + Pε,ξ [k + 1] where

Fk = f ' (ξˆ[k|k]) ∈ Rn×n

Pξ [k +1|k +1] = (In −K[k]Hk )Pξ [k +1|k](In −K[k]Hk )T +K[k]Pε,y [k +1]K[k]T

Applications to deep recurrent neural networks (DNN). The basic setup for the output signal model is z1 [t] = σ(W1 x[t] + b1 + U z1 [t − 1]),

202

General Relativity and Cosmology with Engineering Applications z2 [t] = σ(W2 z1 [t] + b2 ),

and in general, zk+1 [t] = σ(Wk zk [t] + bk ), k = 1, 2, ..., N, y[t] = zN +1 [t] + εz [t] x[t] is the input W1 , .., WN are the weights and z1 , ..., zN are the intermediate states. The state vector at time t is z[t] = [z1 [t], .., zN +1 [t]]T ∈ RN +1 We wish to express the above equations in state variable form, ie, as z[t] = f (z[t − 1], x[t], W ), W = [W1 , ..., WN ]T For that purpose, we define fk (ξ) = σ(ξ + bk ) Then, we can express the above equations as after taking into account weight evolution, z1 [t] = f1 (W1 [t − 1]x[t] + U z1 [t − 1]) z2 [t] = f2 (W2 [t − 1]z1 [t]) = f2 (W2 [t − 1]f1 (W1 [t − 1]x[t] + U z1 [t − 1])) and in general, zk+1 [t] = fk+1 (Wk+1 [t−1]fk (Wk [t−1]fk−1 (Wk−1 [t−1]fk−1 ...f1 (W1 [t−1]x[t]+U z1 [t−1]))...),

k = 1, 2, ..., N Equivalently, in state variable form, writing ψk+1 (W1 [t], ..., Wk+1 [t], x, ξ) = fk+1 (Wk+1 [t]fk (Wk [t]fk−1 (Wk−1 [t]...f1 (W1 [t]x + U ξ))...), k = 1, 1, ..., N, ψ1 (W1 [t], x, ξ) = f1 (W1 [t]x + U ξ) we have taking process noise into account, [z1 [t+1], ..., zN +1 [t+1]]T = z[t+1] = [ψ1 (W1 [t], x[t+1], z1 [t]), ψ2 (W1 [t], W2 [t], x[t+1], z1 [t]), ...,

ψN +1 (W1 [t], ..., WN +1 [t], x[t + 1], z1 [t]]T + εz [t + 1] = g(x[t + 1], z1 [t], W [t]) + εz [t + 1] The measurement model is simple: y[t] = zN +1 [t] + εy [t]

General Relativity and Cosmology with Engineering Applications

203

The dual EKF for this system can be implemented as follows. ˆ 1 [t], ..., W ˆ k [t], x[t + 1], U zˆ1 [t|t])), k = 1, 2, ..., N + 1 zˆk [t + 1|t] = ψk (W zˆk [t + 1|t + 1] = zˆk [t + 1|t] + K[t](y[t + 1] − zˆN +1 [t + 1|t]) Pz [t + 1|t] = Ft Px [t|t]FtT + Pε,z [t + 1] ˆ [t])[g1 , 0, 0, .., 0] ∈ RN +1×N +1 z [t + 1|t], W Ft = g,z (ˆ ˆ [t]), ..., gN +1,z (ˆ ˆ [t])]T g1 = [g1,z1 (ˆ z1 [t + 1|t], W z1 [t|t], W 1 K[t] = (Pz [t + 1|t]−1 + HtT Pε,z [t + 1]−1 Ht )−1 HtT Pε,y [t + 1]−1 where Thus,

Ht = [1, 0, ..., 0] = uT ∈ R1×(N +1 K[t] = Pε,y [t + 1]−1 (Pz [t + 1|t]−1 + Pε,y [t + 1]uuT )−1 u

Note that Pε,y [t + 1] is a scalar, ie a 1 × 1 matrix. We note that g1,z1 (x, z1 , W ) = ((

∂ψk (W1 [t], ..., Wk [t], x, z1 ) N +1 ))k=1 ∈ RN +1×1 ∂z1

Now, ∂ψ1 (W1 , x, z1 ) = U f1' (W1 x + U z1 ) ∂z1 and for k = 1, 2, ..., N , ∂ψk+1 (W1 , ..., WN +1 , x, z1 ) ∂z1 = (∂/∂z1 )fk+1 (Wk+1 fk (Wk fk−1 (Wk−1 ...f1 (W1 x + U z1 ))...) = Wk+1 fk' +1 (Wk+1 ψk (W1 , ..., Wk , x, z1 ))Wk fk' (Wk ψk−1 (W1 , .., Wk−1 , x, z1 ) ...W2 f2' (W2 ψ1 (W1 , x, z))U f1' (W1 x + U z1 )

Remark on polynomial minimization for the computation of the weight es­ timate in the dual EKF. We have to determine the weight w ˆ[k] in the form w ˆ[k − 1] + δw. To this end, we replace w ˆ[m − 1] by w ˆ[m − 1] + δw for all m ≤ k + 1. The function to be minimized is ψk+1 (δw) =

k+1 ∑

λk+1−m || y[m] − hm (ˆ x[m|m − 1], w ˆ[m − 1] + δw) ||2

m=0

We approximate: hm (ˆ x[m|m − 1], w ˆ[m − 1] + δw) ≈

204

General Relativity and Cosmology with Engineering Applications hm (ˆ x[m|m − 1], w ˆ[m − 1]) +

q ∑ ∂ r hm (ˆ x[m|m − 1], w ˆ[m − 1])

∂wr

r=1

(δw)⊗r /r!

∂ r hm (x,w) ∂wr

r

we mean a row vector whose entries are r!(r1 !...rp !)−1 ∂wr1 ∂...∂wrp p 1 ∑p with r1 , ..., rp varying over 0, 1, ..., r such that j =1 rj = r. So the problem of estimating δw with this truncated Taylor approximation amounts to minimizing

where by

ψk+1 (δw) =

k+1 ∑

λ

k+1−m

|| ey [m|m − 1] −

m=0

q ∑

Hm,w,r (δw)⊗r ||2

r=1

Now, || ey [m|m − 1] −

q ∑

Hm,w,r (δw)⊗r ||2

r=1

=|| ey [m|m−1] ||2 +

∑

q T (δw)⊗sT Hm,w,s Hm,w,r (δw)⊗r −2

r,s=1

∑

q

ey [m|m−1]T Hm,w,r (δw)⊗r

r=1

Setting the gradient of this quantity w.r.t.δw to zero gives us a highly nonlinear equation whose roots are not to be determined easily. So we adopt the approx­ imate gradient descent algorithm for estimating δw. This gradient loop is to be run within the outer loop of state estimation. It reads δw[n + 1] = δw[ n] − gradψk+1 (δw[n]) where gradψk+1 (δw) = 2

k+1 ∑

q ∑

s ∑

T Hm,w,r (δw)⊗r [(δw)⊗(l−1)T ⊗ Ip ⊗ (δw)⊗(s−l)T ]Hm,w,s

m=0 r,s=1 l=1

−2

k+1 ∑

q ∑ s ∑

T (δw)⊗(l−1)T ⊗ Ip ⊗ (δw)⊗(s−l)T Hm,w,s ey [m|m − 1]

m=0 r=1 m=1

An alternate approximate way to determine δw is to assume that δw is a ran­ dom vector and that its moments E(δw⊗m ) = ξm , m ≥ 1 can all be varied independently of each other. Further, the expression ψk+1 (δw) is to be replaced by E(ψk+1 (δw)) with the resulting mimimization being carried out w.r.t all the statistical moments of δw appearing in this expectation: E(ψk+1 (δw)) = E(

k+1 ∑

λk+1−m || ey [m|m − 1] −

m=0

=|| ey [m|m−1] ||2 +

∑ r,s=1

q ∑

Hm,w,r (δw)⊗r ||2 )

r=1 q

T E[(δw)⊗sT Hm,w,s Hm,w,r (δw)⊗r ]−2

∑ r=1

q

ey [m|m−1]T Hm,w,r E[(δw)⊗r ]

General Relativity and Cosmology with Engineering Applications Now,

205

T Hm,w,r (δw)⊗r ] E[(δw)⊗sT Hm,w,s T = (V ec(Hm,w,s Hm,w,r ))T E[(δw)⊗(r+s) ]

Thus, the objective function to be minimized can be expressed as a linear func­ tion of the moment vectors ξm , m = 1, 2, ..., N . Now minimizing a linear function is trivial, so we must put a quadratic energy constraint on these moments of the form N ∑ T ξkT Qkm ξm = E, Qkm = Qmk k,m=1

With this constraint, the objective function to be minimized has the form ψ(λ, ξm , m = 1, 2, ..., N ) =

N ∑

N ∑

c[m]T ξm − λ(

m=1

ξkT Qkm ξm − E)

k,m=1

The optimal equations are then c[k] − 2λ

N ∑

Qkm ξm , k = 1, 2, ..., N

m=1

which with obvious notations, has the solution ξ = (2λ)−1 Q−1 c, ξ = ((ξm )), Q = ((Qkm )), c = ((c[k])) and

E = (2λ)−2 cT Q−1 c √ λ = (cT Q−1 c)1/2 /2 E

If we do not wish to incorporate any energy constraint and yet formulate a meaningful solvable optimization problem, we can take a partial expectation assuming independence of the tensor powers δw⊗s and δw⊗r appearing on the two sides of the quadratic term or more precisely replace Eψk+1 (δw) by φk+1 (ξm , m = 1, 2, ...q) = =

k+1 ∑

[

q ∑

T Hm,w,r ξr − 2 ξsT Hm,w,s

m=0 r,s=1

q ∑

ey [m|m − 1]T Hm,w,r ξr ]

r=1

The optimal minimizing equations obtained by setting the gradients of φk+1 w.r.t. ξm , m = 1, 2, ..., q to zero are q k+1 k+1 ∑ ∑ ∑ T T ( Hm,w,s Hm,w,r )ξr = Hm,w,s ey [m|m − 1] r=1 m=0

m=0

which is easily solved by matrix inversion.

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General Relativity and Cosmology with Engineering Applications

Estimating the weights in the presence of white non-Gaussian measurement noise: The measurement model is y[k] = hk (x[k], w[k]) + εy [k] where εy [k], k = 1, 2, ... is an iid sequence of random vectors with pdf p(ε). Based on the idea of maximum likelihood estimation, the objective function to be minimized for calculating δw = w[k] ˆ −w ˆ[k − 1] is given by ψk+1 (δw) =

k+1 ∑

λk+1−m log(p(y[m] − hm (ˆ x[m|m − 1], w ˆ[m − 1] + δw))

m=0

If we make an Edgeworth expansion of p(ε), taking as our base pdf the multi­ variate normal density φ(ε) = (2π)−d/2 |A|−1/2 exp(−εT A−1 ε/2) then we get ∑

p(ε) = φ(ε)(1 +

c[k1 , ...kd ]Hk1 ((A−1/2 ε)1 )...Hkd (A−1/2 ε)d )

k1 +...+kd ≥1

where Hk , k = 1, 2, ... are the standard Hermite polynomials. We note that q(ε) = |A|1/2 p(A1/2 ε) = ∑ c[k1 , ..., kd ]Hk1 (ε1 )...Hkd (εd )) = (2π)−d/2 exp(− || ε ||2 /2)(1 + k1 +...+kd ≥1

q(ε) represents a multivariate Edgeworth expansion in which the different com­ ponents of the random vector are statistically independent while p(ε) represents a multivariate Edgeworth expansion in which the different components of the random vector are statistically dependent and in particular correlated. Continuous time formulation of the weight update equation: A continuous time model for the state and weight evolution equations along with continuous time nonlinear filters for estimating these variables has the advantage of giv­ ing simple approximate proofs of convergence since Ito’s formula can be used which is not the case with the discrete time scenario. Keeping this in mind, we model the state, weight and output evolution equations as stochastic differential equations: dx(t) = ft (x(t), w(t))dt + dεx (t), dw(t) = dεw (t), dy(t) = ht (x(t), w(t))dt + dεy (t) The state is estimated using the EKF and the weight at time t is estimated by minimizing ∫ t || dy(s) − hs (ˆ x(s), w ˆ(s) + δw)ds ||2 0

General Relativity and Cosmology with Engineering Applications ∫ ≈

t

0

207

|| dey (s) − Hs,w δwds ||2

with dey (s) = dy(s) − hs (ˆ x(s), w ˆ(s))ds, Hs,w = hs,w (ˆ x(s), w ˆ(s)) where εx , εw and εy are independent multivariate Brownian motion processes. The minimization results in ∫ t ∫ t T −1 T δw = ( Hs,w Hs,w ds) ( Hs,w dey (s)) 0

Define

∫

0

∫

t

Q(t) = 0

T Hs,w ds, q(t) = Hs,w

Then, and so

t 0

T Hs,w dey (s)

δw = Q(t)−1 q(t) w ˆ(t) = w(t ˆ − dt) + δw = w(t ˆ − dt) + Q(t)−1 q(t)

or we may write

dw ˆ(t) = Q(t)−1 q(t)

This equation does not make much sense as an sde since the rhs does not have the √ order dt which a stochastic differential should have. We try instead something like the stochastic gradient algorithm: dw ˆ(t) = −μ

∂ || dy(t) − ht (ˆ x(t), w ˆ(t))dt ||2 /dt ∂w

x(t), w ˆ(t)) T ∂ht (ˆ ) (dy(t) − ht (ˆ x(t), w ˆ(t)) ∂w This algorithm for weight estimation is to be carried out hand in hand with the continuous time EKF algorithm for state estimation: = −μ(

−2 dx ˆ(t) = ft (ˆ x(t), w ˆ(t))dt + σε,x Px (t)(dy(t) − ht (ˆ x(t), w ˆ(t))dt)

Px' (t) = ( Px (t)(

x(t), w ˆ(t)) ∂ft (ˆ )Px (t)+ ∂x

x(t), w ˆ(t)) T ∂ft (ˆ 2 ) + σε,x I/2 ∂x

x(t), w ˆ(t)) T ∂ht (ˆ x(t), w ˆ(t)) ∂ht (ˆ ) ( )Px (t) ∂x ∂x The convergence analysis of this algorithm can be performed using the standard mean-variance propagation equations for sde’s. −2 Px (t)( −σε,y

This set of sde’s for (x(t), x ˆ(t), w(t), w ˆ(t), Px (t))

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General Relativity and Cosmology with Engineering Applications

can be expressed by replacing dy(t) − ht (ˆ x(t), w ˆ(t))dt with (ht (x(t), w(t)) − x(t), w ˆ(t)))dt + dεy (t) as ht (ˆ ( ) x(t) | | w(t) | | | x ˆ (t) d| | | ( w ˆ() t) V ec(Px (t)) ( | | | =| (

ft (x(t), w(t)) 0 −2 ft (ˆ x(t), w ˆ(t)) + σε,x Px (t)(ht (x(t), w(t)) − ht (ˆ x(t), w ˆ(t))) − μHw,t (ˆ x(t), w ˆ(t))T (ht (x(t− ), w(t)) ht (ˆ x(t), w ˆ(t)))

2 −2 T (I ⊗ Ft (ˆ x(t), w(t)) ˆ + Ft (ˆ x(t), w(t)) ˆ ⊗ I)V ec(Px (t)) + σε,x u − σε,y V ec(Px (t)Hx,t (ˆ x(t), w(t)) ˆ Hx,t (ˆ x(t)

(

) dεx (t) | | dεw (t) | | +( x(t), w ˆ(t))dεy (t) ) −μHw,t (ˆ 0 where Hx,t =

x(t), w ˆ(t)) ∂ht (ˆ , ∂x

x(t), w ˆ(t)) ∂ht (ˆ ∂w are the Jacobian matrices of ht respectively w.r.t its two arguments x and w evaluated at the estimates of these vectors at time t. Hw,t =

Other literature survey: [6] Eric A.Wan and R.Van der Merwe. This paper introduces the unscented Kalman filter which we can apply to our neural state and weight estimation problem. Consider the dynamical system x(k+1) = Fk (x(k))+εx (k) with the measurement model y(k) = hk (x(k))+εy (k). x(k|k)) which is In the EKF, the state predictor is constructed as x ˆ(k+1|k) = Fk (ˆ an approximation to E(Fk (x(k))|Yk ). This approximation is exact only when Fk is a linear function, ie, a matrix. In the general case, if x(k) is approximated by a Gaussian random vector, we still cannot use such an approximation effectively because Gaussian proceses when propagated through a nonlinear system will become non-Gaussian. Hence, to effectively approximate the above conditional expectation, the unscented Kalman filter uses x ˆ(k + 1|k) = N −1

N ∑

Fk (ˆ x(k|k) + ξr )

r=1

where ξr are vectors chosen so that they approximately represent independent samples of x(k) − x ˆ(k|k). Likewise, to compute the state estimate update after

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a measurement has been taken, the UKF uses the standard Gaussian formula for the conditional expectation x ˆ(k+1|k+1) ˆ(k+1|k)+Pxy (k+1|k)Pyy (k+1|k)−1 (y(k+1)−yˆ(k+1|k)) = E(x(k+1)|Yk+1 ) ≈ x but to compute Pxy and Pyy , we use N ∑ Pxy (k+1|k) = N −1 (Fk (ˆ x(k|k)+ξr )−x ˆ(k+1|k))(hk+1 (ˆ x(k+1|k)+ηr )−yˆ(k+1|k)) r=1

where yˆ(k + 1|k) = N

−1

N ∑

hk+1 (ˆ x(k + 1|k) + ηr )

r=1

and further, Pyy (k+1|k) = N −1

N ∑

[(hk+1 (ˆ x(k+1|k)+ηr )−yˆ(k+1|k))(hk+1 (ˆ x(k+1|k)+ηr )−yˆ(k+1|k))T ]

r=1

We note that to calculate the samples ξr of e(k|k) = x(k) − x ˆ(k|k), the UKF makes use of the corresponding error covariance P (k|k) while to compute sam­ ˆ(k + 1|k), the UKF makes use of the error ples ηr of e(k + 1|k) = x(k + 1) − x covariance P (k + 1|k). So we also need update formulas for these error covari­ ances. The UKF once again takes nonlinearity of the system into consideration while approximating these update equations. Specifically, P (k +1|k) = N −1

N ∑

(Fk (ˆ x(k|k)+ξr )−x ˆ(k +1|k))(Fk (ˆ x(k|k)+ξr )−x ˆ(k +1|k))T

r=1

P (k + 1|k + 1) = P (k + 1|k) − Pey (k + 1|k)Pyy (k + 1|k)−1 Pey (k + 1|k)T Pey (k + 1|k) = Pxy (k + 1|k) With reference to our work here, it is easily possible to adapt the UKF for state estimation combined with RLS and pruning for weight estimation. [7] Gintaras V Puskorius and Lee Feldkamp. A recurrent neural network has been considered described by the 2-D differ­ ence equation yij [n] = Fij (yi−1 [n], yi−2 [n], ..., y1 [n], yi [n − 1], yi−1 [n], ..., y1 [n − 1], wi ) ∂y [n]

The partial derivatives ∂wijgh for this model have been computed in a time and layer recursive way so as to enable training of the neural network for given inputoutput data using a time recursive gradient optimization scheme. The authors then use these recursive formulas for the partial derivatives in an EKF based training algorithm. The recurrent structure of the neural network enables the above partial derivatives to be computed using a back propagation scheme which

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allows low computational complexity implementation of the EKF. We could in principle compare this method with our dual EKF. The advantage with the dual EKF method discussed in our paper is that it is model independent and does not require measurements of the internal layer signals. Further, in our algorithm, state and weight updated run in a decoupled way. [8] Ercheng Pei, Xiaohan Xia, Le Yang, Dongmei Jiang and Hichen Sahli. The notion of a switched Kalman filter applied to neural feature extrac­ tion has been presented for the first time here. The main idea is to treat the observed sequence (output) as coming from a probabilistic mixture of linear Gaussian models. To be precise, if we have p linear state variable models with corresponding linear output models, then by applying the KF to each of these models with the given observation sequence as input, we can estimate the cor­ responding state. The final state estimate is then a probabilistic mixture of all these states and appears to produce better results in many cases, especially in speech where the governing vocal tract parameters make Markov transitions between two phoenemes. The state estimate for each model undergoes a change when one switches between the models in accordance with a Markov chain, with the KF state estimate applied after each switching. This switching concept can be applied also to the EKF used in our paper. Basically, we run the EKF x[k + 1] = F (l) (x[k], w) + ε(xl) [k + 1], y[k] = h(l) (x[k], w) + ε(yl) [k] for a duration of N samples and then at the end, switch over to another model with l replaced by l + 1. The model and output functions (F (l) , h(l) ) change to (F (l+1) , h(l+1) ) in accord with a Markov chain law, ie, we have a set of M models say (Fr , hr ), r = 1, 2, ..., M and if the lth block model is (F (l) , h(l) ) = (Frl , hrl ), then the rl → rl+1 transition law is governed by a Markov chain. It would be interesting to see how such a switched EKF can be tuned with an RLS weight estimator and how the performance of such a dual switched EKF would be as compared to an unswitched dual EKF. [9] S.Horvath and H.Neuner. Learning models for modeling deformation processes by applying the EKF for weight learning or training has been performed here. The forward step of the learning process has the form ∑ ∑ wnm ym (t)), ym (t) = φm ( wml xl (t)) yˆn (t) = φn ( m

where xl (t)' s constitute the input processes and the φ'n s are basis functions. We start with some weights wmn , compute ym (t) and then readjust these weights so that yˆn (t) matches a desired output. Then, we continue the recursive process by using these updated weights to again compute ym (t) followed by a readjustment of weights to match yˆn (t) to the desired output. The matching is done using a

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version of the gradient search algorithm in a time recursive way just as in the classical LMS algorithm used for system identification in the signal processing literature. The authors follow this up with a discussion of an EKF method for weight updation based on the following state model for the weights and output model for the measured output: wnm (t + 1) = wnm (t) + pnm (t + 1), ∑ ∑ yn (t) = φn ( wnm (t)φm ( wml (t)xl (t))) + εm (t) m

l

They show that for a certain class of gradient search algorithms for weight update, the results are equivalent to the EKF results. We could therefore adapt this method to our dual EKF based method basically by partitioning the set of weights into two disjoint sets in the above model and updating the weights in the first set using the EKF and those in the second set using the RLS with the same basis function model for all the weights and outputs. Since the RLS has lower complexity, than the EKF, but the EKF is more accurate, it follows that the significant weights will be updated using the EKF while the not so significant weights will be updated using the RLS. For example, to apply the RLS to this formalism, we use the linearized version: ∑ ∑ ∑ ∑ wnk φk ( wkl xl )) φm ( wml xl (t)))δwnm δyn (t) = φ'n ( k

+φ'n (

∑ k

wnk φk (

m

l

∑ l

wkl xl ))

∑ m

' wnm φm (

l

∑

wmr xr )

r

∑

xl (t)δwml

l

[10] Roger J.Williams. First the author shows how to apply the EKF to a feedforward (ie, non­ recurrent) NN. Such a network is described by only an output equation y[k] = h(u(k), W ) = hk (W ) where u(k) is the known input sequence and W is the weight vector. To apply the EKF here for weight estimation, we introduce trivial dynamics for the weights with small noise thus getting a line state variable Markov model for the weight evolution and we also introduce a small output noise. In short, the EKF is applied to the following state and output model: W [k + 1] = W [k] + εW [k + 1], y[k] = hk (W [k]) + εy [k] The author then discusses and alternate EKF algorithm for weight estimation in a recurrent NN. The idea is that the state vector x[k] of the RNN is de­ scribed by the signals at the different interior nodes. Assume that x[k] = [x1 [k]T , ..., xN [k]T ]T where xm [k] denotes the signal vector at the mth layer.

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The RNN model for the evolution of the state vector is that the states at the m + 1th layer are related to the states at the mth layer through a recurrent model of the form T xm [k]), m = 0, 1, ..., N − 1 xm+1 [k + 1] = f (Wm

where N is the number of layers, the zeroth layer is the input layer, ie, x0 [k] is the input signal vector and xN [k] form the output of the NN. To apply the EKF to this model, we introduce trivial dynamics for the weights: Wm [k + 1] = Wm [k], m = 0, 1, ..., N − 1. The system of state equations for the extended state vector [x1 [k]T , ..., xN [k]T , W0 [k]T , ..., WN −1 [k]T ]T is then defined by the above equations and the output/measurement vector is simply the signal at the last layer: y[k] = xN [k] The author suggests that instead of this output equation, we use something like y[k] = [g1 (W1 [k]T x1 [k], ..., gN (WN [k]T xN [k])]T ie, the output consists of some set of nonlinear memoryless transformations {gi } of the signal vectors at the input of the various internal nodes, rather than the output at these nodes (which are the sigmoidal functions f applied to the input). In order to apply the EKF to this model, linearization of the gi' s must be performed, ie, their differentials must be computed. The results are bound to be better that the usual EKF model because, more measurements are made and further the nonlinear functions gi can be chosen at our discretion. It would be interesting to see how the same model can be coupled with the RLS for the weight vector and EKF for the state vector rather than use the joint EKF for both. Computational complexity is bound to be less if we do so. [11] Herbert Jaeger A short tutorial on back propagation through time, real time recurrent neural networks and training RNN’s using the EKF is presented here. Here, we present the same material in a slightly more generalized framework involving a rigorous computation of the gradients of the error energy w.r.t. the weights and how exactly back propagation occurs in a recurrent network as opposed to a feed forward network. Let y(n) denote the output of the feed-forward NN at time n and d(n) the corresponding desired output. We write y(n) = g(WNT xN −1 (n))

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213

where f is a scalar valued map, WN is the weight vector at the (N −1)th stage of the network and xN −1 (T ) is the state signal vector at the (N − 1)th layer. y(n) is a scalar signal and xk (n) is the vector signal at the k th layer. To minimize the error energy T ∑ E(T ) = (y(n) − d(n))2 n=1

with respect to the weight vector WN , we compute the partial derivative (∂/∂WN )E(T ) = (∂/∂WN )

T ∑

(y(n) − d(n)) =

n=1

2

T ∑

(y(n) − d(n))g ' (WNT xN −1 (n))xN −1 (n)

n=1

According to the feed-forward layered structure, we can write xN −1 (n) = f (WN −1 xN −2 (n)) where f is a vector valued map and WN −1 is a weight matrix. We thus have (∂/∂WN −1 )E(T ) = T ∑

[(∂/∂WN −1,rs )(y(n) − d(n))2 ] =

n=1

2

T ∑

(y(n) − d(n))g ' (WNT xN −1 (n))WNT f,r (WN −1 xN −2 (n))xN −2,s (n)

n=1

Define the signals δ0 (n) = g ' (WNT xN −1 (n))(y(n) − d(n)), η1,r (n) = WNT f,r (WN −1 xN −2 (n)) so that

η1 (n) = WNT f ' (WN −1 xN −2 (n))

η1 (n) is a row vector. Then we can write (∂/∂WN )E(T ) = T ∑

δ0 (n)xN −1 (n),

n=1

(∂/∂WN −1,rs )E(T ) =

T ∑

δ0 (n)η1,r (n)xN −2,s (n)

n=1

Likewise, (∂/∂WN −2,rs )E(T ) =

214 T ∑

General Relativity and Cosmology with Engineering Applications (y(n)−d(n))g ' (WNT xN −1 (n))WNT f ' (WN −1 xN −2 (n))WN −1 f,r (WNT −2 xN −3 (n))xN −3,s (n)

n=1

=

T ∑

δ0 (n)η1 (n)η2,r (n)xN −3,s (n)

n=1

where η2,r (n) = WN −1 f,r (WN −2 xN −3 (n)) or equivalently,

η2 (n) = WN −1 f ' (WN −2 xN −3 (n))

so that η2 (n) is a matrix. This recursion can be continued. Note that WN is a column vector while WN −1 , WN −2 , .. are all matrices. So far no back-propagation has been used. However, when we use backpropagation, then the model for state evolution is of the form x(n + 1) = f (W1 (n)x(n) + W2 (n)u(n + 1) + W3 (n)y(n)) where u(n) is the input layer, x(n) is the total state vector of all the layers and y(n) is the output vector of different layers. We have y(n) = g(W4 (n)x(n)) In the previous equation, the term W3 (n)y(n) represents the back propagation term. This term causes the neural network to become recurrent. If this term were absent, the network would be a feed-forward network. The weight matri­ ces Wj (n), j = 1, 2, 3, 4 are assumed to vary with time in accordance with an adaptation procedure that we describe below. For example, suppose we try to minimize the output error energy upto time T : E(T ) =

T ∑

|| y(n) − d(n) ||2

n=1

To minimize this using the gradient algorithm, we compute ∂E(T )/∂W1rs (n−1) = 2(y(n)−d(n))T g ' (W4 (n)x(n))W4 (n)∂x(n)/∂W1rs (n−1) = 2(y(n)−d(n))T g ' (W4 (n)T x(n))W4 (n)f,r (W1 (n−1)x(n−1)+W2 (n−1)T u(n−1) +W3 (n − 1)T y(n − 1))xs (n − 1) Equivalently, replacing n by n + 1 in this equation gives us ∂E(T )/∂W1rs (n) = 2(y(n+1)−d(n+1))T g ' (W4 (n+1)x(n+1))W4 (n+1)∂x(n+1)/∂W1rs (n)

= 2(y(n+1)−d(n+1))T g ' (W4 (n+1)T x(n+1))W4 (n+1)f,r (W1 (n)x(n)+W2 (n)T u(n) +W3 (n)T y(n))xs (n)

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215

Further, ∂E(T )/∂W2rs (n) = 2(y(n + 1) − d(n + 1))T g ' (W4 (n + 1)x(n + 1))W4 (n + 1) ×f,r (W1 (n)x(n) + W2 (n)u(n + 1) + W3 (n)y(n))us (n + 1) and ∂E(T )/∂W3rs (n) = 2(y(n + 1) − d(n + 1))T g ' (W4 (n + 1)x(n + 1))W4 (n + 1) ×f,r (W1 (n)x(n) + W2 (n)u(n + 1) + W3 (n)y(n))ys (n) and finally, ∂E(T )/∂W4rs (n) = 2(y(n) − d(n))T g,r (W4 (n)x(n))xs (n) The weight update formulas are derived from these gradients by using the stan­ dard gradient descent scheme. The phenomenon of back-propagation is seen to follow easily from these formulae, ie the gradient of the error energy w.r.t the weights at time n depends on one step future values of the output process and the state process.

5.26

Application of neural networks to the grav­ itational metric estimation problem

The dynamics of the gravitational field described by the metric tensor gμν (t, r) is described by a second order nonlinear partial differential equation in its four space-time variables, namely the Einstein field equations Rμν = 0. We can discretize the spatial coordinates into pixels and then regard the spatial pixel components of the metric tensor gμν (t, n1 δ, n2 δ, n3 δ), n1 , n3 , n3 = −N, −N + 1, ..., N − 1, N, 0 ≤ μ ≤ ν ≤ 3 as forming a big column vector at time t g(t) and by replacing the spatial partial derivatives with finite differences, the vector g(t) satisfies a second order nonlinear differential equation g'' (t) = F (g(t), g' (t)) These equations constitute the discretized Einstein field equations. In the pres­ ence of a noisy energy momentum tensor of matter and radiation, these equa­ tions assume the form of a system of nonlinear stochastic differential equations: dg(t) = h(t)dt, dh(t) = F (g(t), h(t))dt + dε(t) with the measured metric being a noisy version of some nonlinear function of g(t): dy(t) = ψ(g(t), h(t))dt + dεy (t) In order to apply the theory of neural networks to the problem of estimating the metric, we assume that the metric estimated by the neural network satisfies

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General Relativity and Cosmology with Engineering Applications

a simpler set of nonlinear differential equations with weights being continuously updated in order that the estimated metric is close to the measured metric, ie, if g1 (t) is the estimated metric, then, g1'' (t) = F2 (g1 (t), g1' (t), W (t)) where W (t) are the neural weights and F2 is the overall mapping generated by the multilayered neural network. We adapt W (t) for some time so that g1 (t) follows the EKF observed estimate of g(t) based on Yt = {y(s) : s ≤ t} and after a steady state has been reached, we stop the adaptation. References: [1] Ritika Agarwal, Vijyant Agrawal and H.Parthasarathy, ” The dual ex­ tended Kalman filter for Neural networks with nonlinear measurement models, Technical Report, NSIT, 2017.

5.27

Problems in quantum scattering theory

Discuss asymptotic completeness of the wave operators ie, if A, B are two selfadjoint operators in a Hilbert space H, then, the wave operators are defined by Ω+ (A, B) = slimt→∞ exp(iBt).exp(−iAt), Ω− (A, B) = slimt→−∞ exp(iBt).exp(−iAt) with domains D+ and D− respectively. Let EA (dx) and EB (dx) denote respec­ tively the spectral measures of A and B. Let f ∈ H be such that the measure < f, EA (dx)f > is absolutely continuous w.r.t the Lebesgue measure. Then since Ω+ (A, B)exp(itA) = exp(itB)Ω+ (A, B) it follows that EB (dx)Ω+ (A, B) = Ω+ (A, B)EA (dx) and hence || EB (dx)Ω+ (A, B)f ||2 =|| EA (dx)f ||2 which implies that the measure < Ω+ (A, B)f, EB (dx)Ω+ (A, B)f > is also ab­ solutely continuous w.r.t. the Lebesgue measure. In other words, Ω+ (A, B)Hac (A) ⊂ Hac (B) We say that Ω+ (A, B) is complete iff the above inclusion becomes an equality, ie, Ω+ (A, B) maps Hac (A) onto Hac (B).

General Relativity and Cosmology with Engineering Applications

5.28

217

Compact Operators

Define the notion of a compact operator and a relatively compact operator with an example. Define the notion of a bounded operator and a relatively bounded operator with an example. Show that a compact operator can be uniformly (ie in the operator norm) be approximated by a sequence of finite rank operators. Define the following notions in operator theory: [a] Principle of uniform boundedness [b] The Hahn-Banach theorem on extension of linear functionals in infinite dimensional Banach spaces. [c] Graph of an operator. [d] Closed graph theorem and open mapping theorem. [e] Closed operators, closable operators, closure of a closable operator. [f] Symmetric and selfadjoint operators. [g] Deficiency indices of a symmetric operator. [h] Maximal extension of a symmetric operator. [i] Self adjoint extension of a symmetric operator. [j] Cayley transform of a self-adjoint operator.

5.29 Estimating the metric parameters from geodesic measurements Suppose that the metric of space-time depends on a parameter vector θ, ie gμν (x, θ). We wish to estimate the parameter θ by taking measurements on the geodesic trajectories of a test particle. Suppose that we have available with us an initial guess of θ0 of this parameter. We set θ = θ0 + δθ. Then the linearized geodesic equation of a particle is d2 δxμ )/dτ 2 + Γμαβ (x, θ0 )((dxα /dτ )(dδxβ /dτ ) +(dδxα /dτ )(dxβ /dτ )) + δθr Γμαβ,θr (x, θ0 )(dxα /dτ )(dδxβ /dτ ) = 0 We know the unperturbed trajectory xμ (τ ). The above equation is therefore a linear second order differential equation for the perturbation δxμ (τ ). After discretizing the unperturbed proper time variable τ , we obtain a linear second order difference equation for δx which is of the general form δx[n + 1] = A1 [n]δx[n] + A2 [n]δx[n − 1] + Γ[n]δθ where A1 [n], A2 [n] are known 4 × 4 matrix valued functions of the discretized proper time index n and Γ[n] is a known 4×4 matrix dependent upon the proper time index n and these dependences on n are known from the unperturbed motion, ie, the solution to the geodesic equation with δθ = 0. The above second

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General Relativity and Cosmology with Engineering Applications

order linear difference equation is approximate and hence we can estimate δθ using the RLS method, ie a time recursive minimization of ψN (δθ) =

N −1 ∑

λN −n−1 || δx[n + 1] − A1 [n]δx[n] − A2 [n]δx[n − 1] − Γ[n]δθ ||2

n=0

Problem: Carry out the above minimization and cast it in RLS form by using the matrix inversion lemma.

5.30

Perturbations to the Band structure of a Semiconductor

Perturbation of the band structure of a semiconductor by small aperiodic po­ tentials tsking into account general relativistic corrections.

5.31

Scattering into cones for Schrodinger Hamil­ tonians

C is a cone in position space. Let Ω+ = limt→∞ Ut∗ Ut0 with Ut0 = exp(−itH0 ), Ut = exp(−itH), H = H0 + V . Let |f > be a free particle state, ie, it evolves accord­ ing to the Hamiltonian H0 and Ω+ |f >= |g > the correpsonding out scattered state which means that it evolves according to the Hamiltonian H. χC (Q) is the indicator function of the cone C. Q is the 3-D position operator. ∫ ∞ || χC (Q)Ut g ||2 dt 0

is the average total time spent by the scattered particle inside the cone Q. ∫ ∞ || χC (Q)Ut g ||2 dt T

is the average total time spent by the scattered particle in the cone C after time T . Now, || χC (Q)Ut g ||2 =|| Ut0∗ χC (Q)Ut0 Ut0∗ Ut g ||2 Ut0∗ χC (Q)Ut0 = exp(iP 2 t/2m)χC (Q)exp(−itP 2 /2m) = χC (exp(iP 2 t/2m)Q.exp(−itP 2 /2m)) [iP 2 t/2m, Q] = −P t/m implies

exp(iP 2 t/2m)Q.exp(−itP 2 /2m) = Q − P t/m

General Relativity and Cosmology with Engineering Applications

219

and hence, for large and negative t, χC (exp(iP 2 t/2m)Q.exp(−itP 2 /2m)) ≈ χC (−P t/m) = χC (P ) since C is a cone. Thus, as t → −∞, || χC (Q)Ut g ||2 ≈|| χC (P )Ω∗− g ||2 =|| χC (P )Ω∗− Ω+ f ||2 =|| χC (P )S ∗ f ||2 Some other useful identities in quantum scattering theory: Let B be a Borel subset of R3 . The rate at which the probability of the scattered particle spending within the set B is given by d || χB Ut g ||2 = dt d < g, Ut∗ χB Ut g >=< g, iUt∗ [H0 + V, χB ]Ut g > dt = i < Ut g, [H0 , χB ]Ut g >= (i/2m) < Ut g, [P 2 , χB ]Ut g > = (i/2m)(< Ut g, (P 2 χB − χB P 2 )Ut g > = −(1/m)Im(< Ut g, P 2 χB Ut g >) = (i/2m) < P 2 ht , χB ht > −(i/2m) < ht , χB P 2 ht > (ht = Ut g) ∫

(ht (Q)∗ ∇2 ht (Q) − ht (Q)∇ht (Q)∗ )d3 Q

= (i/2m) B

∫

(ht (Q)∗

= (i/2m) ∂B

5.32

∂ht (Q) ∂ht (Q)∗ − ht (Q) )dS(n) ∂n ∂n

study projects involving conventional field theory in curved background metrics

[a] Quantum Boltzmann equation in general relativity [b] Quantization of a field theory with noise with the example of general relativity. Let L(φ, φ,μ ) be the noiseless Lagrangian density of the field φ(x). To quantize it, we must first determine the Hamiltonian density using the Legendre transformation: ∂L π(x) = ∂φ,0 H(φ, ∇φ, π) = πφ − L

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General Relativity and Cosmology with Engineering Applications

is the Hamiltonian density. The noiseless classical Hamilton equations for the field that are equivalent to the above Euler-Lagrange field equations are φ,0 = π,0 = −

∂H , ∂π

∂H ∂H + div ∂φ ∂∇φ

We now quantize this field using the canonical commutation relations: [c] Approximate analysis of plasmonic waveguide in a background gravita­ tional field described by a metric. First consider the flat space-time situation of special relativity, ie, in the absence of a gravitational field. Rectangular waveguide filled with plasma of charge q per particle. Dimensions of the guide along the x and y axes are a and b. T M (m0 , n0 ) mode of propagation. √ Ez = exp(−γz)(2/ ab)sin(m0 πx/a)sin(n0 πy/b), Hz = 0, √ h(m0 , n0 )2 − ω 2 με √ E⊥ = (−γ/h2 )∇⊥ Ez , h = h(m0 , n0 ) = π m2 /a2 + n2 /b2 γ = γ(m0 , n0 ) =

H⊥ = (jωε/h2 )∇⊥ Ez × zˆ So √ Ex = (−γ/h2 )(2/ ab)(m0 π/a)cos(m0 π/a)sin(n0 πy/b)exp(−γz), √ Ey = (−γ/h2 )(2/ ab)(n0 π/b)sin(m0 π/a)cos(n0 πy/b)exp(−γz)

√ Hx = (jωε/h2 )Ez,y = (jωε/h2 )(2/ ab)(n0 π/b)sin(m0 πx/a)cos(n0 πy/b).exp(−γz) √ Hy = −(jωε/h2 )Ez,x = −(jωε/h2 )(2/ ab)(m0 π/a)cos(m0 πx/a)sin(n0 πy/b).exp(−γz)

Boltzmann kinetic transport equation in the frequency domain: f (ω, r, v) = f0 (r, v) + δf (ω, r, v) f0 (r, v) = K.exp(−(qΦ0 (x, y) + mv 2 /2)/kT ) This is the unperturbed equilibrium Gibbs distribution function. The unper­ turbed potential Φ0 exist inside the guide and may be assumed to be generated by electrostatic plates outside the guide. This potential is independent of z. The Boltzmann equation (approximate) for the distribution function in the frequency domain given the above em fields in the guide (ie the zeroth approximation of the em fields in which the plasma is absent) is given by jωδf (ω, r, v)+(v, ∇r )δf (ω, r, v)+(q/m)(E(ω, r)+v×B(ω, r), ∇v )f0 (r, v) = −δf (ω, r, v)/τ −−−(1)

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Here we are assuming the guide fields E, B to be small, ie, of the first order of smallness, ie, of the same order as δf . f0 is of the zeroth order of smallness, ie, much larger in magnitude that E, H, δf . Once we solve the above equation for δf , we can calculate the induced charge density and current density inside the guide as ∫ ∫ ρ(ω, r) = q δf (ω, r, v)d3 v, J(ω, r) = q vδf (ω, r, v)d3 v and then calculate the first order corrections to the em field in the guide by applying the retarded potential method. To solve the above equation, we write E(ω, r) = E1 (ω, x, y)exp(−γz), H(ω, r) = H1 (ω, x, y)exp(−γz) where E1 , H1 have been determined above and also assume that δf (ω, r, v) = δf1 (ω, x, y, v)exp(−γz) Then (1) becomes jωδf1 (ω, x, y, v) + vx δf1,x (ω, x, y, v) + vy f1,y (ω, x, y, v) −γvz f1 (ω, x, y, v) − (q/kT )(E1 (ω, x, y), v)f0 (x, y, v) = −δf1 (ω, x, y, v)/τ This equation is solved by the method of moments. First note that E1 (ω, x, y) = √ x ˆ(−γ/h2 )(2/ ab)(m0 π/a)cos(m0 π/a)sin(n0 πy/b)exp(−γz)+ √ yˆ(−γ/h2 )(2/ ab)(n0 π/b)sin(m0 π/a)cos(n0 πy/b)exp(−γz) √ zˆ(2/ ab)sin(m0 πx/a)sin(n0 πy/b)exp(−γz) and H1 (ω, x, y) = √ x ˆ(jωε/h2 )(2/ ab)(n0 π/b)sin(m0 πx/a)cos(n0 πy/b).exp(−γz) √ −yˆ(jωε/h2 )(2/ ab)(m0 π/a)cos(m0 πx/a)sin(n0 πy/b).exp(−γz) So our Boltzmann equation can be expressed by writing ∑ √ δf (ω, m, n, v)(2/ ab)sin(mπx/a)sin(nπy/b) δf1 (ω, x, y, v) = m,n≥1

(based on the assumption that the particle distribution function perturbation vanishes at the boundary of the guide) as ∫ a√ ∑ δf (ω, m' , n, v) 2/a(m' π/a)cos(m' πx/a)sin(mπx/a)dx jωδf (ω, m, n, v)+vx m'

0

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∑

δf (ω, m, n' , v)

∫

b

√

2/b(n' π/b)cos(n' πy/b)sin(nπy/b)dy

0

n'

∫

−γvz δf (ω, m, n, v)−(q/kT )(v,

0

a

∫

b 0

√ E1 (ω, x, y)f0 (x, y, v)(2/ ab)sin(mπx/a)sin(nπy/b)dxdy

+δf (ω, m, n, v)/τ = 0 This gives us a sequence of linear algebraic equations for δf (ω, m, n, v), m, n ≥ 1 which are solved by truncation and matrix inversion. [d1] Klein-Gordon field in a background metric-Hamiltonian form [d2] Einstein field equations for gravitation in Hamiltonian form. We express the metric of space-time as dτ 2 = (N 2 + Xi X i )dt2 + 2Xi dxi dt + qij dxi dxj where xi , i = 1, 2, 3 are the spatial variables and Xi = qij X j , X i = g ij Xj , ((q ij )) = ((qij ))−1 This metric can be expressed in matrix form as ( 2 N + X T q −1 X ((gμν )) = X

XT q

)

where X = ((Xi )), q = ((qij )) Assume that the inverse of this metric is given by ( ) a bT ((g μν )) = b C Then the equation ((gμν ))((g μν )) = I4 gives us (N 2 + X T q −1 X)a + X T b = 1, (N 2 + X T q −1 X)bT + X T C = 0, XbT + qC = I3 , C = C T Thus,

C = q −1 (I − XbT ), (N 2 + X T q −1 X)b = −CX, (N 2 + X T q −1 X)b = −q −1 (I − XbT )X = −q −1 (X − XX T b)

so that or

(N 2 + X T q −1 X)X T b = −X T q −1 X(1 − X T b) X T b = −X T q −1 X/N 2

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General Relativity and Cosmology with Engineering Applications so that

(N 2 + X T q −1 X)b = −q −1 X(1 + X T q −1 X/N 2 )

or

b = −q −1 X/N 2

We then get C = q −1 (I + XX T q −1 /N 2 ) = N −2 (N 2 q −1 + q −1 XX T q −1 ) Finally,

a = (1 − X T b)/(N 2 + X T q −1 X) = N −2

Thus finally, we get ( ((g

μν

)) =

N −2 −1 −q X/N 2

We note that

−X T q −1 /N 2 −2 2 −1 N (N q + q −1 XX T q −1 )

)

q −1 X = ((X i )), X = ((Xi ))

So the above equations can also be expressed as g 00 = 1/N 2 , g 0i = −X i /N 2 , g ij = q ij + X i X j /N 2 A simple calculation also shows that g = det((gμν )) = N 2 |q|, |q| = det(q) Thus, the invariant four volume element is √ √ −gd4 x = N ( − |q|)d4 x We now express the Einstein-Hilbert Lagrangian density β α β L = g μν (Γα μν Γαβ − Γμβ Γνα )

in terms of the functions N = N (x), q = q(x) = ((qij (x))), X = X(x) = (Xi (x)) First let us set up the Klein-Gordon Lagrangian density and then Hamiltonian density in this background metric: √ √ LKG (φ, φ,μ ) = (1/2)g μν φ,μ φ,ν −g − (1/2)m2 φ2 −g The corresponding field equation is given by √ √ (g μν φ mu −g),ν + m2 φ −g = 0 We can write √ LKG = [(1/2)g ij φ,i φ,j + g 0i φ,0 φ,i + g 00 φ2,0 − (m2 /2)φ2 ] −g

224

General Relativity and Cosmology with Engineering Applications = [(1/2)(N q ij + X i X j /N )φ,i φ,j − X i φ,0 φ,i /N + φ2,0 /N − (m2 /2)φ2 N ]

√ −|q|

Remark: An alternate parametrization of the metric is to use Y i = X i /N in place of X i and get √ LKG = [(1/2)N (q ij + Y i Y j )φ,i φ,j − Y i φ,0 φ,i + φ2,0 /2N − (m2 /2)N φ2 ] −|q| We now set up the Hamiltonian density corresponding to this KG Lagrangian density. The canonical momentum density is ∂LKG = ∂φ,0 √ [−Y i φ,i + φ,0 /N ] −|q| φφ =

and so the Hamiltonian density is HKG = πφ φ,0 − LKG = [φ2,0 /2N − (N/2)(q ij + Y i Y j )φ,i φ,j ] = (N/2)

5.33

√

−|q|

√ √ −|q|[(πφ / −|q| + Y i φ,i )2 − (q ij + Y i Y j )φ,i φ,j ]

Intuitive explanation of an invariance prin­ ciple in scattering theory

Let A, B be self-adjoint operators in a Hilbert space with spectral measures EA (dx) and EB (dx) respectively. The wave operators are Ω+ (A, B) = limt→∞ exp(itB)exp(−itA), Ω− (A, B) = limt→−∞ exp(itB)exp(−itA) Now we use the intuitive fact that exp(itx) weakly converges to zero as t → ∞ provided x /= 0 in which case, it equals one. This fact is known as the RiemannLebesgue Lemma: If f ∈ L( R), then ∫ fˆ(t) = f (x)exp(itx)dx is square integrable by the Parseval theorem and hence limt→∞ fˆ(t) = 0 which states that exp(itx) weakly converges to zero as t → ∞. Thus, we can intuitively write ∫ Ω+ (A, B) = limt→∞ exp(it(y − x))EB (dy)EA (dx) ∫ =

EB (dx)EA (dx)

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It follows that if φ(x) is one-one function on R, then ∫ Ω+ (A, B) = limt→∞ exp(it(φ(y) − φ(x)))EB (dy)EA (dx) ∫ =

EB (dx)EA (dx)

This has to be made more precise by defining the domains appropriately. In other words, we have proved the invariance principle: For a large class of func­ tions φ on R, we have Ω± (φ(A), φ(B)) = Ω± (A, B) A rigorous statement of this result with a proof has been given in ”T.Kato, Perturbation theory for linear operators”, Springer.

5.34

Scattering theory for the Dirac Hamilto­ nian in curved space-time

We recall that for a general metric gμν , the Dirac equation taking into account the gravitational connection Γμ expressed in terms of the tetrad Vaμ has the form [γ a Vaμ (i∂μ + eAμ + iΓμ ) − m]ψ = 0 From this expression, we obtain the following Lagrangian density √ L = Re[ψ ∗ [γ 0 γ a Vaμ (i∂μ + eAμ + iΓμ ) − m]ψ] −g We wish to obtain the Hamiltonian density corresponding to this Lagrangian density and thereby derive formulas for the wave operators and scattering op­ erator of a projectile by a nucleus in the presence of external electromagnetic and gravitational fields. The canonical momenta corresponding to the position fields ψ and ψ¯ respectively are πψ =

∂L = ∂ψ,0

(i/2)ψ ∗ γ 0 γ a Va0 )T = (i/2)Va0 (γ 0 γ a )T ψ¯ Likewise, π ˜ψ =

∂L = ∂ψ¯,0

(iψ T γ 0 γ¯ a Va0 )T = iVa0 γ 0 γ a ψ Problem: Now express the Hamiltonian density of the Dirac field in curved space-time in terms of the position fields ψ, ψ¯, their spatial derivatives and the ˜ψ . momenta πψ , π

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5.35

Derivation of the approximate Schrodinger Hamiltonian for a particle in curved spacetime with corrections upto fourth order in the space derivatives

hint: p0 is the energy and pr , r = 1, 2, 3 are the momenta. We have the relation g μν pμ pν = m2 If in addition, there is an external electromagnetic field, then this equation should be replaced by g μν (pμ + eAμ )(pν + eAν ) = m2 We can write the first equation as g 00 p20 + 2g 0r p0 pr + g rs pr ps − m2 = 0 and solving this quadratic equation for p0 gives √ p0 = −g 0r pr /g00 + (g 0r g 0s − g rs )pr ps /g 002 + m2 /g 002 √ = hr pr + m2 /g 002 + γ rs pr ps where

γ rs = (g 0r g 0s − g rs )/g 002

We can then approximate the Hamiltonian by p0 ≈ hr pr + (m/g 00 )(1 + g 002 γ rs pr ps /2m2 − (g 004 /8m4 )(γ rs pr ps )2 ) and then solve the Schrodinger equation, time independent or time dependent using the substitution pr = i∂/∂xr .

5.36

Quantum scattering theory in the presence of time dependent Hamiltonians arising in general relativity

Suppose a time dependent gravitational field gμν (t, r) is present along with a charge Q at the origin. The general relativistic Schrodinger equation is obtained by approximating the equation g μν (pμ + eAμ )(pν + eAν ) − m2 = 0 with A0 = Q/|r| and Ak = 0, k = 1, 2, 3. Thus, A0 = g00 A0 = g 00 (t, r)Q/|r|, Ak = gk0 A0 = gk0 Q/|r|

General Relativity and Cosmology with Engineering Applications

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We may choose our coordinate system so that g00 = 1, gk0 = 0 so that A0 = Q/|r|, Ak = 0, k = 1, 2, 3 Thus, we get

(p0 + eA0 )2 + g rs pr ps − m2 = 0

or p0 + eA0 = or

√ m2 − g rs pr ps

p0 ≈ −eA0 + m(1 − g rs pr ps /2m2 − (g rs pr ps )2 /8m4 )

so the approximate time dependent Schrodinger equation reads iψ,t (t, r) = p0 ψ(t, r) = −eA0 (r)ψ(t, r)+∂s g rs ∂s ψ(t, r)/2m−(∂s g rs ∂r )2 ψ(t, r)/8m3 Note that we have taken our time dependent Hamiltonian arising from the time varying nature of the background gravitational field as H(t) = −eA0 (r) − ps g rs (t, r)pr /2m − (ps g rs (t, r)pr )/8m3 We write g rs (t, r) = −δrs + εhrs (t, r) where ε is a small perturbation parameter. Then, H(t) = H0 + εV1 (t) + ε2 V2 (t) where

H0 = −eA0 + p2 /2m − (p2 )2 /8m3

V1 (t) = −ps hrs (t, r)pr /2m + (1/8m3 )(p2 pr hrs (t, r)ps + ps hrs (t, r)pr p2 ) V2 (t) = −(ps hrs (t, r)pr )2 /8m3 When the gravitational field is quantized approximately using creation and an­ nihilation operators we can write ∑ ∗ rs hrs (t, r) = (ak (t)χrs ¯k (r)) k (r) + ak (t) χ k ∗

where ak (t) and ak (t) are respectively the annihilation and creation processes. They can be regarded in the formalism of the Hudson-Parthasarathy quantum stochastic calculus as white noise operators. Then, quadratic functions of these processes will have to be regarded as appropriate conservation processes in the language of the Hudson-Parthasarathy quantum stochastic calculus. We note that ∗ ¯rs V1 (t) = −ps χkrs (r)pr ak (t)/2m − ps χ k (r)pr ak (t) /2m 2 ∗ +(1/8m3 )(p2 pr χrs ¯rs k (r)ps ak (t) k (r)ps ak (t) + p pr χ 2 ∗ +ps χkrs (r)pr p2 ak (t) + ps χ ¯rs k (r)pr p ak (t)

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General Relativity and Cosmology with Engineering Applications

Further, ∗ 2 ¯rs V2 (t) = −(1/8m3 )(ps χrs k (r)pr ak (t) + ps χ k (r)pr ak (t) )

(We are using the Einstein summation convention implying summation over repeated indices). Problem: Express V2 (t) explicitly as quadratic functions of ak (t), ak (t)∗ , k = 1, 2, ... with coefficients being system operators, ie, functions of position and mo­ mentum r, ps and then write down the Hudson-Parthasarathy noisy Schrodinger equation by replacing ak (t)dt, ak (t)∗ dt respectively by the annihilation and cre­ ation process differentials dAk (t), dAk (t)∗ and ak (t)∗ am (t)dt by dΛkm (t), ie, the conservation process differentials occurring in the Hudson-Parthasarathy theory.

5.37

Band structure of a semiconductor altered by a massive gravitational field

The semiconductor crystal has nuclei located at the sites of a periodic lattice ie, at n1 a1 +n2 a2 +n3 a3 , n1 , n2 , n3 ∈ Z. The resulting potential in which an electron moves is a function V (r) = V (x, y, z) having periods ak , k = 1, 2, 3. Denote by bk , k = 1, 2, 3 the reciprocal lattice vectors, ie, ak .bm = δkm , k, m = 1, 2, 3. Then the potential can be expanded as a Fourier series ∑ V (r) = V [n1 , n2 , n3 ]exp(2πi(n1 b1 + n2 b2 + n3 b3 , r)) It is easy to verify that this expression satisfies V (r + ak ) = V (r), k = 1, 2, 3 Assume now that the Hamiltonian is the Dirac Hamiltonian for the electron in such a periodic potential so that the stationary state Dirac equation reads [−i(α, ∇) + βm − eV (r)]ψ(r) = Eψ(r) with α = (α1 , α2 , α3 ), β being the standard 4 × 4 Dirac matrices satisfying the standard anticommutation relations: αa αb + αb αa = 2δab , αa β + βαa = 0, a, b = 1, 2, 3 In the presence of a static gravitational field, the Hamiltonian must be derived from the Dirac Lagrangian density taking into account the tetrad term Vaμ (r) and the gravitational spinor connection term Γμ (r). We leave this as an exercise to the student. Now, in the above Dirac equation where ψ(r) is a four compo­ nent wave function, replacement of r by r + ak , k = 1, 2, 3 leaves the equation invariant. Further, if we assume that the lattice size is L1 , L2 , L3 along the

General Relativity and Cosmology with Engineering Applications

229

three directions a1 , a2 , a3 respectively, then we must apply the periodic bound­ ary conditions ψ(r + Nj aj ) = ψ(r), j = 1, 2, 3 where Lj = Nj aj , Nj ∈ Z+ , j = 1, 2, 3 Thus, iot is thus easy to see that this stationary state wave function ψ(r) must satisfy ψ(r + ak ) = Ck ψ(r), k = 1, 2, 3 where CkNk = 1, k = 1, 2, 3 or equivalently, Ck = exp(2πisk /Nk ) where sk ∈ {0, 1, ..., Nk − 1}. Hence, we can write ψ(r) = exp(2πi(s1 b1 /N1 + s2 b2 /N2 + s3 b3 /N3 , r))φs (r), s = (s1 , s2 , s3 ) where φs is periodic with periods ak , k = 1, 2, 3, ie, φs (r + ak ) = φs (r), k = 1, 2, 3 It follows that φs can also like V , be developed into a Fourier series: ∑ φs (r) = φs [n1 , n2 , n3 ]exp(2πi(n1 b1 + n2 b2 + n3 b3 , r)) n1 ,n2 ,n3 ∈Z

Problem: Substitute this Fourier series representation of the wave function into the Dirac equation with the potential also expanded as a Fourier series and derive the infinite order linear difference equations satisfied by the coefficients φs [n1 , n2 , n3 ] ∈ C4 .

5.38

Design of quantum gates using quantum physical systems in a gravitational field

The typical example here is to perturb Dirac’s equation in a gravitational field by a control electromagnetic field and allow the system to evolve for time T . The control em field is then chosen so that the resulting evolved unitary gate af­ ter time T after appropriate truncation is as close as possible in Frobenius norm distance to a given unitary gate. This is a natural model since any quantum physical system according to general relativity, will be affected by the gravita­ tional field. We may also choose to control the gravitational field, ie, the metric tensor of the background space-time in such a way that a desired unitary gate is formed after time T .

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5.39

General Relativity and Cosmology with Engineering Applications

Quantum phase estimation

Assume that U is a unitary matrix with one of its eigenvalues being exp(2πiφ) and |u > as the corresponding eigenvector. |u > is known but φ is unknown and we wish to estimate φ by a quantum algorithm. We first prepare a pure state of the form t 2∑ −1 −t/2 |ψ >= 2 |k > |u > k=0

which means that the first t qubits of |ψ > are in the state 2−t/2 Here, the integer k is represented in binary form as

∑2t −1 k=0

|k >.

k = a0 + a1 .2 + a2 .22 + ... + at−1 2t−1 , a0 , a1 , ..., at−1 ∈ {0, 1} Now we apply the control unitary V =

t−1 ∑

|k >< k| ⊗ U k

k=0

to |ψ > which results in the state −t/2

V |ψ >= 2

t 2∑ −1

−t/2

|k > U |u >= 2 k

k=0

t 2∑ −1

exp(2πikφ)|k > |u >

k=0

We then follow this up by applying the quantum Fourier transform F to the first t qubits of V |ψ >. The resulting state is −t

F V |ψ >= 2

t 2∑ −1

exp(2πik(φ − n/2t ))|n > |u >

k,n=0 2∑ −1 t

= 2−t

n=0

exp(2πi2t (φ − n/2t )) − 1 |n > |u > exp(2πi(φ − n/2t )) − 1

We then measure the first t qubits. The probability of obtaining |n > after this measurement is exp(2πi2t (φ − n/2t )) − 1 2 | Pt (n) = 2−2t | exp(2πi(φ − n/2t )) − 1 sin2 (2t π(φ − n/2t )) sin2 (π(φ − n/2t )) In the limit as t → ∞, this probability becomes Pt (n) = 1 if φ = n/2t and zero otherwise. More precisely, if t is large and we assume that φ ∈ [0, 1) may well be approximated as φ = n/2t = 2−2t

for some n = 0, 1, ..., 2t − 1, then the probability of the measurement given / 2t φ. This is the essence of the phase estimation n = 2t φ is one and zero if n = algorithm.

General Relativity and Cosmology with Engineering Applications

5.40

231

Noisy Schrodinger equations, pure and mixed states

[a] Schrodinger dynamics preserves the purity of a state although it can also be made to act on initially mixed states. Noisy Schrodinger dynamics does not preserve the purity of a state. If we have a system interacting with a bath and noise processes that are measurable w.r.t the bath variables corrupt the Schrodinger dynamics of the system, then if initially, the system and the bath are in pure states so that the overall state of the system and bath is the tensor product of two pure states, then under noisy Schrodinger dynamics, after time t, we get again a pure state for the system and bath, but if we trace out this state over the bath variables, the resulting state of the system becomes a mixed state and its dynamics is described by the GKSL (Gorini, Kossakowski, Sudarshan, Lindblad) equation. [b] Consider the HP noisy Schrodinger equation: dU (t) = (−(iH + P )dt + L1 dA + L2 dA∗ + SdΛ)U (t) Suppose that the initial state of the system and bath is the pure state |ψ(0) >= |f > ⊗|φ(u) > where |φ(u) > is a normalized coherent state of the bath. We know that dA|φ(u) >= u(t)dt|φ(u) >, dΛ|φ(u) >= (dA∗ dA/dt)|φ(u) >= u(t)dA∗ |φ(u) > and dA∗ |φ(u) >= dB|φ(u) > −dA|φ(u) >= (dB − u(t)dt)|φ(u) > where B(t) = A(t) + A(t)∗ is a classical Brownian motion. It can be easily checked that [B(t), B(s)] = 0, (dB(t))2 = dt, the second equation being a consequence of quantum Ito’s formula. Thus we get on defining |ψ(t) >= U (t)|ψ(0) >, that d|ψ(t) >= [−(iH+P )dt+u(t)L1 dt+L2 (dB(t)−u(t)dt)+S(u(t)dB(t)−u(t)2 dt)]|ψ(t) >

= [−[(iH + P ) + u(t)L1 + u(t)2 S]dt + [L2 + u(t)S]dB(t)]|ψ(t) >

232

5.41

General Relativity and Cosmology with Engineering Applications

Constructions using ruler and compass

Write down all the steps involved in the following constructions using ruler and compass only: [a] Drawing the perpendicular bisector of a line segment AB. [b] Drawing the perpendicular from a point P onto a line segment AB when P does not fall on the line AB even after extension. [c] Given a line AB and a point P lying outside AB, draw a line passing through P that is parallel to AB. [d] Drawing an equilateral triangle given the side length and hence drawing the angle 60 degrees. [e] Drawing the angle 30 degrees. [f] Bisecting an angle. √ √ [g] Drawing the length 2 and 5 using Pythagoras’ theorem.

5.42

Application of the Jordan canonical form for matrices in general relativity

Consider the problem of solving the geodesic equations of motion of a particle in a piecewise constant gravitational field, ie, the entire space is partitioned into three dimensional pixels and over each pixel, the space time first partial derivatives of the metric are indepndent of the space-time coordinates. The geodesic equations read dv μ /dτ + Γμαβ (x)v α v β = 0 where v μ = dxμ /dτ We try a solution of the form v μ (τ ) = V μ + δv μ (τ ) where V μ is a constant and δv μ (τ ) is a small perturbation. The zeroth order terms give Γμαβ (x)V α V β = 0 Over a fixed pixel, Γμαβ (x) is a constant and we assume that the above equations have a solution for V μ . Then the first order terms give dδv μ μ + 2Γαβ V α δv β (τ ) = 0 dτ The solution to this equation over a fixed pixel involves exponentiating the 4 × 4 matrix A = ((aμβ )), aμβ = −2Γμαβ V α More generally in n-dimensional space-time, this will involve exponentiating an n × n matrix which can be done using the Jordan canonical form.

General Relativity and Cosmology with Engineering Applications

5.43

233

Application of the Jordan canonical form in solving fluid dynamical equations when the velocity field is a small perturbation of a constant velocity field v(t, r) = V0 + δv(t, r)

The linearized Navier-Stokes equations are δv,t (t, r) + (V0 , ∇)δv(t, r) = −∇δp(t, r) + ν∇2 δv(t, r) By discretizing space into pixels, this becomes an n × n linear state variable equation for the spatial components of δv(t, r) and is solved by exponentiating a matrix using the Jordan Canonical form.

5.44

The Jordan Canonical Form

Let N be an n × n Nilpotent matrix. Then there exists a unique positive integer m such N m = 0, N m−1 /= 0. Choose vectors x1 , , ..., xk1 so that N m−1 xl , l = 1, 2, ..., k1 forms a basis for R(N m−1 ). Then obviously the union of the sets {N m−1 xl , l = 1, 2, ..., k1 } and {N m−2 xl , l = 1, 2, ..., k1 } form a linearly in­ dependent set and can therefore easily be extended to a set {N m−1 xl , l = 1, 2, ..., k1 } ∪ {N m−2 xl , l = 1, 2, ..., k1 + k2 } in such a way that this set forms a basis for R(N m−2 ) in such a way that N m−1 xl = 0, l = k1 + 1, ..., k1 + k2 . Again, {N m−1 xl : l = 1, 2, ..., k1 } ∪ {N m−2 xl : 1 ≤ l ≤ k1 + k2 } ∪ {N m−3 xl , l = 1, 2, ..., k1 + k2 } forms a linearly independent set and hence can be extended to a set {N m−1 xl : 1 ≤ l ≤ k1 } ∪ {N m−2 xl , l = 1, 2, ..., k1 + k2 } ∪ {N m−3 xl : 1 ≤ l ≤ k1 + k2 + k3 } in such a way that this set forms a basis for R(N m−3 ) in such a in this way, we way that N m−2 xl = 0, l = k1 +k2 +1, ..., k1 +k2 +k3 . Continuing ∪m−2 get a linearly independent set {N xl : 1 ≤ l ≤ k1 + ... + km−2 } ∪ r=1 {N m−r xl : 1 ≤ l ≤ k1 + ... + kr } which can be extended to a basis m−1 ∐

{N m−r xl : 1 ≤ l ≤ k1 + ... + kr }

r=1

for R(N ) in such a way that N m−p xl = 0, l = k1 +..+kp +1, ..., k1 +...+kp+1 , p = 1, 2, ..., m − 2. Finally, we can choose linearly independent vectors xl , l = k1 + ...+km−1 +1, ..., k1 +...+km in the nullspace of N so that these vectors along with the vectors N m−p−1 xl , l = k1 + ... + kp + 1, ..., k1 + ... + kp+1 , p = 1, 2, ..., m − 2 form a basis for the nullspace of N . It is easily seen that the set of vectors ∪m−1 {xl : k1 + ...km−1 + 1, ..., k1 + ... + km } ∪ r=1 {N m−r−1 xl : 1 ≤ l ≤ k1 + ... + kr } forms a basis for the entire vector space Cn and that the matrix of N relative to this basis has the standard Jordan canonical form for a nilpotent matrix. By combining this result with the primary decomposition theorem, the Jordan

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canonical form for an arbitrary complex square matrix is derived. Recall the primary decomposition theorem: If T is a linear operator over an N dimensional complex vector space V with minimal polynomial p(t) = Πrk=1 (t − λk )mk , mk ≥ 1 so that λk , k = 1, 2, ..., r are the distinct eigenvalues of T , then V =

r ⊕

N ((T − λk )mk )

k=1

and N ((T − λk )mk ) coincides precisely with the set of all vectors x for which (T − λk )m = 0 for some m ≥ 1.

5.45

Some topics in scattering theory in L2 (Rn )

∑n H0 = P 2 = a=1 Pa2 is the projectile Hamiltonian and V (Q), Q = (Q1 , ..., Qn ) is the interaction potential. We write P = −i∇Q = (P1 , ..., Pn ). Let f ∈ L2 (Rn ). Let ∫ f˜(k) = f (q)exp(ik.q)dn q/(2π)n/2 denote its Fourier transform. We write dn k = |k|n−1 d|k|dΩ(kˆ) with obvious meanings. Let

λ = |k|2 = k 2

λ represents the kinetic energy H0 of the projectile. We write with kˆ = ω ∈

S n−1 ,

√ f˜(k) = f˜(|k|ω) = f˜( λω)

Then, 2

|| f || =|| ˜ ||f ||2 =

∫

√ √ λ(n−1)/2 d λdω|f˜( λω)|2

where dω = dΩ(kˆ) Thus,

√ || f ||2 = int(1/2)λ(n−2)/2 |f˜( λω)|2 dλdω

For λ > 0, define a map Uλ : L2 (Rn ) → L2 (S n−1 ) by √ √ (Uλ f )(ω) = (1/ 2)λ(n−2)/4 f˜( λω) Then, the above formula can be expressed as ∫ ∫ || f ||2 = || Uλ f ||2 dλ = |(Uλ f )(ω)|2 dωdλ

General Relativity and Cosmology with Engineering Applications Define

235

√ η(λ) = (1/ 2)λ(n−2)/4

It is easy to see that Uλ∗ : L2 (S n−1 ) → L2 (Rn ) is given by (Uλ∗ ψ)(q) = ψ1 (q) where

∫

∫ ¯ψ1 (√μω)φ˜(√μω)η(μ)dμdω = ¯˜ψ1 (k)φ˜(k)dn k ˜ ∫ ∫ n ¯ = ψ1 (q)φ(q)d q = ψ¯(ω)(Uλ φ)(ω)dω ∫ √ = ψ¯(ω)η(λ)φ˜( λω)dω

Thus we must have

√ ¯ ¯ ˜ψ1 ( μω)δ(μ − λ) = ψ(ω)

Equivalently,

√ ψ˜1 ( μω)δ(μ − λ) = ψ(ω)

Thus, formally, we can write √ ψ˜1 ( μω) = ψ(ω)/δ(μ − λ) In particular, we must have √ √ ψ˜1 ( λω) = 0, ψ˜1 ( μω) = ∞, μ /= λ We now derive some other interesting formulae in scattering theory. Let W (q) be a function of the position variables only. Consider f (q) ∈ L2 (Rn ). Define the operator MW (λ) : L2 (Rn ) → L2 (S n−1 ) by (MW (λ))f (ω) = (Uλ W f )(ω) ∫ √ ˜ ( λω − k)f˜(k)dn k = η(λ) W ∫ = Kλ (ω, k)f˜(k)dn k where

√ ˜ ( λω − k) Kλ (ω, k) = η(λ)W

It follows that the Hilbert-Schmidt norm of MW (λ) is given by ˜ ||2 || MW (λ) ||2HS = η(λ)2 Θ || W = η(λ)2 Θ || W ||2 Let us find the adjoint of MW (λ). For f ∈ L2 (Rn ) and g ∈ L2 (S n−1 ), ∫ < g, MW (λ)f >= g¯(ω)Kλ (ω, k)f˜(k)dn kdω

236

General Relativity and Cosmology with Engineering Applications ∫ =

g¯(ω)Kλ (ω, k)exp(ik.q)f (q)dn kdn qdω/(2π)n/2

from which, it immediately follows that ∫ ¯ λ (ω, k)exp(−ik.q)g(ω)dn kdω/(2π)n/2 (MW (λ)∗ g)(q) = K This means that the kernel ofMW (λ)∗ is given by ∫ ¯ λ (ω, k)exp(−ik.q)dn k MW (λ)∗ (q, ω) = (2π)−n/2 K Now, H = H0 + V (Q) = P 2 + V (Q) is the Hamiltonian of the projectile taking into account its interaction with the scattering centre. Let E0 (dλ) denote the spectral measure of H0 and E(dλ) that of H. Then, the scattering matrix at energy λ Sλ has the representation Sλ = I + Rλ where Rλ dλ = 2πiE0 (dλ)(V − V (H − λ)−1 V )E0 (dλ) With ω denoting the initial direction of the projectile momentum before scat­ tering and ω ' the final direction of the projectile momentum after scattering, we thus obtain the following kernel for Rλ as a mapping from L2 (S n−1 ) into itself. Let V (Q) = U (Q)W (Q). Then, (2πi)−1 Rλ = MU (λ)(I − W (Q)(H − λ)−1 W (Q))MU (λ)∗ Reference:W.O.Amrein, ”Hilbert Space Methods in Quantum Mechanics”.

5.46

MATLAB problems on Applications of Lin­ ear Algebra to Signal Processing

[1] Generate an n × p real matrix X of random numbers with n > p. Calculate the orthogonal projection PX = X(X T X)−1 X T Verify by taking sample vectors w ∈ Rn , z ∈ Rp that || w − PX w ||≤|| w − Xz ||

General Relativity and Cosmology with Engineering Applications

237

[2] Write a program to Gram-Schmidt orthonormalize the set of p vectors x1 , ..., xp that are the columns of X in Problem [1]. Denote the resulting vectors by e1 , ..., ep . Verify that eTi ej = δij . Verify that PX =

p ∑

ei eTi

i=1

where the LHS is computed as in Problem [1]. [3] Verify the projection operator update formula: If x1 , ..., xp are the columns of the n × p matrix X having full column rank p as in Problem [1] and xp+1 is ˜ = [X, xp+1 ]. Then the another n × 1 vector that is not in R(X), the define X update formula is PX˜ = PX + PPX⊥ xp+1 or equivalently, PX˜ ξ = PX ξ + where

⊥ xp+1 ⊥ ξ T PX P xp+1 ⊥ || PX xp+1 ||2 X

⊥ = In − PX PX

and ξ ∈ Rn is arbitrary. Let Y = [x0 |X|xp+1 ], [X|xp+1 ] = U, [x0 |X] = V where X ∈ Rn×p is as in Problem [1] and the columns of Y are linearly inde­ pendent, ie, Y has full column rank. Show that ⊥ x0 − P{PX ⊥xp+1 } x0 PU⊥ x0 = PX ⊥ = PX x0 −

⊥ xp+1 ⊥ xT0 PX P xp+1 , ⊥ || PX xp+1 ||2 X

⊥ PV⊥ xp+1 = PX xp+1 − P{PX⊥ x0 } xp+1 ⊥ = PX xp+1 −

⊥ x0 ⊥ xTp+1 PX P x0 ⊥ || PX x0 ||2 X

Explain these results in terms of lattice filters for order updates of forward and backward prediction errors of a process and verify these results using MATLAB simulations. [4] Given an (n + 1) × (n + 1) matrix A in the following block structured form ( ) A1 b A= cT d where A1 ∈ Cn×n , b, c ∈ Cn×1 , d ∈ C

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General Relativity and Cosmology with Engineering Applications

1 evaluate A−1 in terms of A− and b, c, d. Verify this ”matrix inverse update 1 formula” by MATLAB examples. Now apply this to construct time and order updates for the problem of estimating h0 , h1 , ..., hp as h0 (n, p), ..., hp (n, p) chosen so that p n ∑ ∑ λn−r || yn − hk z −k xn ||2 k=0

k=0

is minimized. Here, xn = [x[n], x[n − 1], ..., x[0]]T , z −k xn = [x[n − k], x[n − k − 1], ..., x[0], x[−1], ..., x[−k]]T , k = 1, 2, ..., p and x[n] = 0f orn < 0, yn = [y[n], y[n − 1], ..., y[0]]T

5.47

Applications of the RLS lattice algorithms to general relativity

The approximate linearized geodesic equations around a constant four velocity V μ are given by dδxμ (τ ) = δv μ (τ ), dτ dδv μ (τ ) = −2Γμαβ (V τ )V α δv β (τ ) dτ These constitute a set of linear differential equations with time varying coeffi­ cients −2Γμαβ (V τ ). The aim is to estimate the Christoffel field and hence the first order partial derivatives of the metric tensor. This can be done approxi­ mately as follows: Suppose that the metric gμν (x, θ) depends on the space-time point xμ and a set θ ∈ Rp of vector parameters. Then we can write Γμαβ = Γμαβ (x, θ) ≈

Γμαβ (x, θ0 )

+ δθk

∂Γμαβ (x, θ0 ) ∂θk

where θ0 is our guess value for θ and δθ is the guess error θ − θ0 to be estimated. We assume that our guess value of the parameter is a good approximation so that δθ can be taken as small. Now, the above differential equations can be approximated by the following difference equation: δxμ [n + 1] − δxμ [n] − Δ.δv μ [n] = 0, μ δv μ [n + 1] − δv μ [n] + 2Δ(Γμαβ (nV Δ, θ0 ) + δθk Fkαβ (nV Δ, θ0 ))V α δv β [n] = 0

General Relativity and Cosmology with Engineering Applications where μ Fkαβ =

239

∂Γμαβ

∂θk This gives us an LIP linear difference equation, where LIP stands for ”Linear in Parameters”. The RLS lattice algorithm can be immediately applied to estimate δθk , k = 1, 2, ..., p based on the data δxμ [n], n = 0, 1, ..., N, .... We leave it as a problem to the interested reader to work out the details. Denoting by δk [N, p], k = 1, 2, ...., p respectively the estimates of δθk , k = 1, 2, ..., p based on the measured data δxμ [n], n = 0, 1, ..., N The RLS lattice algorithm will tell us how to arrive at δθk [p + 1, N ], k = 1, 2, ..., p + 1 from δθk [p, N ], k = 1, 2, ..., p and at δθk [p, N + 1], k = 1, 2, ..., p from δθk [p, N ], k = 1, 2, ..., p and the extra data δxμ [N + 1] or more precisely δxμ [N + 1 − k], k = 0, 1, ..., p.

5.48

Knill-Laflamme theorem on quantum cod­ ing theory, a different proof

Let |ψk >, k = 0, 1, ..., p be an onb for the code subspace C. Let N be the noise subspace of operators in the given Hilbert space. Choose E1 , ..., Eq ∈ N so that Ea |ψ0 >, a = 1, 2, ..., q forms an onb for N |ψ0 >. Then it is clear that since < ψ0 |Eb∗ Ea |ψ0 >=< ψk |Eb∗ Ea |ψk > for all k = 1, 2, ..., p and all a, b (by the assumptions of the Knill-Laflamme theorem), it follows that Ea |ψk >, a = 1, 2, ..., q is an onb for N |ψk > for each k = 0, 1, 2, ..., p. Note that the map E|ψ0 >→ E|ψk > for E ∈ N is a unitary isomorphism between N |ψ0 > and N |ψk > for each k by virtue of the assumptions of the theorem, namely that for any N1 , N2 ∈ N , < ψk |N2∗ N1 |ψk > does not depend on k. We therefore get the result that Ea |ψk >, k = 0, 1, 2, ..., p, a = 0, 1, ..., q are (p + 1)q orthonormal vectors in the underlying Hilbert space H. Note that by the assumptions of the ∗ |Eb∗ Ea |ψk >= 0 whenever m = / k. Let H0 denote Knill-Laflamme theorem, < ψm the span of all the (p + 1)q orthonormal vectors {Ea |ψk >: a = 1, 2, ..., q, k = 0, 1, 2, ..., p}. In other words, these vectors form an onb for H0 . Then it follows that we can define operators Rj , j = 1, 2, ...l, in H such that Rj Ea |ψk >= λjk |ψk >, a = 1, 2, ..., q, k = 0, 1, ..., p, j = 1, 2, ..., l and = 0, j = 1, 2, ..., l Rj |H⊥ 0 We then have ¯ jm λjk < ψm |ψk >= λ ¯ jm λjk δmk < ψm |Eb∗ Rj∗ Rj Ea |ψk >= λ Summing over j gives us < ψm |Eb∗ (

l ∑ j=1

Rj∗ Rj )Ea |ψk >= μmm δmk

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General Relativity and Cosmology with Engineering Applications

where μmk =

l ∑

¯ jm λjk = (Λ∗ Λ)mk λ

j=1

We choose μmm = 1, m = 0, 1, ..., p This is possible for example by choosing l ≥ p + 1 and Λ ∈ Cl×(p+1) so that

Λ∗ Λ = Ip+1 . It then easily follows that (

l ∑

Rj∗ Rj )|H0 = IH0 ,

j=1

and (

l ∑

=0 Rj∗ Rj )|H⊥ 0

j=1

Define Rl+1 in H by

= IH ⊥ Rl+1 |H0 = 0, Rl+1 |H⊥ 0 0

Then, it easily follows that l+1 ∑

Rj∗ Rj = I

j=1

in H. Further, for a, b = 1, 2, ..., q, we have l+1 ∑

(Rj Ea |ψk >< ψk |Eb∗ Rj∗ )

j=1

=

l ∑

Rj Ea |ψk >< ψk |Eb∗ Rj∗

j=1

=

l ∑

|λjk |2 |ψk >< ψk | = μkk |ψk >< ψk | = |ψk >< ψk |

j=1

showing that {Rj }l+1 j=1 are recovery operators for the noise subspace spanned by Ea , a = 1, 2, ..., q when the code subspace is C. Now let E ∈ N be arbitrary. Then E|ψk > is expressible as a linear combination of Ea |ψk >, a = 1, 2, ..., q and hence we get from the above equation that l ∑

Rj E|ψk >< ψk |E ∗ Rj∗ = |ψk >< ψk |

j=1

which proves that C is an N error correcting code.

General Relativity and Cosmology with Engineering Applications

5.49

241

Ashtekar’s quantization of gravity

Ashtekar introduced the su(2) connection field in which the curvature of the con­ nection can be expressed in the Yang-Mills formalism. Essentially, this means that we start with a Lie group G, construct its Lie algebra g and then define }-valued gauge fields Aμ (x) = Abμ (x)τb where τa , a = 1, 2, ..., N are generators of g with structure constants C(abc): [τa , τb ] = C(abc)τc The curvature of this connection is given by Fμν = [∂μ + Aμ , ∂ν + Aν ] = Aν,μ − Aμ,ν + [Aμ , Aν ] a = Fμν τa a The components of this curvature Fμν are linearly transformed versions of the Riemann-Christoffel curvature tensor obtained by using the gravitational spin connection. Unlike the Yang-Mills field, however, the Lagrangian density for the gravitational field is the scalar curvature R, ie, a linear function of the √ a , multiplied by the invariant scalar density −g. More precisely, curvature Fμν a the gravitational connection Aμ (x) must be expressed as a quadratic function of the tetrad field Vμa (x) and its covariant derivatives. An example of this was already found earlier when we introduced Dirac’s relativistic wave equation for V = ((Vμa )). Thus, the an electron in curved space-time. −g = det(V )2 where √ Lagrangian density of the gravitational field, namely R −g assumes the form F.det(V ) where F is a quadratic function of the tetrad. The canonical position fields which are Aaμ are quadratic functions of the tetrad and the canonical momenta are derived by considering partial derivatives of the Lagrangian density for the gravitational field w.r.t. the partial derivatives of the Lagrangian density w.r.t. the position fields. This formalism of quantum gravity is usually called ”Loop quantum gravity” for the reason that the connection components are the position fields and parallel transport w.r.t the connection around a closed loop gives the curvature tensor. The canonical approach however regards expressing the action Lagrangian density of the gravitational field as a function of qab , 1 ≤ a ≤ b ≤ 3, N a , a = 1, 2, 3 and N where we embed a three dimensional time dependent spatial surface Σt at time t with coordinates xa , a = 1, 2, 3 inside our four dimensional space-time with coordinates X μ . Let g˜μν denote the metric w.r.t. the space-time coordinates (xa , t) of the embedded surface. Then, μ ν g μν = g˜αβ X,α X,β μ ν μ ν = g˜ab X,a X,b + g˜00 X,0 X,0 μ μ ν +˜ g a0 (X,a X,ν0 + X,a X,0 )

The idea is to choose the coordinates xa that define the embedded surface Σt so that if we write μ μ = N nμ + N a X,a X,0

242

General Relativity and Cosmology with Engineering Applications μ , a = 1, 2, 3, ie, where N a is chosen so that nμ is orthogonal to X,a μ ν gμν (X,μ0 − N a X,a )X,b =0

then the following decomposition occurs: μ ν g μν = g˜ab X,a X,b + nμ nν

where g˜μν nμ nν = 1. Indeed, we have from the above on setting μ N μ = N a X,a ,

that μ μ ν ν g μν = g˜ab X,a X,b +˜ g 00 (N nμ +N μ )(N nν +N ν )+˜ g a0 (X,a (N nν +N ν )+X,a (N nμ +N μ )

μ ν = g˜ab X,a X,b + g˜00 N 2 nμ nν + g˜00 N (nμ N ν + nν N μ ) μ ν μ ν ν μ N ν + X,a N μ ) + g˜a0 N (X,a n + X,a n ) +˜ g a0 (X,a

+˜ g a0 N μ N ν To get the above decomposition, we therefore require that μ g˜00 N μ + g˜a0 X,a =0

or equivalently,

mu μ + g˜a0 X,a =0 g˜00 N a X,a

This is in turn equivalent to showing that g˜00 N a + g˜a0 = 0 − − − (a) But we already know that μ ν gμν (X,μ0 − N a X,a )X,b =0

ie, g˜0b − g˜ab N a = 0 − − − (b) From (b), we get g0b − g˜ab N a ) = 0 g˜bc (˜ or equivalently,

−g˜0c g˜00 − (δac − g˜c0 g˜a0 )N a = 0

or g˜00 g˜c0 + N c − g˜c0 g˜a0 N a = 0 or g00 − g˜a0 N a ) + N c = 0 − − − (b1) g˜c0 (˜

General Relativity and Cosmology with Engineering Applications From (b), we also get or

243

g0b − g˜ab N a ) = 0 g˜0b (˜ (1 − g˜00 g˜00 ) + g˜00 g˜0a N a = 0

This is the same as g0a N a − g˜00 ) + 1 = 0 − − − (b2) g˜00 (˜ From (b1) and (b2), we easily deduce g˜00 N c + g˜c0 = 0 which is precisely (a). We then define our ten canonical position variables to be qab = g˜ab , N a , N Also let μ ν X,b q μν = q ab X,a

where

((q ab )) = ((qab ))−1

It is easy to show that qab = g˜ab Then we have the decomposition g μν = q μν + nμ nν Define Kμν = qμρ qνσ ∇μ nν Note that q μν nν = 0 We claim that Kμν = Kνμ To see this, we first note that nμ is a unit normal to the surface Σt by its very construction, ie, μ μ = N nμ + N μ = N nμ + N a X,a X,0 where N a has been chosen so that μ μ ν − N a X,a )X,b =0 gμν (X,0

or equivalently, g˜0b = g˜ab N a

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General Relativity and Cosmology with Engineering Applications

and N has been chosen so that gμν nμ nν = 1. We now observe that nμ is the unit normal to the spatial surface Σt and hence, can be expressed as the normalized gradient of a scalar field ψ: nμ = F φ,μ where

F = (g μν φ,μ φ,ν )−1/2

We then have ∇ν nμ = nμ,ν − Γρμν nρ and by the symmetry of the Christoffel symbols, ∇ν nμ − ∇μ nν = nμ,ν − nν,μ = F,ν φ,μ − F,μ φ,ν = (logF ),ν nμ − (logF ),μ nν and hence qρμ qσν (∇ν nμ − ∇μ nν ) = 0 where we have used q μν nν = 0, qνμ nμ = 0 This proves that Kμν = Kνμ Note: qνμ = gνρ q μρ = qνρ q μρ since gνρ = qνρ + nν nρ is an orthogonal decomposition as a sum of a tensor tangential to Σt , ie, a spatial tensor and a tensor normal to Σt . Now we define μ ν X,b Kμν Kab = X,a

Then we get Kab = Kba Now we look at the problem of decomposing the curvature tensor into a spatial part, ie, tangential to Σt and a normal part, ie, normal to Σt . To this end, first let uμ be a spatial tensor, ie, nμ uμ = 0. Then, we may define its spatial covariant derivative by Dν uμ = qνα qμβ ∇α uβ Clearly, this is a spatial tensor, ie, nν Dν uμ = 0, nμ Dν uμ = 0

General Relativity and Cosmology with Engineering Applications

245

We now calculate Dρ Dν uμ = qρβ qνσ qμα ∇β Dσ uα The spatial curvature tensor is defined by α uρ = Dμ D ν uρ − D ν D μ uρ Sμνρ α where uμ is a spatial vector. Clearly, Sμνρ is the ideal formula for the spatial curvature tensor since the curvature tensor is defined by α Aα = ∇μ ∇ν Aρ − nablaν ∇μ Aρ Rμνρ

5.50

Example of an error correcting quantum code from quantum mechanics

Given any quantum system with Hamiltonian H, let |ψk >, k = 1, 2, ... denote an onb for the underlying Hilbert space consisting of eigenvectors of H. Define operators Ea , Fa , a = 1, 2, ..., q in the underlying Hilbert space H such that Ea |ψk >= |ψN (a−1)+k >, a = 1, 2, ..., q, k = 1, 2, ..., N Fa |ψk >= |ψq(k−1)+a >, k = 1, 2, ..., N, a = 1, 2, ..., q and Ea , Fa acting on |ψk > for k > N give zero. Then {Ea |ψk >: 1 ≤ a ≤ q, 1 ≤ k ≤ N } and {Fa |ψk >: 1 ≤ a ≤ q, 1 ≤ k ≤ N } are both orthonormal sets each having kN elements. Therefore, if we define noise spaces as N = span{Ea : a = 1, 2, ..., q} M = span{Fa : a = 1, 2, ..., q} then it is easy to see using the Knill-Laflamme theorem that C = span{|ψk >: k = 1, 2, ..., N } is an N correcting quantum code as well as an M correcting quantum code. An example can be found using the general relativistic KleinGordon Hamiltonian √ p0 = H = hr pr + m2 /g00 + pr γ rs ps where

hr = −g r0 /g 00 , γ rs = (g 0r g 0s − g rs )/g002 , pr = i∂r

which is obtained by solving the quadratic equation m2 = g μν pμ pν = g 00 p20 + 2g 0r pr + g rs pr ps and then Hermitianizing the solution.

246

5.51

General Relativity and Cosmology with Engineering Applications

An application of the Jordan Canonical form to noisy quantum theory

Consider the qsde dU (t) = (−(iH + P )dt + LdA(t) − L∗ dA∗ (t))U (t) where P = LL∗ /2 We can get a Dyson series expansion for U (t) by using the following integral version of this qsde: U (t) = U0 (t)W (t), U0 (t) = exp(−(iH + P )t) W ' (t) = (L1 (t)dA(t) − L2 (t)dA(t)∗ )W (t) = L1 (t)W (t)dA(t) − L2 (t)W (t)dA(t)∗ where L1 (t) = U0 (−t)LU0 (t), L2 (t) = U0 (−t)L∗ U0 (t) Now A = (iH + P ) is not a normal matrix in general and hence not generally diagonable w.r.t an onb or w.r.t any basis. Then we would have to use the Jordan canonical form of A to compute U0 (t).

5.52

An algorithm for computing the Jordan canon­ ical form

Let A be an n × n complex matrix. Compute all its distinct eigenvalues as roots of det(z − A). Denote these eigenvalues by λk , k = 1, 2, ..., r. Pick an eigenvalue, say λ1 . Compute a basis {e1 , ..., em } for N (λ1 − A). Let e11 = e1 and compute vectors e12 , ..., e1l such that (A − λ1 )e1,j+1 = e1j , j = 1, 2, ..., l − 1 where l is such that (A − λ1 )e1,l+1 = e1,l has no solution for e1,l+1 . Note that e1k k = 1, 2, ..., l are linearly independent and hence the process has to terminate. Exercise:Prove that e1k , k = 1, 2, ..., l are linearly independent.

General Relativity and Cosmology with Engineering Applications

5.53

247

Rotating blackhole analysis using the tetrad formalism

Problem: Consider a rotating blackhole with metric dτ 2 = P (r, θ)dt2 − B(r, θ)dr2 − C(r, θ)dθ2 − D(r, θ)(dφ − ω(r, θ)dt)2 Write down the Einstein field equations Rμν = 0 for this metric and derive the Kerr solution for P, B, C, E, ω. Equivalently, using Cartan’s equations of structure, write down the Einstein field equations in the tetrad basis √ √ √ √ e0 = P dt, de1 = Bdθ, e2 = Cdθ, e3 = D(dφ − ωdt) and solve for P, B, C, ω. Note that the metric is diagonal in this tetrad basis: g = e0 ⊗ e0 −

3 ∑

e r ⊗ er

r=1

5.54

Maxwell’s equations in the rotating blackhole metric

Write down the Maxwell equations in the Kerr metric of the previous problem assuming azimuthal symmetry. hint:The four vector potential is assumed to have the form A0 = A0 (r, θ), A1 = A1 (r, θ), A2 = A2 (r, θ), A3 = A3 (r, θ) ie, Aμ = Aμ (r, θ) In other words, the four vector potential does not depend on t, φ. We calculate the components of the antisymmetric field tensor. First note that the compo­ nents of the metric tensor are g00 = P − Dω 2 , g11 = −B, g22 = −C, g33 = −D, g03 = Dω = g30 and an all the other covariant components of the metric tensor vanish. We have the following expressions for the covariant components of the electromagnetic four potential: A0 = g00 A0 + g03 A3 = (P − Dω 2 )A0 + DωA3 , A1 = g11 A1 = −BA1 , A2 = g22 A2 = −CA2 , A3 = g30 A0 +g33 A3 = DωA0 −DA3

248

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General Relativity and Cosmology with Engineering Applications

Some notions on operators in an infinite/finite dimensional Hilbert space

Let T be a Hermitian operator in H. This means that T is densely defined and T ∗ = T . In other words, Cl(D(T )) = H, and T ∗ = T . Note that for any operator A in H, A∗ is uniquely defined by the requirement < Ax, y >=< x, A∗ y >, x ∈ D(A), y ∈ D(A∗ ) iff D(A) is dense in H, ie, iff A is densely defined. We can then choose D(A∗ ) as the set of all y ∈ H for which there exists z ∈ H such that < Ax, y >=< x, z > for all x ∈ D(A). In this case, we set A∗ y = z. Since D(A) is assumed to be dense, then the condition < Ax, y >=< x, z >=< x, z ' >, x ∈ D(A) implies < x, z − z ' >= 0, x ∈ D(A) implies < x, z − z ' >= 0, x ∈ H implies z = z ' showing that A∗ is uniquely defined. If D(A) were not dense in H, then D(A)⊥ /= {0} and we could replace z by z + u for any u ∈ D(A)⊥ without affecting the equation < Ax, y >=< x, z >, x ∈ D(A) and we would get a different definition for A∗ . Note that if A is densely defined, then A∗ is uniquely defined but may not be densely defined. An operator A in H is said to be closed if Gr(A) = {(x, Ax) : x ∈ D(A)} is closed in H × H, ie xn ∈ D(A), xn → x, Axn → y implies x ∈ D(A) and y = Ax. An operator A in H is said to be closable if Cl(Gr(A)) is also the graph of a linear operator in H. In other words, if there exists an operator B in H such that xn ∈ D(A) is a sequence such that xn → x, Axn → y imply y = Bx. It is easy to verify that this happens iff xn ∈ D(A) and xn → 0, Axn → y imply y = 0. Indeed the ”only if” part is immediate. The ”if” part is verified as follows. Let (x, y) ∈ Cl(Gr(A)). Then, there exists a sequence xn ∈ D(A) such that xn → x, Axn → y. We define Bx = y. B is a well defined linear operator in H for the following reason. Suppose x'n → x, Ax'n → z. Then xn − x'n → 0, A(xn − x'n ) → y − z. Hence, by hypothesis, y − z = 0, ie, z = y, proving that B is well defined. If A is closable, then we define its closure A¯ as the unique operator in H for which Cl(Gr(A)) = Gr(A¯) It is easily verified that A¯ is the smallest closed extension of A, ie, if B is any other operator such that A ⊂ B (ie, D(A) ⊂ D(B), B|D(A) = A) and B is closed, then A¯ ⊂ B. Now define the Hilbert space isomorphism P :H×H→H×H by P (x, y) = (y, −x) Note that H × H is the same as H ⊕ H. Let A be densely defined. Then we have < (x, Ax), (u, v) >=< x, u > + < Ax, v >= 0∀x ∈ D(A)

General Relativity and Cosmology with Engineering Applications iff

249

v ∈ D(A∗ ), A∗ v = −u

iff

(v, −u) ∈ Gr(A∗ )

Hence,

Gr(A)⊥ = P.Gr(A∗ )

or equivalently,

Gr(A∗ ) = −P.(Gr(A)⊥ )

which proves that Gr(A∗ ) is closed and hence A∗ is a closed operator. In other words, we have proved that the adjoint of any densely defined operator is closed. If A is a densely defined operator such that there exists a closed operator B satisfying A ⊂ B, then A is closable. Indeed, in this case, we have xn ∈ D(A), xn → 0, Axn → y imply Bxn = Axn → y and hence y = B.0 = 0 proving the claim. As an example, suppose A is a symmetric operator, ie, A ⊂ A∗ . Then since A∗ is closed as shown earlier, we have that A is closable. In other words, any symmetric operator is closable. We have from the above for any densely defined operator A such that A∗ is also densely defined (For example a symmetric operator), ¯ Gr(A) = (Gr(A)⊥ )⊥ = (P.Gr(A∗ ))⊥ = P.Gr(A∗ )⊥ = Gr(A∗∗ ) and hence A is a closable operator with A ⊂ A¯ ⊂ A∗∗ Now suppose A is a symmetric operator. Then, we have seen that A is clos­ able. We say that A is essentially selfadjoint if A¯ is selfadjoint (Selfadjoint and Hermitian mean the same). Suppose A is essentially self-adjoint. Then (A¯)∗ = A¯ But then we have Gr(A¯) = Cl(Gr(A)) and so

−P.Gr(A¯) = −P.(Gr(A)⊥ )⊥ = (−P.Gr(A)⊥ )⊥ Gr(A∗ )⊥

or equivalently, since A∗ is closed, Gr(A∗ ) is closed and hence, Gr(A∗ ) = −P.Gr(A)⊥ On the other hand, Gr(A¯) = Gr((A¯)∗ ) = −P.Gr(A¯)⊥ = −P.Cl(Gr(A))⊥

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General Relativity and Cosmology with Engineering Applications = −P.Gr(A)⊥

Combining these two equations gives us Gr(A¯) = Gr(A∗ ) which implies that

A¯ = A∗

It follows from the self-adjointness of A¯ that A∗∗ = A¯ Conversely suppose A is symmetric with A∗∗ = A¯. Then, we have A ⊂ A∗ so that

A∗∗ ⊂ A∗

and hence

A¯ ⊂ A∗

Thus, (P ⊂ Q implies Q∗ ⊂ P ∗ ) A¯ = A∗∗ ⊂ (A¯)∗ ie, A¯ is also symmetric. Now, Gr((A¯)∗ ) = Gr(A∗∗∗ ) = −P.Gr(A∗∗ )⊥ = Gr(A∗ )⊥ = P Gr(A¯) = P.(Gr(A)⊥ )⊥ = P Gr(A∗ )⊥ = Gr(A)⊥ )⊥ = Gr(A¯) so that

(A¯)∗ = A¯

ie, A is essentially self-adjoint. Thus, we have proved that a symmetric operator A is self-adjoint iff A∗∗ = A¯.

5.56

Some versions of the quantum Boltzmann equation

Let H = ⊗N a=1 Ha be the tensor product of N identical copies Ha , a = 1, 2, ..., N of a Hilbert space. The Hilbert space Ha is to be regarded as the Hilbert space of the ath particle and all the particles are identical so that the total Hamiltonian of this system can be expressed as H=

N ∑ a=1

Ha +

∑ 1≤a 0 by virtue of the invariance of the Hamiltonian under particle interchanges. ρ(t) satisfies the quantum Liouville, or Von-Neumann or mixed state version of the Schrodinger equation: iρ' (t) = [H, ρ(t)] It follows by taking partial traces that iρ'1 (t) = [H1 , ρ1 (t)] + (N − 1)T r2 [V12 , ρ12 (t)] iρ'12 (t) = [H1 + H2 + V12 , ρ12 (t)] + (N − 2)T r3 [V13 + V23 , ρ123 (t)] and more generally, iρ'123...r (t) = [H1 + ... + Hr +

∑

Vab , ρ123...r (t)]+

1≤a of photons, electrons and positrons having definite four momenta and spin/polarizations and likewise a final state |f >. These initial and final states can be obtained by acting on the vacuum appropriate creation operators, then writing the interaction terms in the total Hamiltonian between photon field and electron current as∫ integrals expanding the time or­ ∞ dered exponential U (∞, −i∞) = T {exp(−i −∞ H(t' )dt' )} in the interaction ∫ picture as a power series in the interaction energy − J μ (x)Aμ (x)d3 x where J μ (x) = −eψ ∗ (x)γ 0 γ μ ψ(x), then expressing ψ(x) and Aμ (x) in terms of cre­ ation and annihilation operators of the electron-positron and photon field and then using the standard commutation relations, evaluate the scattering ma­ trix elements < f |U (∞, − inf ty)|i > in the interaction picture. This operator formalism was developed by Schwinger-Tomonaga and Dyson while Feynman developed the path integral approach to such calculations with finally Dyson proving the equivalence of the theories of Feynman and Schwigner-Tomonaga. The importance of the propagator computation in quantum field theory has been highlighted in this book. Namely, if φ(x) is a collection of fields and the action for such a collection of fields is expressed as S[φ] = SQ [φ] + εSN Q [φ] where SQ is a quadratic functional and SN Q is a cubic and higher degree func­ tional, the latter coming from interactions, then the scattering matrix using FPI

General Relativity and Cosmology with Engineering Applications

261

can be expressed as ∫ S[φ∞ , φ−∞ ] = ∫ =

exp(iS[φ])Dφ

ex(iSQ [φ])(1 + iεSN Q [φ] − ε2 SN Q [φ]2 /2 + ...)Dφ

It is clear that each term in this infinite series is equivalent to calculating the moments of an infinite dimensional Gaussian distribution with complex variance and we know from basic probability theory, that even higher order moments of a Gaussian vector can be expressed as sums over products of the second order moments. The second order moment appearing here is the propagator of the field: ∫ Dφ [x, y] = exp(iSQ [φ])φ(x)φ(y)Dφ These aspects of quantum electrodynamics have been covered in this book. We give explicit computations of the photon propagator in different gauges (Feynman, Landau and Coulomb gauges) and also for the electron propaga­ tor. These expressions are derived using both the methods, first the operator theoretic method and second, using the Feynman path integral for quadratic La­ grangians combined with the standard formulas for the second order moments of a multivariate Gaussian distribution. We also present modern developments in quantum field theory, starting with the standard Yang-Mills group theo­ retic generalization of the Dirac or Klein-Gordon equations for matter fields interacting with the photon field. In this generalization, we assume that the wave function takes values in C4 ⊗ CN and a subgroup G of the unitary group U (N ) acts on this wave function. We introduce a Lie algebra theoretic co­ variant derivative which is an ordinary four gradient plus a connection field which takes values in the Lie algebra of the group G. This gauge covariant derivative acts on the matter field wave function and if the matter field wave function undergoes a local (ie space-time dependent) transformation g(x) ∈ G, then accordingly the gauge connection field in the covariant derivative has to undergo a certain transformation so that the covariant derivative of the mat­ ter field transforms simply by multiplication by g(x). This ensures that the Lagrangian density constructed out of functions of the matter field and its co­ variant derivative will be invariant under local G-transformations of the matter field and the gauge field provided that the Lagrangian is a G-invariant func­ tion of its agruments. Owing to the non-commutativity of the gauge fields, the gauge field tensor defined as the commutator of the covariant derivatives contains an extra quadratic nonlinear term in the gauge potentials apart from its four dimensional curl. This is a characteristic feature of non-Abelian gauge theories in contrast to the case of the electromagnetic field where the gauge group is U (1). In the electromagnetic field case, the matter field is the Dirac field since the gauge group is U (1), the local group transformation element g(x) is simply a modulus one complex number depending on the space-time vari­ ables. The corresponding gauge transformation of the gauge field potentials,

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General Relativity and Cosmology with Engineering Applications

namely, the electromagnetic four potential simplifies to the standard Lorentz gauge transformation Aμ → Aμ + ∂μ φ. Based on the Yang-Mills non-Abelian generalization of electromagnetism, Salam,Weinberg and Glashow were able to unify the electromagnetic forces and the weak forces, calling it the electro-weak theory. Their idea was to start with a non-massive vector gauge field Lagrangian corresponding to the weak forces in the nucleus, coupled to the matter field of Leptons (which include electrons) and then add symmetry breaking terms to this Lagrangian by coupling a scalar field to the Leptons. Symmetry breaking occurs when the scalar field is in its ground state which is a constant and this leads to mass terms involving quadratic non-derivative terms of the electronic field as well as to the gauge Bosonic fields which are components of the vector gauge field. In this context, we give a brief review of symmetry breaking both spontaneous and local versions. Symmetry breaking can occur when Hamilto­ nian/Lagrangian is invariant under a group G and the ground state is degener­ ate. If the Hamiltonian is invariant under G, then the subspace of ground states is invariant under G but any given ground state may not be invariant under G. If we take such a ground state and look upon the quantum state as this ground state plus a quantum perturbation, then the resulting Lagrangian will not be invariant under G since the ground state is not, but will be invariant under a subgroup H of G also called a broken subgroup. This sort of symmetry breaking produces massless particles called Goldstone Bosons. We can demonstrate this fact that symmetry breaking produces massless particles much better and in a more generalized framework using groupt theory. To do so, we first express the wave function of the system in terms of a unbroken H part and a broken G/H part. The unbroken part transforms according to H and the broken part can be viewed as a field with values in the coset space G/H. When the Lagrangian is expressed in terms of these components, it turns out that it does not contain any non-derivative quadratic components of the broken part while it contains non-derivative quadratic components of the unbroken part. This demonstrates that the broken part describes massless Goldstone Bosons while the unbroken part of the wave function describes massive particles. We also show that the Gsymmetry of a Lagrangian can be broken by adding perturbative terms that are not G-invariant. Sometimes symmetry breaking can lead to massless particles being massive as in the electroweak theory of Salam, Weinberg and Glashow. This happens because of the coupling of the gauge fields to a scalar field. The gauge fields initially are massless but the coupling to the scalar field followed by evaluation of the Lagrangian in the ground state of the scalar field causes terms in the Lagrangian involving quadratic non-derivative terms of the gauge field to be present. These terms cause the gauge fields to become massive. The elec­ troweak theory is an example of such a situation which gives masses to the gauge fields other than the electromagnetic field. All this is quantum field theory from the physical standpoint. We next explain how certain aspects of quantum field theory including the theory of quantum noise can be developed using rigorous mathematics, ie, functional analysis. In particular, we show how certain ma­ jor stochastic processes in classical probability theory like the Brownian motion and Poisson processes are special cases of quantum stochastic processes, ie, a

General Relativity and Cosmology with Engineering Applications

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family of non-commuting observables in a special kind of Hilbert space, namely the Boson Fock space when viewed in specific states. The notion of a quantum probability space as a triplet (H, P (H), ρ) where H is a Hilbert space, P (H) is the lattice of orthogonal projection operators in H and ρ is a state in H, ie, a positive semidefinite unit trace operator in H is introduced as compared to a classical probability space (Ω, F, P ). After doing so, we construct the Boson Fock space which can describe an arbitrary number of Bosons using the sym­ metric tensor product of a particular Hilbert space. The Boson Fock space is the substratum for constructing basic quantum noise processes, ie, noncommuting family of operators which specialize to Brownian motion and Poisson processes when the state is appropriately chosen. The Boson Fock space or noise Bath space is coupled to the system Hilbert space via a tensor product. Then we follow the marvellous approach of R.L.Hudson and K.R.Parthasarathy of con­ struting the creation, annihilation and conservation operator fields in the Boson Fock space. We show via physical arguments that the creation and annihilation operator fields can also be viewed fromt the standpoint of an infinite sequence of Harmonic oscillators, by constructing the creation and annihilation operator for each oscillator and defining the coherent vectors in terms of a superposion of the energy eigenstates of these oscillators and proving that the coherent states are eigenstates of the annihilation operator field now constructed as superpos­ tions of the annihilation operators for the different oscillators. The creation operator is the adjoint of the annihilation operator and turns out to have the same action on coherent vectors as the complex derivative of the latter with respect to the complex numbers used to construct the coherent vector from the oscillator energy eigenstates. In the work of Hudson and Parthasarathy, co­ herent vectors in the Boson Fock space were constructed as a weighted direct sum of multiple tensor products of a fixed vector in the Hilbert space with itself and the creation, conservation and annihilation operators were defined using the generators of one parameter unitary groups derived from the Weyl operator by restricting to translation and unitary rotation. The Weyl operator in the Hudson-Parthasrathy (HP) theory is itself described by its action on coherent vectors. Coherent vectors without normalization in the HP theory were called exponential vectors. This approach to the construction of the basic noise fields is highly mathematical and we provide in this book some physical insight into this correspondence by making an isomorphism between the coherent vectors of the HP theory and coherent vectors constructed using eignstates of harmonic oscil­ lators. Further, in our book, we construct the unitary rotation operators needed for constructing the conservation process by resorting to quadratic forms of the creation and annihilation operators of the harmonic oscillators. The theory of quantum stochastic calculus developed by Hudson and Parthasarathy introduces time dependent creation, annihilation and conservation processes which statisfy a quantum Ito formula for products of time differentials of these processes. The classical Ito formula for Brownian motion and the Poisson process is shown to arise as a special commutative version of the quantum Ito formula of HP. The HP theory says more, namely that in the general noncommutative case, the products of creation and annihilation operator differentials with conservation

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General Relativity and Cosmology with Engineering Applications

differentials need not be zero. This unlike the classical probability case where if B is Brownian motion and N is Poisson process, then dB.dN = 0. Our anal­ ogy of the HP theory with the quantum harmonic oscillator theory provides us with a direct route to quantum optics as described in the celebrated book by Mandel and Wolf on optical coherence and quantum optics. The idea is to first set up the famous Glauber-Sudarshan non-orthogonal resolution of the identity operator in the Boson Fock space as a complex integral of the coherent states |e(u) >< e(u)|, then to express the Hamiltonian of the system interacting with a photon bath as the sum of the system Hamiltonian which consists of spin matrices interacting with a constant classical magnetic field, the bath photon field Hamiltonian expressed as a quadratic form in the creation and annihila­ tion operators and an interaction Hamiltonian expressed as a time varying linear combination of the products(tensor) between the atomic spin observables and the bath annihilation and creation variables, then assume that the density of ρ(t) of the system and bath can be expanded as a Glauber-Sudarshan integral: ∫ ρ(t) = ρA (t, u) ⊗ |e(u) >< e(u)|du where ρA (t, u) is a finite dimensional matrix (of the same order as the spin observables of the atomic system). Finally, we substitute this expansion into the quantum equation of motion iρ' (t) = [HA + HF + HAF (t), ρ(t)] where HA is the atomic Hamiltonian, HF is the bath field Hamiltonian and HAF (t) is the atomic-bath field interaction Hamiltonian. Using properties of cre­ ation and annihilation operators acting on the exponential/coherent vectors and integration by parts, we then derive a partial differential equation for ρA (t, u) which may be called the fundamental equation of quantum optics. After these discussions, we proceed to one of the most modern techniques in time depen­ dent quantum measurement theory. The time dependent HP theory, ie quantum stochastic calculus also has a nice physical interpretation. When the average val­ ues of the creation, and annihilation processes in a coherent state are calculated, we get time integrals of the product of the coherent state defining vector with the creation/annihilation process defining vector upto time t. This result can be interpreted physically by saying that the annihilation (creation) process acts on a coherent state and yields the total amplitude and phase of photons annihi­ lated (created) upto time t taking into account the relative polarization of the photons in the coherent state with respect to that of the annihilation (creation) process defining vector. On the other hand, the conservation process average in a coherent state gives a time integral of a quadratic product of the coherent state defining vector upto time t which has the interpretation as being the number of photons in the state present upto time t. The conservation process in the HP cal­ culus has the Poisson process interpretation in classical probability theory just as the creation and annihilation process are linked to Brownian motion. The whole subject of quantum probability and quantum stochastic processes can be

General Relativity and Cosmology with Engineering Applications

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viewed as an linear-algebra-functional analytic approach to probability theory with generalizations to the non-commutative case. As an important application of the HP stochastic calculus, we present the celebrated work of V.P.Belavkin on quantum filtering. To do so, we first note that in quantum mechanics two or more observables may not be simultaneously measurable when they do not commute but when they commute, they can be simultaneously diagonalized and hence simultaneously measured. Measurement on a quantum system causes the state of the system to collapse to a state dictated by the outcome of the mea­ surement or of the measurement is made by a set of projection valued operators (pvm) or more generallyn by a set of positive operators (povm), then the state of the system collapses to a state formed by superposing the collapsed states corresponding to each measurement outcome. Belavkin proposed a scheme of constructing a filtration on an Abelian Von-Neumann algebra of observables that satisfies the non-demolition property, ie, the algebra generated by the elements of the filtration upto time t is Abelian and also commutes with the states of the HP noisy Schrodinger equation at times s ≥ t. Such measurements, he called non-demolition measurements. The HP noisy Schrodinger equation determines a unitary evolution in the joint system and bath space h ⊗ Γs (L2 (R+ ) ⊗ Cd ). The dynamics of the unitary evolution is dictated by the system Hamiltonian, the fundamental creation, annihilation and conservation processes of the HP quantum stochastic calculus which are operators in the bath space and con­ nect to the system dynamics via system operators. There is also a quantum Ito correction term to the system Hamiltonian in the form of an additive skew Hermitian operator that ensures unitarity of the evolution. It is known that if one computes the Heisenberg dynamics of a system observable using these unitary evolution operators, then the standard Heisenberg equations of motion are obtained along with noise correction terms which and the system observ­ ables after finite time t evolves to an observable in the tensor product of the system Hilbert space and the noise bath space. Specifically, if U (t) denotes the evolution operator of the HP noisy Schrodinger equation and X is a system observable, then after time t it evolves to jt (X) = U (t)∗ XU (t) which satisfies the noisy Heisenberg equations of motio. Belavkin proposed that if we take an input noise process Yin (t) which is a commuting family of operators in the bath space and define the output noise process Yout (t) = U (t)∗ Yin (t)U (t), then by virtue of the fact that the unitarity of U (t) depends only on system operators which commute with bath operators, it follows that Yout (t) = U (T )∗ Yin (t)U (T ) for all T ≥ t from which it is easy to see that Yout (t) commutes with Yout (s) for all t, s ≥ 0 and further that Yout (t) commutes with js (X) for all s ≥ t. This commutativity enables us to jointly measure jt (X), Yout (s), s ≤ t and hence de­ fine the conditional expectation πt (X) = E(jt (X)|Yout (s), s ≤ t). The family πt (X), t ≥ 0 of operators forms an Abelian family since πt (X) is a function of Yout (s), s ≤ t and the latter form a commuting family of operators. Belavkin de­ rived a stochstic differential equation for πt (X) driven by the process Yout (.) and its derivation was greatly simplified using quantum versions of the KallianpurStriebel formula and other methods based on orthogonality properties of the conditional expectation estimate by John Gough and Kostler. The resulting

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General Relativity and Cosmology with Engineering Applications

equation for πt (X) is the fundamental quantum filtering equation and is the non-commutative generalization of Kushner’s equation for classical non-linear filtering. These aspects have been discussed in this book. It should be noted that although this version of the filtering equations takes place in the observable domain, it could be directly transformed to the density domain. This is anal­ ogous to the situation of classical filtering where we can describe the evolution of the conditional moments of the state or more generally of the conditional expectation of any function of the state at time t given measurements upto time t or equivalently of the conditional probability density of the state at time t given measurements upto time t. To obtain the Belavkin stochastic differential equation for the conditoinal density of the state at time t given output measure­ ments upto time t, we must simply note that we can write πt (X) = T r(ρt X) where ρt can be viewed as a density matrix in the system Hilbert space that is a function of the output measurement process upto time t, or equivalently, simply as a classical random process with values in the space of system space density operators. Classical random because the measurement operators commute. We substitute πt (X) = T r(ρt X) into the Belavkin observable version of the filtering equation and using the arbitrariness of X, we derive a classical stochastic dif­ ferential equation for the system state valued classical random process ρt driven by Yout (t). Such equations are called ”Stochastic Schrodinger equations” (Luc Bouten, Ph.D thesis on quantum optics filtering and control). We present a generalization of Belavkin’s work by first constructing a family of p commuting inputmeasurement processes which are expressible as linear combinations of the creation, annihilation and conservation processes. Such processes have inde­ pendent increments in coherent states and the quantum Ito’s formula leads in general to the fact that any integer power of the differentials of such processes is non zero just as in the classical case (dN )k = dN, k = 1, 2, ... where N (.) is a Poisson process. The Belavkin filter is constructed by assuming it to have the form ∑ Gmkt (X)(dYmout (t))k dπt (X) = Ft (X)dt + k≥1,m=1,2,...,p

with Ft (X)andGmkt (X) being measurable with respect to the algebra generated by the output measurments upto time t. The coefficients Ft (X), Gmkt (X) are determined by applying the quantum Ito formula to the orthogonality equation E((jt (X) − πt (X))Ct ) = 0 where dCt =

∑

fmk (t)Ct (dYmout (t))k , t ≥ 0, C0 = 1

m,k

and using arbitrariness of the complex valued functions fmk (t). It should be noted that in the absence of any measurement, ie, when the system evolves according to the HP noisy Schrodinger equation, we can take a system observ­ able X, obtain its noisy evolution jt (X) and then choose a pure states of the form |fk ⊗ φ(u) >, k = 1, 2 of the system and the bath where |fk > is a system state and |φ(u) > is a bath coherent state, then compute the matrix elements

General Relativity and Cosmology with Engineering Applications

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< f1 ⊗ φ(u)|jt (X)|f2 ⊗ φ(u) > and write down the evolution of this quantity. More generally, we can compute the system state at time t defined via the duality equation T r(ρs (t)X) = T r((ρs (0) ⊗ |φ(u) < φ(u)|)jt (X)) or equivalently, as ρs (t) = T rB (U (t)(ρs (0) ⊗ |φ(u) >< φ(u)|)U (t)∗ ) or equivalently, as < f1 |ρs (t)|f2 >= T r(ρs (t)|f2 >< f1 |) = T r((ρs (0) ⊗ |φ(u) >< φ(u)|)jt (|f2 >< f1 |)) where |fk >, k = 1, 2 are system states. The resulting differential equation for the system state ρs (t) is the generalized GKSL equation (Gorini, Kossakowski, Sudarshan, Lindblad). If u = 0, ie, the bath is in vaccuum, then the ordinary GKSL equation is obtained. Using the dual of the GKSL equation, we get a description of the evolution of system observables when corrupted by bath noise in such a way that the system observable always remains a system observable, ie, averaging out over the bath noise variables is being performed at each time. The dual GKSL can be used to describe non-Hamiltonian quantum dissipative systems, for example, damped quantum harmonic oscillators or lossy quantum transmission lines. We then introduce the notion of quantum control of the Belavkin stochastic Schrodinger equation in the sense of Luc Bouten. This involves taking non-demolition Belavkin measurements on a quantum system evolving according to the HP noisy Schrodinger equation, then considering at each time point t, the Belavkin filtered and controlled state ρc (t), applying the Belavkin filter evolution by making a non-demolition measurement dY (t) in time [t, t + dt] to obtain the Belavkin filtered state at time t + dt as ρ(t + dt) = ρc (t) + Ft (ρc (t))dt + Gt (ρc (t))dY (t) c and then applying a control unitary Ut,t+dt = exp(iZdY (t)) in the time interval [t, t + dt] to the Belavkin filter output ρ(t + dt) to get the controlled state at time t + dt as c c∗ ρ(t + dt)Ut,t+dt ρc (t + dt) = Ut,t+dt

Here, Z is a system observable that commutes with dY (t). More precisely, Z should be replaced by U (t + dt)∗ ZU (t + dt) to make it commute with dY (t). We can then show by application of the quantum Ito formula that the system operator Z can be selected so that the evolution from ρc (t) to ρc (t + dt) is such that a major portion of the noise in the Belavkin filter is removed and the evolution of ρc (t) nearly follows that of the noiseless Belavkin equation, ie, the GKSL equation. The next topic discussed in this book is that of designing quantum gates using scattering theory experiments. The basic idea is to realize

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a unitary quantum gate using the quantum scattering matrix. The system consists of a free projectile having Hamiltonian H0 = P 2 /2m and the interaction potential energy between the projectile and the scattering centre is V (Q) so that the total Hamiltonian of the projectile interacting with the scattering centre is H = H0 + V . The projectile arrives from the infinite past (t → −∞) from the input free particle state |φin >. This free state evolves according to H0 , ie, at time t this free particle state is exp(−itH0 )|φin >. After interacting with the scatterer, it goes to the input scattered state |ψin > which evolves according to H, ie, at time t, this input scattered state is exp(−itH)|ψin >. It follows that these two states coincide in the remote past, ie, limt→−∞ (exp(−itH)|ψin > −exp(−itH0 )|φin >) = 0 from which, we easily deduce that |ψin >= Ω− |φin > where Ω− = slimexp(itH).exp(−itH0 ) After interacting with the scatterer for a sufficient long time, we ask the ques­ tion, what is the probability amplitude of the projectile being in the free particle state |φout >?. To evaluate this, we must first determine the scattered state, ie, the output scattered state |ψout > that develops as t → ∞ to |φout >. It is clear that the condition required for this is that limt→∞ (exp(−itH)|ψout > −exp(−itH0 )|φout >) = 0 or equivalently, |ψout >= Ω+ |φout > where Ω+ = slimt→∞ exp(itH).exp(−itH0 ) The domains of Ω− and Ω+ will not in general be the entire Hilbert space L2 (R3 ). In fact, determining the domains of definition of Ω± is a hard prob­ lem in operator theory and nice treatments of this delicate problem have been given in the masterful monographs of T.Kato (Perturbation theory for linear operators) and W.O.Amrein (Hilbert space methods in quantum mechanics). The scattering matrix S is an operator that maps free input particle states to free output particle states so that the scattering amplitude for the process |φin >→ |φout > is given by < φout |S|φin >=< ψout |ψin >= < φout |Ω∗+ Ω− |φin > or equivalently,

S = Ω∗+ Ω−

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We define R=S−I and derive a formula for the operator R in the from < λ, ω ' |R|λ, ω >=< λ, ω ' |V − V (H − λ)−1 V |λ, ω > where λ is the energy of the total system or equivalently of the free projectile coming from t = −∞. The energy is conserved during the scattering process. The state |λ, ω > represents the incident free projectile having energy P 2 /2m = λ and with incoming momentum P being directed along the direction ω ∈ S 2 and likewise |λ, ω ' > represents the free projectile state having energy λ and outgoing momentum P being directed along ω ' ∈ S 2 . It should be noted that the scattering operator S conserves the energy since it commutes with H0 . This follows from the relations Ω− exp(−itH0 ) = exp(itH)Ω− , Ω+ exp(−itH0 ) = exp(itH)Ω+ for all t ∈ R, from which follows ∗ Ω− exp(−itH0 ) = Ω∗+ Ω− exp(itH0 )Ω+

which gives formally, on differentiating w.r.t t at t = 0, [H0 , S] = 0 The design of the quantum gate is based on choosing a potential V so that for a given energy λ, the matrix ((< λ, ω ' |R|λ, ω >))ω,ω' is as close as possilble to Ug − I where Ug is a given N × N unitary matrix and N is the number of dis­ cretized directions ω chosen on the unit sphere. Another topic of importance dis­ cussed in this book is a rough idea about how a unified quantum field theory can be developed encompassing gravitation, electromagnetism, the electron-positron field of matter and if possible other Yang-Mills fields. The theory developed re­ lies heavily on identifying the correct connection for the covariant derivative of spinor fields in the presence of a curved space-time metric. The construction of this gravitational connection for Dirac spinors can be achieved in terms of the tetrad components of the gravitational metric and the Lorentz generators in the Dirac spinor representation. This construction can be found in Wein­ berg’s book (Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity). This connection can also be used for Yang Mills field. Once this has been done, it is an easy matter to construct the Lagrangian density of the Dirac or Yang-Mills field in curved space-time. This Lagrangian density is added to the Einstein-Hilbert Lagrangian density of the gravitational field and to the Lagrangian density of the electromagentic and Yang-Mills gauge fields, and to the Lagrangian density of scalar Higgs field taking once again into account the gravitational connection using the Tetrad. This total Lagrangian

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density is a function of the Dirac wave function, the Yang-Mills wave function, the tetrad components of the gravitational metric and the gauge fields, namely the electromagnetic four potential, ie, the U (1) gauge fields for the Dirac equa­ tion and the non-Abelian SU (N ) gauge fields for the Yang-Mills equations. The corresponding four dimensional action integral is then constructed and Feynman diagrammatic rules are formulated using the path integral approach for deter­ mining the probability amplitude for scattering/absorption-emission processes involving gravitons, photons, electrons, positrons, and particles associated with the Yang-Mills fields. The basic idea for calculating these amplitudes is to use Feynman’s trick: Identify the parts Sq of the action S that are quadratic in the fields retain them in the exponent exp(iS). The cubic and higher degree terms of the fields appearing in S denoted by Sc are considered as perturbations thus writing down exp(iS) = exp(iSq )(1 + iSc − Sc2 /2 − iSc3 /6 + ...) and the path integral is evaluated using the basic Wick theorem which roughly states that higher moments of a Gaussian distribution can be decomposed into a sum over products of second order moments, ie, propagators. Other schemes of quantization of all the fields can be found in the textbook by Thiemann (Modern canonical quantum general relativity). Finally, we discuss the modern theory of Supersymmetry which is one of the mathematically successful attempts to unify the various field theories like electromagnetism, Dirac’s relativistic quantum mechanics, the scalar field theories of Klein-Gordon and Higgs, the Yang-Mills gauge theories and even general relativity. The idea here is to introduce four anticommuting Majorana Fermionic variables θ and to define a superfield S[x, θ] as a function of both the bosonic space-time coordinates x = (xμ ) and the four Fermionic variables θ = (θa ). When the superfield is expanded in powers of θ, all terms involving five or more θ' s vanish owing to the anticommutativity of the four θ' s. Hence, the superfield S is a fourth degree polynomial in θ and the coefficients of θa , θa θb , θa θb θc , θ0 θ1 θ2 θ3 , 0 ≤ a < b < c ≤ 4 are functions of x only. Then following Salam and Stratadhee, we introduce supersymmetric ¯ a a = 0, 1, 2, 3 that are vector field vector field in super-space generators La , L (x, θ). These generators satisfy the standard anticommutation relations required for supersymmetry generators, ie, their anticommutators are bosonic generators: ¯ b } = γ μ ∂/∂xμ {La , L ab The general form of these generators can be obtained using standard group theoretic arguments: Define the composition of superspace variables as '

'

'

(xμ , θa ).(xμ , θa ) = (xμ + xμ' + θT Γμ θ, θa + θa ) Then, the supersymmetry generators (which are supervector fields) are calcu­ lated by expressing ∂ f (x, θ).(x' , θ' )) ∂xμ'

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∂ f ((x, θ).(x' , θ' )) ∂θa' evaluated at x' = 0, θ' = 0 in terms of ∂x∂μ and ∂θ∂a . The supersymmetry gen­ erators when acting on a superfield induce transformations on the component superfield and it is noted that only the coefficient of the fourth degree term in θ suffers a change that is a total differential. In fact, if the coefficient of the fourth degree in θ is split into a sum of two components, one we call the D component and the other we write as a constant times the D’Alembertian acting on the scalar superfield component, then the D component suffers a change that is a perfect divergence. Hence, the four dimensional space-time integral of the D component remains invariant under a supersymmetry transformation and hence this integral can be used as a candidate action for a supersymmetric field theory. It is also noted that if we impose conditions that one part of the θ3 component, namely the λ component (called the gaugino) and the D component is zero, and further if the gauge component Vμ (x) which appear as coefficients of the θ2 portions is a perfect gauge, ie, a perfect divergence, then after a supersymmetry transformation again λ and D vanish. Hence we obtain a subclass of the class of all superfields, namely the Chiral superfields which can be expressed as the sum of a left Chiral and a right Chiral superfield. Chiral fields are characterized by the property that their λ and D components vanish and the Vμ component is a perfect gauge. We further note following the exposition of Wienberg that a left Chiral superfield can be expressed as a function of xμ+ and θL only and likewise, a right Chiral superfield can be expressed as a function of xμ− and θR where T εγ μ θR and θL = (1 + γ5 )θ/2, θR = (1 − γ5 )θ/2 are respectively the xμ± = xμ ± θL left chiral and right chiral projections of the Fermionic parameter θ. There are just two linearly independent left chiral Fermionic parameters and likewise two linearly independent right chiral Fermionic parameters. This means that cubic and higher degree terms in θL vanish and likewise cubic and higher terms in θR vanish. Further, left chiral superfields are characterized by the property that certain ”right” superdifferential operators DR acting on these superfields give zero and right chiral superfields are characterized by the property that certain ”left” superdifferential operators DL acting on these superfields give zero. Here DR and DL are defined as respectively the right chiral and left chiral projections (mutliplication by (1+γ5 )/2 and (1−γ5 )/2) of a super vector field D obtained by changing a sign in the expression of the supersymmetry generators. The proofs of these facts follow by showing first that DR xμ+ = 0, DL xμ− = 0 and then not­ ing that DR (anylef tsuperf ield) = 0, DL (anyrightsuperf ield) = 0 since left superfields are functions of x+ , θL only and right superfields are functions of 2 (called the x− , θR only. Further it is readily shown that the coefficient of θL F -term) in a left chiral superfield ΦL (x, θ) suffers change by a perfect divergence under a supersymmetry transformation and hence its space-time integral is a candiadate supersymmetric action. After this, we come to the crucial point: A kinematic Lagrangian density should canonically by a quadratic function of component superfields just as kinetic energies are quadratic functions of mo­ menta/velocities and potential energies of harmonic oscillators are quadratic

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functions of positions, or equivalently, the Klein-Gordon Lagrangian density is a quadratic function of the space-time derivatives of the field minus a mass term which is a quadratic function of the field itself. Now, if we start with a supefield S[x, θ] and consider the D-component of the superfield S[x, θ]∗ S[x, θ], then we get terms that are bilienar in the D and C terms but no term that is quadratic or higher in the D term. Likewise, we get terms that are bilinear in the λ and ω terms (λ is a cubic component and ω is a first degree component) but no terms that are quadratic or higher in the λ term. Now if we build our action from these terms, then standard Gaussian path integral considerations show that we must evaluate our path integrals by setting the variational derivatives of the action with respect to the various component fields to zero. But the variational derivative of the above mentioned action w.r.t. the D term of S ∗ S is the C term and likewise, the variational derivative of S ∗ S w.r.t the λ term is the ω terms. So we are led to the disastrous consequence that C = 0 and ω = 0, ie, we cannot have scalar fields like the Klein-Gordon field or Fermionic fields like the Dirac field. To rectify this problem, we assume that D = 0, λ = 0 in S while constructing the action [S ∗ S]D . In other, words, to construct matter field Lagrangians, we take our basic matter superfield S as a Chiral field. For a given superfield S[x, θ], the component superfields C(x) (the scalar field) (ze­ roth power of θ coefficient), the Ferminoic field ω(x) (first power of θ coefficient) and the other fields M (x), N (x) (which are coefficients of the second power of θ) constitute the matter fields and the other components Vμ (x) (one set of com­ ponents of the second power of θ) which is also called the gauge field, λ(x) (one part of the Fermion field appearing as a coefficient of the third power of θ) also called the gaugino field (and interpreted as the superpartner of the gauge field) and the auxiliary field D(x) appearing as a coefficient of the fourth power of θ constitute the gauge part of the superfield. To get a gauge invariant theory of supefields, first we must form out of a left Chiral superfield Φ[x, θ] (built out of the matter fields C, ω, M, N ) and a matrix Γ[x, θ] built out of the gauge superfields Vμ , λ, D the D component [Φ∗ ΓΦ]D of Φ∗ ΓΦ (which will of course be supersymmetry invariant since it is the D component of a superfield) and the transformation law of the matter part Φ and the gauge part Γ under a gauge transformation is defined in such a way so that [Φ∗ ΓΦ]D remains invariant. This forms the part of the total Lagrangian density that describes matter like the scalar field, the Dirac Fermionic field of electrons and positrons etc., and the interaction of the matter fields with the gauge fields. Finally, another superfield W [x, θ] is constructed out of the gauge and auxiliary components Vμ , λ, D such that W [x, θ] is a left Chiral field (and hence its F -component is supersymmetry invariant) and its F component has the form ¯ T γ μ Dμ λ + c 3 D 2 c1 Fμν F μν + c1 λ which guarantees gauge invariance of the F -part of W . Here, iFμν = [∂μ + iVμ , ∂ν + iVν ] where in an Abelian gauge theory, Vμ is simply a function of x ¯ T Dμ λ so that Fμν = Vν,μ − Vμ,ν is the electromagnetic field of photons and λ T μ ¯ is the sum of λ γ λ,μ which is like a kinematic part of the Dirac Lagrangian ¯ T iγ μ λVμ which is like the interaction Lagrangian between the density and λ

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Dirac field of Fermions and the photon electromagnetic four potential field Vμ . In a non-Abelian gauge theory, Vμ is a Lie algebra valued function of x and / 0. It follows then that the structure constants of this Lie al­ then [Vμ , Vν ] = gebra will enter into the picture while defining and Fμν , Dμ λ. In this case, the above gauge Lagrangian density will describe particles arising in the YangMills theories like Electroweak theories etc. When the matter and matter-gauge interaction Lagrangian [Φ∗ ΓΦ]D is added to the above gauge Lagrangian, we obtain a Lagrangian density that is both supersymmetry and gauge invariant. In such a theory, if path integrals are evaluated with respect to certain auxiliary fields, we obtain equations like the Dirac relativistic wave equation with mass dependent on the scalar field, the Maxwell photon equations, the Yang-Mills equations and even gravitational fields can be included by more additions to the superfield. In short, supersymmetry provides the ideal ground for unifying all the known theories into a single framework, studying interactions between the particles associated with each theory and finally quantizing such a unified field theory using the Feynman path integral. We do not give all the details here for they can be found in the masterpiece of Wienberg (Supersymmetry, Cambridge University Press). [1] The De-Broglie Duality of particle and wave properties of matter. A plane wave in one dimension is expressed by the following complex amplitude: ψ(t, x) = A.exp(i(kx − ωt)) where k = 2π/λ with λ as the wavelength and ν = ω/2π is the frequency. According to De-Broglie, we can associate a particle having momentum p = h/λ = hk/2π with such a wave where h is Planck,s constant. According to Planck, we can associate a quantum of energy E = hν = hω/2π with such a wave. Thus, we have (ih/2π)

∂ψ = (hω/2π)ψ, = Eψ ∂t

(−ih∇/2π)ψ = (hk/2π)ψ = pψ In the presence of an external potential V (x), E should be taken as p2 /2m+V (x) and hence, Eψ = (p2 /2m + V )ψ = (−h2 ∇2 /8π 2 m + V (x))ψ Although the above plane wave ψ cannot satisfy this equation for general V (x), we assume that the actual De-Broglie matter wave ψ associated with the par­ ticle is a solution to the above equation.. This is called the one dimensional Schrodinger equation. In short, by putting together De-Broglie’s matter-wave duality principle and Planck’s quantum hypothesis, we have heuristically de­ rived the Schrodinger wave equation. [2] Bohr’s correspondence principle.

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Bohr’s correspondence principle is that associated with each observable in classical mechanics is an observable in quantum mechanics that follows certain quantization axioms which can be used to explain the energy spectra of atoms and quantum oscillators. For example, consider an electron of charge −e and Its mometum mass m moving in a circular orbit of radius r around the nucleus. ∫ is p = mv and the correspondence principle implies that Γ pdq = nh, n ∈ Z where q is the position coordinate and the lhs is the line integral around a complete orbit of the electron. Thus, we derive 2πpr = nh, p = nh/2πr, n ∈ Z Thi is the same as mvr = nh/2π According to the centripetal force law that keeps a particle in an orbit, mv 2 /r = KZe2 /r2 , K = 1/4πε0 Thus, we get or and

n2 h2 /4π 2 mr3 = KZe2 /r2 r = n2 h2 /4π 2 mKZe2 v 2 = KZe2 /mr = 4π 2 K 2 Z 2 e4 /n2 h2

and finally, we get the energy spectrum of the particle: E == En = p2 /2m−KZe2 /r = mv 2 /2−Ze2 /r = KZe2 /2r−KZe2 /r = −KZe2 /2r = −2π 2 mK 2 Z 2 e4 /n2 h2 , n = 1, 2, ... This spectrum first derived by Bohr, agreed with experiments. [3] Bohr-Sommerfeld’s quantization rules. If q is a canonical position coordinate vector and p the canonical momentum vector, then for cyclic motion, the Bohr-Sommerfeld’s quantization rules state that ∫ p.dq = nh, n ∈ Z Γ

where Γ is a closed loop. A special case of this rule was applied earlier by Bohr to derive the spectrum of the Hydrogen atom. For example, if the system is described by action-angle variables I, θ, then we get ∫ 2π I(θ)dθ = nh, n ∈ Z 0

Sommerfeld applied this to relativistic quantum mechanics according to which the Hamiltonian of the particle is given by √ H(q, p) = c p2 + m2 c2 − eV (q)

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Writing H = E and solving for p, we get p = ±[(E + eV (q))2 + c2 p2 + m2 c4 ]1/2 = ±p(q, E) If p given by this expression vanishes at q = q1 , q2 , then the Bohr-Sommerfeld quantization rule states that the energy spectrum of this relativistic particle is given by ∫ q2

2

p(q, E)dq = nh, n ∈ Z

q1

Sommerfeld used this to derive the relativistic spectrum of the Hydrogen atom which improves upon the non-relativistic spectrum given by Bohr. [4] The principle of superposition of wave functions and its application to the Young double slit diffraction experiment. In quantum mechanics, a pure state in the position representation is de­ scribed by a complex valued wave function ψ(x), x ∈ R3 . Given two such wave functions ψ1 (x), ψ2 (x), we can construct another wave function ψ(x) = c1 ψ1 (x) + c2 ψ2 (x), where c1 , c2 are two complex numbers. |ψk (x)|2 is propor­ tional to the intensity of the wave ψk at x, k = 1, 2 and |ψ(x)|2 = |c1 |2 |ψ1 (x)|2 + |c2 |2 |ψ2 (x)|2 + 2Re(¯ c1 c2 ψ¯1 (x)ψ2 (x)) is proportional to the intensity of the wave ψ at x. In particular, we can take c1 = c2 = 1, then the intensity of ψ is the in­ tensity of ψ1 plus the intensity of ψ2 plus an interference term 2Re(ψ¯1 (x)ψ2 (x)). This last term is a purely quantum mechanical effect. This fact has been mar­ vellously illustrated by Feynman using the Young double slit experiment: We may two slits in a cardboard sheet and place an electron gun behind this sheet. On the other side of the sheet is a screen than can record the impact of elec­ trons. If the first slit is open and the second closed, then the electron intensity pattern on the screen shows a maximum value at the portion of the screen di­ rectly in front of the first slit and decays down as the distance from the first slit increases on both the sides. Likewise, if slit one is closed and slit two is open, then the electron intensity on the screen in front of slit two shows a maximum. We denote the former intensity pattern by I1 (x) and the latter intensity pattern by I2 (x). Now, when both the slits are open, if the electron were to behave as classical particles, we should expect the intensity pattern on the screen to be I1 (x)+I2 (x), ie, maxima at both the points on the screen in front of the first and second slit respectively. But this is not what we observe. Instead we observe a maximum on the screen at a point in front of the middle between the two slits, ie at a point on the screen that is equidistant from the two slits. Further, as we move away from this intensity maximum point, the intensity shows a sinusoidal variation. This can be explained only by the presence of the above interference term. For example, if ψ1 (x) = A1 .exp(ik1 x), ψ2 (x) = A2 .exp(ik2 x) where x is the distance on the screen from the central point P that is equidistant from both the slits, then the intensity pattern on the screen when both the slits are open is given by I(x) = |ψ1 (x) + ψ2 (x)|2 = |A1 |2 + |A2 |2 + 2|A1 ||A2 |cos((k1 − k2 )x + φ)

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where φ = arg(A1 ) − arg(A2 ) It follows that if φ = 0 (which will be the case when the two slits are equidistant from the electron gun, then I(x) will show a maximum at x = 0 and its spatial variation will be sinusoidal with the distance between two successive maxima or between two successive minima being given by 2π/|k1 − k2 |. Thus we are forced to conclude that electrons exhibit wave behaviour at some time which is completely in accord with the De-Broglie matter-wave duality principle. The Heisenberg uncertainty principle can also be explained using this setup. The difference between the two electron momenta parallel to the screen is Δp = h|k1 −k2 |/2π according to De-Broglie and the position measurement uncertainty is the distance between an intensity maximum and an intensity miniumum on the screen, ie, Δx = π/|k1 − k2 |. Thus, Δp.Δx ≈ h/2 This means that if we attempt to measure the momentum difference accurately then the position measurement will become less accurate and vice versa. [5] Schrodinger’s wave mechanics and Heisenberg’s matrix mechanics. In wave mechanics, the state of a quantum system at time t is defined by a normalized vector |ψ(t) > in a Hilbert space H and this vector satisfies a ”Schrodinger wave equation”: i

d|ψ(t) > = H(t)|ψ(t) > dt

where H(t) is a possibly time varying self-adjoint operator in H. If H(t) = H is time independent, then the solution can be expressed as |ψ(t) >= exp(−itH)|ψ(0) > where exp(−itH) is a unitary operator in H and may be defined via resolvents: exp(−itH) = limn→∞ (1 + itH/n)−n We note that even if H is an unbounded operator, this definition of the expo­ nential may make sense using the theory of resolvents and spectra. (A complex number z is said to be in the resolvent set ρ(H) of H if (z − H)−1 exists and is a bounded operator. The complement of ρ(H) denoted by σ(H) is called the spectrum of the operator H. Since here H is Hermitian, we have the spectral representation ∫ H=d

xE(dx) R

∫

so that exp(−itH) =

exp(−itx)E(dx) R

General Relativity and Cosmology with Engineering Applications and (1 + itH/n)−n = so for any |f >∈ H, we have || (1+itH/n)−n f −exp(−itH)f ||2 =

∫

∫ R

277

(1 + itx/n)−n E(dx)

|(1+itx/n)−n −exp(−itx)|2 < f |E(dx)|f >

which converges by the dominated convergence theorem to zero as n → ∞. We note that if T is any densely defined operator in H, bounded or unbounded, and if for z ∈ ρ(T ), we have an inequality of the form || (z − T )−1 ||≤ f (z) then

|| (1 + itT /n)−n ||≤|| (1 + itT /n)−1 ||n ≤ nf (in/t)/|t|

and this may remain bounded as n → ∞. On the other hand, (1 + itT /n)n is unbounded if T is unbounded and hence we cannot defined exp(itT ) as the limit of this as n → ∞ (Reference: T.Kato, Perturbation theory for linear operators, Springer). In wave mechanics, the observables like position, momentum, angular mo­ mentum, energy etc. are represented by self-adjoint operators in the Hilbert space H and these do not vary with time while the state |ψ(t) > varies with time. Hence, if X is an observable, then its average at time t is given in the Schrodinger wave mechanics picture by < ψ(t)|X|ψ(t) >. On the other hand, in the Heisenberg matrix mechanics picture, observables change with time while the state remains the same and hence, the average of X at time t in the Heisen­ berg picture is < ψ(0)|X(t)|ψ(0) >. Since the physics must be the same no matter which model we use, we must have < ψ(t)|X|ψ(t) >=< ψ(0)|X(t)|ψ(0) > − − −(1) and differentiating this equation w.r.t. time gives i < ψ(t)|[H(t), X]|ψ(t) >=< ψ(0)|X ' (t)|ψ(0) > Writing |ψ(t) >= U (t)|ψ(0) > then gives us X ' (t) = iU (t)∗ [H(t), X]U (t) = [iU (t)∗ H(t)U (t), X(t)] We could also directly write (1) as < ψ(0)|U (t)∗ XU (t)|ψ(0) >=< ψ(0)|X(t)|ψ(0) > and hence

X(t) = U (t)∗ XU (t)

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from which we obtain on differentiation, X ' (t) = iU (t)∗ [H(t), X]U (t) = i[U (t)∗ H(t)U (t), X(t)] This is Heisenberg’s equation of matrix mechanics. In particular, if H(t) = H does not vary with time, we get U (t)∗ HU (t)∗ = H and Heisenberg’s equation of matrix mechanics becomes X ' (t) = i[H, X(t)] with solution, X(t) = exp(itH).X.exp(−itH)

[6] Dirac’s replacement of the Poisson bracket by the quantum Lie bracket. The Poisson bracket between two observables u(q, p) and v(q, p) satisfies {u, vw} = {u, v}w + v{u, w} and likewise for the first argument. If we agree that an analogous bracket [.] exists for non-commutative quantum observables with the same ordering pre­ served, then we must have [uv, wz] = [uv, w]z + w[uv, z] = u[v, w]z + [u, w]vz + wu[v, z] + w[u, z]v on the one hand and on the other hand, [uv, wz] = [u, wz]v + u[v, wz] = [u, w]zv + w[u, z]v + u[v, w]z + uw[v, z] Equating these two expressions gives [u, w]vz + wu[v, z] = [u, w]zv + uw[v, z] Hence [u, w](vz − zv) = (uw − wu)[v, z] It follows from the arbitrariness of the four observables u, v, w, z that the quan­ tum bracket [.] should have the form [u, w] = c(uw − wu) where c is a complex constant. In other words, the quantum bracket that replaces the classical Poisson bracket must be proportional to the Lie bracket. [7] Duality between the Schrodinger and Heisenberg mechanics based on Dirac’s idea. In Schrodinger’s wave mechanics, the state of the system at any time t is described by a density operator ρ(t) in a Hilbert space H. In other words,

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ρ(t) ≥ 0, T r(ρ(t)) = 1. Further, it satisfies the Schrodinger-Liouville- VonNeumann equation of motion: iρ' (t) = [H(t), ρ(t)] This follows by writing the spectral decomposition of ρ(t) as ∑ ρ(t) = |ψk (t) > pk < ψk (t)| k

in diagonal form with p'k s constant and the Schrodinger wave equation

∑ k

pk = 1, pk ≥ 0. The ψk (t)' s satisfy

i|ψk' (t) >= H(t)|ψk (t) > and hence

−i < ψk' (t)| =< ψk (t)|H(t)

so that iρ' (t) =

∑ (i|ψk' (t) > pk < ψk (t)| + i|ψk (t > pk < ψk' (t)|) k

=

∑ (H(t)|ψk (t) > pk < ψk (t)| − |ψk (t)pk < ψk (t)|H(t)) k

= [H(t), ρ(t)] We can write its solution as ρ(t) = U (t)ρ(0)U (t)∗ where U (t) is a unitary operator satisfying the Schrodinger equation U ' (t) = −iH(t)U (t) In this picture, observables do not vary with time. Thus, the average value of an observable X at time t is given by ∑ < X > (t) = T r(ρ(t)X) = pk < ψk (t)|X|ψk (t) > k

In the Heisenberg picture, the observable X changes at time t to X(t) while the state ρ(0) does not change. Hence, in this picture the average of the observable at time t is < X > (t) = T r(ρ(0)X(t)) and the two pictures must give the same average. Thus, T r(ρ(0)X(t)) = T r(ρ(t)X) = T r(U (t)ρ(0)U (t)∗ X) = T r(ρ(0)U (t)∗ XU (t))

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and hence from the arbitrariness of ρ(0), it follows that dynamics of observables in the Heisenberg picture must be given by X(t) = U (t)∗ XU (t) and hence ˜ (t), X(t)] X ' (t) = iU (t)∗ (H(t)X − XH(t))U (t) = i[H where

˜ (t) = U (t)∗ H(t)U (t) H

In particular, if H(t) = H is time independent, then U (t) = exp(−itH) and we get X ' (t) = i[H, X(t)] for the Heisenberg dynamics and iρ' (t) = [H, ρ(t)] for the Schrodinger dynamics. [8] Quantum dynamics in Dirac’s interaction picture. Suppose H(t) = H0 + V (t) is the system Hamiltonian. The Schrodinger evolution operator U (t) satisfies iU ' (t) = H(t)U (t) We write U (t) = U0 (t)W (t), U0 (t) = exp(−itH0 ) Then, substituting this into the above Schrodinger evolution equation gives iW ' (t) = V˜ (t)W (t), V˜ (t) = U0 (t)∗ V (t)U0 (t) This leads us to the Dirac interaction picture where a state |ψ > evolves ac­ cording to the Hamiltonian V˜ (t) while an observable X evolves according to the Hamiltonian H0 . This keeps the physics invariant since if X is an observ­ able and |ψ > is the state at time 0, then in the Dirac interaction picture, the average of the observable in the state at time t is given by (The subscript i stands for evolution of states or observables in the interaction picture. Thus |ψi (t) >= W (t)|ψ =< ψ|W (t)∗ (U0 (t)∗ XU0 (t))W (t)|ψ > where

∫ W (t) = T {exp(−i

t

V˜ (τ )dτ )} 0

General Relativity and Cosmology with Engineering Applications

281

Now, as a check d (U0 (t)W (t)) = U0' (t)W (t) + U0 (t)W ' (t) = dt −iH0 U0 (t)W (t) + U0 (t)(−iV˜ (t))W (t) = −iH0 U0 (t)W (t) − iU0 (t)U0 (t)∗ V (t)U0 (t)W (t) = −iH0 U0 (t)W (t) − iV (t)U0 (t)W (t) = −iH(t)U0 (t)W (t) ˜ (t) satisfies the same equation as U (t) and hence must In other words, U0 (t)U equal U (t). Thus, U0 (t)W (t) = U (t) as expected and this shows that < ψi (t)|Xi (t)|ψi (t) >=< ψ|U (t)∗ XU (t)|ψ > which is in agreement with the Schrodinger and Heisenberg pictures. [9] The Pauli equation: Incorporating spin in the Schrodinger wave equation in the presence of a magnetic field. We describe how the Pauli equation can be used to calculate the average electric dipole moment and average magnetic dipole moment of an atom in an external electromagnetic field. This calculation enables us to calculate the electric polarization field and the magnetization field of a material. It will be a consequence of this calculation that the average electric dipole moment and hence polarization depends on both the external electric and magnetic fields and likewise for the average magnetic dipole moment and magnetization. The electric dipole moment observable is −er, r = (x, y, z) being the electron position relative to the nucleus. The magnetic dipole moment observable is μ = (e/2m)(L + gσ/2) where L = (Lx , Ly , Lz ) is the angular momentum observable vector and σ = (σx , σy , σz ) is the vector of Pauli spin matrices. A heuristic justification of this fact [10] The Zeeman effect: Let H0 be the Hamiltonian of an atom that com­ mutes with the orbital and spin angular momentum operators (Lx , Ly , Lz ) = L (ie, H0 is rotation invariant as happens when H0 = p2 /2m0 + V (r) where V de­ pends only on the radial coordinate) and (σx , σy , σz ) = σ. Then the eigenstates of the Hamiltonian operator H = H0 + e(L + gσ, B)/2m0 where B is a constant magnetic can be calculated as follows. We may assume without loss of generality that B is along the z axis. Then, H = H0 + (eB/2m)(Lz + gσz ) and since L2 , Lz , σz , H0 mutually commute, the energy eigenstates of H are labelled by four quantum numbers |n, l, m, s > where l(l + 1) is an eigenvalue of L2 , m is an eigenvalue of Lz and s is an eigenvalue of σz . If E(n, l) is an energy

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eigenvalue of H0 ie of H without the magnetic field, then this eigenvalue has a degeneracy of 2(2l + 1) corresponding to the 2l + 1 eigenvalues of Lz for a given eigenvalue l(l + 1) (ie,+ m = −l, −l + 1, ..., l − 1, l)of L2 and the two eigenvalues ±1/2 of σz . When the magnetic field is turned on, these 2(2l + 1) degenerate eigenstates split into the the same number of non-degenerate eigenstates with eigenvalues E(n, l, m, s) = E(n, l) + (eB/2m0 )(m + gs), m = −l, −l + 1, ..., l − 1, l, s = ±1/2. The eigenstate of H corresponding to the eigenvalue E(n, l, m, s) is denoted by |n, l, m, s >. [11a] The spectrum of the Hydrogen atom. This is described by the stationary Schrodinger equation ∇2 ψ(r, θ, φ) + 2m(E + e2 /r)ψ(r, θ, φ) = 0 Writing ψ(r, θ, φ) = R(r)Ylm (θ, φ) where Ylm are the spherical harmonics, we get using ∇2 = r−2

∂ 2 ∂ r − L2 /r2 ∂r partialr

where L2 is the squared angular momentum operator: L2 = −

1 ∂ ∂ 1 ∂2 sin(θ) − 2 sin(θ) ∂θ ∂θ sin (θ) ∂φ2

Ylm is an eigenfunction of both the commuting operators L2 and Lz : L = r × p = −ir, L2 = −(r × ∇)2 Lz Ylm = mYlm , L2 Ylm = l(l + 1)Ylm Exercise: Verify that L2 given by the above differential expression coincides with −(r × ∇)2 = (ypz − zpy )2 + (zpx − xpz )2 + (xpy − ypx )2 , px = −i∂x , py = −i∂y , pz = −i∂z The radial equation for R(r) thus becomes R'' (r) + 2R' (r)/r − l(l + 1)R(r)/r2 + 2m(E + e2 /r)R(r) = 0 or equivalently, r2 R'' (r) + 2rR' (r) + 2m(Er2 + e2 r − l(l + 1)/2m)R(r) = 0 As r → ∞, this equation approximates to r2 R'' (r) + 2mEr2 R(r) ≈ 0

General Relativity and Cosmology with Engineering Applications or equivalently,

283

R'' (r) = −2mER(r)

Since for bound states, E√must be negative, it follows that as r → ∞, we have R(r) ≈ C.exp(−αr), α = −2mE. So, writing the exact solution as √ R(r) = f (r)exp(−αr), α = −2mE we get R' = (f ' − αf )exp(−αr), R'' = (f '' − 2αf ' + α2 f )exp(−αr) The Schrodinger equation now becomes r2 (f '' − 2αf ' + α2 f ) + 2r(f ' − αf ) + (−α2 r2 + 2me2 r − l(l + 1))f = 0 or

r2 f '' + 2r(1 − αr)f ' + (2me2 r − l(l + 1))f = 0

We solve this equation by the power series method. Substitute ∑ f (r) = c(n)rn+s n≥0

Then ∑ (n + s)(n + s − 1)c(n)rn+s n≥0 ∑ ∑ (n + s)c(n)rn+s+1 +2 (n + s)c(n)rn+s − 2α n≥0

+2me2

n≥0

∑

c(n)rn+s+1 − l(l + 1)

n≥0

∑

c(n)rn+s = 0

n≥0

or equivalently, ∑

((n + s)(n + s + 1) − l(l + 1))c(n)rn+s

n≥0

+

∑

(2me2 − 2(n + s)α)c(n)rn+s+1 = 0

Equating coefficients of equal powers of r gives (s(s + 1) − l(l + 1))c(0) = 0, and for all n ≥ 1, ((n + s)(n + s + 1) − l(l + 1))c(n) + (2me2 − 2(n + s − 1)α)c(n − 1) = 0 We must assume c(0) = / 0 for otherwise, this recursion would imply that c(n) = 0∀n ≥ 0 which would mean that the wave function vanishes. Then, we get s(s + 1) = l(l + 1)

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so s = l since s = −l − 1 would give a singularity at r = 0 and the wave function would fail to be square integrable when l > 0. Further, for n large, ie, n → ∞, the above difference equation asymptotically is equivalent to c(n) ≈ 2αc(n − 1)/n or equivalently, c(n) ≈ (2α)n /n! and hence f (r) =

∑

c(n)rn+l ≈ rl exp(2αr)

n

so f (r)exp(−αr) ≈ rl exp(αr) which is not square integrable, in fact, it diverges exponentially as r → ∞. The only way out is that c(n) = 0 for all n ≥ n0 for some n0 ≥ 1. Let n0 be the smallest such integer. Then we get from the above recursion by putting n = n0 , and c(n0 − 1) /= 0, me2 = (n0 + s − 1)α = (n0 + l − 1)α Thus,

E = −me4 /2(n0 + l − 1)2

Thus, we get the result that the possible energy levels of the hydrogen atom are En = −me4 /2n2 , n = max(1, l), l + 1, l + 2, ..., This result was first derived rigorously by Schrodinger in this way although it was earlier obtained by Niels Bohr using adhoc arguments like the correspon­ dence principle. [11b] The spectrum of particle in a 3 − D box. This is described by the stationary state Schrodinger equation −∇2 ψ(r)/2m = Eψ(r), r = (x, y, z) ∈ [0, a] × [0, b] × [0, c] with boundary conditions that ψ(r) vanishes on the boundary, ie when either x = 0, a or y = 0, b or z = 0, c. The solution to this boundary valued problem is obtained by separation of variables and the result is the set of orthonormal wave functions ψnmk (r) = ((2/a)(2/b)(2/c))1/2 sin(nπx/a)sin(mπy/b)sin(lπz/c), n, m, k = 1, 2, 3, ... with the corresponding energy eigenvalues E = π(n2 /a2 + m2 /b2 + k 2 /c2 )1/2

General Relativity and Cosmology with Engineering Applications

285

[11c] The spectrum of a quantum harmonic oscillator. A quantum Harmonic oscillator has the Hamiltonian H = p2 /2m + mω 2 q 2 /2 where [q, p] = i We define the annihilation and creation operators by √ √ a = (p − imωq)/ 2m, a∗ = (p + imωq)/ 2m Then, a∗ a = p2 /2m + mω 2 q 2 /2 − ω/2 = H − ω/2, aa∗ = H + ω/2 Thus,

[a, a∗ ] = ω

Now let |E > be any normalized eigenstate of H with eigenvalue E. Then we have 0 ≤< E|a∗ a|E >=< E|H − ω/2|E >= E − ω/2 Hence E ≥ ω/2 ie, the minimum energy level of the oscillator is ω/2. This is attained when and only when a|E >= 0. In other words, we have a|ω/2 >= 0 and hence (d/dq + mωq) < q|ω/2 >= 0 or equivalently,

< q|ω/2 >= C.exp(−mω 2 q 2 /2)

with C being the normalizing constant chosen so that ∫ ∫ 1 =< ω/2|ω/2 >= | < q|ω/2 > |2 = |C|2 exp(−mω 2 q 2 )dq R

R

= |C|2 = (π/m)1/2 ω −1 so we may take

C = (π/mω 2 )1/4

[12] Time independent perturbation theory: Calculation of the energy levels and eigenfunctions of non-degenerate and degenerate systems using perturbation theory.

286

General Relativity and Cosmology with Engineering Applications The perturbed Hamiltonian is H = H0 +

∞ ∑

ε k Vk

k=1

The stationary state wave functions for H are expanded as ∑ εk |ψk > |ψ >= |ψ0 > + k≥1

and the corresponding perturbed energy level is also expanded as a power series: ∑ ε k Ek E = E0 + k≥1

Substituting these expressions into the eigenvalue problem H|ψ >= E|ψ > and equating coefficients of εk , k = 0, 1, 2, ... gives us the series H0 |ψ0 >= E0 |ψ0 >, (H0 − E0 )|ψk > +

k ∑

Vr |ψk−r > +

r=1

k ∑

Er |ψk−r >, k ≥ 1

r=1

For each eigenvalue E0 = E0 (m) of H0 , let |m, r >, r = 1, 2, ..., d(m) denote an orthonormal basis of eigenstates of N (H0 − E0 (m)). Thus, the eigenvalue E0 (m) has a degeneracy of d(m). So we can write ∑

d(m)

|ψ0 >=

c(m, r)|m, r >

r=1

for E0 = E0 (m). The O(ε) equation (H0 − E0 (m))|ψ1 > +V1 |ψ0 >= E1 |ψ0 > − − −(1) then gives on premultiplying by < m, r|, ∑ c(m, r) < m, k|V1 |m, r >= E1 c(m, r)δ[k − r] r

and hence, it follows that the for the unperturbed energy level E0 (m), the possible first order perturbations E1 = E1 (m, s), s = 1, 2, ..., d(m) to the en­ ergy levels are given by the eigenvalues of the d(m) × d(m) secular matrix ((< d(m) m, k|V1 |m, r >))1≤k,r≤d(m) . Further, if cs (m) = ((cs (m, r)))r=1 is the eigenvec­ tor of this secular matrix corresponding to the eigenvalue E1 (m, s), then then ∑d(m) the corresponding unperturbed state is given by r=1 cs (m, r)|m, r >. We note

General Relativity and Cosmology with Engineering Applications

287

∑d(m) that the constants cs (m, r) can be chosen so that the states r=1 cs (m, r)|m, r > , s = 1, 2, ..., d(m) form an orthonormal basis for the d(m) dimensional vector space N (H0 − E0 (m)). Further, we get from (1) by premultiplying by < l, r| for l /= m, (E0 (l) − E0 (m)) < l, r|ψ1 > + < l, r|V1 |ψ0 >= E1 < l, r|ψ0 >= 0 so that

< l, r|V1 |ψ0 > E0 (m) − E0 (l) ∑ which implies that the unperturbed state |ψ0 >= r cs (m, r)|m, r > gets perturbed to |ψ0 > +ε|ψ1 > +O(ε2 ) < l, r|ψ1 >=

where |ψ1 >=

∑

|l, r >< l, r|V1 |ψ0 > /(E0 (m) − E0 (l))

l/=m,r=1,2,...,d(m)

and ∑

d(m)

< l, r|V1 |ψ0 >=

cs (m, k) < l, r|V1 |m, k >

k=1

We now calculate the second order pertrubation to the energy levels and the the corresponding eigenfunctions. [13] Time dependent perturbation theory: Calculation of the transition prob­ abilities of a quantum system from one stationary state of the unperturbed sys­ tem to another stationary state in the presence of a time varying interaction potential. The perturbed Hamiltonian has the form H(t) = H0 +

∞ ∑

εm Vm (t)

m=1

The Schrodinger evolution operator U (t) of this system is expanded as a per­ turbation series: ∑ εm Um (t) U (t) = U0 (t) + m≥1

Substituting this into the evolution equation iU ' (t) = H(t)U (t) and equating equal powers of the perturbation parameter ε gives us the sequence of differential equations: iU0' (t) = H0 U0 (t),

288

General Relativity and Cosmology with Engineering Applications ' iUm (t) − H0 Um (t) =

m ∑

Vk (t)Um−k (t), m ≥ 1

k=1

Thus

Um (t) = −i

U0 (t) = exp(−itH0 ),

m ∫ ∑ k=1

t 0

U0 (t − s)Vk (s)Um−k (s)ds, m ≥ 1 − − − (1)

Let now |n > be an eigenstate of the unperturbed system with energy E(n): H0 |n >= E(n)|n > Then when the perturbation is switched on, the transition probability amplitude / from state |n > to state |m > in time [0, T ] with m = n is given by < m|U (T )|n >= −i

r ∑

εk < m|Uk (T )|n > +O(εr+1 )

k=1

where U1 (T ), ...., Ur (T ) are successively determined from (1). For example, ∫ t U1 (t) = −i U0 (t − s)V1 (s)U0 (s)ds, 0

∫ U2 (t) = −i ∫ =−

0 be the tensor product of the state |n1 >, ..., |np > where |nk > is√the state in the k th copy of H0 such that √ ak |nk >= nk |nk − 1 >, a∗k |nk >= nk + 1|nk + 1 > and a∗k ak |nk >= nk |nk > , ak a∗k |nk >= (nk + 1)|nk >. Define for z ∈ Cp , the state ∑ √ |e(z) >= z1n1 ...zpnp |n1 , ..., np > / n1 !...np ! n1 ,...,np ≥0

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General Relativity and Cosmology with Engineering Applications =

∑

z1n1 ...zpnp a1∗n1 ...ap∗np |0 > /n1 !...np !

n1 ,...,np ≥0

= exp(z.a∗ )|0 >, z.a∗ = z1 a∗1 + ... + zp a∗p Show using < n1 , ..., np |m1 , ..., mp >= Πpk=1 δ[nk − mk ] = δ[n − m] that for z, u ∈ Cp , < e(z)|e(u) >= exp(< z, u >) Now, define for z ∈ C , n ≥ 0, p

ψ(z ⊗n ) =

√ n! √ z1n1 ...zpnp |n1 , ..., np > n !...n ! 1 p n1 +...+np =n ∑

Then show using the multinomial theorem that for z, u ∈ Cp , < ψ(z ⊗n ), ψ(u⊗n ) >=< z, u >n Now extend the map ψ linearly to the Boson Fock space ⊗ (Cp )⊗s n Γs (Cp ) = n≥0

by defining for |f (z) >=

⊗ z ⊗n √ ∈ Γs (Cp ) n! n≥0

ψ(|f (z) >) = |e(z) > or equivalently,

√ ψ(z ⊗n / n!) = ∑

n

z n1 ...zp p √1 |n1 , ..., np >, n ≥ 0 n1 !...np ! n1 +...+np =n Show that ψ defines a Hilbert space isomorphism between Γs (Cp ) and L2 (Rp ). This gives a physical interpretation of the coherent states in Boson Fock space in terms of the p-dimensional Harmonic oscillator algebra. Problems [25]-[29] are study projects related to the relatively more recent field of quantum stochastic processes as founded by R.L.Hudson and K.R.Parthasarathy. The interested reader should study this material from the book K.R.Parthasarathy, ”An introduction to quantum stochastic processes”, Birkhauser, 1992. [25] Creation, conservation and annihilation operators in the Boson Fock space. [26] The general theory of quantum stochastic processes in the sense of Hud­ son and Parthasarathy.

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301

[27] The quantum Ito formula of Hudson and Parthasarathy. [28] The general theory of quantum stochastic differential equations. [29] The Hudson-Parthasarathy noisy Schrodinger equation and the deriva­ tion of the GKSL equation from its partial trace. [30] The Feynman path integral for solving the Schrodinger equation. Let X(t), t ≥ 0 be a Markov process with values in R having generator K, ie E[φ(X(t + dt))|X(t) = x] = φ(x) + Kφ(x)dt + o(dt) Note that by including generalized functions such as the Dirac Delta function and its derivatives, we may represent K as an integral kernel: ∫ Kφ(x) = K(x, y)phi(y)dy We define ∫

t

V (X(s))ds)|X(s) = x], 0 ≤ s ≤ t

u(s, t, x) = E[f (X(t))exp( s

Then an easy application of the Markov property shows that u(s, t, x) = (1 + V (x)ds)E[(u(s + ds, t, X(s + ds))|X(s) = x] + o(ds) ∫ = (1 + V (x)ds)(u(s, t, x)ds + u,s (s, t, x) + ds

K(x, y)u(s, t, y)dy) + o(ds)

and hence, ∫ ∂s u(s, t, x) + V (x)u(s, t, x) +

K(x, y)u(s, t, y)dy = 0

Further, u(t, t, x) = f (x) In particular, if X(t) is Brownian motion, we have K(x, y) = (1/2)δ '' (x − y) and the above becomes ∂s u(s, t, x) + V (x)u(s, t, x) + (1/2)∂x2 u(s, t, x) = 0 which is the Feynman-Kac formula. Since we are assuming that the Markov process is time homogeneous, ie its generator K is time independent, it follows that u(s, t, x) = u(0, t − s, x). We denote this by v(t − s, x) and then we get ∫ ∂t v(t, x) = V (x)v(t, x) + K(x, y)v(t, y)dy, t ≥ 0, v(0, x) = f (x)

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Feynman formally replaced t by it, i = above formula to get

√

−1, ie define w(t, x) = v(it, x) in the ∫

i∂t w(t, x) = −V (x)w(t, x) −

K(x, y)w(t, y)dy

Now replace V by −V to get the generalized Schrodinger equation ∫ i∂t w(t, x) = V (x)w(t, x) − K(x, y)w(t, y)dy, w(0, x) = f (x) where

∫ w(t, x) = E[exp(− ∫ = E[exp(−i

it

V (X(s))ds)f (X(it))|X(0) = x] 0

t

V (X(is)ds)f (X(it))|X(0) = x] 0

Formally, this formula can be interpreted as follows: v(t, x) is approximated as ∫ t v(t, x) = E[exp(− V (X(s))ds)f (X(t))|X(0) = x] ≈ 0

∫ exp(

n ∑

(−V (xk )) + (log(K))(xk , xk+1 ))δsk )f (sn )Π0≤k≤n dxk

k=0

where 0 = s0 < s1 < ... < sn = t and log(K) is the operator logarithm of K (not log(K(x, y))). [31] Comparison between the Hamiltonian (Schrodinger-Heisenberg) and La­ grangian (path integral) approaches to quantum mechanics. The Schrodinger-Heisenberg approaches to quantum mechanics are Hamilto­ nian approaches. Feynman proposed an alternative approach to non-relativistic quantum mechanics that is based on the Lagrangian. To see how this proceeds, assume that the Hamiltonian of the system is H(t, q, p) . The Lagrangian is then L(t, q, q ' ) = (p, q ' ) − H(t, q, p) with p=

∂L ∂q '

Let |t, q ' > be the position space wave function of the quantum system at time t. Then at time t + dt, the position space wave function is |t + dt, q ' >= exp(−iH(t, q, p)dt)|t, q ' > Thus,

< t + dt, q '' |t, q ' >=< t, q '' |exp(−iH(t, q, p)dt)|t, q ' >

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303

We can applying the commutation relations between q and p, assume that in the Hamiltonian H(t, q, p), all the p' s appear to the left of all the q ' s. Then we get ∫ '' ' < t + dt, q |t, q >= < t, q '' |t, p' > dp' < t, p' |exp(−iH(t, q, p)dt)|t, q ' > ∫

exp(i(q”, p' ))exp(−iH(t, q ' , p' )dt)|t, q ' > dp'

=C ∫ =C

exp(i((q '' , p' ) − H(t, q ' , p' )dt)).exp(−i(q ' , p' ))dp' ∫

=C

exp(i[(q '' − q ' , p' )/dt − H(t, q ' , p' )]dt))dp'

By composing these infinitesimal amplitudes, we get the transition amplitude for finite time as K(t2 , q2' |t1 , q1' ) =< t2 , q2' |t1 , q1' >= ∫ ∫ t2 C exp(i (p(t), q ' (t)) − H(t, q(t), p(t)))dt)Πt1 ≤t≤t2 dq(t)dp(t) q(t1 )=q1' ,q(t2 )=q2'

t1

This is indeed the formula for the time evolution operator kernel in the position representation, ie ∫ t2 H(t)dt)}|t1 , q1' > K(t2 , q2' |t1 , q1' ) =< t2 , q2' |T {exp(−i t1

where T {} is the time ordering operator. In the special case, when the Hamil­ tonian has the form H(t, q, p) = p2 /2m + V (q) the integral over p becomes a Gaussian integral and therefore it can be replaced by evaluating the action integral at the stationary point, ie at p(t) given by d ((p(t), q ' (t)) + p2 (t)/2m) = 0 dp(t) ie

q ' (t) = p(t)/m, p(t) = mq ' (t)

Thus, in this special case, ∫ K(t2 , q2' |t1 , q1' ) = C

q(t1 )=q1' ,q(t2 )=q2'

∫

t2

exp(i t1

(mq ' (t)2 /2−V (q(t)))dt)πt1 at time t2 in which the field is exactly another given function φf (r), r ∈ R3 of the spatial variables. The corresponding Hamiltonian will then have the form ∫ ∫ 3 H(t) = H0 (φ(x), ∇φ(x), π(x))d x − f (x)V (φ(x))d3 x (Note: x = (t, r), d3 x = d3 r, d4 x = d3 xdt = d3 rdt) where H0 is the Hamiltonian density corresponding to the Lagrangian density L0 (ie obtained by applying the Legendre transform to L0 ): ∫ H0 = (1/2) (π 2 + (∇φ)2 + m2 φ2 )d3 x The transition probability amplitude from |φi >→ |φf > in the time duration [t1 , t2 ] can be calculated using the Feynman path integral formula: ∫ C ∫ =C

φ(t1 ,.)=φi ,φ(t2 ,.)=φf

∫

exp(iS0 )(1+i

S0 =

[t1 ,t2 ]×R3

f (x)V (φ(x))dtd3 x+(i2 /2!)

∫

where

< φf |S[t2 , t1 ]|φi >= ∫ exp(−( Ldtd3 x))Πr∈R3 ,t∈(t1 ,t2 ) dφ(x)

L0 dtd3 x = (1/2)

∫

∫

f (x)f (x)' φ(x)φ(x' )dtdt' d3 xd3 x' +..)Πdφ(x)

(∂μ φ∂ μ φ − m2 φ2 )dt d3x

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By expanding V (φ(x) as a power series in φ(x), the computation of the above path integral reduces to computing the moments of a complex infinite dimen­ sional zero mean Gaussian distribution sinc S0 is a quadratic functional of φ. In particular, we note that the odd moments of a symmetric Gaussian distribution are zero and the even moments can be computed by summing the products of the second moments taken over all partitions of the product fields into pairs. Thus, computation of the second moments of such a Gaussian distribution becomes significant, ie, ∫ D(x, y) = C exp(iS0 )φ(x)φ(y)Πz∈R4 dφ(z) if we are interested in transitions from t = −∞ to t = +∞. From standard methods in quantum mechanics, it is easily seen that D(x, y) =< 0|T {φ(x)φ(y)}|0 > provided that we use the interaction representation which removes the effect of the unperturbed Hamiltonian H0 . If we use the Schrodinger representation, then we would have to compute D as D(x, y) =< 0|T {U (∞, −∞)φ(x)φ(y)}|0 >= < 0|U (∞, tx )φ(x)U (tx , ty )φ(y)U (ty , −∞)|0 > assuming tx ≥ ty and where U is the unperturbed Schrodinger evolution oper­ ator. Here |0 > is the vacuum state of the field. The function D(x, y) is called the propagator. The complete propagator taking into account interactions is defined as ∫ Dc (x, y) = C exp(iS)φ(x)φ(y)Πz dφ(z) ∫

where S = S0 +

4

∫

f (x)V (φ(x))d x =

Ld4 x

We can write a perturbative expansion for Dc as ∫ Dc (x, y) = exp(iS0 )(1 + iS1 + i2 S12 /2! + ..)φ(x)φ(y)Πz dφ(z) ∫

where S1 =

f (x)V (φ(x))d4 x

is the perturbation to the action caused by external field coupling. Even if there is no external field, but there is a small perturbation to the Lagrangian density/Hamiltonian density, the above series expansion can be used to deter­ mine the complete propagator It was Feynman’s genius to recognize that the various perturbation terms in Dc can be calculated easily using a diagrammatic method which could be applied to more complex situations like quantum elec­ trodynamics wherein the quantum fields are the electromagnetic four potential

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Aμ (x) and the Dirac four component spinor wave function ψ(x). Let us now formally compute the propagator of the unperturbed KG field: ∫ S0 = φ(x)[(1/2)∂μ ∂ μ − m2 /2)δ 4 (x − y)]φ(y)d4 xd4 y ∫ =

φ(x)K(x, y)φ(y)d4 xd4 y

and hence a simple Gaussian second moment evaluation gives ∫ D(x, y) = exp(iS0 [φ])φ(x)φ(y)Πz dφ(z) = C1 (det(iK))−1/2 .K −1 (x, y) In other words D(x, y) is proportional to K −1 (x, y) where K −1 is the inverse Kernel of K: ∫ K −1 (x, y)K(y, z)d4 y = δ 4 (x − z) We can write K(x, y) = K(x − y) and then defining its four dimensional Fourier transform: ∫ ˆ (p) = K(x)exp(−ip.x)d4 x, p.x = pμ xμ = p0 x0 − p1 x1 − p2 x2 − p3 x3 K we get Clearly, and hence

K −1 (p) = 1/K(p) K(x) = (1/2)∂μ ∂ μ − m2 /2)δ 4 (x) ˆ (p) = (pμ pμ − m2 )/2 K

Thus, ˆ D(p) = where

C0 p2 − m2

p2 = pμ pμ = p02 − p12 − p22 − p32

Finally, D(x, y) = D(x − y) = C0 /(2π)4

∫

exp(ip.x) 4 d p p2 − m2

The corrected (complete) propagator: ∫ ∫ Dc (x, y) = exp(iS0 [φ])φ(x)φ(y)(1 + i f (x)V (φ(z))d4 z + ...)Πu dφ(u) Clearly, we can write this in operator kernel notation as Dc = D + DΣD + DΣDΣD + ... using the property of moments of a Gaussian distribution. For

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example, ∫if V (φ) = φ4 and f = c0 , then in the Gaussian average of the product φ(x)φ(y) f (z)φ(z)4 d4 z, we get the coupling terms 4 < φ(x)φ(z) >< φ(z)2 >< φ(z)φ(y) > so if we define Σ(z) = 4f (z)0 < φ(z)2 >, we can write ∫ ∫ 4 4 < φ(x)φ(y)( f (z)φ(z) d z) >= D(x − z)Σ(z)D(z − y)d4 z Likewise, for the next perturbation term ∫ < φ(x)φ(y)( f (z)φ(z)4 d4 z)2 >= ∫

f (z1 )f (z2 ) < φ(x)φ(y)φ4 (z1 )φ4 (z2 ) > d4 z1 d4 z2

Again, this can be expressed using the Gaussian moments formula as a sum of terms of the form ∫ f (z1 )f (z2 ) < φ(x)φ(z1 ) >< φ(z1 )3 φ(z2 )3 >< φ(z2 )φ(y) > d4 z1 d4 z2 and

∫

f (z1 )2 < φ(x)φ(z1 ) >< φ(z1 )2 φ(z2 )4 >< φ(z1 )φ(y) > d4 z1 d4 z2

etc. Now, each term < φ(z1 )m φ(z2 )m > is a product of propagators D(z1 − z2 ) and D(0) so the above general form is valid. [b] Quantization of the electromagnetic field. The Lagrangian density is 1 LF = − Fμν F μν , Fμν = Aν,μ − Aμ,ν 4 Thus, F0r = Er , F12 = −B3 , F23 = −B1 , F31 = −B2 We get 1 2 (E − B 2 ) 2 as required. We compute the canonical momenta: LF =

πr =

∂LF = −F0r ∂Ar,0

and π0 =

∂LF =0 ∂A0,0

This is inconsistent with the canonical commutation relations [Aμ (t, r), πν (t, r' )] = iδνμ δ 3 (r − r' )

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Hence we must use the Dirac brackets which are modifications of the Poisson/Lie bracket when constraints are taken into account. The first constraint is π0 = 0 and the second constraint is obtained from the equations of motion: ∂r

∂LF = −J 0 ∂A0,r '

where J μ the four current density is a matter field unlike the Aμ s which are the Maxwell fields. These equations can be expressed as χ1 = ∂r πr + J 0 = 0 and since time derivatives do not appear here, this equation should be regarded as a constraint, ie, a relationship between the matter field and the electromag­ netic field. The above equation is obtained by adding to the field Lagrangian density, the matter-field interaction Lagrangian Lint = −J μ Aμ Now, we work in the Coulomb gauge (we are free to impose a gauge condition on the potentials that leaves the actual electric and magnetic field invariant). In this gauge, divA = 0, ie, χ2 = Ar,r = 0 The Maxwell equations in this gauge imply that ∇2 A0 = −J 0 which has solution 0

A (t, r) =

∫

J 0 (t, r' ) 3 ' d r |r − r' |

and since both sides of the above equation are taken at the same time, we can regard A0 as a matter field. Hence, the quantized electromagnetic field is described by only three position fields Ar , r = 1, 2, 3 and once we impose the Coulomb gauge condition, there are only two degrees of freedom for the position fields. We now calculate the field Hamiltonian when it interacts with matter. The Hamiltonian density is with L = LF + Lint , H = πr Ar,0 − L = −F0r Ar,0 + (1/4)Fμν F μν + J μ Aμ 1 F0r (Ar,0 + A0,r ) + (1/4)Frs Frs + J μ Aμ 2 Thus, making use of the matter equation A0,rr = −J 0 , the constraint χ1 and neglecting a 3-dimensional divergence (which will not affect the total Hamilto­ nian), we get H=

1 F0r F0r + F0r A0,r + (1/4)Frs Frs + J μ Aμ 2

312

General Relativity and Cosmology with Engineering Applications = π 2 /2 + (∇ × A)2 /2 − F0r,r A0 + J μ Aμ = (1/2)(π 2 + (∇ × A)2 ) + πr,r A0 + J μ Aμ = (1/2)(π 2 + (∇ × A)2 ) − J 0 A0 + J μ Aμ

The term J 0 A0 is a pure matter field while the two terms within the bracket are pure field terms. This simplifies to H = (1/2)(π 2 + (∇ × A)2 ) − J.A where J.A = J r Ar There is no pure matter term in this Hamiltonian. We define π⊥ = π − ∇A0 ie

π⊥r = πr − A0,r

Then,

divπ⊥ = πr,r − A0,rr = −J 0 + J 0 = 0

Thus π⊥ is a solenoidal field. Then, we can express H=

1 2 (π + (∇ × A)2 ) − J.A + (∇A0 )2 /2 − − − (1) 2 ⊥

since the term (∇A0 , π⊥ ) on performing a 3 − D integration is zero because it is a perfect 3-D divergence: (∇A0 , π⊥ ) = div(A0 π⊥ ) (1) is our final form of the Hamiltonian of the electromagnetic field interacting with an external current source. We note that the last term (∇A0 )2 /2 is a pure matter field. Hence, if we are bothered only about the electromagnetic field and its interaction with matter, the Hamiltonian density is HF = (1/2)(π 2 + (∇ × A)2 ) where we have renamed π⊥ as π for convenience of notation. Our constraints are divπ = 0, divA = 0. Dirac brackets for constraints: Suppose Q1 , ..., Qn , P1 , ..., Pn are the uncon­ strained positions and momenta of a system. The constraints are Qj = Pj = 0, j = n+1, ..., n+p. Without loss of generality, we are choosing our constrained variables as new positions and momenta. The Poisson bracket relations are {f, g} =

n+p ∑ i=1

f,Qi g,Pi − f,Pi g,Qi )

General Relativity and Cosmology with Engineering Applications

313

In particular, we get the contradiction f,Qi = −f,Pi , {f, Pi } = f,Qi , i > n since Qi = Pi = 0, i > n. In order to rectify this problem, Dirac introduced a new kind of bracket defined as follows: Let χij = {ηi , ηj } = Jij J is the standard symplectic matrix of size 2p × 2p. where η = [Qn+1 , ..., Qn+p , Pn+1 , ..., Pn+p ]T Qn+i , Pn+i , i = 1, 2, ..., p, ie ηi are functions of Qi , Pi , i = 1, 2, ..., n and the bracket {., .}ef f is calculated using Qi , Pi , i = 1, 2, ..., n and regarding Qn+i , Pn+i as functions of Qj , Pj , j ≤ n. The bracket {f, g}P is computed using Qi , Pi , i ≤ n and taking Qn+i = 0, Pn+i = 0: {f, g}P =

n ∑

(f,Qi g,Pi − f,Pi g,Qi )

i=1

We have

C = χ−1 = −J

as 2p × 2p matrices. Then, the Dirac bracket is defined as ∑ {f, g}D = {f, g} + {f, ηi }Jij {ηj , g} i,j

We see that for k ≤ n, {f, Qk }D = {f, Qk } = −f,Pk since {ηj , Qk } = 0, k ≤ n Note that {., .} is the unconstrained Poisson bracket. Again, we note that {f, Pk }D = {f, Pk } = f,Qk since {Pk , ηj } = 0, k ≤ n. Further, for i, j ≥ 1, we have ∑ {f, ηk }Jkl {ηl , ηi } {f, ηi }D = {f, ηi } + k,l

= {f, ηi } −

∑

{f, ηk }Ckl χli = 0

k,l

since

∑ l

Ckl χli = δki

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General Relativity and Cosmology with Engineering Applications

We note that {f, g}D = {f, g} − =

∑

(f,Qi +

+ =

i≤n

i,j

(f,ηj ηj,Qi ))(g,Pi +

∑

j

i≤n

∑

∑

∑ {f, ηi }Jij {ηj , g}

∑

g,ηj ηj,Pi )

j

−interchangeof f andg ∑ {f, ηi }Jij {ηj , g} f,ηi Jij g,ηj +

i

(f,Qi +

∑

(f,ηj ηj,Qi ))(g,Pi +

∑

j

g,ηj ηj,Pi )

j

This formula tells us that the Dirac bracket between two observables is cal­ culated using the Poisson bracket w.r.t. the unconstrained variables and by regarding the constrained variables as functions of the unconstrained variables. More generally, suppose Q = (Q1 , ..., Qn ), P = (P1 , ..., Pn ) are arbitrary canonical coordinates and the constraints are of the form χi (Q, P ) = 0, i = 1, 2, ..., r Define the Poisson bracket {., .} as usual w.r.t Q, P , and then define the Dirac bracket r ∑ {f, g}D = {f, g} − {f, χi }Cij {χj , g} i,j=1

where

((Cij )) = (({χi , χj }))−1

Then, {f, χk }D = {f, χk } −

∑

{f, χi }Cij {χj , χk }

i,j

= {f, χk } −

∑

{f, χi }δik = 0

i

as required. Further, {f, Qk }D = −f,Pk −

∑

f,Pi Cij χj,Pk

i,j

It is then not hard to show that the rhs is the same as −dPk f , ie, the partial derivative of f w.r.t. Pk where we regard f as an independent function of Q, P and the χ'i s and then defined ∑ dPk f = f,Pk + f,χj χj,Pk j

Now, we evaluate the Dirac bracket between πi and Aj taking into account the constraints: χ1 = πi,i = 0, χ2 = Ai,i = 0

General Relativity and Cosmology with Engineering Applications We get

315

{χ1 (t, r), χ2 (t, r' )} = i∇2 δ 3 (r − r' )

The inverse kernel of the rhs is K(r − r' ) = i/4π|r − r' |. Further, 3 3 (r − r' ) = iδ,m (r − r' ) {Am (t, r), χ1 (t, r' )} = −iδkm δ,k

{Am (t, r), χ2 (t, r' )} = 0, {χ1 (t, r), πm (t, r' )} = 0, k 3 3 δ,k (r − r' ) = iδ,m (r − r' ) {χ2 (t, r), πm (t, r' )} = iδm

Hence, {Am (t, r), πk (t, r' )}D ∫ = iδkm δ 3 (r−r' )− d3 r'' d3 r''' {Am (t, r), χ1 (t, r'' )}K(r'' −r''' ){χ2 (t, r''' ), πk (t, r' )} ∫ 3 3 = iδkm δ 3 (r − r' ) + d3 r'' d3 r''' δ,m (r − r'' )K(r'' − r''' )δ,k (r''' − r' ) = iδkm δ 3 (r − r' ) −

∫ (

∂2 K(r'' − r''' ))δ 3 (r − r'' )δ 3 (r''' − r' )d3 r'' d3 r''' ∂xm'' ∂xk'''

= iδkm δ 3 (r − r' ) −

∂2 K(r − r' ) ∂xm ∂xk'

= iδkm δ 3 (r − r' ) + K,mk (r − r' ) where K(r) = i/4π|r| Quantum electrodynamics using creation and annihilation operators for pho­ tons, electrons and positrons: We work in the Coulomb gauge so that divA = 0 and this implies ∇2 A0 = −J 0 , ie, A0 is a matter field. The Maxwell wave equation for A in the absence of matter, ie charge and current densities is given by ∇2 A − A,0 = 0 and the general solution to this is ∫ Ak (t, r) =

er (K, σ)[a(K, σ)exp(−i(|K|t−K.r))+¯ er (K, σ)a(K, sigma)∗ exp(i(|K|t−K.r))]d3 K

Here, the summation is over σ = 1, 2 corresponding to only ∑3 two linearly inde­ pendent polarizations of the photon, ie, divA = 0 implies r=1 K r er (K, σ) = 0. The energy of the electromagnetic field in the Coulomb gauge is ∫ ∫ HF = (1/2) (E 2 + B 2 )d3 x = (1/2) [(A2,t + (∇ × A)2 ]d3 x

316

General Relativity and Cosmology with Engineering Applications ∫ =

2|K|2 |e(K, σ)|2 a(K, σ)∗ a(K, σ)d3 K

once we make use of the fact that |K × e(K, σ)| = |K||e(K, σ)|. For this to be interpretable as the sum of energies of harmonic oscillators, each oscillator in the spatial frequency domain having energy |K|, ie, the frequency of the wave. This means that we must have |e(K, σ)| = (2|K|)−1/2 in order to ensure that ∫ HF =

|K|a(K, σ)∗ a(K, σ)d3 K

We can cross check this result as follows. Assuming that the a(K, σ)' s satisfy the canonical commutation relations: [a(K, σ), a(K ' , σ ' )∗ ] = δ 3 (K − K ' )δσ,σ' it follows from the Heisenberg equations of motion that a(t, K, σ),t = i[HF , a(t, K, σ)] = −i|K|a(t, K, σ), a∗ (t, K, σ),t = i[HF , a(t, K, σ)∗ ] = i|K|a∗ (t, K, σ) These equations imply a(t, K, σ),t = −|K|2 a(t, K, σ) a∗ (t, K, σ),tt = −|K|2 a∗ (t, K, σ) which are the correct equations for the spatial Fourier transform of the vector potential arrived from the wave equation. Another way to check these commu­ tation relations which we leave as an exercise, is to start with the Lagrangian density LF = (1/2)(A,t )2 − (1/2)(∇ × A)2 so that the momentum density is πk (t, r) =

∂LF = Ak,t ∂Ak,t

then apply the canonical commutation relations k 3 δ (r − r' ) [Ak (t, r), πm (t, r' )] = iδm

and verify that these relations are satisfied by the above Fourier integral repre­ sentation of A assuming the canonical commutation relations between a(K, σ) and a(K ' , σ ' ). We leave this verification as an exercise to the reader.

General Relativity and Cosmology with Engineering Applications

317

Now consider the second quantized Dirac field described by the four compo­ nent field operators ψ(x), ψ(x)∗ where x = (t, r), t ∈ R, r ∈ R3 . In the absence of any classical or quantum electromagnetic field ,ψ satisfies the Dirac equation [iγ μ ∂μ − m]ψ(x) = 0 or equivalently, [γ μ pμ − m]ψ = 0, pμ = i∂μ The solutions to ψ are plane waves: ∫ ψ(x) = (u(P, σ)a(P, σ)exp(−ip.x) + v(P, σ)b(P, σ)∗ exp(ip.x))d3 P where p.x = pμ xμ = E(P )t−P.r, E(P ) =

√ m2 + P 2 , p = (π μ ) = (E, P ), u(P, σ), v(P, σ) ∈ C4

Here, the summation is over σ = ±1/2 corresponding to the fact that Dirac’s equation can be expressed as [i∂0 − (α, P ) − βm]ψ(x) = 0, P = −i∇ and hence if P denotes an ordinary 3-vector (not an operator), then u(P )exp(−ip.x) satisfies the Dirac equation iff [p0 − (α, P ) − βm]u(P ) = 0 and likewise, v(P )exp(ip.x) satisfies the Dirac equation iff (−p0 + (α, P ) − βm)v(P ) = 0 Thus, u(P ) is an eigenvector of the matrix HD (P ) = (α, P ) + βm with eigenvalue p0 and v(−P ) is an eigenvector of HD (P ) with eigenvalue p0 . Now since HD (P ) is a 4×4 Hermitian matrix, it has four real eigenvalues √ taking all multiplicities into account. These eigenvalues are ±E(P ), E(P ) = m2 + P 2 with each one have a multiplicity of two. We denote the corresponding mutually orthogonal eigenvectors by u(P, σ), v(−P, σ), σ = ±1/2. On applying second quantization, the free Dirac Hamiltonian becomes ∫ HDQ = ψ(x)∗ ((α, −i∇) + βm)ψ(x)d3 x and it is easy to verify that the normalizations of u(P, σ) and v(P, σ) are chosen so that ∫ HDQ = E(P )(a(P, σ)∗ a(P, σ) + b(P, σ)b(P, σ)∗ )d3 P and if we postulate the anticommutation relations {a(P, σ), a(P ' , σ ' )∗ } = {b(P, σ), b(P ' , σ ' )∗ } = δσ,σ' δ 3 (P − P ' ) then and only then we can ensure the canonical anticommutation relations (CAR) {ψl (t, r), πm (t, r' )} = iδlm δ 3 (r − r' )

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where πm is the canonical momentum associated with the canonical position field ψm . From the free Dirac Lagrangian density LD = ψ(x)∗ (i∂0 − (α, −i∇) − βm)ψ(x) = ψ(x)∗ γ 0 (iγ μ ∂μ − m)ψ(x) we infer that πm (x) =

∂LD = iψl (x)∗ ∂ψl,0

so that the CAR gives {ψl (t, r), ψm (t, r' )∗ } = δlm δ 3 (r − r' ) Thus in particular, we can subtract an infinite constant from the second quan­ tized Dirac Hamiltonian to get ∫ HDQ = E(P )(a(P, σ)∗ a(P, σ) − b(P, σ)∗ b(P, σ)) − − − (1) This equation has the following nice interpretation: a(P, σ)∗ creates an electron with momentum P and spin σ, a(P, σ) annihilates an electron with momentum P and spin σ. b(P, σ)∗ creates positron with momentum P and spin σ while b(P, σ) annihilates a positron with momentum P and spin σ. a(P, σ)∗ a(P, σ) is the number operator density for electrons and b(P, σ)∗ b(P, σ) is the number operator for positrons. Since the presence of an additional electron increases the energy of the Dirac sea of electrons by E(P ) while the presence of an additional positron decreases the energy of the Dirac sea by E(P ), equn (1) has the correct physical interpretation for the energy of the second quantized Dirac field. Now suppose we have a collection of photons, electrons and positrons. The total Lagrangian density is then L = LEM + LD + Lint = (−1/4)Fμν F μν + ψ ∗ γ 0 (γ μ (i∂μ + eAμ ) − m)ψ so that

LEM = (−1/4)Fμν F μν , LD = ψ ∗ γ 0 (iγ μ ∂μ − m)ψ, Lint = −J μ Aμ , J μ = −eψ ∗ γ 0 γ μ ψ

J μ is the Dirac four current density. It is easily verified to be conserved even when an electromagnetic field is present. In other words, we can verify using the Dirac equation [γ μ (i∂μ + eAμ ) − m]ψ = 0 that ∂μ J μ = 0

General Relativity and Cosmology with Engineering Applications

319

ie, the current is conserved. We can further show that the matrices K μν = (−1/4)[γ μ , γ ν ] satisfy the same commutation relations as do the standard skew-symmetric gen­ erators of the Lorentz group do. Hence these matrices furnish a representation of the Lie algebra of the Lorentz group. Let D denote the corresponding repre­ sentation of the Lorentz group. D is called the Dirac spinor representation of the Lorentz group and if Λ is any Lorentz transformation, we write D(Λ) = exp(ωμν K μν ) where Λ = exp(ωμν Lμν ) with ω a skew symmetric matrix and Lμν the standard generators of the Lorentz group: (Lμν )αβ = η μα η νβ − η μβ η να Further, we note the following: D(Λ)γ μ D(Λ)−1 = Λμν γ ν and hence, the Dirac equation is invariant under Lorentz transformations ie if xμ → Λμν xν and ψ(x) → D(Λ)ψ(x), Aμ → Λμν Aν , then the Dirac equation remains invariant. Further, the existence of the positron follows from the fact that if we start with the Dirac equation, conjugate it and multiply by the unitary matrix iγ 2 , then we get γ μ )(iγ 2−1 )(−i∂μ + eAμ ) − m]iγ 2 ψ¯ = 0 [(iγ 2 )(¯ It is easily verified that this equation is the same as [γ μ (i∂μ − eAμ ) − m]ψ˜ = 0 where

ψ˜ = iγ 2 ]ψ¯

In other words ψ˜ satisfies the Dirac equation in an electromagnetic field but with the charge e replaced by −e or equivalently, −e replaced by e. This observation led Dirac to conclude the existence of the positron, namely the antiparticle of the electron, having the same mass but opposite charge as that of the electron. The positron was discovered in an accelerator later by Anderson. Another property of the Dirac equation in an external electromagnetic field is obtained by considering (γ μ (i∂μ + eAμ ) + m)(γ ν (i∂ν + eAν ) − m)ψ = 0 If Aμ = 0, this reduces to the free KG equation [∂μ ∂ μ + m2 ]ψ = 0

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since

{γ μ , γ |nu } = 2η μν

In case however Aμ /= 0, we get [γ μ γ ν (i∂μ + eAμ )(i∂ν + eAν ) − m2 ]ψ = 0 or equivalently, ((1/2){γ μ , γ ν } + (1/2)[γ μ , γ ν ])[(i∂μ + eAμ )(i∂ν + eAν ) − m2 ]ψ = 0 or since {γ μ , γ ν } = 2η μν , we get [(∂μ − eAμ )(∂ μ − ieAμ ) + m2 + (1/4)[γ μ , γ ν ][∂μ − ieAμ , ∂ν − ieAν ]]ψ = 0 or since [∂μ − ieAμ , ∂ν − ieAν ] = −i(Aν,μ − Aμ,ν ) = −iFμν we can write [(∂μ − eAμ )(∂ μ − ieAμ ) + m2 − (i/4)[γ μ , γ ν ]Fμν ]ψ = 0 This equation is the same as the KG equation in an external electromagnetic field obtained by replacing ∂μ by ∂mu − ieAμ except for the last term which dis­ plays explicitly the interaction of the electromagnetic field Fμν (whose non-zero components are the electric and magnetic fields) with the spin of the electron described by the antisymmetric ”spin tensor” (−i/4)[γ μ , γ ν ]. Exercise: Write down explicitly in terms of the components of the electric and magnetic fields, the spin-field interaction component and display in par­ ticular, the spin magnetic dipole moment and the spin electric dipole moment.

The photon and electron propagator: We make this calculation using first the Feynman path integral for fields and then leave as an exercise to demonstrate the same result using the operator expansion of the fields. First, note that the photon propagator is defined in space-time as Dμν (x, y) =< 0|T {Aμ (x)Aν (y)}|0 > and the electron propagator as Slm (x, y) =< 0|T {ψl (x)ψm (y)∗ }|0 > Here we are using the Lorentz gauge in which case even A0 is a component of the electromagnetic field potential, not a matter field. In the absence of matter, Aμ (x) satisfies the wave equation ∂μ ∂ μ Aα = 0

General Relativity and Cosmology with Engineering Applications

321

and this equation has solutions ∫

√ √ [a(K, σ)eμ (K, σ)exp(−ik.x)/ 2|K|+a(K, σ)∗ e¯μ (K, σ)exp(ik.x)/ 2|K|]d3 K

Aμ (x) =

where k.x = kμ xμ = |K|t − K.r The Lorentz gauge condition ∂μ Aμ = 0 implies kμ eμ (K, σ) = 0 which means that eμ has three degrees of freedom. Formally, we do not take the canonical momentum π0 as zero, even though the Lagrangian density LEM of the electromagnetic field does not depend on A0,0 and hence implies π0 = 0. The way out is to introduce a small perturbing Lagrangian density to the Lagrangian density of the electromagnetic field involving A0,0 replace LEM by this perturbed Lagrangian density and define the canonical momenta as πμ =

∂LEM ∂Aμ,0

Then, we introduce the commutation relations [Aμ (t, r), Aν (t, r' )] = δνμ δ 3 (r − r' ) This is satisfied provided [a(K, σ), a(K ' , σ ' )∗ ] = δσ,σ' δ 3 (K − K ' ) Then, we find that since a(K, σ)|0 >= 0, < 0|a(K, σ)∗ = 0, we have ∫ eν (K ' , σ ' ) Dμν (x, x' ) = θ(t − t' ) < 0|a(K, σ)a(K ' , σ ' )∗ |0 > eμ (K, σ)¯

∫ +

×(exp(−i(k.x − k ' .x' ))/2|K|)d3 Kd3 K ' θ(t' −t) < 0|a(K ' , σ ' )a(K, σ)∗ |0 > eν (K ' , σ ' )¯ eμ (K, σ)(exp(i(k.x−k ' .x' ))/2|K|)d3 Kd3 K '

= θ(t−t' )

∫

+θ(t' −t)

eμ (K, σ)¯ eν (K ' , σ ' )δσ,σ' δ 3 (K−K ' )/2|K|)(exp(−i(k.x−k ' .x' ))/2|K|)d3 Kd3 K '

∫ ∫

=

eν (K ' , σ ' )¯ eμ (K, σ)δσ,σ' δ 3 (K−K ' )(exp(i(k.x−k ' .x' ))/2|K|)d3 Kd3 K ' [eμ (K, σ)¯ eν (K, σ)θ(t − t' )exp(−i|K|(t − t' ) + iK.(r − r' ))+

eμ (K, σ)θ(t' − t)exp(i|K|(t − t' ) − iK.(r − r' ))]d3 K/2|K| eν (K, σ)¯

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The normalization condition er (K, σ)¯ es (K, σ) = δrs , r, s = 1, 2, 3 (summation over σ = 1, 2, 3) had to be imposed (look at the treatment in the Coulomb gauge) in order to guarantee the correct commutation relations for the creation and annihilation operators following from the canonical commutation relations (CCR) for the position and momentum fields. From this we get ∫ Drs (x, x' ) = δrs (2|K|)−1 [θ(t−t' )exp(−ik.(x−x' ))+θ(t' −t)exp(ik.(x−x' ))]d3 K where k 0 = |K|. This is a function of only x − x' and so, we can denote it by Drs (x − x' ). Now, consider k 0 to be a variable and consider the identity ∫ dk 0 /(k 02 − |K|2 ) = πi/|K| Γ

where Γ is a contour along the real axis from −∞ to +∞ making a small encirclement of the pole at |K| above the real axis but excluding the pole at −|K| and then completed into a big infinite semicircle below the real axis (so that on this semicircle, k 0 has a negative imaginary part). Then it is clear that for t > t' , the contour integral ∫ exp(−ik.(x − x' ))dk 0 /(k 02 − |K|2 ) = iπexp(−i|K|(t − t' ) + iK.(r − r' ))/|K| Γ

and likewise for the other term t' > t. Thus, it follows from the above formula that ∫ −δrs Drs (x).exp(−ik.x)d4 x = 02 k − |K|2 + iε and to preserve Lorentz invariance, we may assume ∫ ημν ημν ˆ Dμν (k) = Dμν (x).exp(−ik.x)d4 x = 02 = 2 k − |K|2 + iε k + iε where

k 2 = k 02 − |K|2 = kμ k μ

We can repeat this calculation for the electron propagator and show that ∫ Sˆlm (p) = Slm (x)exp(−ip.x)d4 x = (γ 0 γ μ pμ − m)−1 These results can be derived directly from the FPI: ∫ ∫ Dαβ (y, z) = exp((−i/4) Fμν (x)F μν (x)d4 x)Aα (y)Aβ (z)DAμ Fμν (x)F μν (x) = (Aν,μ − Aμ,ν )(Aν,μ − Aμ,ν )

General Relativity and Cosmology with Engineering Applications

323

= 2Aν,μ Aν,μ − 2Aν mu Aμ,ν = −2Aν ∂μ ∂ μ Aν + 2Aν ∂ ν ∂μ Aμ on ignoring perfect divergences which do not contribute to the action integral. Thus, ∫ S[A] = (−1/4) Fμν (x)F μν (x)d4 x = ∫

K μν (x − y)Aμ (x)Aν (y)d4 xd4 y

or equivalently in the Fourier domain ∫ ˆ μν (k)Aˆμ (k)Aˆν (k)∗ d4 k S[A] = K where

ˆ μν (k) = k 2 η μν − k μ k ν K

ˆ (k) is singular and hence its inverse cannot be evaluated. However, The matrix K ˆ μν (k)kν = we evaluate its pseudo-inverse and use it as the propagator, or since K 0, we use as the photon propagator, the solution to the equation ˆ (k) = P (k) ˆ (k)D K where P (k) is the orthogonal projection onto the spatial variable subspace, ie, Pμν (k) = I − kμ kν /k 2 or equivalently,

P μν (k) = I − k μ k ν /k 2

We are using the fact that the second moment matrix of a Gaussian distribution is its covariance matrix, ie, in this case, the inverse/pseudo-inverse of the matrix ˆ (k). For the electron propagator, we find via the operator formalism that K taking ∫ ψl (x) = (ul (p, σ)a(p, σ)exp(−ip.x) + vl (p, σ)b(p, σ)∗ exp(ip.x))d3 P with

p0 = E(P ) =

√ P 2 + m2

and u(P, .), v(−P, .) eigenfunctions of the free Dirac Hamiltonian (α, P ) + βm, αr = γ 0 γ r , β = γ 0 , normalized in such a way that the CAR for ψl , πl = iψl imply {a(P, σ), a(P ' , σ ' )∗ } = δ 3 (P − P ' )δσ,σ' , {b(P, σ), b(P ' , σ ' )∗ } = δ 3 (P − P ' )δσ,σ' , {a(P, σ), b(P ' , σ ' )} = 0, {a(P, σ), b(P ' , σ ' )∗ } = 0

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General Relativity and Cosmology with Engineering Applications

the electron propagator is given by Slm (x, y) =< 0|T (ψl (x)ψm (y)∗ )|0 >= ∫ ∑ [ (ul (P, σ)um (P, σ)∗ exp(−ip.(x−y))+vl (P, σ)vm (P, σ)∗ )exp(ip.(x−y))]d3 P l

∫ =

where

Sˆlm (p)exp(ip.(x − y))d4 p

Sˆ(p) = (γ 0 γ μ pμ − m + iε)−1

We leave this derivation as an exercise to the reader. Alternatively using the FPI, ∫ ∫ Slm (x, y) = exp(i ψ(x)∗ (iγ 0 γ μ ∂μ − m)ψ(x)d4 x)ψl (y)ψm (z)∗ DψDψ ∗ gives directly the answer using the standard formula for the second moment of a Gaussian distribution. Now we discuss interactions. Suppose the initial state of the field is |i >= |pim , σim , p'il , σil' , kin , sin , m = 1, 2, ..., Ni1 , l = 1, 2, ..., Ni2 , n = 1, 2, ..., Ni3 > ' are where pim , σim are the four momenta and spins of the mth electron, p'il , σil the four momenta and spins of the lth positron and kin , sin are the four momenta and helicities of the nth photon. Then, we can write

|i >= Πm,l,n a(pim , σim )∗ b(p'il , σil' )∗ c(kin , sin )∗ |0 > where we are using the notation c(k, s) for the photon annihilation operators. Likewise for the final state. Since transitions in the state occur only because of interactions, we work in the interaction representation in which the Hamiltonian of the electron-positron-photon field is given by ∫ HI (t) = −e Aμ (x)ψ(x)∗ (iγ 0 γ μ ∂μ ψ(x)d3 x

Substituting the operator expressions for the quantum fields Aμ , ψ, ψ ∗ , it follows that HI can be expressed as a trilinear functional of (c(k, s), c(k, s)∗ ), (a(P, σ), b(P, σ)∗ ), (a(P, σ)∗ , b(P, σ)).

The transition probability amplitude from |i > at time t → −∞ to |f > at time t → +∞ is given by ∫ ∞ < f |T {exp(−i HI (t)dt)}|i > = δ(f − i) +

∞ ∑ n=1

∫

−∞

(−i)n −∞ as Fourier integrals of the position ∫ space field operators. To proceed further, we observe that (LEM + LD )d4 x is a quadratic functional of the position fields and Lint is small. So perturbation theory gives for the above FPI, ∫

∫ exp(i

(LEM +LD )d4 x)(1+i(

∫

∫ Lint d4 x)+(i2 /2!)( Lint d4 x)2 +...)F (ψ, ψ ∗ , Aμ )DψDψ ∗ DAμ

∫

where Lint = e

ψ ∗ (iγ 0 γ μ ∂μ − m)ψAμ d4 x ∫ =−

J μ Aμ d4 x

Then the various terms in the integral are evaluated using the standard ∫ formu­ lae for the moments of a Gaussian distribution on noting that exp(i (LEM + LD )d4 x) is a Gaussian density functional, it being a quadratic functional of

326

General Relativity and Cosmology with Engineering Applications

ψ, ψ ∗ , Aμ . The propagators then enter naturally into the picture when we ex­ press the higher moments of even order as products of second moments, ie propagators. Renormalization, an example: Consider the Hamiltonian H = H0 − g|e >< e| Let |k >, k = 1, 2, ... denote the energy eigenstates of H0 with H0 |k >= Ek |k > Let |ψ > denote an energy eigenstate of H with energy eigenvalue E. Then, H|ψ >= E|ψ > gives H0 |ψ > −g|e >< e|ψ >= E|ψ > or

|ψ >= g(H0 − E)−1 |e >< e|ψ > ∑ =g |k > (Ek − E)−1 < k|e >< e|ψ > k

The normalization condition < ψ|ψ >= 1 then implies ∑ g2 (Ek − E)−2 | < k|e > |2 | < e|ψ > |2 = 1 k

This implies that | < e|ψ > |2 = (g 2

∑

(Ek − E)2 | < k|e > |2 )−1

k

If the above sum is divergent, then we would get < e|ψ >= 0 which may nor be the case. To avoid such divergences, we make an ultraviolet cutoff meaning thereby that the sum over k is truncated to k such that Ek ≤ Λ where Λ is a finite positive constant. For a given ultraviolet cutoff Λ, we may thus define a renormalized coupling constant g = g(Λ) so that ∑ g 2 (Λ) = ( (Ek − E)2 < k|e > |2 )−1 k:Ek ≤Λ

and get with this cutoff imposed that | < e|ψ > |2 = 1 Thus, divergence problems are avoided by redefining the coupling constant. In quantum field theory, when we calculate the transition probability amplitudes like vacuum polarization, self energy of the electron or anomalous magnetic

General Relativity and Cosmology with Engineering Applications

327

moment of the electron, the integrals obtained using the Feynman diagrams for the various Dyson series terms diverge. So to ge meaningful answers, we renormalize the fields and coupling constants like charge and mass, by scaling these with constant factors and then split the resulting Lagrangian density into a term not involving the coupling constants Z and an interaction term involv­ ing the Lagrangian terms scaled by Z − 1. The latter terms are regarded as perturbations and we expand the resulting exponential in powers of Z − 1 and calculate the modified matrix elements or propagators. Finally, Z may be made to tend to infinity in such a way so as to cancel out the infinities arising in the matrix elements or propagators computed without the Z. This method was first demonstrated by Dyson to lead to the experimentally correct values for the above phenomena. [33] Dirac’s wave equation in a gravitational field. [34] Canonical quantization of the gravitational field. Let Λ(x) be a local Lorentz transformation and let Λ → D(Λ) be the Dirac spinor representation of the Lorentz group. Let gμν (x) be the metric of curved space-time and ηab the Minkowski metric of flat space-time. Let Vμa (x) be the associated tetrad, ie, ηab Vμa (x)Vνb (x) = gμν (x) ((Vμa (x))) can be regarded as a locally inertial frame, ie, ξ μ → Vμa ξ μ = ξ a transforms a vector field ξ μ (x) in curved space-times to a Minkowski vector, ie each component ξ a is a scalar. Let ((γ a )) be the Dirac Gamma matrices. They determine the Dirac equation in flat space-time: (iγ μ ∂μ − m)ψ = 0 orin the presence of an electromagnetic field, (γ μ (i∂μ + eAμ (x)) − m)ψ = 0 This equation is invariant under global Lorentz transformations ,ie, if Λ is a constant 4 × 4 Lorentz transformation matrix, then D(Λ)(γ μ (i∂μ + eAμ ) − m)ψ = 0 implies [D(Λ)γ μ D(Λ)−1 (i∂μ + eAμ ) − m]D(Λ)ψ = 0 which implies [Λμν γ ν (i∂μ + eAμ ) − m]D(Λ)ψ = 0 or equivalently, [γ ν (i∂ν' + eA'ν (x' )) − m]ψ ' (x' ) = 0 where

'

x μ = Λμν xν ,

328

General Relativity and Cosmology with Engineering Applications

so that

i∂ν' = Λμν ∂μ

and

A'ν (x' ) = Λμν Aμ (x), ψ ' (x' ) = D(Λ)ψ(x)

ie, ψ ' (x' ) = D(Λ)ψ(x) satisfies Dirac’s equation in the Lorentz transformed sys­ ' tem (x μ ) with the electromagnetic field also transformed in accordance with the Lorentz transformation Λ. Note that the Lorentz generators of the Dirac representation are given by J μν = (i/4)[γ μ , γ ν ]. We wish to define a Dirac equa­ tion in curved space-time that is invariant under local Lorentz transformations in accordance with the equivalence principle of general relativity. To do this, we must use the tetrad which will transform the non-inertial metric to the inertial metric. So we assume our curved space-time Dirac equation to be given by [γ a Vaμ (i∂μ + iΓμ (x) + eAμ (x)) − m]ψ(x) = 0 where Γμ (x) is a 4×4 matrix interpreted as the gravitational connection of spacetime in the Dirac spinor representation. Applying a local Lorentz transformation D(Λ(x)) gives [D(Λ(x))γ a D(Λ(x))−1 Vaμ (x)(iD(Λ(x))(∂μ + Γμ (x))D(Λ(x))−1 +eAμ (x) − m]D(Λ(x))ψ(x) = 0 or [Λab (x)γ b Vaμ (x)(i(∂μ + D(Λ(x))Γμ (x)D(Λ(x))−1 ) + iD(Λ(x))(∂μ D(Λ(x))−1 ) +eAμ (x)) − m]D(Λ(x))ψ(x) = 0 If we define

'

Vaμ (x) → Λba (x)Vbμ (x) = Vbμ (x) as the transformation of the tetrad under the local Lorentz transformation Λ(x), then we can express the above equation as '

[γ b Vbμ (x)(i(∂μ + Γμ' (x)) + eAμ (x)) − m]D(Λ(x))ψ(x) = 0 where Γμ' (x) is the transformed gravitational connection under the local Lorentz transformation Λ(x): Γ'μ (x) = D(Λ(x))Γμ (x)D(Λ(x))−1 + D(Λ(x))(∂μ D(Λ(x))−1 ) Equivalently, if ω(x) = ((ωab (x))) is an infinitesimal local Lorentz transforma­ tion so that Λ(x) = I + dD(ω(x)) = I + ωab (x)J ab and with neglect of O(|| ω(x) ||2 ) terms, D(Λ(x))−1 = I − ωab (x)J ab

General Relativity and Cosmology with Engineering Applications

329

then under the infinitesimal local Lorentz transformation Λ(x), the gravitational connection Γμ (x) transforms to Γ'μ (x) = (I + ωab (x)J ab )Γμ (x)(I − ωcd (x)J cd ) −(I + ωab (x)J ab )ωcd,μ (x)J cd = Γμ (x) + ωab (x)[J ab , Γμ (x)] − ωab,μ (x)J ab We need to look for a Dirac gravitational connection Γμ (x) that satisfies such a transormation law under an infinitesimal local Lorentz transformation I + ω(x). It is easily seen that Γμ (x) = J ab Vaν (x)Vνb,μ (x) does the job. Indeed, under I + ω(x), this transforms to J ab (Vaν + ωac V νc )(Vνb + ωbd Vνd ),μ d = J ab Vaν Vνb,μ + J ab (Vaν Vν,μ ωbd + V νc Vνb,μ ωac ) d = Γμ + J db ωbd,μ + J ab (Vaν Vν,μ ωbd + V νc Vνb,μ ωac )

This is seen to coincide with Γμ − ωab,μ J ab + ωab [J ab , Γμ ] on noting that [J ab , Γμ ] = [J ab , J cd ]Vcν Vνd,μ and using the Lie algebra commutation rules for the Lorentz group generators ((J ab )). Finally, we should replace ordinary partial derivatives of the tetrad field by covariant derivatives, ie, Γμ (x) = J ab Vaν (x)Vνb:μ (x)

[35] The scattering matrix for the interaction between photons, electrons, positrons and gravitons. Calculating the scattering matrix using the Feynman path integral and also using the operator formalism with the Feynman diagram­ matic rules. [36] Atom interacting with a Laser; The general theory based on quantum electrodynamics. Representing the quantum electromagnetic field using finite sets of creation and annihilation operators. Representing any density operator for the quantum electromagnetic field via the diagonal Glauber-Sudarshan rep­ resentation. Representing any state of the laser interacting with the spin of an atom using the Glauber-Sudarshan representation having matrix coefficients. Expressing the evolution equation for the density operator of the laser-atom system using the Glauber-Sudarshan representation.

330

General Relativity and Cosmology with Engineering Applications The Hamiltonian of the field is given by HF =

p ∑

ωk ak∗ ak , a = (a1 , ..., ap ), a∗ = (a∗1 , ..., a∗p ),

k=1 ∗ [ak , am ] = δkm

and all the other commutators vanish. The Hamiltonian of the atom is HA an N × N Hermitian matrix and finally, the interaction Hamiltonian between the atom and the field is given by HI (t) =

p ∑

(Ak (t)Fk (a, a∗ ) + Ak (t)∗ Fk (a, a∗ )∗ )

k=1

where Ak (t) is a time varying N × N matrix and Fk' s are ordinary complex valued functions which become field operators when their complex arguments are replaced by the field operators a, a∗ . The evolution of the joint state ρ(t) of the atom and field follows the Schrodinger equation iρ' (t) = [H(t), ρ(t)], H(t) = HF + HA + HI (t) Note that HA and HF commute. For obtaining the interaction representation, we define ρ˜(t) = U0 (t)∗ ρ(t)U (t) where U0 (t) = U0A (t)U0F (t), U0A (t) = exp(−itHA ), U0F (t) = exp(−itHF ) We note that U0F (t)∗ ak U0F (t) = exp(−iωk t)ak = ak (t), U0F (t)∗ a∗k U0F (t) = exp(iωk t)a∗k = ak (t)∗

Thus, U0F (t)∗ HI (t)U0F (t) =

∑

(Ak (t)Fk (a(t), a(t)∗ ) + Ak (t)∗ Fk (a(t), a(t)∗ ))

k

and defining the N × N complex matrices Bk (t) = U0A (t)∗ Ak (t)U0A (t) we get the interaction picture Schrodinger equation for the atom and field ˜ I (t), ρ˜(t)] iρ˜' (t) = [H where ˜ I (t) = U0 (t)∗ HI (t)U0 (t) = H

∑ k

Bk (t)Fk (a(t), a(t)∗ ) + Bk (t)∗ Fk (a(t), a(t)∗ )∗

General Relativity and Cosmology with Engineering Applications

331

We write Fk (a(t), a(t)∗ ) = F ({ak exp(−iωk t)}, {a∗k exp(iωk t)}) = Fk (t, a, a∗ ) so ˜ I (t) = H

∑

(Bk (t)Fk (t, a, a∗ ) + Bk (t)∗ Fk (t, a, a∗ )∗ )

k

and the Schrodinger equation in the interaction picture becomes ˜ I (t), ρ˜(t)] iρ˜' (t) = [H To solve this differential ∑ equation, we adopt ∑ the Galuber-Sudarshan diagonal represention: Let a(z) = k z¯k ak , a(z)∗ = k zk a∗k where z = (zk ) ∈ Cp . Then write ∑ |e(z) >= z n a∗n |0 > /n! = exp(a(z)∗ )|0 > n

with the obvious p-tuple notation. The normalized energy eigenstates of the field are √ |n >= a∗n |0 > / n! and hence |e(z) >=

∑

√ z n |n > / n!

n

Thus, < e(u)|e(z) >= exp(< u|z >) We can normalize |e(z) > by multiplying it by a function φ(z) of z so that ∫ I = |e(z) > φ(z) < e(z)|d2n z We have a(u)|e(z) >=< u|z > |e(z) >, ak |e(z) >= zk |e(z) > Also, a∗k |e(z) >=

∑

z n a∗k a∗n|0 > /n! =

n

∂ |e(z) > ∂z

We can evaluate ak |e(z) >< e(z)| = z¯k |e(z) >< e(z)|, a∗k |e(z) >< e(z)| =

∂ |e(z) >< e(z)| ∂zk

Thus, ∂ )|e(z) >< e(z)| ∂z assuming that in the expression F (t, a, a∗ ), all the a' s appear to the left of all ' the a∗ s. Such a representation is possible in view of the commutation relations ' between the a' s and the a∗ s. Likewise, we have F (t, a, a∗ )|e(z) >< e(z)| = F (t, z,

|e(z) >< e(z)|F (t, a, a∗ )

332

General Relativity and Cosmology with Engineering Applications = |e(z) > (F (t, a, a∗ )∗ |e(z) >)∗

and F (t, a, a∗ )∗ |e(z) >= F¯ (t, a∗ , a)|e(z) > where now in the expression for F¯ (t, a∗ , a) all the a∗ s will appear to the left of all the a' s. We thus get ∂ , z¯)|e(z) >< e(z)| |e(z) >< e(z)|F (t, a, a∗ ) = F¯ (t, ∂z¯ Here, F¯ (t, u, v) is the function obtained by conjugating all the coefficients in the Taylor series expansion of F (t, .). Now, with this understanding, we note that ˜ I (t) = H

∑

(Bk (t)Fk (t, a, a∗ ) + Bk (t)∗ Fk (t, a, a∗ )∗ )

k

can be written as ˜ I (t) = H

∑

Ck (t)Gk (t, a, a∗ )

k

G'k s

'

'

where in the all the a s appear to the left of all the a∗ s. Then, by the above observation, [Gk (t, a, a∗ ), |e(z) >< e(z)|] = [Gk (t, z,

∂ ∂ ¯ k (t, z, )−G ¯ )]|e(z) >< e(z)| ∂z ∂z¯

∂ ∂ It should be noted that ∂z acts only on the factor |e(z) > while ∂z ¯ acts only on the factor < e(z)| in the term |e(z) >< e(z)|. This is because z → |e(z) > is an analytic Hilbert space valued function of the complex variable z and z¯ →< e(z)| is therefore an analytic function of the complex variable z¯. Using

< e(u), e(z) >= exp(< u, z >) it is easy to show that φ(z) = π −p exp(|| z ||2 ) Indeed, then a simple Gaussian integral evaluation gives ∫ ∫ 2p < e(u)| φ(z)|e(z) >< e(z)|d z|e(v) >= φ(z)exp(< u|z > + < z|v >)d2p z = 1 We can express the joint state of the atom and the field in the interaction representation as a Glauber-Sudarshan integral: ∫ ρ˜(t) = ψ(t, z, z¯) ⊗ |e(z) >< e(z)|d2p z where ψ˜(t, z, z¯) ∈ CN ×N

General Relativity and Cosmology with Engineering Applications We then get ρ˜' (t) =

∫

333

∂ψ(t, z, z¯) ⊗ |e(z) >< e(z)|d2p z ∂t

and further, ∑∫

˜ I (t), ρ˜(t)] = [H [Ck (t)ψ(t, z, z¯)⊗Gk (t, a, a∗ )|e(z) >< e(z)|−ψ(t, z, z¯)Ck (t)⊗|e(z) >< e(z)|Gk (t, z, z ∗ )]d2p z

k

∑∫

¯ k (t, z.∂/∂ [Ck (t)ψ(t, z, z¯)⊗Gk (t, z, ∂/∂z)|e(z) >< e(z)|−ψ(t, z, z)C ¯ k (t)⊗G ¯ z)|e(z) ¯ >< e(z)|]d2p z

k

=

∑∫

¯ k (t, z, Ck (t)(Gk (t, z, ∂/∂z)T − G ¯ ∂/∂ z) ¯ T )ψ(t, z, z) ¯ ⊗ |e(z) >< e(z)|d2p z

k

where we have used integration by parts. Here for example, (z1m1 z2m2 ...zpmp = (−1)n1 +...+np

∂ n1 +...+np T n ) ∂z1n1 ...∂zp p

∂ n1 +...+np m1 mp ...zp n z ∂z1n1 ...∂zp p 1

It follows that ψ(t, z, z¯) satisfies the pde i ∑

∂ψ(t, z, z¯) = ∂t

¯ k (t, z, ¯ ∂/∂ z) ¯ T )ψ(t, z, z) ¯ Ck (t)(Gk (t, z, ∂/∂z)T − G

k

=

∑ ¯ k (t, z, ¯ ∂/∂ z) ¯ T )Ck (t)ψ(t, z, z) ¯ (Gk (t, z, ∂/∂z)T − G k

Remark: The following notation has been used here: If G(t, z, u) is a poly­ nomial in variables z, u which may even be noncommuting operators, then by ¯ z, u), we mean the polnomial obatined from G(t, z, u) by replacing its com­ G(t, plex coefficients by their respective conjugates without making any change in the variables z, u. [37] The classical and quantum Boltzmann equations. Quantum Boltzmann equation: Let Hi , i = 1, 2, ..., N be Hilbert spaces and let H=

N ⊗

Hk

k=1

Let ρ(t) be a density operator in H. Let the Hamiltonian according to which ρ evolves have the form H=

N ∑ k=1

Hk +

∑ 1≤k 0, the second marginal state ρ12 (t)will be a small perturbation of ρ1 (t) ⊗ ρ2 (t) (Note that, we are assuming that ρ2 (t) is an identical copy of ρ1 (t)). Thus, if the interaction potential V12 is small, then in (3), we can replace [V12 , ρ12 (t)] by [V12 , ρ1 (t) ⊗ ρ2 (t)]. Then, the approximate solution to (3) is given by ∫ ρ12 (t) = exp(−itad(H1 +H2 ))(ρ12 (0))+

t

0

exp(−i(t−s)ad(H1 +H2 ))ad(V12 )(ρ1 (s)⊗ρ1 (s))ds−−−(4)

and this can be substituted into (2) to get the following nonlinear integro­ differential equation for the first marginal ρ1 (t): iρ'1 (t) = [H1 , ρ1 (t)] + (N − 1)T r2 [ad(V12 )T12 (t)(ρ12 (0))]+ ∫ t ad(V12 )T12 (t − s)ad(V12 )(ρ1 (s) ⊗ ρ1 (s))ds] − − − (5) (N − 1)T r2 [ 0

where T12 (t) = exp(−itad(H1 +H2 )) = exp(−itad(H1 )).exp(−itad(H2 )) = exp(−itad(H2 )).exp(−itad(H1 ))

since H1 and H2 commute. (5) may be termed as the quantum Boltzmann equation. Other versions of this equation exist like we can solve (3) to get ρ12 (t) = exp(−itad(H12 ))(ρ12 (0)) where H12 = H1 + H2 + V12

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(2) then gives iρ'1 (t) = ad(H1 )(ρ1 (t)) + (N − 1)T r2 ad(V12 )exp(−itad(H12 )(ρ12 (0)) − − − (6) The third version is to note that the last term in (4) already contains a mul­ tiplicative factor ad(V12 ) and hence since we are interested in terms only upto linear orders in V12 , we can replace ρ1 (s) in (4) by exp(−is.ad(H1 ))(ρ1 (0)) (4)then becomes ρ12 (t) = exp(−itad(H1 + H2 ))(ρ12 (0))+ ∫

t 0

exp(−i(t − s)ad(H1 + H2 ))ad(V12 )exp(−isad(H1 + H2 ))(ρ1 (0) ⊗ ρ1 (0))ds

Thus, (2) can be approximated by iρ'1 (t) = [H1 , ρ1 (t)] + (N − 1)T r2 [ad(V12 exp(−it.ad(H1 + H2 ))(ρ12 (0))]+ ∫ t exp(−i(t − s)ad(H1 + H2 ))ad(V12 ) (N − 1)T r2 [ad(V12 ) 0

×exp(−isad(H1 + H2 ))(ρ1 (0) ⊗ ρ1 (0))ds]

[38] Bands in a semiconductor: Derivation using the Bloch wave functions in a 3-D periodic lattice. V : R3 → R is the potential of the periodic lattice produced by nuclei located at different sites of the crystal. The Lattice vectors are a1 , a2 , a3 and the periodicity gives V (r + n1 a1 + n2 a2 + n3 a3 ) = V (r + n.a) = V (r), n1 , n2 , n3 ∈ Z V (r) can be expressed as a Fourier series ∑ c(m)exp(2πimT M r) V (r) = m∈Z3

where the reciprocal lattice matrix M is calculated as M [a1 , a2 , a3 ] = I ie

M = A−1 , A = [a1 , a2 , a3 ]

It follows that mT M (n1 a1 + n2 a2 + n3 a3 ) = mT M An = mT n ∈ Z, n ∈ Z3

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This ensures that the above Fourier series for V (r) is periodic along the a1 , a2 , a3 directions. The wave function ψ(r) satisfies [−∇2 /2m + V (r)]ψ(r) = Eψ(r) Changing r to r + An (An = n1 a1 + n2 a2 + n3 a3 ) and using V (r + An) = V (r) gives Hψ(r + An) = Eψ(r + An) so by uniqueness (assuming non-degeneracy), ψ(r + An) = C(n)ψ(r), C(n) ∈ C, |C(n)| = 1 We write Ck = C(ak ), k = 1, 2, 3 Then if Nk nuclei are along the ak direction, k = 1, 2, 3, then by imposing periodicity relations for the wave function at the crystal boundaries, we get CkNk = 1, k = 1, 2, 3 and hence Ck = exp(2πilk /Nk ), k = 1, 2, 3 for some lk = 0, 1, ..., Nk − 1. We define the Bloch wave function corresponding to l = (l1 , l2 , l3 ) by ψ(r) = ul (r)exp(2πilT Kr) = where K is some 3×3 matrix. Then, for ul to be periodic with periods a1 , a2 , a3 , we require that ψ(r + ak )exp(−2πilT Kak ) = ψ(r), k = 1, 2, 3 or equivalently, Ck = exp(2πilT Kak ) Thus we require that exp(2πilk /Nk ) = ex[(2πilT Kak ) and so we can take K = N −1 M = N −1 A−1 , N = diag[N1 , N2 , N3 ] We note that lT N −1 A−1 ak = lT N −1 ek = lk /Nk , k = 1, 2, 3 We write b = b(l) = 2πK T l

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337

Then, ψ(r) = u(r)exp(ibT r), u = ul We substitute this into the Schrodinger equation and derive the pde satisfied by u(r). After that since u is periodic with period A, we can expand it in a Fourier series as we did for V (r): ∑ d(m)exp(2πimT M r) u(r) = ul (r) = m∈Z3

and then derive a difference equation for the Fourier coefficients d(m) of u(r). For each l = (l1 , l2 , l3 ) ∈ ×3k=1 {0, 1, ..., Nk − 1} we thus obtain a sequence of solutions ulk (r), k = 1, 2, ... with energy eigenvalues E = E(l, k), k = 1, 2, .... We say that each l defines an energy band. [39] The Hartree-Fock apporoximate method for computing the wave func­ tions of a many electron atom. The Hamiltonian of the system comprising N particles has the form H=

N ∑

Hk +

∑

Vkj

1≤k= ⊗N k=1 |ψk > where |ψk >∈ Hk . We substitute this into the expression < ψ|H|ψ > for the average energy and extremize this w.r.t. the component wave functions |ψk >, k = 1, 2, ..., N subject to the constraints < ψk |ψj >= δkj . Incorporating these constraints using Lagrange multiplier λ(k, j) gives us the functional to be extremized as ∑ S[{ψk , λ(k, j)}] =< ψ|H|ψ > − λ(k, j)(< ψk |ψj > −δkj ) 1≤k≤j≤N

We observe that < ψ|H|ψ >=

∑

< ψk |Hk |psik > +

∑

< ψk ⊗ ψj |Vkj |ψk ⊗ ψj >

1≤k −δkj )

1≤k≤j≤N

The variational equations

δS/δψk∗ = 0

gives Hk |ψk > +

∑ j:j>k

< Ik ⊗ ψj |Vkj |ψk ⊗ ψj > +

∑ j:j

338

General Relativity and Cosmology with Engineering Applications −λ(k, k)|ψk > −

∑

λ(k, j)|ψj >= 0

j:k |ψ >= (n!)−1/2 σ∈Sn

Carry out for this trial wave function the above extremization of < ψ|H|ψ > subject to the constraints < ψk |psij >= δkj and specialize to the position representation. Next, take into account the spin of each electron so that the component wave function |ψk > depends on both the position rk and spin variable sk = ±1/2. The trial wave function is then ∑ ψσ1,sσ1 (r1 ) ⊗ ... ⊗ ψσn,sσn (rn ) ψ(r1 , s1 , ..., rn , sn ) = (n!)−1/2 σ∈Sn

Note that the constraints are < ψk,s |ψk' ,s' >= δkk' δss'

[40] The Born-Oppenheimer approximate method for computing the wave functions of electrons and nuclei in a lattice. Nucleon positions are Rk , k = 1, 2, ..., N . electron positions associated with the k th nucleus are rkl , l = 1, 2, ..., Z. Nucleon mass is M , electron mass is m. Total Hamiltonian of the system is H = TN + Te + Vee + VN N + VeN

General Relativity and Cosmology with Engineering Applications where TN = TN (R) = −

∑

339

∇2Rk /2M

k

is the total kinetic energy operator of all the nucleons. ∑ Te = Te (r) = − ∇2rkl /2m k,l

is the total kinetic energy of all the electrons. ∑ e2 /2|rkl − rmj | Vee = (k,l)/=(m,j)

is the total electron-electron interaction potential energy. ∑ VN N = Z 2 e2 /|Rk − Rm | k/=m

is the total nucleon-nucleon interaction potential energy. ∑ Ze2 /|Rk − rml | VeN = − k,m,l

is the total electron-nucleon interaction potential energy. We first solve (Te + Vee + VeN )Φ(r, R) = Ee (R)Φ(r, R) ie, the eigenfunctions for the electrons with fixed values of the nuclear positions. The energy levels of the electrons then depend on the nucleon positions R. We then Assume that the total wave function of the electrons and nucleons is Ψ(r, R) = Φ(r, R)χ(R) Substituting this into the complete electron-nucleon eigenvalue equation gives (Te + TN + Vee + VN N + VeN )(Φ(r, R)χ(R)) = EΦ(r, R)χ(R) or using the above electron eigenvalue equation, (TN + VN N + Ee (R))(Φ(r, R)χ(R)) = E(Φ(r, R)χ(R)) or equivalently, Φ(r, R)−1 TN (Φ(r, R)χ(R)) + (VN N + Ee (R))χ(R) = Eχ(R) Now, TN (Φ(r, R)χ(R)) = ∑ (− ∇2Rk /2M )(Φ(r, R)χ(R)) k

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General Relativity and Cosmology with Engineering Applications

= −(2M )−1

∑

[χ(R)(∇2Rk Φ(r, R))+Φ(r, R)∇2Rk χ(R)+2(∇Rk Φ(r, R), ∇Rk χ(R))]

k

[41] The performance of quantum gates in the presence of classical and quan­ tum noise. Suppose we design a quantum gate by perturbing the Hamiltonian H0 of a ∑ quantum system to H0 + δ k fk (t)Vk and running the unitary evolution for a duration of T seconds. Upto O(δ 2 ), the evolution gate at time T is given by U (T ) = U0 (T )W (T ), U0 (T ) = exp(−iT H0 ), W (T ) = I − iδ

∑∫ k

T 0

fk (t)V˜k (t)dt − δ 2

∑∫ k,m

0 where |f >∈ h and |φ(u) >= exp(− || u ||2 /2)|e(u) > where |e(u) > is the exponential vector in the Boson Fock space. Then we claim that E(jt (X)|ηt ) = U (t)∗ Et (X|ηti )U (t) Note that ηti commutes with X and hence ηt commutes with jt (X). To prove the above formula, we observe that if if Z ∈ ηti , then U (t)∗ ZU (t) ∈ ηt and hence E[(jt (X) − U (t)∗ Et (X|ηti )U (t))U (t)∗ ZU (t)] = E[U (t)∗ XZU (t) − U (t)∗ Et (X|ηti )ZU (t)] = E[U (t)∗ (XZ − Et (X|ηti )Z)U (t)] = Et (XZ − Et (XZ|ηti )) = 0 by the definition of conditional expectation. This proves the claim. Now suppose F (t) is a system operator, ie, in L(h) such that Et (X) = E(U (t)∗ XU (t)) = E(F (t)∗ XF (t)) = EF (t) (X) Note that [F (t), ηti ] = 0. It follows clearly that E(F (t)∗ F (t)) = E(U (t)∗ U (t)) = 1

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Then, we claim that EF (t) (X|ηti ) = E(F (t)∗ XF (t)|ηti )/E(F (t)∗ F (t)|ηti ) To prove this, we observe that the lhs and the numerator and denominator of the rhs all commute with each other since ηti is an Abelian algebra. Further, we have for any operator Z ∈ ηti that EF (t) [(X − (E(F (t)∗ XF (t)|ηti )/E(F (t)∗ F (t)|ηti ))Z] = E[F (t)∗ XZF (t) − F (t)∗ (E(F (t)∗ XF (t)Z|ηti )/E(F (t)∗ F (t)|ηti ))F (t)] = E[F (t)∗ XZF (t)] − E[E(F (t)∗ XZF (t)|ηti )F (t)∗ F (t)/E(F (t)∗ F (t)|ηti )] since F (t) commutes with ηti and Z ∈ ηti . Further, conditioning the second term above on ηti and then taking the expectation gives E[E(F (t)∗ XZF (t)|ηti )F (t)∗ F (t)/E(F (t)∗ F (t)|ηti )] = E[E(F (t)∗ XZF (t)|ηti )E(F (t)∗ F (t)|ηti )/E(F (t)∗ F (t)|ηti )] = E(F (t)∗ XZF (t)) We have thus proved that EF (t) [(X − (E(F (t)∗ XF (t)|ηti )/E(F (t)∗ F (t)|ηti ))Z] = 0 and hence the claim. The Hudson-Parthasarathy noisy Schrodinger evolution is described by the qsde β dU (t) = (Lα β dΛα (t))U (t) where summation over the repeated indices α, β ≥ 0 is understood and the basic processes Λα β satisfy quantum Ito’s formula μ μ α dΛα β .dΛν = εν dΛβ

where εμν is zero if either μ = 0 or ν = 0 or μ /= ν and is one otherwise. The system operators Lα β satisfy the following conditions for U (t) to describe a unitary evolution: 0 = d(U ∗ U ) = dU ∗ .U + U ∗ .dU + dU ∗ .dU = β α β β∗ μ β ν U ∗ (Lβ∗ α dΛα + Lβ dΛα + Lα Lν dΛα dΛμ )U

so that β α β β∗ μ β ν (Lβ∗ α dΛα + Lβ dΛα + Lα Lν dΛα dΛμ )U

which gives β∗ ν∗ μ ν + Lα Lα β + Lα Lβ εμ = 0

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359

The Evans-Hudson flow corresponding to this HP equation is obtained by taking a self-adjoint operator X in the system Hilbert space h and setting jt (X) = U (t)∗ XU (t) Then application of quantum Ito’s formula gives djt (X) = dU ∗ XU + U ∗ XdU + dU ∗ XdU = μ α ν ν∗ β U ∗ (Lβ∗ α X + XLβ + εμ Lα XLβ )U.dΛα

= jt (θβα (X))dΛβα where the structure maps θβα are given by θβα (X) = μ α ν ν∗ Lβ∗ α X + XLβ + εμ Lα XLβ

We note that they satisfy the structure equations since jt as defined is a ∗ unital algebra homomorphism. The Belavkin input measurement processes are taken as β Yin,k (t) = cα β [k]Λα , k = 1, 2, ..., r, t ≥ 0 where the α = β = 0 term is omitted. These processes jointly generate an Abelian family of Von-Neumann algebras provided that μ ν β cα β [k]cν [m]dΛα dΛμ

= cβα [k]cνμ [m]εβμ dΛνα is the same when k gets interchanged with m. In other words, we require that β μ cα β [k]εμ cν [m] β μ = cα β [m]εμ cν [k]

which can be expressed in matrix notation as C[k]εC[m] = C[m]εC[k], k, m = 1, 2, ..., r The corresponding output processes are Yout,k (t) = Yk (t) = U (t)∗ Yin,k (t)U (t) and since U (t) is unitary, and Yin,k (t) commute with the system operators, it follows that Yk (t) = U (T )∗ Yin,k (t)U (T ), T ≥ t and hence [Yk (t), js (X)] = 0, s ≥ t

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General Relativity and Cosmology with Engineering Applications

ie the output measurements are non-demolition processes. Let ηt = σ(Yk (s) : s ≤ t, k = 1, 2, ..., r). Then ηt is an Abelian Von-Neumann algebra and belongs to the commutant of the algebra generated by jt (X) as X ranges over the system operators. We write πt (X) = E(jt (X)|ηt ) where the expectation is taken in the state |f φ(u) > with |f >∈ h, < f, f >= 1 and φ(u) = exp(− || u ||2 /2)|e(u) > where u ∈ L2 (R+ ) ⊗ Cd and |e(u) > is the corresponding exponential vector in Γs (L2 (R+ ) ⊗ Cd ). Now we can write dπt (X) = Ft (X)dt + Gkmt (X)(dYm (t))k where the summation in the last term is over m ≥ 1, k ≥ 1 and Ft (X), Gkmt (X) are all ηt measurable operators, ie, they can be regarded as commutative stochas­ tic processes. We can write dYm (t) = dYin,m (t) + dU (t)∗ .dYin,m (t).U (t) + U (t)∗ dYin,m (t).dU (t) = jt (Sβα [m])dΛβα and hence for k ≥ 1 (dYm (t))k = jt (Sβα [m, k])dΛβα (t) where Sβα [k], Sβα [m, k] are system operators, ie, operators in h. These operators are expressible in terms of the system operators {Lα β } and the complex numbers {cβα [k]}. The equations E[(djt (X)−dπt (X))(dYm (t))k |ηt ]+E[(jt (X)−πt (X))(dYm (t))k |ηt ] = 0, k ≥ 1−−−(1) and E[(djt (X) − dπt (X))|ηt ] = 0 − − − (2) follow by taking the differential of the expression E[(jt (X) − πt (X))C(t)] = 0 where C(t) satisfies the qsde ∑ fm,k (t)C(t)(dYm (t))k , C(0) = 1 dC(t) = m,k≥1

and using the arbitrariness of the complex valued functions fm,k (t). We have E[djt (X)(dYm (t))k |ηt ] = E[jt (θβα (X))jt (Sνμ [m, k])dΛβα (t).dΛνμ (t)|ηt ] = εβμ πt (θβα (X).Sνμ [m, k])uν (t)¯ uα (t)dt E[dπt (X)(dYm (t))k |ηt ] =

General Relativity and Cosmology with Engineering Applications

361

Grst (X)E[(dYs (t))r (dYm (t))k |ηt ] = Grst (X)jt (Sβα [s, r]Sνμ [m, k])E[dΛβα (t).dΛμν (t)|ηt ] = Grst (X)πt (Sβα [s, r]Sνμ [m, k])εβμ uν (t)¯ uα (t)dt Further, E[jt (X)(dYm (t))k |ηt ] = E[jt (X)jt (Sβα [m, k]dΛβα |ηt ] uα (t)dt = πt (XSβα [m, k])uβ (t)¯ E[πt (X)(dYm (t))k |ηt ] = πt (X)E[jt [Sβα [m, k]dΛβα |ηt ] = πt (X)πt (Sβα [m, k])uβ (t)¯ uα (t)dt Further, E[djt (X) − dπt (X)|ηt ] = E[jt (θβα (X)dΛβα

− Ft (X)dt − Gkmt (X)(dYm (t))k |ηt ]

uα (t) − Ft (X) = [πt (θβα (X)uβ (t)¯ −Gkmt (X)πt (Sβα [m, k])uα (t)¯ uβ (t)]dt since Ft (X) and Gkmt (X) are ηt -measurable and E[(dYm (t))k |ηt ] = E[jt (Sβα [m, k]dΛβα |ηt ] = πt (Sβα [m, k])uβ (t)¯ uα (t)dt These equations can be substituted into (1) and (2) and solved for the Abelian family of operatorsFt (X), Gkmt (X). [53] Classical control of a stochastic dynamical system by error feedback based on a state observer derived from the EKF. [54] Quantum control using error feedback based on Belavkin quantum filters for the quantum state observer. Let Y (t) be a measurement noise process and V (t) a system operator process say like A(t) + A(t)∗ . Consider a qsde dUc (t) = (−iV (t)dY (t) − V (t)2 dt/2)Uc (t) We are assuming that V (t) is Hermitian. If V (t) = V does not vary with time, we can write the solution to the above qsde as Uc (t) = exp(−iV Y (t)) Uc (t) is a unitary matrix and its application to the state evolved from a qsde can remove the effect of noise if Y (t) is present as a noise in the qsde. For example, consider the following qsde dU (t) = (−(iH + V 2 /2)dt − iV dY (t))U (t)

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General Relativity and Cosmology with Engineering Applications

We can write this approximately as U (t + dt) = (I − dt(iH + V 2 /2) − iV dY (t))U (t) We now apply the control unitary (I − iV (t)dY (t) − V 2 (t)dt/2)−1 = I + iV (t)dY (t) − V 2 (t)dt/2 to U (t + dt) to get (I + iV (t)dY (t) − V 2 (t)dt/2)(I − iHdt − iV (t)dY (t) − V 2 (t)dt/2)U (t) = (I − iHdt)U (t) ie, the effect of the noise is cancelled out. We note that the output measurement is given by Yo (t) = U (t)∗ Y (t)U (t) and it is this process which follows the nondemolition property. We can measure only Yo without disturbing the dynamics generated by U (t) on the system Hilbert space h. We cannot measure the input measurement Y without disturbing the system dynamics. Now Belavkin’s quantum filtering equation can be expressed as dπt (X) = Ft (X)dt + Gt (X)dYt where

jt (X) = U (t)∗ XU (t), πt (X) = E[jt (X)|ηt ] ηt = σ{Ys : s ≤ t}

is the output measurement Abelian algebra at time t. U (t) is generated by the Hudson-Parthasarathy noisy Schrodinger equation and expectations are calcu­ lated in the pure state |f φ(u) > where f is a normalized system vector and φ(u) is a normalized coherent vector (See the paper by John Gough and Kostler). Ft (X), Gt (X) are linear functions of the system observable and belong to the measurement algebra ηt . More precisely, we can express the above filtering equation as dπt (X) = πt (θ0 (X))dt+[πt (L1 X +XL∗1 )−π(L2 )π(X)](dYt −πt (L3 X +XL∗3 )dt) L2 is a Hermitian matrix. For quadrature measurements, Zt = Yt − ∫where t ∗ π (L s 3 X+XL3 )ds is a Brownian motion process. θ0 is the Gorini-Kossakowski­ 0 Sudarshan-Lindblad (GKSL) generator on the system operator space. More generally, for general non-demolition measurements, like photon counting and a combination of quadrature and photon counting measurements, the Belavkin equation has the form ∑ ∑ fk (πt (Mk ))πt (θk (X))](dYt − gk (πt (Nk ))πt (φk (X))dt) dπt (X) = πt (θ0 (X))dt+[ k≥1

k≥1

where θk , φk , k ≥ 1 are linear operators on the Banach space of system operators. If we write ρt for the density operator on the system space conditioned on the measurements upto time t, we have πt (X) = T r(ρt X)

General Relativity and Cosmology with Engineering Applications

363

ρt should be viewed as a random density matrix on the system operator algebra where the randomness comes from conditioning on the measurements upto time t. Thus, the above Belavkin equation reads ∑ ∑ dρt = θ0∗ (ρt )dt + [ fk (T r(ρt Mk ))θk∗ (ρt )][dYt − gk (T r(ρt Nk ))φ∗k (ρt )dt] k≥1

k≥1

It is to be noted that the process Mt = Yt −

∫ t∑ 0 k≥1

gk (T r(ρs Nk ))φ∗k (ρs )ds

is a Martingale (Ph.d thesis of Luc Bouten). The above equation for ρt is called a Stochastic Schrodinger Equation. Now, we wish to remove the noise from the above Belavkin equation by an appropriate control so that the evolution equation of the density has just the first term θ0∗ (ρt ), ie, we wish to recover the GKSL equation. Consider now the control unitary Uc (t) = exp(iW Yt ) where W is a system observable and Yt the above output measurement. Assume for simplicity, that we take quadrature measurements, ie, Mt is a Wiener process. We apply Uc (t) to ρt to get ρc,t = Uc (t)ρt Uc (t)∗ Then, by Quantum Ito’s formula, dρc,t = dUc (t)ρt Uc (t)∗ +Uc (t)dρt Uc (t)∗ +Uc (t)ρt dUc (t)∗ +dUc (t)dρt Uc (t)∗ +Uc (t)dρt dUc (t)∗ [55] Lyapunov’s stability theory with application to classical and quantum dynamical systems. [56] Imprimitivity systems as a description of covariant observables under a group action. Construction of imprimitivity systems, Wigner’s theorem on the automorphisms of the orthogonal projection lattice. (Ω, F, pμ) is a measure space. P is a spectral measure on this space, ie, for each E ∈ F, P (E) is an orthogonal projection operator in a Hilbert space H. The set of all orthogonal projections on H is denoted by P(H). It is also called the projection lattice. Let τ be an automorphism of P(H), ie, if τ ∑ : P(H) → ∑ P(H) is such that if P1 , P2 , ... are mutually orthogonal, then τ ( j Pj ) = j τ (Pj ). More generally, we require that τ (max(P1 , P2 )) = max(τ (P1 ), τ (P2 )) and τ (min(P1 , P2 )) = min(τ (P1 ), τ (P2 )) for any two P1 , P2 ∈ P(H). Then, Wigner proved that there exists a unitary or antiunitary operator U in H such that τ (P ) = U P U ∗ , P ∈ P(H) It follows that if G group that acts on the measure space (Ω, F, P ) and g ∈ G → τg is a homomorphism from G into aut(P(H)) such that τg (P (E)) = P (g.E),

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then there exists a projective unitary-antiunitary representation g → Ug of G into U A(H) such that P (gE) = Ug P (E)Ug∗ , g ∈ G, E ∈ F We note that τg1 τg2 (P (E)) = P (g1 g2 E) = τg1 g2 (P (E)) so the requirement that g → τg be a homomorphism is natural from the view­ point of covariant transformation of observables. Let now H be a subgroup of G. Consider the homogeneous space X = G/H. G acts transitively on X. Let μ be a quasi-invariant measure on X, ie, for any g ∈ G, the measures μ.g −1 and μ are absolutely continuous with respect to each other. Consider f ∈ L2 (X, μ). Consider for f ∈ L2 (X, μ), Ug f (x) = (dμ.g −1 /dμ)1/2 f (g −1 x), x ∈ X Then, ∫

|| Ug f (x) ||2 dμ(x) ==

∫

|f (g −1 x)|2 dμ.g −1 (x) = X

∫

|f (x)|2 dμ(x) X

This proves that Ug is a unitary operator. It is easy to see that U is also a representation: Ug1 g2 f (x) = (dμ.(g1 g2 )−1 (x)/dμ)f (g2−1 g1−1 x) On the other hand, Ug1 (Ug2 f )(x) = (dμ.g1−1 (x)/dμ)1/2 (Ug2 f )(g1−1 x) = (dμ.g1−1 (x)/dμ)1/2 (dμ.g2−1 g1−1 (x)/dμ.g1−1 x)1/2 f (g2−1 g1−1 x) = (dμ.g2−1 g1−1 (x)/dμ)f (g2−1 g1−1 x) Thus, Ug1 g2 = Ug1 .Ug2 , g1 , g2 ∈ G Let A(g, x) be a map from G×X into the algebra of linear operators in a Hilbert space < such that A(g1 g2 , x) = A(g1 , g2 x)A(g2 , x) Then consider the operator Ug defined on L2 (μ, h) Ug f (x) = (dμ.g −1 (x)/dμ)1/2 A(g, g −1 x)f (g −1 x) We see that Ug1 (Ug2 f )(x) = (dμ.g1−1 (x)/dμ)1/2 A(g1 , g1−1 x)(Ug2 f )(g1−1 x) −1.g1−

= (dμ.g1−1 (x)/dμ)1/2 A(g1 , g1−1 x)(dμ.g2

1

(x)/dμ.g1−1 )1/2 A(g2 , g2−1 g1−1 x)f (g2−1 g1−1 x)

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365

= (dμ.g2−1 g1−1 (x)/dμ)1/2 A(g1 g2 , g2−1 g1−1 x)f ((g1 g2 )−1 x) = Ug1 g2 f (x) Thus, g → Ug is a representation of G in L2 (X, h). [57] Schwinger’s analysis of the interaction between the electron and a quan­ tum electromagnetic field. Let A(t, r) denote the vector potential corresponding to a quantum electromagnetic field. The dynamical variables q, p of the elec­ tron bound to the nucleus commute with A. Let Φ(t, r) denote the quantum scalar potential of the quantum ∫ t electromagnetic field. Note that if we adopt the Lorentz gauge, Φ = −c2 0 divAdt while if we adopt the Coulomb gauge, then divA = 0, Φ = 0 where for the latter, we are assuming that there is no externally charged matter to generate the scalar potential. The Hamiltonian of the atom interacting with the quantum electromagnetic field is given by H(t) = (p + eA)2 /2m + V (q) − eΦ + Hem where the field Hamiltonian Hem has also been added. H(t) is thus the total Hamiltonian of the atom interacting with the quantum em field. We have H(t) = p2 /2m + V (q) + ((p, A) + (A, p))/2m − eΦ + Hem We note that (p, A) + (A, p) = 2(A, p) − i.div(A) = −2i(A, ∇) − idiv(A) Schrodinger’s equation for the wave function ψ(t) of the atom and field is given by iψ ' (t) = H(t)ψ(t) Making the transformation ψ(t) = exp(−itHem ))φ1 (t) and assuming the Coulomb gauge gives ˜ p)/m)φ1 (t) iφ'1 (t) = (p2 /2m + V (q) + (A, where A˜ = exp(itHem ).A.exp(−itHem ) Making another transformation φ1 (t) = exp(−iS(t))φ2 (t) where S(t) is linear in the creation and annihilation operators of the em field, we get φ'1 (t) = exp(−iS(t))(−iS ' (t) + [S(t), S ' (t)]/2)φ2 (t) + exp(−iS(t))φ'2 (t)

366

General Relativity and Cosmology with Engineering Applications since [S(t), S ' (t)] is a c-number function of time owing to the commutation relations between the creation and annihilation operators of the em field. Taking ∫t ∫t ˜ ˜ = (p, Z(t)) where Z(t) = 0 Adt/m gives S(t) = (p, 0 Adt)/m iφ'2 (t) = exp(iS(t))(p2 /2m−S ' (t)−i[S(t), S ' (t)]/2+V (q)+S ' (t))exp(−iS(t))φ2 (t) = (p2 /2m + V (q + Z(t)))φ2 (t) where the c-number function −i[S(t), S ' (t)] has been neglected as it only gives an additional phase factor to the wave function. (1)

[58] Quantum control: The system observable X evolves to jt (X) after (2) time t. The desired system observable Xd evolves to jt (Xd ). Here, (k)

jt

= (Hk (t), Lk ), k = 1, 2

which means that (k)

jt (Z) = Uk (t)∗ ZUk (t), k = 1, 2, Z = X, Xd where Uk (t) satisfies the HP equation dUk (t) = (−(iHk (t) + Qk )dt + Lk dA(t) − L∗k dA(t)∗ )Uk (t), Qk = Lk L∗k (

Define ˜= X ( ˜) = jt (X

X 0

(1)

jt (X) 0

0 Xd

) , )

0 (2) jt ( Xd )

Let h denote the system Hilbert space and Γs (L2 (R+ )) the Boson Fock space. jt is a ∗ unital homomorhism from the algebra B(h) ⊕ B(h) into the algebra (B(h ⊗ Γs (L2 (R+ ))) ⊕ B(h × Γs (L2 (R+ ))). Define the operators H(t) = diag[H1 (t), H2 (t)], L = diag[L1 , L2 ], U (t) = diag[U1 (t), U2 (t)], Q = diag[Q1 , Q2 ]

Then, the HP equations for Uk , k = 1, 2 can be expressed as a single qsde: dU (t) = [−(iH(t) + Q)dt + LdA(t) − L∗ dA(t)∗ ]U (t) ˜ ). ˜ in B(h) ⊕ B(h) evolve to U (t)∗ XU ˜ (t) = jt (X The corresponding observables X ˜ A non demolition measurement for the process jt (X ) in the sense of Belavkin is given by Y o (t) = U (t)∗ Y i (t)U (t), Y i (t) = A(t) + A(t)∗ It satisfies the sde dY o (t) = dY i (t) + dU (t)∗ dY i (t)U (t) + U (t)∗ dY i (t).dU (t)

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367

= dY i (t) − U (t)∗ (L + L∗ )U (t)dt = dY i (t) + jt (S)dt = dA(t) + dA(t)∗ + jt (S)dt where

S = −(L + L∗ )

Let ηt = σ(Yso : s ≤ t). Then ηt is an Abelian Von-Neumann algebra and we define the conditional expectation ˜ )|ηt ) = πt (X ˜) Ejt (X We can assume that dπt (Z) = Ft (Z)dt + Gt (Z)dY o (t) where Ft (Z) and Gt (Z) are in ηt . Then, its evident [See the paper by Gough and Koestler] that E[(jt (Z) − πt (Z))dY o (t)|ηt ] + E[(djt (Z) − dπt (Z))dY o (t)|ηt ] = 0 − − − (1) E[(djt (Z) − dπt (Z))|ηt ] = 0 − − − (2) We can write djt (Z) = jt (θ0 (Z))dt + jt (θ1 (Z))dA(t) + jt (θ2 (Z))dA(t)∗ where for Z = diag[Z1 , Z2 ], we have (1)

(2)

jt (Z) = diag[jt (Z1 ), jt (Z2 )] and (1)

(1)

(1)

(1)

(2)

(2)

(2)

(2)

djt (Z1 ) = jt (θ10 (Z1 ))dt + jt (θ11 (Z1 ))dA(t) + jt (θ12 (Z1 ))dA(t)∗ djt (Z2 ) = jt (θ20 (Z2 ))dt + jt (θ21 (Z1 ))dA(t) + jt (θ22 (Z2 ))dA(t)∗ Thus, θk (Z) = diag[θ1k (Z1 ), θ2k (Z2 )], k = 0, 1, 2

[59] Quantum error correcting codes: Let H = CN be the Hilbert space of the quantum system and let C be a subspace of H. C is called the code subspace. If ρ is a density operator in H, we say that ρ ∈ C if Range(ρ) ⊂ C. If N is a subspace of the space CN ×N of all linear operators in H, then we say C is an ∈ CN ×N such that N correcting code, if there exist operators R0 , ..., Rr ∑ ∑ error ∗ ∗ k Rk Rk = I and whenever E1 , ..., Er ∈ N are such that k Ek Ek = I, and ρ ∈ C, then ∑ Rk E(ρ)Rk∗ = cρ k

where c ∈ C and E(ρ) =

∑ k

Ek ρEk∗

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is the output state of the noisy channel {Ek }. {Rk } are called recovery operators for the noise subspace N and the code subspace C. If there exist such recovery operators, then we say that C is an N error correcting code. Theorem (Knill-Laflamme): C is an N error correcting code iff P N2∗ N1 P = λ(N2 ∗ N1 )P for all N1 , N2 ∈ N and some complex scalar λ(N2∗ N1 ) (dependent on N2∗ N1 ). Here, P is the orthogonal projection onto C. Proof: Suppose first that C is an N error correcting code with recovery operators R0 , ..., Rr . Let Ek , k = 1, 2, ..., m be a quantum channel in N . Then we have for all |ψ >∈ C the relation ∑ Rk Es |ψ >< ψ|Es∗ Rk∗ = λ(ψ)|ψ >< ψ| k,s

where λ(ψ) is a complex scalar possibly dependent on |ψ >. It follows that for all |ψ >⊥ |ψ >, ie, < φ|ψ >= 0, we have ∑ | < φ|Rk Es |ψ > |2 = λ| < φ|ψ > |2 = 0 k,s

Thus, Rk Es |ψ >⊥ |φ > ∀|φ >⊥ |ψ ie, Rk Es |ψ >= βks (ψ)|ψ >, ∀|ψ >∈ C and some complex numbers βks (ψ). By linearity of the operators, it is clear that βks cannot depend on |ψ >. We get therefore, Rk Es P = βks P, ∀k, s Thus,

P Eq∗ Rk∗ Rk Es P = βks β¯kq P ∑ and summing this equation over k and using k Rk∗ Rk = I, we get P Eq∗ Es P = aqs P, aqs ∈ C

We have thus proved that P N2∗ N1 P =∝ P, ∀N1 , N2 ∈ N Conversely, suppose that this relation holds. Then, let N0 = {N ∈ N : λ(N ∗ N ) = 0} It is clear that N0 is a subspace of N . Indeed, suppose N1 , N2 ∈ N0 . Then, P Nk∗ Nk P = λ(Nk∗ Nk )P = 0, k = 1, 2 and hence Nk P = 0, k = 1, 2

369

General Relativity and Cosmology with Engineering Applications Thus, P Nj∗ Nk P = 0, j, k = 1, 2 and hence, P (c1 N1 + c2 N2 )∗ (c1 N1 + c2 N2 )P = 0, .c1 , c2 ∈ C Now, define for N ∈ N , [N ] = N + N0 ∈ N /N0 We define < [N1 ], [N2 ] >= λ(N1∗ N2 ), N1 , N2 ∈ N This definition is valid since for N0 ∈ N0 and N ∈ N we have λ(N ∗ N0 ) = λ(N0∗ N ) = 0 This is because, P N0∗ N0 P = λ(N0∗ N0 )P = 0 implies P N0∗ = N0 P = 0 and hence, 0 = P N0∗ N P = λ(N0∗ N )P, 0 = P N ∗ N0 P = λ(N ∗ N0 )P

It is clear therefore that < ., . > defines an inner product on N /N0 . So, we can choose an onb {[Nk ], k = 1, 2, ..., r} for N /N0 w.r.t. this inner product. Now, define r ∑ Rk = P Nk∗ , Pk = Nk P Nk∗ , k = 1, 2, ..., r, R0 = I − Pk k=1

Then, Pk Pj = Nk P Nk∗ Nj P Nj∗ = λ(Nk∗ Nj )Nj P Nj∗ = δkj Pj proving that Pj , j = 1, 2, ..., r for a set of mutually orthogonal orthogonal pro­ jections in H. Thus R0 is also an orthogonal projection. Now, suppose ρ ∈ C. Then ρ = P ρ = ρP = P ρP . Then, for N ∈ N , ∑ ∑ P Nk∗ N P ρP N ∗ Nk P + R0 N ρN ∗ R0 Rk N ρN ∗ Rk∗ = k

k≥1

=

∑

|λ(Nk∗ N )|2 ρ + R0 N ρN ∗ R0

k≥1

= λ(N ∗ N )ρ + R0 N ρN ∗ R0

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General Relativity and Cosmology with Engineering Applications

Here, we have used the Bessel equality/Parseval relation for orthonormal bases for a Hilbert space. Further, ∑ ∑ R0 N P = N P − Pk N P = N P − Nk P Nk∗ N P k

= NP −

∑

k

λ(Nk∗ N )Nk P

k

= NP − NP = 0 by the generalized Fourier series in a Hilbert space and the relation N0 P = 0 for all N0 ∈ N . Thus, we finally get r ∑

Rk∗ N ρN ∗ Rk∗ = λ(N ∗ N )ρ

k=0

and further, r ∑

Rk∗ Rk =

k=0

∑

Nk P Nk∗ + R0 =

k≥1

∑

Pk + I −

k≥1

∑

Pk = I

k≥1

This completes the proof of the Knill-Laflamme theorem. Construction of quantum error correcting codes using imprimitivity systems: Let A be a finite Abelian group and define the onb |x >, x ∈ A for L2 (A) so that |x >= [δx,0 , ..., δx,N −1 ]T where we are assuming A = {0, 1, ..., N − 1} with addition modulo N . with each n ∈ A, define the character χn hx) = exp(2πinx/N ), x ∈ A We may identify this character with n ∈ A. In this way, we get an isomorphism of Aˆ with A and we write < n, x >=< x, n >= χn (x) Define the unitary operators Ux , Vx on L2 (A) by Ua |x >= |x + a >, Va |x >=< a, x > |x > Then define the Weyl operator W (x, y) = Ux Vy , x, y ∈ A on L2 (A). These are N 2 unitary operators and in fact, as we shall soon see, they form an orthogonal basis for L2 (A×A), for the space of all linear operators in L2 (A). We derive the Weyl commutation relations: W (x, y)|a >= Ux Vy |a >= Ux < y, a > |a >=< y, a > |a + x >

General Relativity and Cosmology with Engineering Applications

371

Vy Ux |a >= Vy |a + x >=< y, a + x > |a + x >=< y, a >< y, x > |a + x > Thus, Vy Ux =< y, x > Ux Vy =< x, y > Ux Vy Also, W (x, y)W (u, v) = Ux Vy Uu Vv =< y, u > Uu+x Vv+y =< y, u > W (u + x, v + y) So (x, y) → W (x, y) is a projective unitary representation of the Abelian group G = A × A into the space of operators in L2 (A). Now, W (x, y)|a >=< y, a > |a + x > Hence, T r(W (x, y)) =

∑

< a|W (x, y)|a >=

∑

a

< y, a > δa+x,a

a

It follows that ∑ T r(W (x, y)) = 0 if x /= 0 and if x = 0, then T r(W (x, y)) = T r(W (0, y)) = a < y, a >= N δy,0 Thus, for all x, y ∈ A, T r(W (x, y)) = N δx,0 δy,0 It follows that W (x, y)∗ W (u, v) = V−y U−x Uu Vv = V−y Uu−x Vv =< y, x−u > Uu−x Vv−y =< y, x−u > W (u−x, v−y) Thus, T r(W (x, y)∗ W (u, v)) =< y, x − u > T r(W (u − x, v − y)) = 0, (u, v) /= (x, y) and further,

T r(W (x, y)∗ W (x, y)) = T r(I) = N

√ Thus, {W (x, y)/ N : x, y ∈ A} forms an orthonormal basis for L2 (A × A) and in particular, this is a set of N 2 linearly independent operators. We can express this orthonormality relations as T r(W (x, y)∗ W (u, v)) = N δ[x − u]δ[y − v] Now let H be a Gottesman subgroup of G = A × A. This means that for each pair (u, v), (x, y) ∈ H, the identity < u, y >∗ < v, x >= 1 holds. For arbitrary (u, v) ∈ G, we define the map ωH (u, v) : H → C such that ωH (u, v)(x, y) =< u, y >∗ < v, x > Then we have ωH (u, v) = 1, (u, v) ∈ H

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In other words, H ⊂ Ker(ωH ) = K say. Now we have that for (x, y), (u, v) ∈ H, W (x, y)W (u, v) =< y, u > W (x+u, y+v), W (u, v)W (x, y) =< v, x > W (u+x, v+y) and since

< y, u > / < v, x >=< y, u >< v, x >∗ = 1

it follows that [W (x, y), W (u, v)] = 0∀(x, y), (u, v) ∈ H Thus, there exists an onb for L2 (A) such that relative to this basis, W (x, y) is diagonal for all (x, y) ∈ H. W (x, y) = diag[αk (x, y), k = 1, 2, ..., N ], (x, y) ∈ H ˜ (x, y), (x, y) ∈ G by the equation Define W ˜ (x, y) W (x, y) = α1 (x, y)W Note that |αk (x, y)| = 1∀k and also for (x, y), (u, v) ∈ H we have ˜ (x, y)W ˜ (u, v) = (α1 (x, y)α1 (u, v))−1 W (x, y)W (u, v) = W (α1 (x, y)α1 (u, v))−1 < y, u > W (x + u, y + v) =

< y, u > α1 (x + u, y + v) ˜ W (x + u, y + v) α1 (x, y)α1 (u, v)

Now the relation W (x, y)W (u, v) =< y, u > W (x + u, y + v) implies using the diagonal representation when restricted to H that α1 (x, y)α1 (u, v) =< y, u > α1 (x + u, y + v) and hence ˜ (x, y)W ˜ (u, v) = W ˜ (x + u, y + v), (x, y), (u, v) ∈ H W ie, the projective unitary representation W of G reduces to an ordinary unitary representation when restricted to H. An example: Let F be a finite field and consider the vector spaces V = Fp . Let L : V → V and M : V → V be two linear transformations such that LT M : V → V is symmetric. Define N : V → V by LT M = N + N T . For x, y ∈ V , we define the Weyl operator W (x, y) : L2 (V ) → L2 (V ) in the usual way. We note that V is a finite vector space over F and if F consists of a elements, then V will consist of ap elements and dimL2 (V ) = ap . Now, let χ0 be a character of the field F viewed as an Abelian group under addition. Consider for a fixed a ∈ V ˜ (u) = χ0 (aT u + uT N u)W (Lu, M u), u ∈ V W

General Relativity and Cosmology with Engineering Applications

373

Now ˜ (u + v) = χ0 (aT (u + v) + uT N u + v T N v)W (Lu, M u)W (Lv, M v) W χ0 (aT (u + v) + uT N u + v T N v)χ0 (uT M T Lv)W (L(u + v), M (u + v)) = χ0 (aT (u + v) + uT N u + v T N v + uT (N + N T )v)W (L(u + v), M (u + v)) = χ0 (aT (u + v) + (u + v)T N (u + v))W (L(u + v), M (u + v)) ˜ (u + v) =W provided that we assume that the Weyl operator W (x, y) has been defined so that in the expression W (x, y)W (u, v) =< y, u > W (x + u, y + v), < y, u >= χ0 (y T u) or in other words, the character of V corresponding to any y ∈ V when V is viewed as an Abelian group under addition, is given by u → χ0 (y T u). Thus ˜ (u) is a unitary representation of V . We note that H = {(Lu, M u) : u ∈ u→W V } is a Gottesman subgroup of V × V . This is because, for u, v ∈ V , < Lu, M v >∗ < M u, Lv >= χ ¯0 (uT LT M v)χ0 (uT M T Lv) = χ0 (uT (M T L−LT M )v) = 0

since

χ ¯0 (x) = χ0 (x)−1

Construction of quantum error correcting codes using the Imprimitivity the­ orem of Mackey: Let G be a finite group acting on a finite set X. For each be an orthogonal projection onto a Hilbert space H such that x ∈ X let Px ∑ Px Py = δx,y I, x∈X Px = I. Assume that g → Ug is a unitary representation of G in H satisfying the Imprimitivity condition Ug Px Ug∗ = Pgx , g ∈ G, x ∈ X Let E ⊂ X and then we have Ug P (E)Ug∗ = P (gE), g ∈ G Now let N be a linear space of linear operators in H spanned by {Ug P (E) : g ∈ G} where E ⊂ X is fixed. We choose x ∈ X and ask the question when does the quantum code Px detect N . This happens iff Px Ug P (E)P (E)Ug∗ Px is proportional to Px for all g, h ∈ G. This is the same as saying that Px P (gE)Px = λPx

374

General Relativity and Cosmology with Engineering Applications This happens iff either x ∈ / gE in which case, the lhs is zero, or if gE = x for / E. Now we take all g ∈ G which is impossible. Thus, Px corrects N iff g −1 x ∈ F ⊂ X and derive the conditions for P (F ) to detect N . This happens iff P (F )Ug P (E)P (E)Ug∗ P (F ) is proportional to P (F ), ie, iff P (F ∩gE) is proportional to P (F ). This happens iff either gE ∩F = φ or gE ⊂ F . Now we consider the correction problem. P (F ) corrects N iff for all g, h ∈ G, P (F )Ug P (E)P (E)Uh∗ P (F ) = λP (F ) where λ may depend on g, h. This happens iff P (F )P (gE)P (gh−1 F )Uhg−1 = λP (F ) or equivalently, iff P (F ∩ gE ∩ gh−1 F )Uhg−1 = λP (F ) Post-multiplying both sides of this equation by their adjoints gives us P (F ∩ gE ∩ gh−1 F ) = |λ|2 P (F ) This is a necessary condition for P (F ) to correct N and happens only if for each g, h either F ∩ gE ∩ gh−1 F = φ or F ⊂ gE ∩ gh−1 F . As a special case, choosing / gE or or F = {x}, we see that Px corrects N only if for each g, h either x ∈ x = gh−1 x. We also derive the following conclusions from the above discussion. / gE or x = gh−1 x and Px corrects N iff for each g, h, either x ∈ Ugh−1 Px = λPx We also observe that if Ua Px = λPa for some a ∈ G, then, Ua Px Ua∗ = |λ|2 Px or equivalently,

Pax = |λ|2 Px

which implies that |λ| = 1 and ax = x. Thus if we define Gxeig = {g ∈ G : Ug Px = λPx } and Gxiso = {g ∈ G : gx = x} then we find that Gxeig ⊂ Gxiso and that Px corrects N = span{Ug P (E) : g ∈ G} iff for each g, h ∈ G (either x∈ / gE or gh−1 ∈ Gxeig }) In particular, Px corrects span{Ug Py : g ∈ G} iff for

General Relativity and Cosmology with Engineering Applications

375

x ). As a special case, Px corrects each g, h ∈ H, (either x = / gy or gh−1 ∈ Geig span {Ug Px : g ∈ G} if for each g, h ∈ G either g ∈ / Gxiso or gh−1 ∈ Gxeig . x . Then Let C be a cross section of G/Gxiso . Let c1 , c2 ∈ C, g, h ∈ Geig

Px Uc∗1 g Uc2 h Px = Px Ug−1 c−1 c2 h Px = λPx 1

x since Gxeig ⊂ Gxiso . Hence, Px corrects span{Ug : g ∈ CGeig } for each x ∈ G. A linear algebraic example of a quantum error correcting code. Let the code subspace projection P be given by ( ) Ir 0 P = ∈ Cn×n 0 0

Let U be a fixed n × n unitary matrix and let the noise subspace of operators N be the span of the adjoints of all n × n matrices Nk having the block structure ( ) λkj Ak λkj Bk Nkj = , k = 0, 1, ..., n/k − 1 Ckj Dkj where [Ak |Bk ] ∈ Cr×(n−r) is obtained as the kr + 1 to (k + 1)r rows of U arranged one below the other (k = 0, 1, ..., (n/k) − 1) and the λkj are arbitrary complex numbers. We find that since ∗ ∗ + B k Bm = δkm Ir Ak Am

(obtained using U U ∗ = In ), we have ( ¯ kl δkm Ir λkj λ ∗ = Nkj Nml F2

F1 F3

)

where F1 , F2 , F3 are matrices of size r × (n − r), (n − r) × r and (n − r) × (n − r) respectively. Thus, ∗ ¯ kl P P = δkm λkj λ P Nkj Nml and hence the quantum code P corrects N . [60] Quantum hypothesis testing: Let A, B be two density matrices and let 0 ≤ T ≤ I. The probability of making a correct decision when the density is A and the measurement is T is given according to quantum mechanical rules by T r(AT ). The probability of making a correct decision when the density is B is given by T r(B(I − T )). Now we wish to choose T such that T r(AT ) is a maximum subject to the constraint T r(BT ) ≤ α where α ∈ [0, 1) is given. Such a test corresponds to minimizing the error probability under A subject to the constraint that the error probability under B is smaller than a prescribed threshold. The function c → T (c) = {A ≥ cB} is assumed to be continuous on [0, ∞). We note that T (0) = I, T (∞) = 0 (By {U ≥ V } for Hermitian operators U, V , we mean the orthogonal projection onto the space spanned by

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all vectors v for which (U − V )v = λv for some λ ≥ 0.) Hence, T r(BT (c)) takes all values in [0, 1) as c varies over [0, ∞). Let α ∈ [0, 1) and choose c such that T r(BT (c)) = α. Then, we claim that T (c) is an optimal test. To see this, suppose 0 ≤ T ≤ I is any other test (ie measurement or positive operator) such that T r(BT ) ≤ α. Then T r(AT ) = T r((A − cB)T ) + cT r(BT ) ≤ T r((A − cB)T ) + cα = T r((A − cB)T (c)T ) + T r((A − cB){A < cB}T ) + cα ≤ T r((A − cB)T (c)) + cα = T r(AT (c)) Here, we make use of the fact that since A − cB commutes with T (c) and T (c)2 = T (c), we have T r((A−cB)T (c)T ) = T r(T (c)(A−cB)T (c)T ) = T r(T 1/2 T (c)(A−cB)T (c)T 1/2 ) ≤ T r(T (c)(A − cB)T (c)) = T r((A − cB)T (c)) since 0 ≤ T 1/2 ≤ I and T (c)(A − cB)T (c) ≥ 0. Now, we can derive bounds on the error probabilities. We have for s ≥ 0, P1 (c) = T r(A(I − T (c)) = T r(A{A < cB}) ≤ cs T r(A1−s B s ) Thus, log(P1 (c)) ≤ s.log(c) + log(T r(A1−s B s ) = s(R + log(A1−s B s )/s) where R = log(c) Consider now the function f (s) = sR + log(T r(A1−s B s ), s ≥ 0 We wish to select R so that f (s) has a minimum at s = 0. For this, we require that f ' (0) = 0, f '' (0) ≥ 0. Now, f ' (0) = R − D(A|B) where D(A|B) = T r(A(logA − logB)) is the relative entropy between A and B. Thus, the first condition gives R = D(A|B). The second condition f '' (0) ≥ 0 gives d [T r(−A1−s log(A)B s + A1−s B s log(B))/T r(A1−s B s )]|s=0 ≥ 0 ds or equivalently, T r(A(log(A))2 − 2Alog(A)log(B) + A(log(B)2 ) + (T r(Alog(A) − Alog(B)))2 ≥ 0

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which is true since by the Schwarz inequality, |T r(Alog(A)log(B))|2 ≤ T r(A(log(A))2 ).T r(A.(log(B))2 ) It follows that P1 (eR ) = 0 if R = D(A|B) and for the second kind of error probability, we get P2 (c) = T r(BT (c)) = T r(B{A < cB}) = 1 − T r(B{A > cB}) Also, for s ≤ 1 so

T r(B{A > cB}) ≤ cs−1 T r(A1−s B s ) log(1 − P2 (c)) ≥ (s − 1)log(c) + log(T r(A1−s B s ))

We now find on letting s = 0 that log(1 − P2 (c)) ≥ −log(c) + 1 so if we put R = log(c), we get log(1 − P2 (eR )) ≥ 1 − R or equivalently, P2 (eR ) ≤ 1 − exp(1 − R) We also note that lims→0 log(T r(A1−s B s )/s = −D(A|B) So, taking log(c) = D(A|B) − δ gives log(1 − P2 (c)) ≥ −log(c) + s(log(c) + log(T r(A1−s B s ))/s) ≥ −D(A|B) + δ + s(D(A|B) − δ − D(A|B)) = −D(A|B) + (1 − s)δ for s ≤ 1. We also recall that log(P1 (c)) ≤ s(log(c) + log(T r(A1−s B s ))/s) = s(D(A|B) − δ + D(A|B) + ε(s)) = −s(δ − ε(s)) where ε(s) → 0, s → 0 Hence, if s is sufficiently small, say s ≤ s0 , then ε(s) < δ/2 and for all such s, we have log(P1 (c)) ≤ −sδ/2 ≤ −s0 δ/2 Note: Consider the function g(s) = log(T r(A1−s B s ))/s

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We have g ' (s) = T r(−A1−s log(A)B s +A1−s B s log(B))/(s.T r(A1−s B s ))−log(T r(A1−s B s ))/s2

The limit of this as s → 0 is the same as the limit of −D(A|B)/s + D(A|B)/s as s → 0 which is zero. We also note that the limit of g '' (s) as s → 0 is the same as the limit of T r(A(log(A))2 + B(log(B))2 − 2Alog(A)log(B))/s+D(A|B)/s2 +(D(A|B)2 /s)+(D(A|B)/s2 )+(2/s3 )log(T r(A1−s B s )

which is same as the limit of T r(A(log(A))2 + B(log(B))2 − 2Alog(A)log(B))/s + D(A|B)/s2 + (D(A|B)2 /s) +(D(A|B)/s2 ) − 2D(A|B)/s2 The above equals T r(A(log(A))2 + B(log(B))2 − 2Alog(A)log(B))/s + (D(A|B)2 /s which is positive. Hence, g(s) for s ≥ 0 attains its minimum value of D(A|B) at s = 0. [61] The Sudarshan-Lindblad equation for observables on the quantum sys­ tem has the following general form: X ' = i[H, X] −

1∑ ∗ (Lk Lk X + XL∗k Lk − 2L∗k XLk ) 2 k

Assume that H=

1 T 1 p F1 p + q T F2 q 2 2

and Lk = λTk p + μTk q, k ≥ 1 We have p = ((pn )), q = ((qn )), [qn , pm ] = iδn,m L∗k Lk X + XL∗k Lk − 2L∗k XLk = L∗k [Lk , X] + [X, L∗k ]Lk = θk (X) say. We thus have [Lk , qn ] = [λTk p, qn ] = −iλk [n], ¯ T p] = iλ ¯ k [n] [qn , L∗k ] = [qn , λ k so ¯T p + μ ¯ k [n](λT p + μT q) ¯Tk q) + iλ θk (qn ) = −iλk [n](λ k k k

General Relativity and Cosmology with Engineering Applications so

∑ k

θk (q) =

∑

379

¯ k λT )p + (−iλμ∗ + iλ ¯ k μT )q] [(−iλk λ∗k + iλ k k k

k

=

∑

[2Im(λk λ∗k )p + 2Im(λk μ∗k )q]

k

Likewise, ¯Tk q] = −iμ ¯k [n] [Lk , pn ] = [μTk q, pn ] = iμk [n], [pn , L∗k ] = [pn , μ so ¯T p + μ ¯Tk q) − iμ ¯k [n](λTk p + μTk q) θk (pn ) = iμk [n](λ k so

θk (p) = −2Im(μk λ∗k )p − 2Im(μk μ∗k )q

We write θ=

∑

θk

k

Then, [H, qn ] = [pT F1 p/2, qn ] = −i

∑

F1 [n, m]pm

m

ie, [H, q] = −iF2 p Likewise, [H, p] = iF1 q [62] The Yang-Mills field and its quantization using path integrals: Let G be a finite dimensional Lie group and g its Lie algebra. For simplicity, assume that G is a subgroup of U (n, C). Then g consists of n × n complex skew-Hermitian matrices. We can thus choose a basis {iτa : a = 1, 2, ..., n} for g with the τa' s Hermitian matrices. Thus, the commutation relations of these basis elements has the form [iτa , iτb ] = −C(abc)iτc where the C(abc) are real constants and summation over the repeated index c is understood. Equivalently, these commutation relations can be expressed as [τa , τb ] = iC(abc)τc We let Aμ : R4 → g be the gauge fields. Thus, the covariant derivatives in terms of these gauge fields are defined by ∇μ = ∂μ − eAμ

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where e is a real constant. We can write Aμ (x) = iAaμ (x)τa where the Aaμ (x)' s are now real valued fields. There are in all 4n of such fields. We assume that the wave function ψ(x) ∈ R4n satisfies an equation of the form [γ μ i∂μ ⊗ In + eAaμ (x)γ μ ⊗ τa − mI4n ]ψ(x) = 0 where γ μ are the four Dirac γ matrices forming a basis for the Clifford algebra in C4×4 . Formally, we can write the above wave equation as γ μ (i∂μ − ieAμ ) − m)ψ = 0 or equivalently as [γ μ (i∂μ + eAaμ τa ) − m]ψ = 0 This eqations can be derived from the variational principle δψ S = 0 ∫

where S[ψ] =

ψ¯∗ γ 0 (γ μ (i∂μ + eAaμ τa ) − m)ψd4 x ∫ =

Ld4 x

Note that γ 0 γ μ are Hermitian matrices and so is γ 0 γ μ ⊗ τa . Hence using inte­ gration by parts, it follows that S[ψ] is real. In terms of the gauge covariant derivative defined above, we have L = ψ ∗ γ 0 (iγ μ ∇μ − m)ψ We consider the following transformation of the wave function ψ(x) → ψ ' (x) = g(x)ψ(x) where g(x) ∈ G is to be interpreted as I4 ⊗ g(x). This is called a local gauge transformation of the wave function. Then we consider a corresponding trans­ formation of the gauge field Aμ (x): Aμ (x) → A'μ (x) so that if ∇'μ = ∂μ − eA'μ then ∇'μ ψ ' = g(x)∇μ ψ

General Relativity and Cosmology with Engineering Applications

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If this happens, the the above Lagrangian density L will be invariant under a local gauge transformations of the wave function and the gauge fields. To get this, we must satisfy g(x)(∂μ − eAμ (x))ψ(x) = (∂μ − eA'μ (x))ψ ' (x) = (∂μ − eA'μ (x))g(x)ψ(x) which is equivalent to A'μ = gAμ g −1 + e−1 (∂μ g)g −1 Note that we get a gauge invariant Lagrangian by considering ψ ∗ γ 0 γ μ (∂μ − eAμ )ψ which is the same as ψ ∗ gamma0 γ μ ∇μ ψ or more precisely, ψ ∗ (γ 0 γ μ ⊗ ∇μ )ψ we note that the gauge transformation g(x) is actually to be interpreted as I4 ⊗ g(x) and it acts only on the second component in the tensor product of C4 ⊗ Cn . [63] A general remark on path integral computations for gauge invariant actions. Suppose that we have an action integral I(φ) of the fields φ(x) and we compute a path integral of the form ∫ S = exp(iI(φ))B[φ]dφ Suppose that the action integral I(φ) is invariant under a Gauge transformation φ → φΛ . Suppose also that the path integral measure dφ = Πx∈R4 dφ(x) is invariant under the same Gauge transformation. Then, we can write ∫ S = exp(iI(φΛ )B[φΛ ]dφΛ ∫ =

exp(iI(φ))B[φΛ ]dφ

If follows that for an function C(Λ) on the gauge group (Λ ∈ G), we have ∫ ∫ ∫ S C(Λ)dΛ = exp(iI(φ))dφ B[φΛ ]C(Λ)dΛ Now let Λ, λ be two gauge group transformations. Then, we have ∫ B[φΛ ]C(Λ)dΛ ∫ =

B[φΛoλ ]C(Λoλ)dλ

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assuming that the measure dλ on the gauge group is a left invariant Haar mea­ sure so that dΛoλ = dλ. Define for any functional F of the fields φ F(φ, x) = (δF (φΛ) )/δΛ)(x)|Λ=Id We have (δF (φΛoλ )/δλ)(x) = (δF (φΛoλ )/δ(Λoλ))(x)(δ(Λoλ)/δλ)(x) It follows that on evaluating both sides at λ = Id, the identity gauge transfor­ mation, we get F˜ (φ, Λ, x) = F (φ, Λ, x)G(Λ) where F˜ (φ, Λ, x) = (δF (φΛoλ )/δλ)(x)|λ=Id , and G(Λ) = (δ(Λoλ)/δλ)|λ=Id F (φ, Λ, x) = (δF (φΛ )/δΛ)(x) ∫

Now consider S=

exp(iI(φ))B[F [φ]]F (φ, x)Πdφ(x)

∫ =

exp(iI(φ))B[F [φΛ ]]Πx F (φΛ , x)dφ(x)

provided that we assume invariance of the path measure exp(iI(φ))dφ under gauge transformations Λ, ie, exp(iI(φΛ ))dφΛ = exp(iI(φ))dφ ∫

It follows that S

C(Λ)dΛ =

∫ exp(iI(φ))B[F [φΛ ]]Πx F (φΛ , x)dφ(x)C(Λ)dΛ ∫ =

exp(iI(φ))B[F [φΛ ]](δF [φΛ ]/δΛ)G(Λ)−1 C(Λ)dΛ

So by choosing C = G, we get ∫ S

C(Λ)dΛ =

∫

∫ exp(iI(φ))B[F [φ]]dF [φ] =

exp(iI(φ))B[φ]dφ

establishing the invariance of the scattering matrix S under the gauge fixing functional F .

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[64] Calculation of the normalized spherical harmonics. Ylm (θ, φ) = Plm (cos(θ))exp(imφ) where f (x) = Plm (x) satisfies the modified Legendre equation ((1 − x2 )f ' )' + (l(l + 1) − m2 /(1 − x2 ))f = 0 or equivalently, (1 − x2 )2 f '' − 2x(1 − x2 )f ' + (l(l + 1)(1 − x2 ) − m2 )f = 0 or

(1 − x2 )2 f '' − 2x(1 − x2 )f ' + (l(l + 1) − m2 − l(l + 1)x2 )f = 0

For simplicity of notation, let λ = l(l + 1) Then, the above equation is the same as (1 + x4 − 2x2 )f '' − 2x(1 − x2 )f ' + (λ − m2 − λx2 )f = 0 Let f (x) =

∑

c(n)xn

n≥0

Then, we get on substituting this into the above differential equation, ∑ ∑ nc(n)(xn − xn+2 )+ c(n)n(n − 1)(xn−2 + xn+2 − 2xn ) − 2 n

n

(λ − m2 )

∑

c(n)xn − λ

n

∑

c(n)xn+2 = 0

n

Equating coefficients of xn gives (n+2)(n+1)c(n+2)+(n−2)(n−3)c(n−2)−2n(n−1)c(n)−2nc(n)+2(n−2)c(n−2)+ (λ − m2 )c(n) − λc(n − 2) = 0 or (n + 2)(n + 1)c(n + 2) + ((n − 1)(n − 2) − λ)c(n − 2) + (λ − m2 − 2n2 )c(n) = 0 or (n+4)(n+3)c(n+4)+(n(n+1)−l(l+1))c(n)+(l(l+1)−m2 −2(n+2)2 )c(n+2) = 0 If l is not a non-negative integer, then there will be nonzero c(n)' s for arbitrarily large n and as n → ∞, we would get c(n + 4) + c(n) − 2c(n + 2) ≈ 0, n → ∞

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General Relativity and Cosmology with Engineering Applications

or equivalently, c(n + 4) − c(n + 2) − (c(n + 2) − c(n)) ≈ 0 which would imply that c(n + 2) − c(n) converges to a constant, say ∑K. Then c(2n) or c(2n + 1) for large n behaves as Kn and since the series n≥0 nx2n behaves as x/(1 − x2 )2 which is not integrable over x ∈ [0, 1] since it has a singularity at x = 1 (ie, at θ = 0), it follows that the series has to terminate at some finite N , ie, there must exist a finite integer N such that c(n) = 0 for all n > N which is equivalent to saying that f (x) must be a polynomial. This can happen only if we impose the condition l = N and c(N + 2) = 0. Assume first that N = 2r is even. Then, we may assume that c(2n + 1) = 0 for all n and (2n+4)(2n+3)c(2n+4)+(2n(2n+1)−2r(2r+1))c(2n)+(2r(2r+1)−m2 −4(n+1)2 )c(2n+2) = 0

for n = 0, 1, ..., r − 1. Putting n = r we then get c(2r + 2) = 0 since we are assuming that c(2r + 4) = 0. Thus, c(2n) = 0 for all n > r and we get that f (x) =

r ∑

c(2n)x2n

n=0

where c(0) is arbitrary, c(2) is obtained by putting n = −1 in the above difference equation and using c(k) = 0, k < 0: c(2) = (m2 − 2r(2r + 1))c(0)/2 and (2n + 4)(2n + 3)c(2n + 4) = −[(2n(2n + 1) − 2r(2r + 1))c(2n) + (2r(2r + 1) − m2 −4(n + 1)2 )c(2n + 2)]/(2n + 4)(2n + 3) for n = 0, 1, ..., r − 2. In a similar way, we can describe the polynomials Plm (x) for l odd, ie l = 2r + 1. We assume that c(2n) = 0 for all n and get using the recursion (n + 4)(n + 3)c(n + 4) + (n(n + 1) − (2r + 1)(2r + 3))c(n) + ((2r + 1)(2r + 3) −m2 − 2(n + 2)2 )c(n + 2) = 0 at n = −1, so that

6c(3) + ((2r + 1)(2r + 3) − m2 − 2)c(1) = 0 c(3) = (m2 + 2 − (2r + 1)(2r + 3))c(1)/6

and then replacing n by 2n + 1 in the above recursion, c(2n + 5) = −[(2n + 5)(2n + 4)]−1 [((2n + 1)(2n + 3) − (2r + 1)(2r + 3))c(2n + 1)+

General Relativity and Cosmology with Engineering Applications

385

((2r + 1)(2r + 3) − m2 − 2(2n + 3)2 )c(2n + 3)] for n = 0, 1, ..., r−2 and we may assume that c(2n+1) = 0 for n = r+1, r+2, .... The values of c(0) and c(1) for the two cases are determined by the normalization condition ∫ 1

2π −1

Plm (x)2 dx = 1

This guarantees that ∫

π 0

∫

2π 0

|Ylm (θ, φ)|2 sin(θ)dθdφ = 1

A technique for calculating the representation matrix πl (R) for R ∈ SO(3). Here, πl (R) is defined by l ∑

ˆ) = Ylm (R−1 n

[πl (R)]m' m Ylm' (ˆ n)

m' =−l

From this equation, it is clear that ∫ ˆ )Y¯lm' (ˆ n)dΩ(ˆ n) [πl (R)]m' m = Ylm (R−1 n For example, taking R = Rx (β) we have

R−1 [cos(φ)sin(θ), sin(φ)sin(θ), cos(θ)]T =

[cos(φ)sin(θ), sin(φ)sin(θ)cos(β)−cos(θ)sin(β), sin(φ)sin(θ)sin(β)+cos(θ)cos(β)]T Denoting, this new point by (θ' , φ' ), we get θ' = cos−1 (sin(φ)sin(θ)sin(β) + cos(θ)cos(β)), φ' = tan−1 [(sin(φ)sin(θ)cos(β) − cos(θ)sin(β))/cos(φ)sin(θ)] We then have ∫ [πl (Rx (β))]m' m =

π 0

∫

2π 0

Y¯lm' (θ, φ)Ylm (θ' , φ' )sin(θ)dθdφ

A MATLAB programme for tabulating the matrix elements [πl (Rx (−β))]m' m would then proceed along the following lines: We store [πl (Rx (−2πk/N ))]m' m with k = 0, 1, ..., N − 1, m' , m = −l, −l + 1, ..., l as a two dimensional array A with matrix elements A[k + 1, (2l + 1)(m' + l) + m + 1] of size N × (2l + 1)2 . Thus, for k = 0 : N − 1 for m' = −l : l

386

General Relativity and Cosmology with Engineering Applications for m = −l : l sum = 0; for r = 0 : N − 1 for s = 0 : N − 1 θ = πr/N φ = 2πs/N β = 2πk/N θ' = cos−1 (sin(φ)sin(θ)sin(β) + cos(θ)cos(β)), φ' = tan−1 [(sin(φ)sin(θ)cos(β) − cos(θ)sin(β))/cos(φ)sin(θ)] sum = sum + conj(Ylm' (θ, φ)) ∗ Ylm (θ' , φ' ) ∗ (π/N ) ∗ (2π/N ); end; end; A[k + 1, (2l + 1)(m' + l) + m + 1] = sum; end; end; end;

[65] Volterra systems in quantum mechanics: The Hammiltonian has the form H(t) = H0 + f (t)V0 Let U (t) be the Schrodinger evolution operator: U ' (t) = −iH(t)U (t), U (0) = I Then, U (t) = U0 (t)W (t), U0 (t) = exp(−itH0 ) and W (t) has the Dyson series ∫ ∑ W (t) = I + (−i)n 0 r, that ∑

c(r)wj (j + r − s, kj ) = 0

m≥r≥s

and hence with s = m we get c(m)wj (j, kj ) = 0 and hence c(m) = 0. Then with s = m − 1 we get c(m − 1) = 0 etc. Thus all the c(r)' s are zero. This proves that S(j, kj ) is a linearly independent set for each j and kj . We now observe that ' = {w1 (m, k1 ), w2 (m, k2 ), ..., wm (m, km ) : 1 ≤ kj ≤ rj , j = 1, 2, ..., m} Bm

General Relativity and Cosmology with Engineering Applications

399

is a basis for Range(N 0 ) = V . Also, as noted above, each vector wj (m, kj ) generates the cyclic subspace span{wj (j + r, kj ) : 0 ≤ r ≤ m − j} = span(S(j, kj )) Remark related to the Jordan decomposition: Let {N m−1 v1,k : k = 1, 2, ..., r1 } be a basis for Range(N m−1 ), let {N m−2 v2,k : 1 ≤ k ≤ r2 } + Range(N m−2 ) be a basis for Range(N m−1 )/Range(N m−2 ) and in general, let {N m−j vj,k : 1 ≤ k ≤ rj } + Range(N m−j+1 ) be a basis for Range(N m−j )/Range(N m−j+1 ) where. j = 1, 2, ..., m. Suppose we assume that N m−j+1 vj,k = 0, 1 ≤ k ≤ rj , 1 ≤ j ≤ m. Then, we claim that B = {N p vj,k : 1 ≤ k ≤ rj , 1 ≤ j ≤ m, 0 ≤ p ≤ m − 1} is a linearly independent set in V where V is the vector space on which N is defined. Indeed suppose ∑ c(p, j, k)N p vj,k = 0 0≤p≤m−1,1≤j≤m,1≤k≤rj

Then applying N m−1 on both sides gives ∑ c(0, 1, k)N m−1 v1,k = 0 k

and hence c(0, 1, k) = 0∀k. Hence, ∑

c(p, j, k)N p vj,k = 0

(p,j)=(0,1),k

and applying N m−2 to both / sides gives ∑ c(0, 2, k)N m−2 v2,k = 0 which implies that c(0, 2, k) = 0∀k. Continuing in this way gives c(0, j, k) = 0∀j, k. Thus, ∑ c(p, j, k)N p vj,k = 0 p≥1,j,k

Applying N

m−2

to both sides gives ∑ c(1, 1, k)N m−1 v1,k = 0 k

and hence c(1, 1, k) = 0∀k. Thus, ∑ p≥1,(p,j)/=(1,1),j,k

c(p, j, k)N p vj,k = 0

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General Relativity and Cosmology with Engineering Applications

Applying N m−3 to both sides gives ∑ c(1, 2, k)N m−2 v2,k = 0 k

and hence c(1, 2, k) = 0∀k. Continuing in this way, we finally get c(p, j, k) = 0∀p, j, k proving linear independence of the set B. A neat proof of the canonical Jordan representation of a nilpotent matrix: Let N m = 0, N m−1 /= 0. Let B1 = {N m−1 x(1, α) : 1 ≤ α ≤ d1 } be a basis for R(N m−1 ). Let {N m−2 x(1, α1 ), N m−2 (x(2, α2 ) : 1 ≤ α1 ≤ d1 , 1 ≤ α2 ≤ d2 } be a basis for R(N m−2 ) etc. In general, {N m−s x(1, α1 ), N m−s x(2, α2 ), ..., N m−s x(s, αs ) : 1 ≤ αk ≤ dk , k = 1, 2, ..., s} is a basis for R(N m−s ). We may assume that N m−1 (x(2, α2 )) = 0∀α2 , ...N m−s+1 x(s, αs ) = 0∀αs , since N m−1 (x(2, α2 )) is expressible as a linear combination of N m−1 (x(1, α1 ), α1 =

1, 2, ..., d1 , ie, N m−1 x(2, α2 ) =

∑

c(k)N m−1 x(1, k)

k

∑ and hence x(2, α2 ) can be replaced by∑ x(2, α2 )− k c(k)x(1, k), ie N m−2 x(2, α2 ) can be replaced by N m−2 x(2, α2 ) − k c(k)N m−2 x(1, k) and these vectors for different α2 along with the vectors N m−2 x(1, k) for different k again form a basis for R(N m−2 ). This argument can be continued. The final result is that the vec­ tors B = {N m−s x(k, αk ) : 1 ≤ αk ≤ dk , k = 1, 2, ..., s, s = 1, 2, ..., m−1} forms a basis for V and the matrix of N relative to B has the Jordan canonical form of a Nilpotent matrix {Ref erence : T.Kato, ”P erturbationtheoryf orlinearoperators”} [3] Evaluation of a function of a matrix using the Jordan canonical form. Let A be a matrix over C. We know that A=D+N where D is diagonable and N is nilpotent. Thus, sum of Jordan blocks of the form ( λ 1 0 0 | 0 1 0 λ | 0 λ .. Jm (λ) = | | 0 ( 0 ..0.. 0.. 0.. 0 0 ..0 ..0

A can be written as a direct ... ... 0.. λ ..

This matrix belongs to Cm×m . We write Zm = ((δ[j − i − 1]))1≤i,j≤m

0 0 0 1 λ

) | | | | )

General Relativity and Cosmology with Engineering Applications

401

Then, Jm (λ) = λIm + Zm We have for any function f that is infinitely differentiable at λ, 2 m−1 f (Jm (λ)) = f (λ)Im + f ' (λ)Zm + f '' (λ)Zm /2! + ... + f (m−1) (λ)Zm /(m − 1)!

since m Zm =0

Then, choosing a basis B such that the matrix A has the Jordan canonical form A=

pk r ⊕ ⊕

Jmk,j (λk )

k=1 j=1

where λk , k = 1, 2, ..., r are the distinct eigenvalues of A. We can write f (A) =

pk r ⊕ ⊕

f (Jm(k,j) (λk )

k=1 j=1

Another way to compute f (A) is by using the Cauchy residue theorem. If A is diagonalble, then T = T DT −1 where D = diag[λ1 , ..., λn ] and (zI − A)−1 = T (zI − D)−1 T −1 so if Γk is a contour enclosing only the eigenvalue λk , then ∫ −1 (2πi) (zI − A)−1 dz = T Ek T −1 = Pk Γk

say, where Ek is a diagonal matrix having a one at those points where D has at the other points. Thus, Pk∑ is the projection onto the entry λk and zeroes ⊕ ) along N (T − λ ). We clearly have N (T − λ k j j=k k Ek = I and hence ∑ / P = I. Clearly E E = 0 for k = j and hence P P k j = 0 for k = j and k j k k / / since Ek2 = Ek , it follows that Pk2 = Pk . Thus, ∑ Pk = I k

the summation being over indices k corresponding to the distinct eigenvalues of T , defines a spectral resolution of identity. Note that ∫ we clearly have that if Γ is a contour enclosing all the eigenvalues of T , then Γ f (z)(zI − A)−1 dz can be replaced by ∑∫ ∑∫ −1 −1 −1 f (z)(zI−A) dz = (2πi) f (z)(zI−A)−1 dz (2πi) Γ Γ k k k k ∑ = f (λk )Pk = f (A) provided that f has no singularity within Γ. By a standard continuity argument, this result is also valid for a non-diagonable matrix A. If A is non-diagonable,

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then since the set of diagonable matrices is dense in the space of matrices, it follows that there exists a sequence εk → 0 such that A + εk I is non-singular for all k and if we assume that f is continuous, then we get ∫ f (A) = limk f (A+εk I) = (2πi)−1 f (z)(zI−A−εk I)−1 dz Γ ∫ −1 = (2πi) f (z)(zI−A)−1 dz where Γ encloses all the eigenvalues of A.

Γ

A.2.Functional Analysis [1] Let S be a symmetric operator in a Hilbert space H, ie S ⊂ S ∗ . (We are assuming that D(S) is dense in H). This means that D(S) ⊂ D(S ∗ ) and S ∗ |D(S) = S. We wish to show that if R(S + i) and R(S − i) are both dense in H, then S is essentially self-adjoint, ie S¯ is self-adjoint. Further if R(S + i) and R(S − i) are both exactly H, then S is self-adjoint. First we prove that S is closable. Indeed, suppose xn ∈ D(S) and xn → 0, Sxn → y. Then, to prove that S is closable, we must show that y = 0. For any z ∈ D(S), we have z ∈ D(S ∗ ) and hence, < Sxn , z >=< xn , S ∗ z >=< xn , Sz >→ 0, n → ∞ Thus, < y, z >= 0∀z ∈ D(S) Since D(S) is dense in H, it follows that y ⊥ H and hence y = 0, proving that S is closable. Remark: S is closable iff xn , zn ∈ D(S), xn → x, zn → x, Sxn → y, Szn → w imply y = w. This is the same as requiring that xn − zn ∈ D(S), xn − zn → 0, S(xn − zn ) → v all imply that v = 0 and this is the same as requiring that xn ∈ D(S), xn → 0, Sxn → y all imply y = 0. Now, let x ∈ D(S ∗ ). Then (S ∗ − i)x = limn (S − i)zn for some sequence zn ∈ D(S) because by hypothesis, R(S − i) is dense in H. Thus, < (S ∗ − i)x, y >=< x, (S + i)y >= lim < (S − i)zn , y >= lim < zn , (S ∗ + i)y >= lim < zn , (S + i)y > ∀y ∈ D(S) where we have used D(S) ⊂ D(S ∗ ). Now, we show that zn is a convergent sequence. We have that lim(S − i)zn = (S ∗ − i)x exists and hence (S − i)zn is a Cauchy sequence, ie (S − i)(zn − zm ) → 0, n, m → ∞ and hence || (S − i)(zn − zm ) ||2 → 0 or equivalently,

|| S(zn − zm ) ||2 + || (zn − zm ) ||2 + 2Im(< S(zn − zm ), zn − zm >) → 0

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Now < Sz, z >=< z, Sz > ∀z ∈ D(S) ∗

since S ⊂ S . Thus, Im(< Sz, z >) = 0∀z ∈ D(S). Thus, Im(< S(zn − zm ), zn − zm >) = 0. This proves that || S(zn − zm ) ||2 + || (zn − zm ) ||2 → 0 and hence || zn − zm ||→ 0 Thus, zn is Cauchy and hence convergent. Let zn → z. Then since (S + i)zn converges, it follows that Szn also converges. Thus, z ∈ D(S¯) and we get < x, (S + i)y >=< z, (S + i)y > ∀y ∈ D(S) Thus, x − z ⊥ R(S + i) and since R(S + i) is dense in H, it follows that x = z ∈ D(S¯). Thus, we have proved that D(S ∗ ) ⊂ D(S¯) Now S ⊂ S ∗ implies S¯ ⊂ S ∗ since S ∗ is closed. Thus, D(S¯) ⊂ D(S ∗ ) ⊂ D(S¯) from which we deduce that D(S ∗ ) = D(S¯) and therefore S¯ = S ∗ . Further, S ⊂ S ∗ impliesS ∗∗ ⊂ S ∗ = S¯ and since S ∗∗ is a closed extension of S, it follows that S ∗∗ = S¯. Thus, S ∗ = S¯ = S ∗∗ which proves that S¯ is self-adjoint, ie, (S¯)∗ = S¯. Remarks: (a) If A, B are operators in H such that A ⊂ B, then B ∗ ⊂ A∗ . Indeed, let x ∈ D(B ∗ ). Then < B ∗ x, y >=< x, By >=< x, Ay >, ∀y ∈ D(A) ⊂ D(B). Hence, x ∈ D(A∗ ) and < B ∗ x−A∗ x, y >= 0∀y ∈ D(A). This proves that B ∗ x−A∗ x ⊥ D(A) and since A is densely defined, it follows that B ∗ x−A∗ x = 0, ie, B ∗ x = A∗ x, proving the claim. (Note that we are assuming without any loss in generality that all operators are densely defined). (b) If A is any operator, then A∗ is closed. Indeed, let xn ∈ D(A∗ ), xn → x, A∗ xn → y. Then for all z ∈ D(A), we have < A∗ xn , z >→< y, z >, < A∗ xn , z >=< xn , Az >→< x, Az > and hence, < y, z >=< x, Az > ∗

This proves that y ∈ D(A ) and A∗ x = y, proving that A∗ is closed. (c) If A is any closable operator and B is a closed operator such that A ⊂ B ⊂ A¯, then B = A¯. Indeed, it suffices to show that A¯ ⊂ B. So let x ∈ D(A¯). ¯ and Then there exists a sequence xn ∈ D(A) such that xn → x, Axn → Ax ¯ and since B is since A ⊂ B, we have xn ∈ D(B), xn → x, Bxn = Axn → Ax ¯ This proves the claim. A related closed, it follows that x ∈ D(B), Bx = Ax. ¯ , then A¯ ⊂ B ¯. statement is that if B is any closed operator such that A ⊂ B ¯ ¯ This follows from the implications A ⊂ B implies Gr(A) ⊂ Gr(B) and hence

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¯ ¯ B). Another related remark is that if A is any closable operator Gr(A) ⊂ Gr(∗ ∗∗ then A is a closed extension of A and hence by the above A¯ ⊂ A∗∗ . For ¯ y = Ax ¯ and we have suppose xn ∈ D(A), xn → x, Axn → y. Then, x ∈ D(A), ∗ for z ∈ D(A ), < y, z >= lim < Axn , z >= lim < xn , A∗ z >=< x, A∗ z > Hence, x ∈ D(A∗∗ ) and y = A∗∗ x. This proves that A¯ ⊂ A∗∗ . Now we come to the last statement. Let S ⊂ S ∗ and R(S ± i) = H. Then, we have to show that S ∗ = S. Indeed, let x ∈ D(S ∗ ). Then there exists a y ∈ D(S) such that (S ∗ − i)x = (S − i)y since R(S − i) = H. Thus for any z ∈ D(S) ⊂ D(S ∗ ), it follows that < (S ∗ −i)x, z >=< x, (S+i)z >=< (S−i)y, z >=< y, (S ∗ +i)z >=< y, (S+i)z > and hence, < x − y, (S + i)z >= 0, z ∈ D(S) ie x−y ⊥H and therefore, x = y ∈ D(S) This proves that and hence so that

D(S ∗ ) ⊂ D(S) ⊂ D(S ∗ ) D(S) = D(S ∗ ) S = S∗

The proof is complete. A.3.Stochastic processes Here we discuss various kinds of noise processes that arise in perturbed quan­ tum systems and explain how to compute transition probabilities of quantum systems in the presence of such noise processes. [1] Brownian motion [2] Poisson process [3] Compound Poisson processes [4] Levy processes [5] Reflected and absorbed Brownian motion Let B(t), t ≥ 0 be a standard Brownian motion starting at any given point and let T0 = inf (t ≥ 0 : B(t) = 0) Obviously, the statistics of T0 will depend on B(0). Absorbed Brownian motion is the Brownian motion process upto time T0 and after time T0 it is set equal to

General Relativity and Cosmology with Engineering Applications

405

zero. We denote this process by X(t), t ≥ 0. We can show that X is a Markov process by the following simple intuitive reasoning: If at time t0 the process X(t0 ) = 0, then obviously X(t) = 0 for all t > t0 . On the other hand, if at time t0 , X(t0 ) = x > 0, then obviously B(t0 ) = x and by the Markovian property of B(.), the statistics of X(t) for all t > t0 will depend only on (t0 , x). It remains to compute the transition probability of X(.). For x, y > 0, we have P (X(t) > x|X(0) = y) = Py (B(t) > x, T0 > t) = Py (B(t) > x, mins≤t B(s) > 0) = P0 (B(t) > x − y, mins≤t B(s) > −y) = P0 (−B(t) > x − y, mins≤t (−B(s)) > −y) = P0 (B(t) < y − x, −maxs≤t (B(s)) > −y) = P0 (B(t) < y − x, St < y) = P0 (B(t) < y − x) − P0 (B(t) < y − x, St > y) where St = maxs≤t B(s) By the reflection principle, P0 (B(t) < y − x, St > y) = P0 (B(t) > y + x) Hence, P (X(t) > x|X(0) = y) = P0 (B(t) < y − x) − P0 (B(t) > y + x) and hence the transition density of X(t) (taking values in [0, ∞)) is given by qt (x|y) = −

d P (X(t) > x|X(0) = y) = (2πt)−1/2 (exp(−(x−y)2 /2t) dx −exp(−(x+y)2 /2t)), x, y > 0

Also P (X(t) = 0|X(0) = y) = Py (T0 < t) = Py (mins≤t B(s) ≤ 0) = P0 (mins≤t B(s) ≤ −y) = P0 (St > y) = 2P0 (B(t) > y) the last step following from the reflection principle. We thus verify that ∫ ∞ P (X(t) ≥ 0|X(0) = y) = qt (x|y)dx + 2P (B(t) > y) 0

= P (B(t) > −y) − P (B(t) > y) + 2P (B(t) > y) = P (B(t) > −y) + P (B(t) > y) = P (B(t) < y) + P (B(t) > y) = 1 [6] Bessel processes [7] The Brownian local time process [1] Let θ(x) be the unit step function. Let B(t) be Brownian motion and L(t) its local time at zero, ie, dL(t) = δ(B(t))dt

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General Relativity and Cosmology with Engineering Applications

Let τ (s) = inf {t > 0 : L(t) ≥ s} then, clearly τ (s) is a stopping time for each s. By Ito’s formula, 1 d(B(t)θ(B(t)) = θ(B(t))dB(t) + δ(B(t))dt + B(t)δ ' (B(t))dt 2 Now,

xδ ' (x) = (xδ(x))' − δ(x) = −δ(x)

Hence, 1 θ(B(t))dB(t) = d(B(t)θ(B(t)) − dL(t) 2 and hence

∫ 0

t

1 θ(B(s))dB(s) = B(t)θ(B(t)) − L(t) 2

Therefore applying Doob’s optional stopping theorem to the exponential mar­ tingale ∫ ∫ t t θ(B(s))dB(s) − (a2 /2) θ(B(s))ds) M (t) = exp(−a 0

0

and the stopping time τ (t) gives 1 = E[M (τ (t))] = exp(at − (a2 /2)

∫

τ (t)

θ(B(s))ds) 0

(Note that B(τ (t)) = 0, L(τ (t)) = t). Thus, ∫ E[exp(−s

τ (t)

√ θ(B(s))ds)] = exp(− 2st)

0

(Reference: Marc Yor, Some aspects of Brownian motion, Birkhauser). [8] Stochastic integration with respect to continuous semi-Martingales. Let Xt be continuous semimartingale. By the Doob-Meyer theorem, it can be decomposed as Xt = At + Mt where At is a process of bounded variation and Mt is a martingale. We can write At = At+ − A− t − where A+ t and At are increasing processes. The integral of a bounded adapted process Yt w.r.t At is defined as an ordinary Riemann-Stieltjes integral or equiv­ alently as a Lebesgue integral. Since Mt is not a process of bounded variation but has a well defined finite quadratic variation, its integral must be defined in ∫T the Ito sense. Specifically, defining 0 Yt dMt as the mean square limit of partial

General Relativity and Cosmology with Engineering Applications sums of the form Sn = We have for n > m,

∑n k=1

Ytn,k (Mtn,k+1 − Mtn,k ) as maxk |tn,k+1 − tn,k | → 0.

E(Sn − Sm )2 = +

∑

407

∑

E(Yt2n,k )E(Mtn,k+1 − Mtn,k )2

k

E(Yt2m,k )(E(Mtm,k+1

− Mtm,k )2 )

k

−2E(Sn Sm ) Suppose we assume that the partition {tn,k } is a refinement of the partition {tm,k }. Consider for example a term like Ys1 ((Ms2 − Ms1 ) in Sm and a term ∑3 like k=1 Ytk (Mtk+1 − Mtk ) in Sn where s1 = t1 < t2 < t3 < t4 = s2 . The expected value of their product is an example of a term in E(Sn Sm ). This term can be expressed as E(Yt21 (Mt2 − Mt1 )2 ) + E(Yt1 Yt2 (Mt3 − Mt2 )2 ) +E(Yt1 Yt3 (Mt4 − Mt3 )2 ) From this observation, it is clear that if the quantity ∫ T E Yt2 d < M >t < ∞ 0

then,

E((Sn − Sm )2 ) → 0, n, m → ∞

and hence by the fact that L2 (Ω, F, P ) is a Hilbert space, it follows that there ex­ ∫T its a random variable S∞ which we denote as 0 Yt dMt and call it the stochastic integral of Y w.r.t M over the interval [0, T ]. We are assuming that the increas­ ∫t ing function t →< M >t = 0 (dMs )2 defines a finite measure on a bounded in­ terval like [0, T ] and that the adapted process Yt is Riemann integrable w.r.t this measure. More precisely, the stochastic integral w.r.t a martinagle can be de­ fined for almost surely progressively measurable processes Yt that are integable w.r.t the random measure d < M >t over finite intervals. For a detailed dis­ cussion of this construction see the book by Karatzas and Shreve on ”Brownian motion and stochastic calculus” or the book by Revuz and Yor on ”Continuous martingales and Brownian motion. Remark: Consider the quantum system dU (t) = [−(iH(t)dt+

p p ∑ 1 ∑ Vk (t)Vm (t)d < Mk , Mm > (t))−i Vk (t)dMk (t)]U (t) 2 k,m=1

k=1

where Mk (t), k = 1, 2, ..., p are Martingales. It is easily verified using Ito’s formula that U (t) is unitary for all t if U (0) = I. We introduce a perturbation parameter ε into the martingale terms and obtain dU (t) = [−(iH(t)dt+ε2

p p ∑ 1 ∑ Vk (t)Vm (t)d < Mk , Mm > (t))−iε Vk (t)dMk (t)]U (t) 2 k,m=1

k=1

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General Relativity and Cosmology with Engineering Applications

We solve for U (t) using perturbation theory: ∑ U (t) = εm Um (t) m≥0

Equating same powers of ε gives successively dU0 (t) = −iH(t)U0 (t)dt, ∑ dU1 (t) = −iH(t)U1 (t) − i Vk (t)U0 (t)dMk (t), k

∑ 1∑ dU2 (t) = −iH(t)U2 (t)− Vk (t)Vm (t)U0 (t)d < Mk , Mm > (t)−i Vk (t)U1 (t)dMk (t) 2 k,m

k

and in general, dUr (t) = −iH(t)Ur (t)−

∑ 1∑ Vk (t)Vm (t)Ur−2 (t)d < Mk , Mm > (t)−i Vk (t)Ur−1 (t)dMk (t), 2 k,m

k

r≥2 After calculating U0 (t) +

N ∑

εr Ur (t)

r=1

we calculate the transition probablity from an initial state |i > to a final state |f > in time T : N ∑ E[| < f |U0 (T ) + εr Ur (T )|i > |2 ] r=1

where the average is taken over the probability distribution of the martingale processes over [0, T ]. Examples of Martingales and the Ito formula for them (a) ∫

∫

t

M (t) =

g(s, x, ω)(N (ds, dx, ω) − λ(s)dF (x)ds)

f (s, ω)dB(s, ω) + 0

s≤t,x∈E

where f, g are progressively measurable functions with B being Brownian motion and N (t, ., ω) a Poisson field with EN (ds, E) = λ(s)dsF (E). Ito’s formula for this martingale is ∫ g(t, x)2 N (dt, dx) (dM (t))2 = d < M > (t) = f (t)2 dt + x∈E

A.4.Syllabus for a short course on Linear algebra and its application to classical and quantum signal processing. [1] Vector space over a field, linear transformations on a vector space.

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[2] Finite dimensional vector spaces, basis for a vector space, matrix of a linear transformation relative to a basis, Similarity transformation of the matrix of a linear transformation under a basis change. [3] Examples of finite and infinite dimensional vector spaces. [4] range, nullspace, rank and nullity of a linear transformation. [5] Subspace, direct sum decompositions of vector spaces, projection opera­ tors. [6] Linear estimation theory in the language of orthogonal projection oper­ ators. [7] Statistics of the estimation error of a vector under a small random per­ turbation of the data matrix, statistics of the perturbation of the orthogonal projection operator under a small random perturbation of the data matrix. [8] Primary decomposition theorem, Jordan decomposition theorem, func­ tions of matrices. [9] Cauchy’s residue theorem in complex analysis and its approach to the computation of functions of a matrix. [10] Norms on a vector space, norms on the space of matrices, Frobenius norm, spectral norm. [11] Notions of convergence in a vector space, calculating the exponential function and inverse of a perturbed matrix using a power series. [12] Recursive least squares lattice algorithms for time and order updates of prediction error filters based on appending rows and columns to matrices and computing functions of the appended matrices, RLS lattice for second or­ der Volterra systems, Statistical properties of the prediction filter coefficients under the addition of a small noise process to the signal process: A statistical perturbation theory based approach. [13] The MUSIC and ESPRIT algorithms for direction of arrival estimation based on properties of signal and noise eigensubspaces. [14] Computing the solution to time varying linear state variable systems using the Dyson series. Convergence of the Dyson series. [15] Computing the approximate solution to nonlinear state varable systems using Dyson series applied to the linearized system. [16] Dyson series in quantum mechanics. [17] Computing transition probabilities for quantum systems with random time varying potentials using the Dyson series. [18] Approximate solution to stochastic differential equations driven by Brow­ nian motion and Poisson fields using linearization combined with Dyson series. Mean and variance propagation equations based on linearization. [19] The spectral theorem for finite dimensional normal operators and infinite dimensional unbounded self-adjoint operators in a Hilbert space. [20] Properties of spectral families in finite and infinite dimensional Hilbert spaces. [21] The general theory of estimating parameters in linear models for Gaus­ sian and non-Gaussian noise. [22] The quantum stochastic calculus of Hudson and Parthasarathy and its application to the modeling of a quantum system coupled to a photon bath.

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[23] Kushner nonlinear filter and its linearized EKF version. [24] The Belavkin quantum filter based on non-demolition measurements and its application to quantum control. Application to quantum control: The Belavkin equation can be expressed as dρt = Lt (ρt )dt + Mt (ρt )dWt where Wt is a classical Wiener process arising from the innovations process of the measurement dWt = dYt − πi (St + St∗ )dt = dYt − T r(ρt (St + St∗ ))dt Note that ρt can be viewed as a classical random process with values in the space of signal space density matrices. St is a system space linear operator. The Belavkin equation is a commutative equation since all the terms appearing in it like ρt , Lt (ρt ), Wt etc. are signal space operator valued functionals of the commutative noise processs {Yy }. Now let Uc (t) be the control unitary satisfying the sde dUc (t) = (−(iH1 (t) + Q1 (t))dt − iK(t)dYt )Uc (t) We have

Y (t) = U (t)∗ Yi (t)U (t) = U (T )∗ Yi (t)U (T ), T ≥ t

Here Yi (t) is an operator on the Boson Fock space and is thus independent of the system Hilbert space operators. We have taking Yi (t) = A(t) + A(t)∗ , dY (t) = dYi (t) + jt (Zt )dt where Zt is a system space operator and jt (Z) = U (t)∗ ZU (t). Thus, dUc (t) = (−(iH1 (t) + Q1 (t))dt − iK(t)(jt (Zt )dt + dYi (t)))Uc (t) We have

d(Uc∗ Uc ) = dUc∗ Uc + Uc∗ dUc + dUc∗ dUc

= Uc (t)∗ (−(Q∗1 + Q1 )dt + idYt K(t) − iK(t)dYt + dY (t)K(t)2 dY (t))Uc (t) If K(t) commutes with dY (t) and Q1 +Q∗1 = K(t)2 , then we would get d(Uc∗ Uc ) = 0 and Uc will be a control unitary operator. Taking K(t) = jt (Pt ) where Pt is a system operator, we have K(t)dY (t) = jt (Pt )(jt (Zt )dt + dYi (t)) = jt (Pt Zt )dt + jt (Pt )dYi (t) and dY (t)K(t) = jt (Zt Pt )dt + jt (Pt )dYi (t) So for Uc (t) to be unitary, we require that [Zt , Pt ] = 0 for all t. Note that Zt , Pt are system Hilbert space operators. We can now define ρc (t) = Uc (t)ρ(t)Uc (t)∗

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Now, dρc (t) = dUc (t)ρ(t)Uc (t)∗ +Uc (t)dρ(t).Uc (t)∗ +Uc (t)dρ(t)dUc (t)∗ +dUc (t)dρ(t)Uc (t)∗ +Uc (t)dρ(t).dUc (t)∗ + dUc (t)ρ(t)dUc (t)∗

[25] Linear algebra applied to the study of the linearized Einstein field equa­ tions in the presence of matter and radiation for the study of galactic evolution as the propagation of small non-uniformities in matter and radiation propagat­ ing in an expanding universe. (0) gμν (x) is the background Robertson Walker (RW) metric. Its perturbation is (0) gμν = gμν + δgμν The coordinate system can be chosen so that δg0μ = 0 Then, the linearized Ricci tensor is α δRμν = δΓα μα,ν − δΓμν,α β(0)

β α −(δΓμν )Γαβ − Γα(0) μν δΓαβ α(0)

β(0)

+Γμβ δΓβνα + (δΓα μβ )Γαβ This expression can be expressed as

α δRμν = (δΓα μα ):ν − (δΓμν ):α

where the covariant derivatives are computed using the unperturbed RW metric. The energy momentum tensor of the matter field is Tμν = (ρ + p)vμ vν − pgμν and that of the radiation field is Sμν =

1 Fαβ F αβ gμν − Fμα Fνα 4

For example, computing in a flat space-time S00 = Now

1 Fαβ F αβ − F0α F0α 4

Fαβ F αβ = −2F0r F0r + Frs Frs = −2|E|2 + 2|B|2 F0α F0α = −F0r F0r = −|E|2

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General Relativity and Cosmology with Engineering Applications

so

1 1 (−|E|2 + |B|2 ) + |E|2 = (|E|2 + |B|2 ) 2 2 which is the correct expression for the energy density of the electromagnetic field. Likewise, S 0r = S r0 defines the energy flux as well as the momentum density and S rs = S sr defines the momentum flux. We can write the expression for δRμν in the general form S00 =

δRμν = C1 (μναβρ, x)δgαβ,ρ (x) + C2 (μναβρσ, x)δgβρ,σ (x) where C1 , C2 are functions of x determined completely from the background (0) gravitational field gμν (x) The perturbation to the energy momentum tensor of matter is given by δTμν = (δρ + δp)Vμ(0) Vν(0) + (ρ(0) + p(0) )(Vμ(0) δvμ + Vν(0) δvμ ) (0) −δpgμν + p(0) δgμν

We note that

Vμ(0) = (1, 0, 0, 0),

and p(0) , ρ(0) are functions of time only. Further, (0)

(0)

(0)

(0)

g00 = 1, g11 = −S 2 (t)f (r), g22 = −S 2 (t)r2 , g33 = −S 2 (t)r2 sin2 (θ), 1 1 − kr2 with k = 1 for a spherical universe, k = 0 for a flat universe and k = −1 for a hyperbolic universe. We now consider a flat unperturbed universe for which the metric has the form dτ 2 = dt2 − S 2 (t)(dx2 + dy 2 + dz 2 ) f (r) =

Thus,

g00 = 1, grr = −S 2 (t), r = 1, 2, 3

The Ricci tensor components are: α α β R00 = Γα 0α,o − Γ00,α − Γ00 Γαβ β +Γα 0β Γ0α

Now, r Γα 0α = Γ0r =

So

1 rr g grr,0 = S ' /S, r = 1, 2, 3 2 ' ' Γα 0α,0 = (S /S)

Γα 00,α = 0

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413

β Γα 00 Γαβ = 0 β Γα 0β Γ0α = ∑ ∑ ' (Γr0r )2 = (g rr grr,0 /2)2 = 3S 2 /S 2 r

So

r '

'

R00 = (S ' /S)' + 3S 2 /S 2 = S '' /S + 2S 2 /S 2 β α α α β Rkk = Γα kα,k − Γkk,α − Γkk Γαβ + Γkβ Γkα ∑ 0 0 k Γkk,0 − Γ0kk Γr0r + 2Γkk Γk0 r

(No summation over k) = −gkk,00 /2 + (gkk,0 /2)(S ' /S) + 2(−gkk,0 /2)(gkk,0 /2gkk ) '

= S '' /2 − S 2 /2S + S ' (S ' /2S) = S '' /2 We have in fact

Rkm = (S '' /2)δkm

Now let us study the perturbed Einstein field equations w.r.t. the above flat space-time metric. First note that we can choose our coordinate system so that δg0μ = 0 and hence δg 0μ = 0. Raising and lowering of indices are carried out w.r.t. the above flat space time metric which is diagonal. The unperturbed space-time is comoving, ie, v μ(0) = (1, 0, 0, 0) define geodesics in the unperturbed space-time. The unperturbed pressure and density p(0) (t), ρ(0) (t) are functions of t only. The unperturbed energy momen­ tum tensor of matter is T μν(0) = (ρ(0) + p(0) )v μ(0) v ν(0) − p(0) g μν(0) so that

T 00(0) = ρ(0) , T kk(0) = p(0) /S 2

with the other components of the energy momentum tensor being zero. The unperturbed Einstein field equations 1 (0) Rμν = K.(Tμν(0) − T (0) gμν(0) ), K = −8πG 2 thus give after noting that (0) μν(0) T (0) = gμν T =

ρ(0) − 3p(0) ,

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Tkk = −S 2 T kk(0) = −p(0) and hence the unperturbed field equations are ' 1 S '' /S + 2S 2 /S 2 = K.(ρ(0) − (ρ(0) − 3p(0) ), 2

S '' /2 = K(−p(0) +

S 2 (0) (ρ − 3p(0) )) 2

These two equations along with an equation of state: p(0) = f (ρ(0) ) determing the three functions of time S(t), ρ(0) (t), p(0) (t). The perturbed equations are δRμν = KδTμν where δTμν = (ρ(0) + p(0) )(vμ(0) δvν + vν(0) δvμ ) + (δρ + δp)vμ(0) vν(0) (0) −δpgμν − p(0) δgμν

Now, α(0)

β α − δΓα δRkm = δΓkα,m km,α − Γkm δΓαβ β(0)

α(0)

β −Γαβ δΓα km + Γkβ δΓmα α +Γβ(0) mα δΓkβ

The perturbed field equations taking into account contributions from the elec­ tromagneti field are expressible in the form C1 (μναβγ, x)δgαβ,γ (x) + C2 (μναβγσ, x)δgαβ,γσ (x) = C3 (μνα, x)δvα (x) + C4 (μν, x)δρ(x) + C5 (μν, x)δp(x) +C6 (μναβ, x)δAα,β (x) = 0 The equations implied by the Bianchi identity are (T μν + S μν ):ν = 0 This is the same as (T μν + S μν ),ν + Γμαν (T αν + S αν ) + Γναν (T μα + S μα ) = 0 We calculate is first order perturbed version: αν (δT μν + δS μν ),ν + Γμ(0) + δS αν )+ αν (δT μα Γν(0) + δS μα )+ αν (δT μ δΓαν (T αν(0) + S αν(0) )+ ν δΓαν (T μα(0) + δS μα(0) ) = 0

General Relativity and Cosmology with Engineering Applications

415

This can be put in the form C7 (μαβ, x)δvα,β (x) + C8 (μα, x)δvα (x) + C9 (μαβγ, x)δAα,βγ (x) +C10 (μαβγ, x)δgαβ,γ (x) + C11 (μ, x)δρ(x) + C12 (μ, x)δp(x) = 0 We note that this last equation contains 3 equations for the velocity components δvr , r = 1, 2, 3 and one equation for δρ(x). δp(x) is determined from δρ(x) using the equation of state. Also δv0 is determined from δvr using 0 = δ(gμν v μ v ν ) (0) μ(0) (v δv ν + v ν(0) δv μ = (δgμν )v μ(0) v ν(0) + gμν

using that the unperturbed dynamics is comoving,ie, v μ(0) = (1, 0, 0, 0) we get from this equation δg00 + 2δv 0 = 0 so

1 δv 0 = − δg00 2

and hence, δv0 = δ(g0μ v μ ) = δg00 (0)

since v r(0) = 0 and g0r = 0. [26] Perturbation theory applied to electromagnetic problems. Here, we discuss the rudiments of the theory of time independent perturbation theory in quantum mechanics and explain how the same techniques can be used to solve waveguide and cavity resonator problems having almost aribtrary cross sections by transforming the boundary into a simpler boundary using the theory of analytic functions of a complex variable. Let D be a flat connected region parallel to the xy plane. D represents the cross section of a waveguide or a cavity resonator. The boundary ∂D is a closed curve that represents the boundary of the guide or resonator. The z direction is orthogonal to the D plane and for the guide, it represents the direction along which the em waves propagate. For the resonator case, we assume that 0 ≤ z ≤ d with the surfaces z = 0 and z = d being perfectly conducting surfaces as is the case with the side walls. We choose a system (q1 , q2 , z) of coordinates so that (q1 , q2 ) are functions of (x, y) alone. Being more specific, we assume that w = q1 (x, y) + jq2 (x, y) = f (z) = f (x + jy) with an inverse z = x + jy = g(w) = g(q1 + jq2 ) and assume that f (z) is an analytic function of the complex variable z and its inverse g(w) is also an analytic function of the complex variable w. We can

416

General Relativity and Cosmology with Engineering Applications regard (z, z¯) as independent variables just as (x, y) are. The relation between the two pairs of variables is z = x + jy, z¯ = x − jy, x = (z + z¯)/2, y = (z − z¯)/2j We have

∂ ∂ ∂ = z,x + z¯,x ∂x ∂z ∂z¯ ∂ ∂ = + ∂z ∂z¯

and likewise, ∂ ∂ ∂ = j( − ) ∂y ∂z ∂z¯ Thus, ∇2⊥ = =(

∂2 ∂2 + 2 2 ∂x ∂y

∂ ∂ 2 ∂ ∂ 2 + ) −( − ) ∂z ∂z¯ ∂z ∂z¯ ∂2 =4 ∂z∂z¯

Now, ∂ ∂ = w' (z) ∂z ∂w ∂ ∂ =w ¯ ' (z) ∂z¯ ∂w ¯ ' also, we clearly have since w (z) is an analytic function of z and w ¯ ' (z) is an ' analytic function of z¯ and hence that 1/w (z) = dz/dw is an analytic function ¯ that of w and 1/w ¯ ' (z) is an analytic function of w 2 =4 ∇⊥

∂2 ∂2 = 4|z ' (w)|−2 ∂z∂z¯ ∂w∂ w ¯

= |z ' (w)|−2 (

∂2 ∂q22 + ∂q12 )

Thus, the eigenvalue problem (∇2⊥ + h2 )ψ(x, y) = 0 with the Dirichlet boundary condition ψ = 0 on ∂D becomes (∇2⊥,q + h2 F (q1 , q2 ))ψ(q1 , q2 ) = 0 with ψ(a, q2 ) = 0 where the boundary ∂D is the same as q1 = a. For example, if we take w = log(z) = log(ρ) + jφ, then q1 = a is the circle q1 = ρ = ea which is a circle and (q1 , q2 ) = |z ' (w)|2 = |g ' (q1 + jq2 )|2

General Relativity and Cosmology with Engineering Applications Also, we have defined ∇2⊥,q =

417

∂2 ∂2 + 2 =L 2 ∂q1 ∂q2

say. Solution by perturbation theory: Let h2 = −λ. Then, we have to solve (L − λ(1 + εG(q1 , q2 )))ψ(q1 , q2 ) = 0 where we are assuming that F (q) = 1 + εG(q) so that the boundary is a small perturbation of the rectangular boundary. [27] Perturbation theory applied to general nonlinear partial differential equations with noisy terms. The gravitational wave equations, fluid dynami­ cal equations, Klein-Gordon and Dirac equations are special cases of this. The field φ : Rn → Rp satisfies a nonlinear pde n ∑

bk (x)φl,k (x) +

k=1

p ∑

akm (x)φl,km (x) + δ.Fl (φl (x), φl,k (x), φl,km (x), x)

k,m=1

+δ.

∑

Glm (φl (x), φl,k (x), φl,km (x), x)wm (x)

m

where wm (x) are Gaussian noise processes. [28] The Knill-Laflamme theorem for quantum error correcting codes: Ex­ plicit construction of the recovery operators for a noisy quantum channel in terms of the code subspace and the noise subspace. [29] Post-Newtonian equations of hydrodynamics. The perturbations are carried out in powers of the velocity. The mass parameter is of the order of the square of the velocity (v 2 = GM/r for the orbital velocity) and the following expansions are valid ρ = ρ2 + ρ4 + ..., p = p4 + p6 + ..., v r = v1r + v3r + ..., v 0 = 1 + v20 + v40 + ... g00 = 1 + g00(2) + g00(4) + ..., g0r = g0r(3) + g(0r(5) + ... grs = −δrs + grs(2) + grs(4) + ... g 00 = 1 + g200 + g400 + ... g 0r = g30r + g50r + ...,

418

General Relativity and Cosmology with Engineering Applications g rs = −δ rs + g2rs + g4rs + ... T μν = (ρ + p)v μ v ν − pg μν

The equation gμν v μ v ν = 1 can be expressed as g00 v 02 + 2g0r v 0 v r + grs v r v s = 1 so the O(1) equation is

v00 = 1

The O(v 2 ) equation is v20 + g00(2) −

∑

v1r2 = 0

r

or equivalently, v20 = −g00(2) +

∑

v1r2

r

We have

T 00 = T200 + T400 + ... T 0r = T30r + T50r + ... T rs = T2rs + T4rs + ...

where T200 = ρ2 , T400 = 2ρ2 v20 , T600 = ρ2 v202 + 2ρ4 v20 + 2p4 v20 − p4 g200 T30r = 2ρ2 v20 v1r , T50r = 2ρ2 v20 v1r + 2ρ2 v3r + 2ρ4 v1r + 2p4 v1r , T2rs =

[30] Lab problems on linear algebra based signal processing: [1] If X is an m × n matrix, then calculate δPX upto O(|| δX) ||) where PX is the orthogonal projection onto R(X) and X gets perturbed to X + δX where δX is a small random perturbation of X. Calculate using this formula, the second order statistics of δPX , ie, E(δPX ⊗ δPX ) in terms of E(δX ⊗ δX). Calculate δPX upto O(||δX||2 ) [2] Take an n × n matrix A. Add a row and a column to this matrix at the end and express the inverse of this matrix B in terms of the inverse of A. Assume now that A gets perturbed to A+δA and correspondingly, the appended row and column get perturbed by small amounts. Then calculate the inverse of the perturbed appended matrix in terms of A−1 and the appended rows and columns and their perturbations upto linear orders in the perturbations. [3] Generate some functionals of the Brownian motion process B(t) like M (t) = max(B(s) : s ≤ t), m(t) = min(B(s) : s ≤ t), Ta = min(t >

General Relativity and Cosmology with Engineering Applications

419

0 : B(t) = a), |B(t)| (reflected Brownian motion), Absorbed Brownian mo­ tion (B(min(t, T0 ) : t ≥ 0 where B(0) = a > 0 and T0 = min(t > 0 : B(t) = 0, Local time process La (t) of the Brownian motion process at the ∫t level a. This is defined as La (t) = 0 δ(B(s) − a)ds and is approximated by ∫ t (2ε)−1 0 χ[a−ε,a+ε] (B(s))ds where ε is a very small positive number. For a bi­ variate standard Brownian motion process (B1 (t), B2 (t)), t ≥ 0, simulate the ∫t area process A(t) = 0 B1 (s)dB2 (s) − B2 (s)dB1 (s) and calculate its statistics. ∑d Simulate the Bessel process of order d, ie X(t) = ( k=1 Bk (t)2 )1/2 , t ≥ 0 and verify by numerical simulations that it satisfies its standard stochastic differen­ tial equation. [4] Verification of the Knill-Laflamme theorem for quantum error correcting codes. Generate a set of r < n linearly independent column vectors {f1 , ..., fr } in H = Cn . Denote the subspace spanned by these r vectors by C. Calculate the orthogonal projection P onto C using the standard formula P = A(A∗ A)−1 A∗ where A = [f1 , ..., fr ] = Cn×r . Generate K n × n matrices having the block structure ( ) C1k C2k , k = 1, 2, ..., K Nk = C3k C4k where C1k is an r × r matrix for each k such that for 1 ≤ k, j ≤ K, we have C1∗j C1k + C3∗j C3k = λjk Ir for some complex numbers λjk . This can be achieved by choosing the r × n ∗ ∗ |C3j ], 1 ≤ j ≤ K as non-overlapping rows of an n × n unitary matrices [C1j matrix U and then multiply the resultant matrices by complex constants. Thus, we must have K ≤ [n/r]. The matrices C2k , C3k , C4k can be chosen arbitrarily. Now take the orthogonal projection P onto C as ( ) Ir 0 P = 0 0 It is then easily seen that P Nk∗ Nj P = λkj P and hence the quantum code P can correct the noise subspace {Nk : 1 ≤ k ≤ K}. [5] (a) Study waves in a plasma influenced by a strong gravitational field and electromagnetic fields by the method of linearization: The Boltzmann particle distribution function f (t, r, v) where v = (v r = dxr /dt)r = 13 satisfies (after approximating the collision term by a linear relaxation term f,t (t, r, v) + v k f,xk (t, r, v) + v,k0 f,vk (t, r, v) = (f0 (r, v) − f (t, r, v))/τ (v) Here the velocity v k satisfies the geodesic equation in an electromagnetic field: dxk /dτ = γv k , dv k /dτ = γd/dt(dxk /dτ ) = γd/dt(γv k ) = γ 2 v,k0 + γγ,0 v k

420

General Relativity and Cosmology with Engineering Applications

where

γ = dt/dτ = (g00 + 2g0r v r + grs v r v s )−1/2

k γ 2 v,k0 + γγ,0 v k + γ 2 Γk00 + 2γ 2 Γk0m v m + γ 2 Γmp v m v p = eγ(F0m + Fsm v s ) − − − (a)

or equivalently, k k k v,0 + (γ,0 /γ)v k + Γ00 + 2Γ0m v m + Γkmp v m v p = eγ −1 (F0m + Fsm v s )

This value of v,k0 is substituted into the above Boltzmann equation to get f,t (t, r, v) + v k f,xk (t, r, v) − (γ,0 /γ)v k + Γk00 + k 2Γk0m v m +Γmp v m v p −eγ −1 (F0m +Fsm v s ))f,vk (t, r, v)−(f0 (r, v)−f (t, r, v))/τ (v) = 0

We note that γ,0 is a function of v,k0 , v k , x. We need to get a Boltzmann equation that does not involve v,k0 . For this purpose, we go back to the equation of motion of the charged particle (a). First observe that −1/2

γ,0 = (g00 +2g0r v r +grs v r v s ),0

= (−γ 3 /2)(g00,0 +g00,k v k +2g0r,0 v r +2g0r,s v r v s +2g0r v,r0

+grs,0 v r v s + grs,m v r v s v m + 2grs v,s0 ) − − − (b) Substituting (b)into (a) gives us a linear algebraic equation for (v,k0 )3k=1 which is inverted to get v,k0 as a function (v k ), xμ , the electromagnetic field F μν (x) and of course the metric gμν (x) and its first order partial derivatives gμν,α . Thus, we get a well defined Boltzmann equation. Now given the particle distribution function f (t, r, v), we need to calculate the energy momentum tensor of matter. We have T μν = (ρ + p)V μ V ν − pg μν ∫

where ρ=m ∫ Ur =

v r f d3 v/

∫

f d3 v,

f d3 v, V r = γ(U )U r

V 0 is calculated using gμν V μ V ν = 1 Equivalently, γ(U )2 g00 + 2γ(U )2 U r g0r + γ(U )2 grs U r U s = 1 or

γ(U ) = (g00 + 2g0r U r + grs U r U s )−1/2 , V 0 = γ(U )

The average internal kinetic energy per particle is ∫ ∑∑ ( (v r − U r )2 )f (t, r, v)d3 v K(t, r) = (m/2) r

General Relativity and Cosmology with Engineering Applications

421

and the pressure field is given by ∫ ∑ p(t, r) = nm < |v − U | > /3 = (m/3) (v r − U r )2 f (t, r, v)d3 v 2

r

where m is the mass of a plasma particle and n is the number of plasma particles per unit volume. This energy momentum tensor of the plasma can be added to the energy momentum tensor of the electromagnetic field and substituted into the right side of the Einstein field equations. Thus, we get a couple system of pde’s for f, gμν , Aμ . An alternate way to define the energy momentum tensor is as ∫ T μν = m f (t, r, v)γ(t, r, v)v μ v ν d3 v − p(t, r)g μν (t, r) where v μ = dxμ /dt so that

v 0 = 1, v r = dxr /dt

and γ = γ(t, r, v) is defined by the equation γ = (g00 + 2g0k v k + gkm v k v m )−1/2 The pressure p is as defined earlier. [31] Quantum image processing: The image field is obtained by passing a quantum em field through a spatio temporal filter having impulse response h(t, τ, r, r' ). Assume the Coulomb gauge with zero charge density. Then the scalar potential A0 = 0 and the vector potential is given by ∫ r A = [(2|K|)−1/2 a(K, σ)er (K, σ)exp(−i(|K|t − K.r))+ (2|K|)−1/2 a∗ (K, σ)¯ er (K, σ)exp(i(|K|t − K.r))]d3 K The Coulomb gauge condition divA = Ar,r = 0 implies K r er (K, σ) = 0 or equivalently, (K, e(K, σ)) = 0 which means that there are only two degrees of polarization which are indexed by σ = 1, 2. The electric field is ∫ Er = −Ar,0 = i [(|K|/2)1/2 a(K, σ)er (K, σ)exp(−i(|K|t − K.r))− (|K|/2)1/2 a∗ (K, σ)¯ er (K, σ)exp(i(|K|t − K.r))]d3 K or equivalently in three vector notation, ∫ E = i [|K|/2)1/2 a(K, σ)e(K, σ).exp(−i(|K|t − K.r))

422

General Relativity and Cosmology with Engineering Applications −(|K|/2)1/2 a∗ (K, σ)¯ e(K, σ)exp(i(|K|t − K.r))]d3 K The magnetic field is given by ∫ B = curlA = i [(2|K|)−1/2 a(K, σ)K × e(K, σ)exp(−i(|K|t − K.r))− (2|K|)−1/2 a∗ (K, σ)K × e(K, σ)exp(i(|K|t − K.r))]d3 K The em field of the image can be regarded as the output of a spatio-temporal filter with this free em field as input. Thus, the output field is ∫ ∫ o ' ' 3 ' o E (t, r) = h(t, τ, r, r )E(τ, r )dτ d r , B (t, r) = h(t, τ, r, r' )B(τ, r' )dτ d3 r' and hence the output field can be expressed as ∫ ¯ E (t, r, K, σ)a∗ (K, σ))d3 K E o (t, r) = (HE (t, r, K, σ)a(K, σ) + H ∫ o

B (t, r) =

¯ B (t, r, K, σ)a∗ (K, σ)]d3 K [HB (t, r, K, σ)a(K, σ) + H

where HE , HB are 3 × 1 functions constructed from the image impulse response h(t, τ, r, r' ). The energy of the em field coming from the image is ∫ HF = (|E o |2 + |B o |2 )d3 r/2 and this can be expressed in the form after discretization of the spatial frequen­ cies p ∑ ¯ 2 (k, m, θ)a∗ a∗ ) (Q1 (k, m, θ)ak∗ am + Q2 (k, m, θ)ak am + Q HF (θ) = (1/2) k m k,m=1

where ¯ 1 (k, m, θ) = Q1 (m, k, θ) Q Here θ is a parameter vector upon which the image impulse response h(t, τ, r, r' ) depends. θ is the parameter which contains all information about the image field and is to be estimated by exciting an atom with the output image field and and taking measurements on the state of the atom at different times. We can simplify the form of the image field Hamiltonian as HF (θ) =

p ∑

Q(k, m, θ)ak∗ am

k,m=1

Here, ∗ ]=0 [ak , a∗m ] = δkm , [ak , am ] = 0, [ak∗ , am

General Relativity and Cosmology with Engineering Applications

423

The Hamiltonian of the atom assumed to be an N state system, is an N × N Hermitian matrix HA and the interaction Hamiltonian between the image field and the atom has the form Hint (t) =

p ∑

(Fk (t, θ) ⊗ ak + Fk (t, θ)∗ ⊗ a∗k )

k=1

[32] Performance analysis of the MUSIC algorithm. X = AS + W, X ∈ CN ×K , A ∈ CN ×p , S ∈ Cp×K , W ∈ CN ×K 2 Rxx = K −1 E(XX ∗ ), Rss = K −1 E(SS ∗ ), σw I = K −1 E(W W ∗ ), E(S⊗W )

= E(S⊗W ∗ ) = 0 All signals are complex Gaussian. Thus, E(X ⊗ X) = 0, E(S ⊗ S) = 0, E(W ⊗ W ) = 0 The stochastic perturbation in the array signal correlation matrix is given by δRxx = K −1 XX ∗ − Rxx = 2 K −1 ASS ∗ A∗ + K −1 W W ∗ + K −1 ASW ∗ + K −1 W S ∗ A∗ − ARss A∗ − σw I

= AδRss A∗ + δRww + K −1 (ASW ∗ + W S ∗ A∗ ) where

2 δRss = K −1 SS ∗ − Rss , δRww = K −1 W W ∗ − σw I

To calculate the mean and covariance of the DOA estimates, we need the mean and covariance of the statistical perturbation δRxx of Rxx . Now, E(δRss ⊗ δRss ) = K −2 E(SS ∗ ⊗ SS ∗ ) − Rss ⊗ Rss Now,

E(SS ∗ ⊗ SS ∗ ) = E[(S ⊗ S)(S ∗ ⊗ S ∗ )]

This is equivalent to calculating E(si sj s¯k s¯m ) = E(si s¯k )E(sj s¯m ) + E(si s¯m )E(sj s¯k ) = [Rss ]ik [Rss ]jm + [Rss ]im [Rss ]jk Also since S and W are independent random matrices, it follows that δRss and δRww are independent zero mean random matrices. The second order moments of δRxx are thus computed as E(δRxx ⊗ δRxx ) = E[AδRss A∗ + δRww + K −1 (ASW ∗ + W S ∗ A∗ ))⊗2 ] = (A ⊗ A)E(δRss ⊗ δRss )(A∗ ⊗ A∗ )

424

General Relativity and Cosmology with Engineering Applications +E(δRww ⊗ δRww ) +K −2 (I + F )E(ASW ∗ ⊗ W S ∗ A∗ )

where F denotes the flip operator: F (x ⊗ y) = y ⊗ x Computing the last expectation is equivalent to computing ¯lk wi' j ' s¯k' j ' a ¯ k ' l' ) E(aij sjk w ¯k' l' E(sjk s¯k' j ' )E(wi' j ' w ¯lk ) = aij a Thus the second order moments of δRxx are easily computed and this can be combined with matrix perturbation theory to obtain the covariance of the signal ˆ xx = Rxx + δRxx . These covariances and noise eigenvalues and eigenvectors of R can in turn be used to calculate the error covariances in the DOA estimates using the MUSIC pseudospectrum. [33] Estimating the quantum image parameters from measurements on the state of an atom excited by the quantum em field coming from the image in the interaction representation. The image em field interacts with an atom described by an N × N Hamiltonian matrix HA . This interaction Hamiltonian can be expressed as Hint (t|θ) =

p ∑

(Gk (t|θ) ⊗ ak + Gk (t|θ)∗ ⊗ a∗k )

k=1 N ×N

, θ is the image parameter vector. The joint density of where Gk (t|θ) ∈ C the atom and the image em field at time t can be expressed using the GlauberSudarshan representation: ∫ ρ(t) = C exp(−|z|2 )A(t, z) ⊗ |e(z) >< e(z)|d2p z where |e(z) >=

∑ n≥0

since

z n a∗n |0 > /n! =

∑

√ z n |n > / n!

n≥0

√ |n >= a∗n |0 > / n!

is the normalized state of the field in which a∗k ak has the eigenvalue nk and n = (nk ). Further C = π −p We have ak |e(z) >= zk |e(z) >, a∗k |e(z) >=

∂ |e(z) > ∂zk

425

General Relativity and Cosmology with Engineering Applications Thus, ak |e(z) >< e(z) >= zk |e(z) >< e(z)|, a∗k |e(z) >< e(z)| = |e(z) >< e(z)|ak =

∂ |e(z) >< e(z)|, ∂zk

∂ |e(z) >, e(z)|, ∂z¯k

|e(z) >< e(z)|a∗k = z¯k |e(z) >< e(z)| Note that z → |e(z) > is an analytic function of z and so z¯ →< e(z)| is an analytic function of z¯. Now, the joint density ρ(t) satifies Schrodinger’s equation in the interaction picture: iρ' (t) = [Hint (t|θ), ρ(t)] and this translates to ∫ i A,t (t, z) ⊗ |e(z) >< e(z)|exp(−|z|2 )d2p z = ∑∫ ( (Gk (t|θ)A(t, z) ⊗ ak |e(z) >< e(z)| + Gk (t|θ)∗ A(t, z) ⊗ a∗k |e(z) >< e(z)| k

−A(t, z)Gk (t|θ)⊗|e(z) >< e(z)|ak −A(t, z)Gk (t|θ)∗ ⊗|e(z) >< e(z)|a∗k )exp(−|z|2 )d2p z) ∑∫ ∂ ( (Gk (t|θ)A(t, z)zk ⊗|e(z) >< e(z)|+Gk (t|θ)∗ A(t, z)⊗( |e(z) >< e(z)|) = ∂zk k

∂ |e(z) >< e(z)|)−A(t, z)Gk (t|θ)∗ z¯k ⊗|e(z) >< e(z)|)exp(−|z|2 )d2p z) ∂z¯k ∑∫ ∂ = ( (zk Gk (t|θ)A(t, z) − (Gk (t|θ)∗ ( − z¯k )A(t, z)

−A(t, z)Gk (t|θ)⊗(

∂zk

k

∂ − zk )A(t, z)Gk (t|θ) − z¯k A(t, z)Gk (t|θ)) ⊗ |e(z) >< e(z)|d2p z) ∂z¯k Thus, we get iA,t (t, z) = (T (t|θ)A)(t, z) +(

where T (t|θ) is a differential operator acting on the space of matrix valued functions of the complex variable z ∈ Cp defined by T (t|θ)X(z) =

∑

(zk Gk (t|θ)X(z) − z¯k X(z)Gk (t|θ) − Gk (t|θ)∗ (

k

∂ − zk )X(z)Gk (t|θ)) ∂z¯k The formal solution to this partial differential equation is ∫ t A(t, z) = τ {exp(−i T (s|θ)ds)}(A(0, z)) +(

0

∂ − z¯k )X(z) ∂zk

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General Relativity and Cosmology with Engineering Applications

where τ is the time ordering operator. The atomic (system) density at time t is given by ∫ ρA (t) = T r2 (ρ(t)) = π −p A(t, z)d2p z A(t, z) and ρA (t) are N × N matrices. [34] Existence and uniqueness of solutions to stochastic differential equations. (a) Kolmogorov’s inequality for discrete time sub-martingales. Let Mk , k = 0, 1, 2, ... be a non-negative sub-martingale. Let τa = min(k ≥ 0 : Mk ≥ a), a > 0 Then, {τa = m} = {M0 < a, M1 < a, ..., Mm−1 < a, Mm ≥ 0} Thus, if Fn is the underlying filtration for {Mn }, it follows that {τa = m} ∈ Fm In particular, τa is a stop-time. Thus, by the submartinagle property, for n ≥ m, E(Mn χτa =m ) ≥ E(Mm χτa =m ) ≥ a.P (τa = m) and summing over m gives E(Mn ) ≥ E(Mn χτa ≤n ) ≥ a.P (τa ≤ n) or P (τa ≤ n) = P (max0≤k≤n Mk ≥ a) ≤ E(Mn )/a This can be generalized to the continuous time scenario. Specifically, if Mt , t ≥ 0 is a continuous martingale, then P (max0≤s≤t |Ms | ≥ a) ≤ E|Mt |/a since |Mt | is a submartingale by Jensen’s inequality. Using this, we can deduce a version Doob’s inequality: E(max0≤t≤T |Mt |2 ) ≤ C1 (E(MT2 ) Now consider the sde dXt = f (t, Xt )dt + g(t, Xt )dMt , X0 = x where f, g satisfy appropriate Lipshitz conditions which will be specified later. We wish to prove the existence and uniqueness of the solution to this sde. We (n) define the processes Xt , n = 1, 2, ... recursively as ∫ t ∫ t (n+1) (n) Xt =x+ f (s, Xs )ds + g(s, Xs(n) )dMs 0

0

General Relativity and Cosmology with Engineering Applications

427

These processes are all adapted to the underlying filtration on which the process M is defined and we have by Doob’s in ∫ T (n+1) (n) 2 −Xt | ) ≤ KT E |Xs(n) −Xs(n−1) |2 ds E(max0≤t≤T |Xt 0 ∫ T (n) (n−1) 2 |Xt −Xt | d < M >t ) +KC1 (E 0

If we assume that the measure < M > is absolutely continuous w.r.t. the Lebesgue measure, with bounded Radon-Nikodym derivative, then we derive from the above that ∫ T (n) 2 (n) (n+1) − Xt | ) ≤ (K0 T + K1 ) |Xt − X (n−1 )t |2 dt E(max0≤t≤T |X 0

from which the existence of a solution to the sde can be inferred using standard arguments based on Gronwall’s inequality. [35] Statistical analysis of the RLS lattice algorithm. Let x[n] be a random process and we form the vector ξn = [x[n], x[n − 1], ..., x[0]]T ∈ Rn+1×1 z −p ξn = [x[n − p], x[n − p − 1], ..., x[−p]]T ∈ Rn+1×1 where x[k] = 0f ork < 0. Define the data matrix Xn,p = [z −1 ξn , ..., z −p ξn ] ∈ Rn+1×p Let Pn,p = PR(Xn,p ) . The forward and backward prediction error sequences or order p at times n and n − 1 are respectively defined by ⊥ ⊥ −p−1 ef [n|p] = Pn,p ξn , eb [n − 1|p] = Pn,p z ξn

We have the obvious formula based on orthogonal direct sum decompositions: ⊥ ⊥ Pn,p+1 = Pn,p − Psp{eb [n−1|p]}

and hence ef [n|p + 1] = ef [n|p] − Kf [n|p]eb [n − 1|p] − − − (1) and likewise, e˜b [n|p + 1] = eb [n − 1|p] − Kb [n|p]ef [n|p − − − (2)] where

⊥ −p−1 e˜b [n|p + 1] = Psp{ξ ξn −1 ξ ,...,z −p ξ } z n ,z n n

It is easy to see that ( eb [n|p + 1] =

e˜b [n|p + 1] 0

)

428

General Relativity and Cosmology with Engineering Applications Here, the forward reflection coefficient is Kf [n|p] =< eb [n − 1|p], ef [n|p > /Eb [n − 1|p] and the backward reflection coefficient is Kb [n|p] =< eb [n − 1|p], ef [n|p] > /Ef [n|p] where

Ef [n|p] =|| ef [n|p] ||2 , Eb [n − 1|p] =|| eb [n − 1|p] ||2

We thus have 0 ≤ Kf [n|p]Kb [n|p] = | < eb [n − 1|p], ef [n|p] > |2 /Ef [n|p]Eb [n − 1|p] ≤ 1 From (1) and (2), we get Ef [n|p + 1] = Ef [n|p] + Kf2 [n|p]Eb [n − 1|p] − 2Kf [n|p]2 Eb [n − 1|p] = Ef [n|p] − Kf2 [n|p]Eb [n − 1|p] = (1 − Kf [n|p]Kb [n|p])Ef [n|p] and likewise, Eb [n|p + 1] = (1 − Kf [n|p]Kb [n|p])Eb [n − 1|p] The time update formulas for Kf and Kp and of Ef , Ep require time update formulas for Pn,p . We have T T (Xn,p Xn,p )−1 Xn,p Pn,p = Xn,p

Let T Xn,p Rn,p = Xn,p

(

Then since Xn+1,p =

T ηn,p Xn,p

)

where ηn,p = x[n], x[n − 1], ..., x[n − p + 1] we get T + Rn,p Rn+1,p = ηn,p ηn,p

and hence −1 −1 Rn+1,p = Rn,p −

−1 T −1 Rn,p ηn,p ηn,p Rn,p T R−1 η 1 + ηn,p n,p n,p

Thus, T T −1 ] (Rn,p − Pn+1,p = [ηn,p |Xn,p

−1 T −1 ηn,p ηn,p Rn,p Rn,p T R−1 η 1 + ηn,p n,p n,p

).

429

General Relativity and Cosmology with Engineering Applications ( | (

T .[ηn,p |Xn,p ] T −1 ηn,p Rn,p ηn,p

−1 1+etaT n,p Rn,p ηn,p −1 Xn,p Rn,p n,p T R−1 η 1+ηn,p n,p n,p

η

T −1 T ηn,p Rn,p Xn,p

Pn,p −

T R−1 η 1+ηn,p n,p n,p −1 T −1 T Xn,p Rn,p ηn,p ηn,p Rn,p Xn,p T R−1 η 1+ηn,p n,p n,p

) | )

[36] Electric dipole moment and magnetic dipole moment of an atom with an electron in a constant electromagnetic field. The unperturbed Hamiltonian of the atom is given by H0 = p2 /2m − eV (r) The perturbing Hamiltonian is −eV1 + e2 V2 where V1 = −(r, E) + (Lz + gσz )B0 /2m V2 = (B0 zˆ × r)2 /2m = B02 (x2 + y 2 )/2m To calculate the eigenfunctions of H0 −eV1 +e2 V2 upto O(e2 ), we need to develop second order time independent perturbation theory for degenerate unperturbed systems. Consider therefore a Hamiltonian H = H0 + δH1 + δ 2 H2 (0)

with the eigenvalues of H0 being En , n = 1, 2, ... and an orthonormal basis for (0) the eigenspace N (H0 − En ) being |ψnk >, k = 1, 2, ..., dn . Let |ψn(0) >=

dn ∑

(0)

c(n, k)|ψnk >

k=1

be the unperturbed state of the system. We note that this state has an eigen­ value En for H0 . The constants c(n, k) are yet to be determined. We write for the perturbed state |ψn >= |ψn(0) > +δ|ψn(1) > +δ 2 |ψn(2) + O(δ 3 ) and correspondingly for the perturbed energy level En = En(0) + δEn(1) + δ 2 En(2) + O(δ 3 ) Substituting these expansions into the eigen-equation and equating coefficients of δ m , m = 0, 1, 2 successively gives (H0 − En(0) )|ψn(0) >= 0 − − − (1) which is already known, (H0 − En(0) )|ψn(1) > +H1 |ψn(0) > −En(1) |ψn(0) >= 0 − − − (2) (H0 − En(0) )|psin(2) > +H1 |ψn(1) > +H2 |ψn(0) >

430

General Relativity and Cosmology with Engineering Applications −En(1) |ψn(1) > −En(2) |ψn(0) >= 0 − − − (3) (0)

From (2), we get on forming the bracket with < ψmk | from the left, (0)

(0) (Em − En(0) ) < ψmk |ψn(1) > +

∑

(0)

(0)

< ψmk |H1 |ψnl c(n, l) − En(1) c(n, k)δmn = 0 − − − (4)

l (1)

Setting m = n gives us the secular equation for the possible values of En that lift the degeneracy of the unperturbed state: ∑ (0) (0) < ψnk |H1 |ψnl > c(n, l) = En(1) c(n, k), 1 ≤ k ≤ dn l (1)

1 Thus, En assumes the values Enj , k = 1, 2, ..., dn which are the eigenvalues of (0)

(0)

the dn × dn secular matrix ((< ψnk |H1 |ψnl >))1≤k,l≤dn with the eigenvector (1) dn corresponding to the eigenvalue Enj being denoted by ((cj (n, k)))k=1 . We may assume that these dn eigenvectors form an orthonormal basis for Cdn . From (4) with m /= n, we get (0) < ψmk |ψn(1) >= ∑ < ψ (0) |H1 |ψ (0) > c(n, l) mk nl (0)

(0)

En − Em

l

(0)

(1)

and hence the first orde perturbation to the eigenvector |ψn >, namely δ.|ψnj > corresponding to the perturbed eigenvalue (1)

|ψnj >=

∑

(0)

(0) En

(0)

+

(1) δ.Enj

is given by

(0)

(0) |ψmk >< ψmk |H1 |ψnl > cj (n, l)/(En(0) − Em )

mkl,m/=n (1)

(1)

(1)

(1)

Turning now to (3), we assume En = Enj and |ψn >= |ψnj >. Taking the bracket of this equation with
+ < ψmk |H1 |ψnj > (0)

(1)

(0)

(1)

+ < ψmk |H2 |ψn(0) > −Enj < ψmk |ψnj > −En(2) cj (n, k)δmn = 0 For m = n, this gives (0)

(1)

< ψnk |H1 |ψnj > +

∑

(0)

< ψnk |H2 |ψnl > cj (n, l)

l (1)

(0)

(1)

−Enj < ψnk |ψnj > −En(2) cj (n, k) = 0 − − − (5) Actually, these equations are not all linearly independent. The only linearly (2) independent equation for En which emerges from this is obtained by forming

431

General Relativity and Cosmology with Engineering Applications

∑ (0) (0) the bracket of (3) with + < ψn | = k c¯j (n, k) < ψnk |. In fact, if we extend the conjugate of this vector to an onb for Cdn and form the bracket with these (2) vectors, then the term involving En disappears. Thus we infer from from (5) that (2)

(

∑

|cj (n, k)|2 )−1 [

k

∑

En(2) = Enj = (0)

(1)

c¯j (n, k) < ψnk |H1 |ψnj > +

k (1) −Enj

∑

∑

(0)

(0)

< ψnk |H2 |ψnl > c¯j (n, k)cj (n, l)

k,l

c¯j (n, k)
]

k

[37] Induced characters: Let G be a finite group and H a subgroup of G. Select one element x∪in each left coset of H. ∩ Let I denote the set of all such / y, x, y ∈ I. Let L be x' s. Thus, we have x∈I xH = G and xH yH = φ, x = a representation of H acting in the vector space V . We shall formally write x.V for the vector space V attached to the element x. The representation space for of G induced by the the representation π = IndG H L (ie π is the representation ⊕ representation L of H) may be denoted by U = x∈I x.V . π(g) acts on this space by mapping x.v to [gx].v where g ∈ G, x ∈ I, v ∈ V and [gx] ∈ I is the element for which [gx]H = gxH. We may use the notation gxV = [gx]V . Let χU denote the character of π and χV that of L. It is clear that g ∈ G will map x.V onto itself iff gxV = xV iff x−1 gxV = V iff x−1 gx ∈ H. For a given g ∈ G, let Xg denote the set of all such x' s. In other words, Xg = {x ∈ I : x−1 gx ∈ H} It follows that χU (g) =

∑

χV (x−1 gx)

x∈Xg

We note that for any x, g ∈ G, x−1 gx ∈ H iff y −1 gy ∈ H for all y ∈ G(g).x and in this case, x−1 gx = y −1 gy, where G(g) is the centralizer of g in G. Thus, we can write for x ∈ Xg , ∑ χV (x−1 gx) = μ(G(g))−1 χV (y −1 gy) y∈G(g)x

where μ(G(g)) denotes the number of elements in G(g). It follows that ∑ χV (y −1 gy) χU (g) = μ(G(g))−1 x∈Xg ,y∈G(g)x

Further, it is clear that h ∈ H and y −1 gy ∈ H implies (yh)−1 gyh = h−1 y −1 gyh ∈ H and hence χV ((yh)−1 gyh) = χV (y −1 gy). Thus, the above formula can also be expressed as ∑ χV (y −1 gy) χU (g) = μ(G(g))−1 μ(H)−1 x∈Xg H,y∈G(g)x

432

General Relativity and Cosmology with Engineering Applications = (μ(G(g))μ(H))−1

∑

χV (y −1 gy)

y∈G(g)Xg H

Now, consider the set of conjugacy classes of g ∈ G: C(g) = {xgx−1 : x ∈ G} C(g) contains μ(C(g)) = μ(G)/μ(G(g)) distinct elements. Reference: Claudio Procesi, ”Lie groups, an approach through invariants and representations”, Springer. A.5. Some more problems in group theory and quantum mechanics. [1] The basic observables of non-relativistic quantum mechanics obtained from the unitary representations of the Galilean group based on Mackey’s theory of semidirect products. Suppose N is a vector space regarded as an Abelian group under addition. Let H be a group that acts on N as n → τh (n). Let G = N ⊗s H. Thus, any element g ∈ G can be uniquely expressed uniquely as g = nh, n ∈ N, h ∈ H and the composition law in G is given by n1 h1 n2 h2 = n1 τh1 (n2 )h2 . We have τh1 h2 = τh1 oτh2 , τh (n + n' ) = τh (n) + τh (n' ) In other words, h → τh is an homomorphism of H into aut(N ) with aut(N ) being the same as End(N ) (aut(N ) is a group theoretic notation while End(N ) is a vector space theoretic notation. The composition of n1 h1 and n2 h2 : Then we may regard N as being normalized by the action τ of H. Equivalently, via the construction of a group isomorphism, we can regard τh as begin given by τh (n) = hnh−1 , so that n1 h1 n2 h2 = (n1 + τh1 (n2 )).h2 Now let B(n1 , n2 ) be a skew symmetric real bilinear form on N that is Hinvariant, ie, B(τh (n1 ), τh (n2 )) = B(n1 , n2 ) Then consider σ(n1 h1 , n2 h2 ) = exp(iB(n1 , τh1 (n2 ))) We claim that σ satisfies the conditions for a multiplier on G, ie, for a projective unitary (pu) representation U of G, ie, σ(g1 , g2 )σ(g1 g2 , g3 ) = σ(g1 , g2 g3 )σ(g2 , g3 ) − − − (1) We leave this verification to the reader. This follows from the fact that if U satsifies by the definition of a pu representation U (g1 )U (g2 ) = σ(g1 , g2 )U (g1 g2 )

General Relativity and Cosmology with Engineering Applications

433

and hence by associativity of linear operator multiplication, (U (g1 )U (g2 ))U (g3 ) = U (g1 )(U (g2 )U (g3 )) we get σ(g1 , g2 )U (g1 g2 )U (g3 ) = U (g1 )U (g2 g3 )σ(g2 , g3 ) or σ(g1 , g2 )σ(g1 g2 , g3 )U (g1 g2 g3 ) = σ(g1 , g2 g3 )σ(g2 , g3 )U (g1 g2 g3 ) ie (1). [2] Let f (r1 , ..., rN ) and g(r1 , ..., rN ) be two functions on (R3 )N . Let R ∈ SO(3) and σ, τ, ρ ∈ SN . Let χλ (σ) be an irreducible character of Sn correspond­ ing to the Young tableaux (ie a partition of n) λ and consider ∑ f (rσ1 , ..., rσn )¯ g (rτ 1 , ..., rτ n )χλ (στ −1 ) I(f, g, r1 , ..., rN ) = σ,τ ∈Sn

We have I(f, g, rρ1 , ..., rρN ) =

∑

f (rρσ1 , ..., rρσn )¯ g (rρτ 1 , ..., rρτ n )χλ (στ −1 )

σ,τ

=

∑

g (rτ1 , ..., rτ n )χλ (ρστ −1 ρ−1 ) f (rσ1 , ..., rσn )¯

σ,τ

= I(f, g, r1 ‘, ..., rN ) since

χ(ρσρ−1 ) = χ(σ)

for any character χ of Sn . More generally, we can define ∑ ' I(f, g, r1 , ..., rN , r1' , ..., rN )= f (rσ1 , ..., rσn )¯ g (rτ' 1 , ...., rτ' n )χλ (στ −1 ) σ,τ ∈Sn

Then, we get ' ' I(f, g, rρ1 , ..., rρn , rρ1 , ..., rρn )=

∑

' ' f (rρσ1 , ..., rρσn )¯ g (rρτ 1 , ..., rρτ n )

σ,τ

=

∑

.χλ (στ −1 ) = g (rτ' 1 , ..., rτ' n ) f (rσ1 , ..., rσn )¯

σ,τ ' ) χλ (στ −1 ) = I(f, g, r1 , ..., rN , r1' , ..., rN ' ∈ R3 . Now,let R ∈ SO(3). The rotated and permuted ∀r1 , ..., rN , r1' , ..., rN image fields obtained from f and g are ' ' ' ) = g(R−1 rρ1 , ..., R−1 rρN ) f1 (r1 , ..., rN ) = f (R−1 rρ1 , ..., R−1 rρN ), g1 (r1' , ..., rN

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General Relativity and Cosmology with Engineering Applications

and we get ' ' I(f1 , g1 , r1 , ..., rN , r1' , ..., rN ) = I(f, g, R−1 r1 , ..., R−1 rN , R−1 r1' , ..., R−1 rN )=

and hence if χl denotes an irreducible character of SO(3), we get ∫ SO(3)×SO(3)

∫ = SO(3)×SO(3)

' I(f1 , g1 , S1 r1 , ...S1 rN , S2 r1' , ..., S2 rN )χl (S1 S2−1 )dS1 dS2

' I(f, g, R−1 S1 r1 , ..., R−1 S1 rN , R−1 S2 r1' , ..., R−1 S2 rN )χl (S1 S2−1 )dS1 dS2

∫

= SO(3)×SO(3)

' I(f, g, S1 r1 , ..., S1 rN , S2 r1' , ..., S2 rN )χl (RS1 S2−1 R−1 )dS1 dS2

∫

= SO(3)×SO(3)

' I(f, g, S1 r1 , ..., S1 rN , S2 r1' , ..., S2 rN )χl (S1 S2−1 )dS1 dS2

It follows that ' ) I0 (f, g, r1 , ..., rN , r1' , ..., rN ∫ ' = I(f, g, S1 r1 , ...S1 rN , S2 r1' , ..., S2 rN )χl (S1 S2−1 )dS1 dS2 SO(3)×SO(3)

is invariant under permutations and rotations. References for A.4 and A.5.: [1] Hoffman and Kunze, Linear Algebra, Prentice Hall. [2] T.Kato, Perturbation theory for linear operators, Springer. [3] K.R.Parthasarathy, ”An introduction to quantum stochastic calculus”, Birkhauser. [4] S.J.Orfanidis, ”Optimum signal processing”, Prentice Hall. [5] C.R.Rao, ”Linear statistical inference and its applications, Wiley. [6] J.Gough and Koestler, ”Quantum filtering in coherent states”. [7] Lec Bouten, ”Filtering and control in quantum optics”, Ph.D thesis. [8] Leonard Schiff, ”Quantum mechanics”. [9] W.O.Amrein, ”Hilbert space methods in quantum mechanics”. [10] K.R.Parthasarathy, ”Coding theorems of classical and quantum infor­ mation theory”. Hindustan Book Agency. [11] Naman Garg, H.Parthasarathy and D.K.Upadhyay, ”Estimating param­ eters of an image field modeled as a quantum electromagnetic field using its interaction with a finite state atomic system”, Technical Report, NSIT, 2016. [12] Naman Garg, H.Parthasarathy and D.K.Upadhyay, ”MATLAB imple­ mentation of Hudson-Parthasarathy noisy Schrodinger equation and Belavakin’s quantum filtering equation with analysis of entropy evolution”, Technical Re­ port, NSIT, 2016. A.6. New Syllabus for a short course on transmission lines and waveguides. [1] Study of non-uniform transmission lines by expanding the distributed parameters and the line voltage and current as Fourier series in the spatial

General Relativity and Cosmology with Engineering Applications

435

variable z. The modes of propagation (propagation constants) appear as the eigenvalues of an infinite matrix defined in terms of the spatial Fourier series coefficients of the non-uniform line impedance and admittance. [2] Study of the statistics of the line voltage and current (spatial correlations) when the distributed parameters of the line have small random fluctuations. The study is based on perturbation theory applied to matrix eigenvalue problems and is very similar to time independent perturbation theory used in quantum mechanics. [3] Analysis of transmission lines when the distributed parameters are ran­ domly fluctuating functions of both space and time. We focus on estimating the distributed parameter statistical correlations from measurements of the line voltage and current and applying the ergodic hypothesis for estimating the line voltage and current correlations and then matching these correlations with the theoretically derived correlations. [4] Analysis of transmission lines with random loading along the line using infinite dimensional stochastic differential equations. The voltage and current loading along the line are assumed to be expandable in terms of basis functions of the spatial variable with the coefficients being white noise processes in time, ie, derivatives of Brownian motion processes. We then calculate the probability law of the line voltage and current [5] Equivalence of transmission lines and waveguides obtained by expanding the guide electric and magnetic fields in terms of basis functions of (x, y) and the coefficients being functions of z. From the Maxwell equations, we derive an infinite series of first order linear differential equations for the coefficient functions of z and compare these equations with an infinite sequence of coupled transmission lines. The basis functions of (x, y) used in the expansion of the electric and magnetic fields must satisfy the boundary conditions, namely, that Ez and the normal derivative of Hz vanish on the boundary. [6] Study of nonlinear hysteresis and nonlinear capacitive effects on the dy­ namics of a transmission line. Hysteresis is related to a nonlinear B − H curve having memory and is a consequence of the Landau-Lifshitz theory of magnetism in which the magnetic moment of an atom precesses in an external magnetic field due to the M × H torque on it. The solution to this equation is a Dyson series for the magnetization M in terms of H and truncated upto second degree terms, this leads to a quadratic expression for the hysteresis voltage term as a function of the line current. In other words, we have a second order Volterra relation between the hysteresis voltage and current. This is a consequence of magnetic properties of the material of which the line is made. Nonlinear capacitive ef­ fects can be explained from the nonlinear-memory relation between the dipole moment/polarization of an electron relative to its nucleus when an external electric is incident on it. The binding of the electron to the atom has harmonic as well as anharmonic terms which causes the differential equation satisfied by the position of the electron to be nonlinear and hence solving this equation us­ ing perturbation theory, we obtain the dipole moment as a Volterra series in the external electric field. When applied to transmission lines, this manifests itself as a Volterra relation between the line charge (which is proportional to

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General Relativity and Cosmology with Engineering Applications

the electric displacement vecto D = ε0 E + P where P is the polarization/dipole moment per unit volume). The time derivative of this charge is the capacitor current and this component is incorporated into the line current equation and analysis is done using perturbation theory. [7] Quantization of the line equations using the Gorini-Kossakowski-SudarshanLindblad (GKSL) formalism. The line equations for a lossless line are dis­ tributed parameter analogs of LC circuits. The lossless line equations like an LC circuit can be derived from a Lagrangian and hence from a Hamiltonian that is a quadratic function of the phase variables. The effect of noise on such a system is obtained by adding a GKSL term to the dynamics of states and observables. These GKSL terms can natually be obtained using the Hudson-Parthasarathy quantum stochastic calculus by tracing out over the bath variables. By choosing our GKSL operators L as complex linear functions of the phase variables, we obtain resistive damping terms in the dynamical equations and hence we are able to obtain a quantum mechanical model for a lossy line. A.7. Creativity in the mathematical, physical and the engineering sciences. In this section, we give a brief history of the various intellectual achieve­ ments in the mathematical, physical and engineering sciences showing how cre­ ativity in these sciences very often comes from a need to understand nature and the working of the world around us. The examples we choose are Newton, Maxwell, Einstein, Planck, Bose, Rutherford, Bohr, Heisenberg, Schrodinger, Dirac, Dyson, Feynman, Schwinger, Tomonaga, Weinberg, Salam, Glashow and Hawking in the physical sciences, Faraday and Edison and the inventor Esaki of the tunnel diode in the engineeering sciences and in the mathematical sci­ ences, Euler, Gauss, Fourier, Fermat, Galois, Abel, Cauchy, Hilbert, Poincare, Kolmogorov, Ramanujan, Harish-Chandra and more recently, Edward Witten, a pioneer in mathematical string and superstring theory. The creation of quantum electrodynamics: After the creation of the quantum theory of atoms and molecules, there remained several gaps in our understand­ ing of the physical world. For example, it was not clear how to explain various experimental observations like the force of an electron on itself, the electron self energy ie, the movement of an electron produces an em field which acts back on the electron, the phenomenon of vacuum polarization according to which a photon propagates in vacuum to produce an electron-positron pair which prop­ agate and again annihilate each other to produce once again a photon, the anomalous magnetic moment of the electron which gives radiative corrections to the magnetic moment caused once again by the the em field generated by the electron acting back on itself. Compton scattering of an electron/positron by a photon also remained unexplained. In other words, a satisfactory quantum theory describing the interactions of electrons, positrons and photons and how to calculate probabilities of scattering processes of these particles remained to be carried out. Thus because of the need to understand these unexplained phys­ ical processes, Feynman, Schwinger and Tomonaga created new mathematical tools which eventually were sharpened by Dyson and the succeeding generation of theoretical physicists like Wienberg, Salam, Glashow, Witten, and more re­

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cently by the Indian physicists Sen and Ashtekar. Feynman proposed a path integral formulation for calculating the scattering matrix for particles. This in­ volved identifying the Lagrangian density L0 [φ] of the electron-positron Dirac field without their interactions ∫ and that of the electromagnetic field and then evaluating the action S[φ] = Ld4 x for these ∫ fields. Feynman followed it with evaluation of the Gaussian path integrals exp(iS[φ])Dφ taking into account the Berezin path integrals. Then the interaction term ∫ change for Fermionic ∫ Sint [φ] = J μ Aμ d4 x = ψ ∗ γ 0 γ μ ψAμ d4 x (φ = (Aμ , ψ)) is considered and its contribution was evaluated by expanding the exponential exp(iSint [φ]) in as a power series in S[φ] and Feynman associated a diagram with each term in this series explaining how each term gives rise to a different order term in the scattering matrix and how these terms can be calculated easily by a diagram­ matic algorithm. On the other hand Schwinger and Tomonaga proposed an operator theoretic approach to calculating the scattering matrix. Their algo­ rithm was based on first quantizing the electromagnetic field using creation and annihilation operators. This had already been observed by Paul Dirac when he wrote down the energy of the electromagnetic field as a quadaratic func­ tion of the four vector potential in the spatial frequency domain and deduced that this quadratic structure meant that the em field should be considered as an ensemble of harmonic oscillators with two oscillators associated with each spatial frequency (two degrees of polarization arise from the fact that in the coulomb gauge, in the absence of charges, the electric scalar potential is zero while divA = 0 for the Coulomb gauge implies that in the spatial frequency do­ main, the magnetic vector potential is orthogonal to the wave vector). The next idea of Schwinger was to substitute this quantum electromagnetic field consist­ ing of a superposition of operators into the atomic Hamiltonian described by position and momentum operators and obtain an interaction term between the position-momentum pair for the atom with the quantum electromagnetic field operators. This interaction Hamiltonian was then used to calculate the scat­ tering matrix elements and deduce corrections to the magnetic moment of the electron. Schwinger and Tomonaga also proposed a Lorentz invariant scheme for writing down the Schrodinger equation (which is not Lorentz covariant) for field theories. The idea basically involved replacing the time t variable by a three dimensional surface variable σ. In other words, just as the t = constt sur­ face is the three dimensional Euclidean space R3 as a subspace of R4 , likewise, σ = constt. could be an arbitrary three dimensional submanifold of R4 . The Schrodinger equation i ∂ψ(t) ∂t = H(t)ψ(t) was replaced by the Lorentz covariant equation δψ(σ) i = H(σ)ψ(σ) δσ and this formalism was used with great power by Schwinger and Tomonaga to calculate the scattering matrix element between two three dimensional surfaces. Unification of the Feynman and Schwinger-Tomonaga theory was performed by Freeman Dyson who simply showed why one should obtain the same results using Feynman diagrams and the operator theoretic approach. For a very long time,

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General Relativity and Cosmology with Engineering Applications Dyson’s notes on this was the standard textbook for all courses in quantum field theory all over the world. Dyson’s work led to renormalization theory developed by himself and subsequently othe researchers. This involved getting rid of the infinities in quantum field theory by renormalizing charge, mass and fields, ie, scaling these quantities with numbers depending on an ultraviolet and infrared cutoff while integrating in the four frequency domain. After the great riddle of construcing a cogent quantum theory of electrodynamics was solved almost completely by these four powerful mathematical physicists, the problem of understanding nuclear weak and strong forces remained and also how to unify these with quantum electrodynamics. The problem of unifying quantum electrodynamics with the weak forces, called the Electroweak theory was successfully solved by Weinberg, Salam and Glashow using group theoretic formalism, more precisely the SU (2) × U (1) formalism. Both the weak forces and electromagnetic forces appeared as gauge fields in this theory. Principles of symmetry breaking were used in this unification giving rise to mass of electrons and other nuclear particle. Goldstone had a say in this unification when he proposed the idea of how when a Lagrangian of fields that is invariant under a group G has a vacuum state that is not G-invariant because of degeneracies of the vacuum state, the symmetry of the Lagrangian as viewed from the vacuum state is broken to s smaller group H ⊂ G and associated with each degree of broken symmetry is a massless particle called a massless Goldstone boson. The unbroken symmetries correspond to massive particles. Symmetry can also be broken by adding a term to the G-invariant Lagrangian density. This is what happens in the electroweak theory. The electroweak-strong unification was achieved by Gell-Mann and Nee-Mann who based their theory on the group SU (3) × SU (2) × U (1). The idea is to derive all the coupling constants of electrdynamics, the weak and the strong theories from one unified theory. The entire idea of unifying gauge fields is based on the basic principle of Yang and Mills, namely that one can construct a covariant derivative ∇μ = ∂μ + iAμ (x) acting on a vector space (Cn ) valued function of x in such a way that the gauge field Aμ (x) takes values in a Lie algebra g of a subgroup G of the unitary group U (n). The wave function on which ∇μ acts is Cn . Further, this covaraiant derivative satisfies the property that under a local G-transformation by g(x) ∈ G, the gauge field Aμ (x) which takes values in g transforms in such a way to A'μ (x) so that g(x)(∂μ + A'μ (x)) = (∂μ + Aμ (x))g(x) or equivalently as

g(x)∇'μ = ∇μ g(x)

where both sides are regarded as first order differential operators acting on wave functions ψ(x) ∈ Cn . This gives iA'μ (x) = g(x)−1 ∂μ g(x) + ig(x)−1 Aμ (x)g(x) This idea of gauge transformation in which the massive field wave function ψ(x) ∈ Cn transforms to g(x)ψ(x) while the massless gauge field Aμ (x) trans­ forms in the above way leads to the conclusion that given a Lagrangian density

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of the form L(ψ(x), ∇μ ψ(x)) that is G − invariant, ie, L(gψ(x), g∇μ ψ(x)) = L(ψ(x), ∇μ ψ(x)) for all g ∈ G, it follows that L is invariant also under local G-actions g(x) provided that the Aμ (x) sitting inside the covariant derivative ∇μ is also subject to the above gauge transformation. When the group G is the Abelian group U (1), Aμ (x) is simply a real valued function for each μ and the above gauge transformation reduces to the Lorentz gauge transformation for the electromagnetic potentials. Thus, the non-commutative Yang-Mills theory pro­ vides a sweeping generalization of the commutative em field theory. It can also be applied to describe the interaction of the Dirac field ψ(x) with a noncommu­ tative gauge field with the electromagnetic potentials also coming as an extra component of the gauge potential. Although this idea of Yang and Mills is a purely group theoretic construction, it turned out to be one of the most fruitful constructs for unifying almost all the quantum particle fields. What remains now is the development of a quantum theory of gravity which would enable one to associate a particle which we may call a graviton and to describe its interac­ tion with other quantum particles like the photon, electron, positron, and the propagators of the weak and strong forces. It should be borne in mind that the action of a classical gravitational field on any quantum particle described by a relativistic wave equation like the Dirac equation can be achieved using the Idea of Yang and Mills, namely by introducing a gravitational connection Γμ (x) which is a matrix. For example, if the gravitational field is described by a tetrad Vaμ (x) so that the metric is g μν (x) = η ab Vaμ (x)Vbν (x) with η ab being the Minkowski metric of flat space-time, then this tetrad can be understood as a lo­ cal transformation of curved space-time to an inertial frame. We then construct a covariant derivative ∂μ + Γμ (x) and transform it locally to an inertial frame by using Vaμ (x)(∂μ + Γμ (x)) and setting up the Dirac equation in a gravitational field as [iγ a Vaμ (x)(∂μ + Γmu (x)) − m]ψ(x) = 0 where γ a , a = 0, 1, 2, 3 are the usual Dirac gamma matrices. To qualify as a valid general relativistic wave equation, this must be invariant under local Lorentz transformations Λ(x). Under such a local Lorentz transformation, ψ(x) transforms to D(Λ(x))ψ(x) where D is the Dirac representation of the Lorentz group and its Lie algebra generators are J ab = 14 [γ a , γ b ]. Suppose that under such a local Lorentz transformation Γμ (x) which is a 4 × 4 matrix changes to Γ'μ (x). Then we should have with ψ ' (x) = D(Λ(x))ψ(x), the equation [iγ a Λba (x)Vbμ (x)(∂μ + Γ'μ (x) − m]ψ ' (x) = 0 where the change of space-time coordinates has been accounted for by the factor matrix elements Λba (x) of the local Lorentz transformation of the inertial frame index a in the tetrad frame Vaμ (x). Using the identity D(Λ)γ b D(Λ)−1 = Λba γ a we get from the above [D(Λ(x))(iγ b Vbμ (x)(D(λ(x))−1 (∂μ + Γ'μ (x)) − m]D(Λ(x))ψ(x) = 0

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This is equivalent to iγ b Vbμ (x)((D(Λ(x))−1 ∂μ D(Λ(x)))+∂μ +D(Λ(x))−1 Γ'μ (x)D(Λ(x)))−m]ψ(x) = 0 It follows that for this to coincide with the untransformed Dirac equation in curved space-time, the connection Γμ (x) of the gravitational field in the Dirac representation should transform to Γ'μ (x) where D(Λ(x))−1 Γ'μ (x)D(Λ(x)) + (D(Λ(x))−1 ∂μ D(Λ(x))) = Γμ (x) or equivalently, Γ'μ (x) = D(Λ(x))Γμ (x)D(Λ(x))−1 − (∂μ D(Λ(x)))D(Λ(x))−1 Such a connection has been constructed and is given by Γμ (x) =

1 ab ν J Va Vbν:μ 2

(Reference: Steven Weinberg, ”Gravitation and Cosmology, Principles and Ap­ plications of the General Theory of Relativity”, Wiley.) Gravity was unified with classical electromagnetism by Einstein in this beau­ tiful theory the general theory of relativity. This theory said that gravity is not a force, it is simply a curvature of space-time and when matter moves in such a curved space time, follows geodesics which are shortest paths on the curved four dimensional manifold of space-time. These shortest paths are curved because any path on a curved surface is curved. By saying that geodesics are curved, we mean that the relation between the spatial and time coordinates of a moving particle is nonlinear and hence the motion appears to us as being accelerated motion. A.8 Classification and representation theory of semisimple Lie algebras. By Serre’s theorem, a complex semisimple Lie algebra L is generated by 3n generators ei , hi , fi , i = 1, 2, ..., n satisyfying the commutation relations [ei , ej ] = [fi , fj ] = [hi , hj ] == [ei , fj ] = 0, [ei , fj ] = δij hi , [hi , ej ] = aij ej , [hi , fj ] = −aij fj where aij are integers called the Cartan integers. This result of Serre follows from the Cartan’s theory which says that L has a maximal Abelian subalgebra h (Called a Cartan Algebra) and that any two maximal Abelian subalgebras are mutually conjugate. This result is not true for real semisimple Lie alge­ bras where there can be more than one non-conjugate Cartan algebras. The reprsentation theory for real semisimple Lie algebras was developed almost sin­ gle handedly by the great Indian mathematician Harish-Chandra who derived generalizations of the character formula of H.Weyl using the theory of distribu­ tions and also obtained the complete Plancherel formula for such algebras by

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introducing in addition to the principal and supplementary series of irreducible representations (introduced by Gelfand for obtaining the Plancherel formula for complex semisimple Lie groups), the discrete series of irreducible representa­ tions. Coming back to the theme of complex semisimple Lie algebras, Cartan proved that the elements of h in the adjoint representation, act in a semsimple way on L, ie, each operator ad(H), H ∈ h acting on the vector space L is diag­ onable. It follows from basic Linear algebra, that the operators ad(H), H ∈ h are simulatneously diagonable and hence we have the following direct sum de­ composition of L: ⊗ L=h⊗ Lα α∈Φ

where Φ is the set of all roots of L, ie, for each α ∈ Φ and X ∈ Lα , we have [H, X] = α(H)X We are here defining Lα = {X ∈ L : [H, X] = α(H)X∀H ∈ h} α is a non-zero linear functional on h, ie, α ∈ h∗ and all the α' s are distinct linear functionals. Moreover, Cartan’s theory states that there exists a subset Δ ⊂ Φ called a set of simple roots such that any α ∈ Φ is either a purely positive or purely negative integer linear combination of elements of Δ. This means that writing Δ = {α1 , ..., αp }, we have that for any α ∈ Φ, there exist integers m1 , ..., mp such that mj ≥ 0∀j or mj ≤ 0∀j and α=

p ∑

mj αj

j=1

and further, no α ∈ Δ has such a decomposition. Actually, there exist many such sets Δ. Further, dimLα = 1∀α ∈ Φ. This follows from the following ele­ mentary argument, From the Jacobi identity, [Lα , Lβ ] ⊂ Lα+β , α, β ∈ Φ. Thus [Lα , L−α ] ⊂ h. Choose an eα ∈ Lα , fα ∈ L−α such that B(eα , fα ) = b(α) where B(X, Y ) = T r(ad(X)ad(Y )) and b(α) is a constant to be chosen appropriately. Cartan proved that B defines an non-degenerate symmetric bilinear form on L (only if L is semisimple). Now define tα = a(α)[eα , fα ] where a(α) is a constant to be chosen appropriately. Then, tα ∈ h and [tα , eα ] = α(tα )eα , [tα , fα ] = −α(tα )fα and more generally, for any α, β ∈ Φ, [tα , eβ ] = β(tα )eβ , [tα , fβ ] = −β(tα )fβ

442

General Relativity and Cosmology with Engineering Applications Consistency is checked as follows using B([X, Y ], Z) = B(X, [Y, Z]): = b(β)β(tα ) = β(tα )B(eβ , fβ ) = B([tα , eβ ], fβ ) = −B(eβ , [tα , fβ ]) = β(tα )B(eβ , fβ ) = b(β)β(tα ) We further have B(tα , tβ ) = a(α)B([eα , fα ], tβ ) = a(α)B(eα , [fα , tβ ]) = a(α)α(tβ )B(eα , fα ) = a(α)b(α)α(tβ ) and we denote this by (α, β). The above formula implies that (α, β) = (β, α) = a(α)b(α)α(tβ ) = a(β)b(β)β(tα ) Now define hα = 2tα /(α, α) Then, [hα , eα ] = 2α(tα )eα /(α, α) = 2eα provided that we choose the a(α)' s and the b(α)' s so that α(tα ) = (α, α) For such a choice of the constants, we also have [hα , fα ] = −2fα , and [eα , fα ] = tα /a(α) = (α, α)hα /(2a(α)) = hα provided that we choose the a(α)' s and the b(α)' s so that (α, α) = 2a(α) = α(tα ) In other words, we have found eα ∈ Lα , fα ∈ L−α , hα ∈ h so that {eα , fα , hα } satisfy the same commutation relations as the standard generators of sl(2, C). We denote the Lie algebra generated by these three elements by slα (2, C). We note that in the adjoint representation, slα (2, C) is (a module for) an irreducible representation of the Lie algebra slα (2, C). We are assuming here that relative to the set of simple roots Δ, α is a positive root, ie it is expressible as a positive integer linear combination of the elements of Δ. We note that if X ∈ Lα , Y ∈ Lβ , then B([H, X], Y ) = −B(X, [H, Y ]) implies (α(H) + β(H))B(X, Y ) = 0, H ∈ h and hence B(X, Y ) = 0 unless β = −α. It is therefore clear from the nondegeneracy of B that for each X ∈ Lα , B(X, Y ) = / 0 for some Y ∈ L−α It is also clear that dimLα = 1

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This can be seen as follows: Consider the sum ⊕ Lcα Mα = h ⊕ c

the sum being over all the irreducible representations of slα (2, C) in which the weights are multiples of α. (By weight β, here we mean that if X is a weightvector with weight β where β is a linear functional on h, then ad(H)(X) = β(H)X∀H ∈ h. We note that Mα is a module for slα (2, C). When H = hα , then β(hα ) becomes an eigenvalue of ad(hα ) and hence β can be regarded as a weight for the Lie algebra slα (2, C). Since slα (2, C) as a Lie algebra is isomorphic to sl(2, C), in any irreducible representation of slα (2, C), either a weight zero or a weight one will occur with a unique weight vector. Clearly, ad(hα ) when acting on Mα has weight zero iff the weight vector is in h. Mα contains the module h+slα (2, C) of slα (2, C) and hence it cannot contain any irreducible even module that has zero intersection with h + slα (2, C) appearing in Mα as a direct summand for any even irreducible module for slα (2, C) must necessarily contain a zero weight vector which must be an element of h and hence will intersect the module h + slα (2, C)(Note that h + slα (2, C) is a module for slα (2, C), ie, it is left invariant by the adjoint action of the latter and hence this module can be decomposed into irreducible modlues for slα (2, C)). Therefore, we must have Mα = h + slα (2, C) and in particular, dimLα = 1, α ∈ Φ By an even module g of slα (2, C), we mean that ad(hα ) has an even eigenvalue when operating on the root vectors in g in the adjoint representation. For exam­ ple, if the eigenvalues {−2q, −2q + 2, ..., 0, 2, 4, ..., 2p}} occured in an irreducible submodule of Mα for the Lie algebra slα (2, C) as a direct summand different from slα (2, C) in the adjoint representation, then the zero weight vector in this represenation would be hα which is a contradiction. Further, an odd summand (ie in which a vector having weight one occurs) also cannot occur, for then α/2 would be a root ((α/2)(hα ) = 1) and hence α = 2(α/2) cannot be a root by the above argument(Note that in an irreducible representation of slα (2, C), only the weights from the set 2Z or only weights from 2Z + 1 can occur. We have thus proved that if α is a root and cα is also a root, then c = ±1. Remark: If g is a semisimple Lie algebra and, then we can decompose g=

N ⊕

gi

i=1

as a direct sum of vector ∩ spaces gi where each gi is an ideal in g, ie [g, gi ] ⊂ gi ∀i and hence [gi , gj ] ⊂ gi gj = {0}, i = / j. A.8.Schrodinger wave equations for quantum general relativity.

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A manifold specified by space-time coordinates x ˜μ is given. Another coor­ μ dinate system for this manifold is X . The metric tensor relative to the former is g˜μν and the metric tensor for the latter is gμν . Thus, we have μ ν X,σ = g˜ρσ gμν X,ρ μ

μ By X,ρ we mean ∂X ∂xρ . We denote the spatial coordinates of the former system a b by x , x ,etc, where a, b = 1, 2, 3. Thus, the spatial components of the metric in the former system are μ ν X,b g˜ab = gμν X,a

We define qab = g˜ab We denote by ((q )) the 3 × 3 matrix that is the inverse of ((qab )). Now, write ab

X,μ0 = T μ = N μ + N nμ where N μ is purely spatial, ie, of the form μ N μ = N a X,a

and the vectors N μ and nμ are orthogonal with nμ normalized by the factor N . This means that N a is selected so that μ ν N μ = N a X,a , gμν nμ X,a =0

The first is the condition for N μ to be a spatial vector and the second is the condition for nμ to be orthogonal to all spatial vectors. We can visualize this by saying that the three dimensional spatial manifold Σt defined by x0 = t = constt is embedded in the four dimensional manifold specified by the coordinates X μ . The vector (nμ ) is the unit normal to the manifold Σt and the vectors μ 3 )μ=0 , a = 1, 2, 3 are tangential to the manifold Σt . We thus get the following (X,a equations for N a : μ ν gμν (X,μ0 − N a X,a )X,b =0 or g˜0b − N a g˜ab = 0 or equivalently, qab N b = g˜0a We now prove the following decomposition: g μν = q μν + nμ nν where q μν nν = 0 In other words, g μν can be decomposed as a sum of a purely spatial part and a purely normal part with regard to the surface Σt . To see this, we write μ ν X,β = g μν = g˜αβ X,α

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ν g˜00 (N μ + N nμ )(N ν + N nν ) + 2˜ g 0a (N μ + N nμ )X,a + μ ν g˜ab X,a + X,b

We have to show that the cross term in this expansion is zero, ie, terms involving μ and the normal part nμ . The cross part here is products of spatial parts X,a ν g 0a N nμ X,a 2N g˜00 N μ nν + 2˜

To prove that this is zero amounts to proving that ν =0 g˜00 N μ nν + g˜0a nμ X,a

To prove this it suffices to show that μ g˜00 N μ + g˜0a X,a =0

or equivalently,

g˜00 N a + g˜0a = 0

Proving this is equivalent to proving that g˜ba g˜0a + g˜00 g˜ab N a = 0 which is the same as

−g˜b0 g˜00 + g˜00 g˜ab N a = 0

(since g˜ba g˜0a + g˜b0 g˜00 = δba = 0). Thus, we have to show that qab N a = g˜b0 But this has already been established using the orthogonality of the normal μ . vector nμ with the spatial vectors X,a A.9. Time travel in the special and general theories of relativity and the revised notions of space-time in quantum general relativity. Gravitational red-shift: Let U (r) be the gravitational potential. Then the approximate (Newtonian) metric of space-time is given by dτ 2 = (1 + 2U (r)/c2 )dt2 − c−2 (dx2 + dy 2 + dz 2 ) We are assuming that U depends only on the radial coordinate relative to a system. The radial null geodesic (radial propagation of photons) is given by 0 = dτ 2 = (1 + 2U (r)/c2 )dt2 − dr2 or equivalently,

dr/dt = (1 + 2U (r)/c2 )1/2

Thus assuming r1 < r2 , a photon pulse starting from r1 at time t1 arrives at r2 at time t2 given by ∫ r2 t2 − t1 = (1 + 2U (r)/c2 )−1/2 dr r1

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In this expression, t1 , t2 are coordinate times, ie, times measured by a clock at a large distance from the gravitiational field, ie, at a point where the gravi­ tatitional field is zero. Now if another pulse starts from r1 at coordinate time t1 + δt1 , then it will arrive at r2 at time t2 + δt2 where ∫ r2 (1 + 2U (r)/c2 )−1/2 dr t2 + δt2 − t1 − δt1 = r1

Note that we are assuming a static gravitational field. It thus follows that δt2 = δt1 The proper time intervals measured by clocks static at r1 and r2 for the pulses are respectively given by dτ1 = (1 + 2U (r1 )/c2 )1/2 dt1 , dτ2 = (1 + 2U (r2 )/c2 )1/2 dt2 It follows therefore that dτ1 /dτ2 = [

1 + 2U1 /c2 1/2 ] 1 + 2U2 /c2

where U1 = U (r1 ), U2 = U (r2 ) Hence, if U2 < U1 , we get dτ1 > dτ2 or in terms of frequencies, ν1 = 1/dτ1 < 1/dτ2 = ν2 or more precisely,

ν1 1 + 2U2 /c2 1/2 =[ ] dτ2 , it follows that clocks run slower in a strong gravitational field, ie when the gravi­ tational potential is more negative. A.10[a].Transmission lines with random fluctuations in the parameters and random line loading: v,z (t, z) + (R0 (z) + δR(t, z))i(t, z) + L0 (z)i,t (t, z) + (δL(t, z)i(t, z)),t = wv (t, z) i,z (t, z) + (G0 (z) + δG(t, z))v(t, z)) + C0 (z)v,t (t, z) + (δC(t, z)v(t, z)),t = wi (t, z)

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In these equations, δR(t, z), δL(t, z), δC(t, z), δG(t, z), wv (t, z), wi (t, z) are ran­ dom Gaussian space-time Gaussian fields. We wish to solve this system of pde’s approximately using first order perturbation theory and hence calculate the approximate statistical correlations of the fluctuations in the line voltage and line current in terms of the correlations in the parameter flucutations and the voltage and current loading terms wv and wi . A.10[b]. Taking non-linear hysteresis and nonlinear capacitive effects into account, generalize the problem of A.9. A.11. Estimating parameters in statistical image models described by linear and nonlinear partial differential equations: First consider the linear case: The model for the image field φ(x, y), (x, y) ∈ D is given by p ∑

A(a, b, θ)∂xa ∂yb φ(x, y) = s(x, y) + w(x, y), (x, y) ∈ D

a,b=1

where w(x, y) is zero mean coloured Gaussian noise. θ ∈ Rm is the parameter vector to be estimated from measurements on φ. Here, s(x, y) is a given input non-random signal field. We assume that an initial guess estimate θ0 of θ is known and that the correction δθ to this estimate is to be made. We write φ(x, y) = φ0 (x, y) + δφ(x, y) where φ0 is the solution with the guess parameter θ0 and zero noise and δφ is the first order correction to φ0 arising from the parameter estimate correction term δθ and the noise w. We regard δφ, δθ, w all as being of the first order of smallness. We define ∑ A(a, b, θ)(jω1 )a (jω2 )b H(ω1 , ω2 , θ) = a,b

Then if two dimensional spatial Fourier transforms are denoted by placing a hat on top of a signal/noise field, we get ˆ 1 , ω2 ) H(ω1 , ω2 , θ)φˆ(ω1 , ω2 ) = sˆ(ω1 , ω2 ) + w(ω Thus to zeroth order, we get H(ω1 , ω2 , θ0 )φˆ0 (ω1 , ω2 ) = sˆ(ω1 , ω2 ), H(ω1 , ω2 , θ0 )δφˆ(ω1 , ω2 ) + (Hr (ω1 , ω2 , θ0 )δθr )φˆ0 (ω1 , ω2 ) = w(ω ˆ 1 , ω2 ) where Hr (ω1 , ω2 , θ0 ) =

∂H(ω1 , ω2 , θ0 ) ∂θr

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Thus, if

g(x, y) = F −1 (H(ω1 , ω2 , θ0 )−1 )

it then follows that

∫

φ0 (x, y) = g(x, y) ∗ s(x, y) =

g(x − x' , y − y ' )s(x' , y ' )dx' dy '

Likewise, if we write hr (x, y) = F −1 (Hr (ω1 , ω2 , θ0 )) ∑ = Ar (a, b, θ0 )δ (a) (x)δ (b) (y) a,b

where Ar (a, b, θ0 ) =

∂A(a, b, θ0 ) ∂θr

then we get δφ(x, y) = −g(x, y) ∗ hr (x, y)δθr + g(x, y) ∗ w(x, y) − − − (2) where summation over the repeated index r is implied. We measure δφ(x, y) as follows: First since we know θ0 and the input signal field s(x, y), we can calculate φ0 (x, y) using (1) and then we measure the actual noise perturbed image field φ(x, y) and calculate δφ(x, y) = φ(x, y) − φ0 (x, y). Now using (2), we calculate the maximum likelihood estimator of δθ as ∫ argminδθ (

ˆ = δθ (δφ(x, y)+gr (x, y)δθr )Q(x, y|x' , y ' )(δφ(x' , y ' )+gs (x' , y ' )δθs )dxdydx' dy ' )

where Q(x, y|x' , y ' ) is the inverse Kernel of R(x, y|x' , y ' ) = E[(g(x, y) ∗ w(x, y)).(g(x' , y ' ) ∗ w(x' , y ' ))]

∫

g(x − x1 , y − y1 )g(x' − x'1 , y ' − y1' )E(w(x1 , y1 )w(x'1 , y1' ))dx1 dy1 dx'1 dy1'

ie,

∫

Q(x, y|x' y ' )R(x' , y ' |x'' , y '' )dx' dy ' = δ(x − x'' )δ(y − y '' )

and gr (x, y) = g(x, y) ∗ hr (x, y) =

∑

Ar (a, b, θ0 )∂xa ∂yb g(x, y)

a,b

We write g(x, y) = ((gr (x, y))) Then, ∫ ˆ =[ δθ

Q(x, y|x' , y ' )g(x, y)g(x' , y ' )T dxdydx' dy ' ]−1 [

∫

Q(x, y|x' , y ' )g(x, y)δφ(x' , y ' )dxdydx' dy ' ]

General Relativity and Cosmology with Engineering Applications

449

A simple computation gives us the covariance of this parameter vector estimator: ∫ Cov(δθˆ) = [ Q(x, y|x' , y ' )g(x, y)g(x' , y ' )T dxdydx' dy ' ]−1 Wavelet based image parameter estimation: Let ψn (x, y), n = 1, 2, ... be a wavelet orthonormal basis for L2 (R2 ). Here, the index n consists of the scaling and translational index in both the dimensions, ie, n corresponds to four ordered integer indices. We define ∫ c(n, φ) =< φ, ψn >= φ(x, y)ψn (x, y)dxdy Then, φ(x, y) =

∑

c(n, φ)ψn (x, y)

n

Substituting this into the image pde model gives ∑ c(n, φ)A(a, b, θ)∂xa ∂yb ψn (x, y) = s(x, y) + w(x, y) n,a,b

Taking the inner product on both sides with ψm (x, y) gives ∑ c(n, φ)A(a, b, θ) < ψm , ∂xa ∂yb ψn >= c(m, s) + c(m, w) n,a,b

Define P (m, n|θ) =

∑

A(a, b, θ) < ψm , ∂xa ∂yb ψn >

a,b

The above equation can then be expressed as ∑ P (m, n|θ)c(n, φ) = c(m, s) + c(m, w) n

Writing θ = θ0 + δθ, φ(x, y) = φ0 (x, y) + δφ(x, y) gives us on applying first order perturbation theory, ∑ P (m, n|θ0 )c(n, φ0 ) = c(m, s), n

∑ ∑ (( Pr (m, n|θ0 )δθr )(¸n, φ0 ) + P (m, n|θ0 )δc(n)) = c(m, w) n

r

where Pr (m, n|θ0 ) =

∂P (m, n|θ0 ) ∂θr

and δc(n) = c(n, φ0 + δφ) − c(n, φ0 )

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General Relativity and Cosmology with Engineering Applications

We write c0 (n) = c(n, φ0 ), Pr (m, n) = Pr (m, n|θ0 ), P0 (m, n) = P (m, n|θ0 ) and thus get

∑ ∑

P0 (m, n)c0 (n) = c(m, s)

n

Pr (m, n)δθr c0 (n) +

∑

n,r

P0 (m, n)δc(n) = c(m, w)

n

A.12 Mackey’s theory on the construction of the basic quantum observables from unitary representations of the Galilean group. (a, v)inV = R3 × R3 . V is the Abelian group of translations and uniform velocity motions acting on the space-time manifold M = {(t, x) : t ∈ R, x ∈ R3 }. This action is given by (a, v)(t, x) = (t + x + vt + a) (τ, g) ∈ R × SO(3). R × SO(3) is the non-Abelian group of time translations and rotations acting on M: (τ, g)(t, x) = (t + τ, gx) The Galiean group G is the semidirect product of V and H: G = V ⊗s H where H acts on V as follows: (τ, g).(a, v).(τ, g)−1 = (a' , v ' ) or equivalently, (τ, g).(a, v) = (a' , v ' ).(τ, g) Acting both sides on (t, x) ∈ R4 gives (τ, g)(t, x + vt + a) = (a' , v ' )(t + τ, gx) or equivalently, (t + τ, gx + tgv + ga) = (t + τ, gx + v ' t + a' + v ' τ ) or equivalently, v ' = gv, a' = g(a − vτ ) Thus, the semimdirect product structure is given by (τ, g).(a, v).(τ, g)−1 = (g(a − vτ ), gv) ∈ V

General Relativity and Cosmology with Engineering Applications

451

Any element of the Galilean group G can be expressed in two ways, one as an element (a, v, τ, g) defined by its action on R4 by (a, v, τ, g)(t, x) = (t + τ, gx + vt + a) and in another way as the element (a, v).(τ, g). The action of this on R4 is given by (a, v).(τ, g)(t, x) = (a, v)(t + τ, gx) = (t + τ, gx + vt + vτ + a) It follows that the relationship between these two methods of expressing an element of the Galilean group is given by (a + vτ, v, τ, g) = (a, v).(τ, g) or equivalently by (a − vτ, v).(τ, g) = (a, v, τ, g) Let now h be a Hilbert space (like C2j+1 for a spin j particle) and H∫ = L2 (R3 , h) the Hilbert space of all measurable functions f : R3 → h for which R3 || f (x) ||2 d3 x < ∞. The projective unitary representations of G are obtained by using the multiplier m((a, v).(τ, g), (a' , v ' ).(τ ' , g ' )) = exp(iB((a, v), (τ, g)[(a' , v ' )]) where B : V × V → R (with V = R3 × R3 ) being a skew symmetric bilinear form invariant under H = {(τ, g) : τ ∈ R, g ∈ SO(3)}. that is invariant under H or equivalently under (τ, g). Note that the action of H on V is defined by (τ, g)[(a, v)] = (τ, g).(a, v).(τ, g)−1 = (g(a − vτ ), gv) We thus find that m((a, v).(τ, g), (a' , v ' ).(τ ' , g ' )) = exp(iB((a, v), (g(a − vτ ), gv))) = exp(iλ((a, gv) − (v, g(a − vτ )))) for some λ ∈ R where (u, v) = uT v. Let U be a projective unitary representation of G with this multiplier. Then, U ((a, v).(τ, g)) = U (a, v)U (τ, g) We can write U (a, 0) = V1 (a), U (0, v) = V2 (v), U (τ, g) = W1 (τ )W2 (g) where V1 , V2 are unitary representations of the Abelian group V = R3 × R3 and W1 and W2 are unitary representations of R and SO(3). By Stone’s theorem on unitary representations of Abelian groups, it follows that there exist Hermitian operators Q = (Q1 , Q2 , Q3 ) and P = (P1 , P2 , P3 ) in L2 (R3 , h) such that V1 (a) = exp(−ia.P ), V2 (v) = exp(−iv.Q)

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General Relativity and Cosmology with Engineering Applications

and also a Hermitian operator H in the same space such that W1 (τ ) = exp(−iτ H) We have U (a1 , v1 )U (a2 , v2 ) = exp(iλ(aT1 v2 − aT2 v1 ))U (a1 + a2 , v1 + v2 ) In particular, V1 (a)V2 (v) = U (a, 0)U (0, v) = exp(iλaT v)U (a, v), V2 (v)V1 (a) = U (0, v)U (a, 0) = exp(−iλaT v)U (a, v) Thus, we get the Weyl commutation relations V1 (a)V2 (v) = exp(i2λaT v)V2 (v)V1 (a) which can be expressed as exp(−ia.P ).exp(−iv.Q) = exp(i2λaT v)exp(−iv.Q).exp(−ia.P ) and hence by considering infinitesimal a and v in R3 . we get −Pi Qj + Qj Pi = 2iλδij or equivalently, [Qi , Pj ] = 2iλδij To get agreement with quantum mechanics that the momentum operators P generate translations and the position operators Q generate uniform velocities, we must take λ = 1/2 and thus, we get [Qi , Pj ] = iδij The Stone-Von-Neumann theorem then implies that the Hilbert space < can be chosen so that the actions of the Q'i s and the Pj' s in L2 (R3 , h) are such that (Qj f )(x) = xj f (x), (Pj f )(x) = −i

∂f (x) ∂xj

In other words, regarding L2 (R3 , h) as L2 (R3 ) ⊗ h (This is a Hilbert space ∂ ⊗ Ih . We note isomorphism), we have that Qj = xj ⊗ Ih and Pj = −i ∂x j compute U (τ, g)U (a, v) = U ((τ, g).(a, v)) Now, (τ, g).(a, v)(t, x) = (τ, g)(t, x + vt + a) = (t + τ, gx + tgv + ga) = (ga, gv, τ, g)(t, x)

General Relativity and Cosmology with Engineering Applications

453

ie, (τ, g).(a, v) = (ga, gv, τ, g) On the other hand, (a, v).(τ, g)(t, x) = (a, v).(t + τ, gx) = (t + τ, gx + vt + vτ + a) = (a + vτ, v, τ, g)(t, x) Thus, (τ, g).(a, v) = (g(a − vτ ), gv).(τ, g) So we get

U (τ, g)U (a, v)U (τ, g)−1 = U (g(a − vτ ), gv)

Taking g = I, this gives W1 (τ )U (a, v)W1 (−τ ) = U (a − vτ, v) Setting a = 0 in this formula gives W1 (τ )V2 (v)W1 (−τ ) = U (−vτ, v) while setting v = 0 gives W1 (τ )V1 (a)W1 (−τ ) = V1 (a) The second equation implies [H, Pj ] = 0, j = 1, 2, 3 We now note that V1 (−vτ )V2 (v) = exp(−iτ |v|2 /2)U (−vτ, v) Thus, we get from the first equation W1 (τ )V2 (v)W1 (−τ ) = exp(iτ |v|2 /2)V1 (−vτ )V2 (v) = exp(iτ |v|2 /2)exp(iτ v.P )V2 (v) For infinitesimal τ , this gives −i[H, exp(−iv.Q)] = (i|v|2 /2 + iv.P )exp(−iv.Q) or equivalently, H − exp(−iv.Q).H.exp(iv.Q) = −|v|2 /2 − v.P − − − (1) The O(v) term of this equation gives i[v.Q, H] = −v.P or equivalently, i[H, Qj ] = Pj

454

General Relativity and Cosmology with Engineering Applications Combining this with the equation [H, Pj ] = 0 we may conclude using the commutation relations [Qi , Pj ] = iδij that H = P 2 /2 + E =

3

1∑ 2 P + E − − − (2) 2 j=1 j

where E is an operator of the form I ⊗E1 in L2 (R3 )⊗h, ie, for f (x) ∈ h, x ∈ R3 , we have (Ef )(x) = E1 f (x) By considering the O(v 2 ) term in (1), we get [v.Q, [v.Q, H]] = −v 2 or equivalently, [[H, Qi ], Qj ] = −δij /2 This is verified by (2): [P 2 /2, Qi ] = −iPi , [[P 2 /2, Qi ], Qj ] = −i[Pi , Qj ] = −δij We now consider the equation U (τ, g)U (a, v)U (τ, g)−1 = U (g(a − vτ ), gv) with τ = 0. We get W2 (g)U (a, v)W2 (g)−1 = U (ga, gv) In particular, we get W2 (g)V1 (a)W2 (g)−1 = V1 (ga), W2 (g)V2 (v)W2 (g)−1 = V2 (gv) These equations are the same as W (2(g)Pi W2 (g)−1 =

3 ∑

gji Pj ,

j=1

W2 (g)Qi W2 (g)−1 =

3 ∑

gji Qj

j=1

Here, g ∈ SO(3). Thus, W2 (g) has the effect of rotating the position and momentum operators. N-particle system: We assume that Hi is the Hilbert space for the ith particle and that the projective unitary representation U of the Galilean group G acting in H = ⊗N i=1 Hi has the form U (a, v, 0, g) = ⊗N i=1 Ui (a, v, 0, g)

General Relativity and Cosmology with Engineering Applications

455

In other words, as regards translation, motion with uniform velocities and rota­ tions, the actions of these on each particle in the system is the same. The above discussion for a single particle implies that each Ui (a, v, 0, g) acts in the same way on the corresponding particle. However, time evolution described by the operator U (0, v, τ, I) acts on the entire system and may not be factorizable into a tensor product of single particle operators. This is because, the3 generator of this group which is the energy/Hamiltonian H is the sum of the individual kinetic energies and an interaction potential energy and the latter depends on some complex combination of all the particle position operators. So for the present, we can let U (0, 0, τ, I) = exp(−iτ H) where H is a Hermitian operator acting in H. We also assume the existence of velocity operators Vi = (Vi1 , Vi2 , Vi3 ), i = 1, 2, ..., N acting in H satisfying the following properies: U (0, v, 0, I)Vij U (0, v, 0, I)−1 = Vij + vj , 1 ≤ j ≤ 3, i = 1, 2, ..., N, Since Ui (0, v, 0, I) = exp(−iv.Q) it follows that i[v.

∑

Qk , Vij ] = −vj

k

where Qk = (Qk1 , Qk2 , Qk3 ) are the position operators acting in Hk and likewise Pk = (Pk1 , Pk2 , Pk3 are the momentum operators acting in Hk . It should be noted that by the theory for one particle discussed above and the separability of U (a, v, 0, g) we have that U (a, 0, 0, I) = exp(−ia.P ) = ⊗k exp(−iaPk ), P = ∑ k Pk and likewise for Q. Thus, we get i[

∑

Qkl , Vij ] = −δjl

k

We also have from the one particle theory and separability of U (a, v, 0, g) that i[Qkl , Pij ] = −δki δlj So, if we postulate that [Qkl , Vij ] = 0, k /= i (this is true if we assume that Vij acts in Hi ), then we get i[Qkl , Vij ] = −δki δij and hence we derive [Qkl , Pij − Vij ] = 0 which implies that Pij − Vij = Aij (Q)

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General Relativity and Cosmology with Engineering Applications

where Aij (Q) is a function of Q = (Qij : 1 ≤ i ≤ N, 1 ≤ j ≤ 3) only. We define H0 =

3

1∑ 2 1 ∑∑ 2 Vk = Vki 2 2 i=1 N

N

k=1

k=1

From the composition theory of Galilean group representations developed above, we have that U (0, 0, τ, I)U (a, 0, 0, I) = U (a, 0, τ, I) = U (a, 0, 0, I)U (0, 0, τ, I) and hence [H,

∑

Pki ] = 0

k

Note that U (a, 0, 0, I) = exp(−ia.

∑

Pk ) = exp(−i

∑

ai Pki )

k,i

This means that the total momentum of the system of N particles is conserved. Now, we consider [H0 , Qki ] = [

3 ∑

2 Vkr /2, Qki ] = [

r=1

3 ∑

(Pkr − Akr (Q))2 /2, Qki ] =

r=1

−i(Pki − Aki (Q)) = −iVki We also assume (by definition of the velocity as the time derivative of the posi­ tion), d U (0, 0, −τ, I).Qki U (0, 0, τ, I)|τ =0 = Vki dt This gives [H, Qki ] = −iVki and hence [H − H0 , Qki ] = 0 and therefore, H − H0 = V0 (Q) + E where E is of the form I ⊗ E1 with E1 acting in h and V0 (Q) and arbitrary function of the positions Q = (Qki : 1 ≤ k ≤ N, i = 1, 2, 3). Thus, we finally get the general form of the total system Hamiltonian: H=

1∑ (Pki − Aki (Q))2 + V0 (Q) + E 2 k,i

(Ref: K.R.Parthasarathy, ”Mathematical Foundations of Quantum Mechanics”, Hindustan Book Agency)

General Relativity and Cosmology with Engineering Applications

457

A.13. A remark on quantum stochastic calculus. Let ut ∈ H, Ut ∈ U (H), t ≥ 0 Let P be a spectral measure on [0, ∞) with values in P(H) and assume that P commutes with all the Ut' s. We can write Ut = exp(iHt ) where Ht is a Hermitian operator in H. Suppose that the Ht' s commute with each other and that Ht acts in Pt H where Pt = P [0, t] (For example we can choose a Hermitian operator H in H that commutes with P and then define Ht = Pt H = HPt ). More generally, we shall assume that for s < t, Ht − Hs acts in P [s, t]H. Then we have dUt = iUt dHt and since dHt acts in dPt H = P [t, t + dt]H while Ut acts in Pt H, it follows that for v, w ∈ H, we have with Γ(Ut ) denoting the second quantization of Ut (ie, Γ(Ut ) = W (0, Ut )), d < e(v)|Γ(Ut )|e(w) >=< e(v)|dΓ(Ut )|e(w) >= i < e(v)|Γ(Ut )|e(w) > .d < v|Ht |w > and hence Γ(Ut ) satisfies the qsde dΓ(Ut ) = Γ(Ut )dΛ(Ht ) where Λ(X) is the second quantization of X defined by < e(v)|exp(Λ(X))|e(w) >=< e(v)|e(exp(X)w) >= exp(< v|exp(X)|w >)

A.14. Scattering theory with time dependent interactions with the scattering centre. The free particle Hamiltonian is H0 and the Hamiltonian after the particle starts interacting with the scattering centre is H(t) = H0 + δ.V (t). We assume that V (t) = V0 is a constant operator for |t| > T . Let |φ1 > be a free particle state evolving according to H0 in the remote past (the ”in state”) Let |ψ1 > be the corresponding scattered state evolving according to H(t). Likewise, let |φ2 > be a free particle state evolving according to H0 in the future, ie, as t → ∞ (ie, the ”out state”) and |ψ2 > the corresponding scattered state evolving according to H(t). Define for t2 > t1 , U0 (t2 − t1 ) = exp(−i(t2 − t1 )H0 ), ∫ U (t2 , t1 ) = T {exp(−i

t2

H(t)dt)} t1

where T {.} denotes the time ordering operator. We must have limt→∞ (U (t, 0)|ψ2 > −U0 (t)|φ2 >) = 0

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General Relativity and Cosmology with Engineering Applications

and hence

|ψ2 >= limt→∞ U (t, 0)−1 U0 (t)|φ2 >

We write

Ω2 (t) = U (t, 0)−1 U0 (t), t ≥ 0,

Then on an appropriate domain of out states, we have the operator Ω2 = limt→∞ Ω2 (t) Thus, |ψ2 >= Ω2 |φ2 > Likewise,

limt→−∞ (U (0, t)−1 |ψ1 > −U0 (−t)|φ1 >) = 0

or equivalently, |ψ1 >= limt→−∞ U (0, t)U0 (−t)|φ1 > = limt→∞ U (0, −t)U0 (t)|φ1 > = Ω1 |φ1 > where Ω1 = limt→∞ Ω1 (t)|φ1 > where Ω1 (t) = U (0, −t)U0 (t) The scattering matrix is given by S = Ω∗2 Ω1 = limt→∞ Ω2 (t)∗ Ω1 (t) = limt→∞ U0 (−t)U (t, 0)U (0, −t)U0 (t) = limt→∞ U0 (−t)U (t, −t)U0 (t) ∫ t = limt→∞ exp(itH0 ).T {exp(−i H(s)ds)}.exp(−itH0 ) −t

Note: In discussing scattering theory with noise, we assume that {V (t)} is an operator valued random process and then compute the average value of the scattering matrix with respect to the probability distribution of {V (t)}. By the Dyson series expansion, ∫ t T {exp(−i H(s)ds)} = −t

exp(−2itH0 )+exp(−itH0 )(

∞ ∑ n=1

∫ (−i)n −t k,m

where

∑ k,m

|I[k, m]|2 = 1

460

General Relativity and Cosmology with Engineering Applications This means that given that we measure pixel number m, the probability of getting the grey scale amplitude and phase level k is given by PI (k|m) = I[k, m]|2 More generally, the image can be in a mixed state ρI defined by ∑ I[k, m, r, s]|k, m >< r, s| ρI = 1≤k,r≤p,1≤m,s≤N

where the condition T r(ρI ) = 1 implies that

∑

I[k, m, k, m] = 1

k,m

Then if the image is in the mixed state ρI , the probability of getting the grey scale level amplitude specified by the index k given that the mth pixel is mea­ sured is given by < k, m|ρI |k, m > PI (k|m) = ∑ r < r, m|ρI |r, m > I[k, m, k, m] =∑ r I[r, m, r, m] We shall now express ρI in the frequency domain of the grey scale amplitudes: The quantum Fourier transform of the grey scale state |k > is given by |k˜ >= p−1/2

p ∑

exp(−i2πkr/p)|r >

r=1

The inverse quantum Fourier transform is thus given by |k >= p−1/2

p ∑

exp(i2πkr/p)|˜ r>

r=1

Then, ∑ ρI = I[k, m, r, s]|k, m >< r, s| ∑ = p−1 I[k, m, r, s]exp(i2π(kk ' −rr' )/p)|k˜' >< r˜' |⊗|m >< s| ∑ = Iˆ[k ' , m, s, r' ]|k˜' >< r˜' | ⊗ |m >< s| where

Iˆ[k ' , m, s, r' ] = p−1

∑

I[k, m, r, s]exp(i2π(kk ' − rr' )/p)

k,r

We may express this as ρI =

∑

ˆ m, r, s]|k˜ >< r| I[k, ˜ ⊗ |m >< s|

General Relativity and Cosmology with Engineering Applications =

∑

461

˜ m >< r, Iˆ[k, m, r, s] = |k, ˜ s|

The average image energy at pixel number m and frequency k˜ is thus given by ˜ m|ρI |k, ˜ m> ˆ m, k, m] < k, I[k, ∑ =∑ ˆ ˜ m|ρI |˜ r, m > r < r, r I [r, m, r, m] Now we consider the processing of the quantum image state ρI in the spatial domain and in the frequency domain using linear filters. Consider first a matrix of size pN × pN defined by ∑ T = Tms ⊗ |m >< s| m,s

where Tms is a p × p matrix for each m, s ∈ {1, 2, ..., N }. We assume that T is a unitary matrix, ie, T ∗ T = IpN This condition is equivalent to requiring that ∑ ∗ Tm' s' ⊗ |s >< m|m' >< s' | = IpN Tms or equivalently,

∑

∗ Tms Tms' ⊗ |s >< s' | = IpN

m,s,s'

or equivalently,

∑

∗ Tms Tms' = δss' Ip

m

Applying the unitary operator T to ρI gives a transformed image field specified by the density matrix ∑ I[k, m, r, s]T (|k >< r| ⊗ |m >< s|)T ∗ ρ'I = T ρI T ∗ = ρI = =

∑

ˆ m, r, s]T (|k˜ >< r˜| ⊗ |m >< s|)T ∗ I[k,

Now,

(

∑

T (|k >< r| ⊗ |m >< s|) = ∑ Tjl ⊗ |j >< l|)(|k >< r| ⊗ |m >< s|)( Tj∗' l' ⊗ |l' >< j ' |) j ' l'

jl

=

∑

Tjl |k >< r|Tj∗' l' ⊗ |j >< l|m >< s|l' > |m >< j ' | =

∑ jj '

Tjm |k >< r|Tj∗' s ⊗ |j >< j ' |

462

General Relativity and Cosmology with Engineering Applications A.16. Quantization of the em fields inside a rectangular waveguide. 2 + h2 )Hz = 0 (∇2⊥ + h2 )Ez = 0, (∇⊥

γ jωμ ∇⊥ Ez − 2 ∇⊥ Hz × z, ˆ h2 h γ jωε H⊥ = − 2 ∇⊥ Hz + 2 ∇⊥ Ez × zˆ h h We wish to select potentials Φ, A such that when the em fields satisfy the above, then E = −nablaΦ − jωA, μH = ∇ × A E⊥ = −

or equivalently, Ez = γΦ − jωAz , E⊥ = −∇⊥ Φ − jωA⊥ , μHz zˆ = ∇⊥ × A⊥ , μH⊥ = ∇⊥ Az × zˆ − γzˆ × A⊥ In addition, we wish that the potentials Φ, A satisfy the Lorentz gauge condition divA = −jωεμΦ, or equivalently, ∇⊥ .A⊥ − γAz = −jωεμΦ It is easily seen that the most general potentials satisfying the above require­ ments are given by A⊥ = −(∇⊥ Φ + E⊥ )/jω, Az = (γΦ − Ez )/jω, where Φ is any scalar field that satisfies the Helmholtz equation (∇2⊥ + h2 )Φ = 0 In particular, we can take Φ = 0 and then A = −E/jω The general solution for the electromagnetic fields in the guide with the bound­ ary conditions that Ez and the normal components of H vanish on the side boundaries is given by ∑ ∑ C(n, m)exp(−γnm z)unm (x, y), Hz = D(n, m)exp(−γnm z)vnm (x, y) Ez = where

γnm = (h2nm − ω 2 με)1/2 , √ unm (x, y) = (2/ ab)sin(nπx/a)sin(mπy/b),

General Relativity and Cosmology with Engineering Applications

463

√ vnm (x, y) = (2/ ab)cos(nπx/a)cos(mπy/b) These functions are normalized: ∫ ∫ a∫ b 2 unm dxdy = 0

0

a 0

∫

b 0

2 vnm dxdy = 1

and further they are orthogonal ∫ unm un' m' dxdy = 0, (n, m) /= (n' , m' ) ∫

vnm vn' m' dxdy = 0, (n, m) = / (n' , m' )

We thus find that E=

∑

zˆ C(n, m)unm (x, y)exp(−γnm z)− ∑ [(γnm /h2nm )C(n, m)∇⊥ unm (x, y)+(jωμ/hnm2 )D(n, m)∇⊥ vnm (x, y)×zˆ]exp(−γnm z)

∫

We note that

(∇⊥ unm , ∇⊥ vn' m' × zˆ)dxdy ∫ (∇⊥ unm × ∇⊥ vn' m' , zˆ)dxdy

= ∫ =

(unm,x vn' m' ,y − unm,y vn' m' ,x )dxdy = 0

on integration by parts (we are left with only boundary terms which vanish). We note that E⊥ = −

∑

[C(n, m)γnm /h2nm )∇⊥ unm (x, y)+D(n, m)(jωμ/hnm2 )∇⊥ vnm (x, y)×zˆ]exp(−γnm z)

and hence

=

∑

∫

a

∫

0

b 0

|E|2 dxdy

2 (|C(n, m)|2 (1 + |γnm |2 /h2nm ) + |D(n, m)|2 (μω)2 /hnm )exp(−2αnm z)

where γnm (ω) = αnm (ω) + jβnm (ω) Thus,

∫

a

∫

b

∫

d

ε ∑ n,m

0

0

0

|E|2 dxdydz =

(λ(n, m)|C(n, m)|2 + μ(n, m)|D(n, m)|2 )

464 where

General Relativity and Cosmology with Engineering Applications 2 )(1 − exp(−2αnm d))/2αnm λ(n, m) = (1 + |γnm |2 /hnm 2 μ(n, m) = ((μω)2 /hnm )(1 − exp(−2αnm d))/2αnm

Note that the energy in the magnetic field is given by ∫ |∇ × A|2 dxdydx/2μ = and (∇ × A, B) = ∇.(A × B) − (A, ∇ × B) The first term on the rhs is a perfect divergence and by Gauss’ theorem, its volume integral over the guide volume is zero if assuming that the fields vanish on the surface. Further, ∇ × B = ∇ × (∇ × A) = ∇(divA) − nabla2 A = −∇2 A since divE = 0 implies divA = 0. Further, ∇2 A = −∇2 (E/jω) = ω 2 εμE/jω = −jωεμE and hence, ∫ (2μ)−1

|∇ × A|2 dxdydz =

∫ (ε/2)

|E|2 dxdydz

In other words, the energy in the magnetic field is same as the energy in the electric field. Thus, the total field energy is given by ∫ ε |E|2 dxdydz =

∑ (λ(n, m)|C(n, m, ω)|2 + μ(n, m)|D(n, m, ω)|2 ) n,m

and this energy can be quantized using creation and annihilation operators in place of C(n, m, ω), D(n, mω) and their conjugates. A.17. Image processing for non-Gaussian noise models based on the Edgeworth expansion. √ The Edgeworth expansion: Let φ(x) = exp(−x2 /2)/ 2π, the standard nor­ mal density. Define the Hermite polynomials by Hn (x) = (−1)n exp(x2 /2)Dn exp(−x2 /2), D = d/dx ∫

The Edgeworth expansion of a density f (x) for which all the moments R |x|k f (x)dx, k = 1, 2, ..., are finite is given by ∑ ∑ c[n]Dn exp(−x2 /2) f (x) = φ(x)(1 + c[n]Hn (x)) = φ(x) + (2π)−1/2 n≥1

n≥1

465

General Relativity and Cosmology with Engineering Applications Generating function and orthogonality of the Hermite polynomials: ∑ ∑ tn Hn (x)/n! = exp(x2 /2)( (−t)n Dn /n!)exp(−x2 /2) n≥0

n≥0 2

= exp(x /2)exp(−tD)exp(−x2 /2) = exp(x2 /2)exp(−(x − t)2 /2) = exp(tx − t2 /2) Thus, ∑

∫

∫

tn sm

n,m≥0

R

Hn (x)Hm (x)φ(x)dx =

φ(x)exp((t + s)x − t2 /2 − s2 /2)dx

= exp(ts) ∫

and hence

Hn (x)Hm (x)φ(x)dx = n!δ[n − m], n, m ≥ 0 √ Thus {Hn (x)/ n! : n ≥ 0} forms an orthnormal basis for the Hilbert space L2 (R, φ(x)dx). It therefore follows that the coefficients c[n], n ≥ 0 for the Edgeworth expansion of f (x) are given by ∫ c[n] = f (x)Hn (x)dx/n!, n ≥ 0 R

Note that H0 (x) = 1. Now consider a multivariate Edgeworth pdf defined by ψ(x) = |A|ΠM i=1 f ((Ax)i ) where A is an M × M matrix and x ∈ RM . We have ψ(x) = (2π)−M/2 |A|exp(−xT AT Ax/2)ΠM i=1 (1 +

∑

c[n]Hn ((Ax)i )

n≥1

We shall calculate is moment generating function: Let X ∈ RM have ψ as its pdf. Then ∫ ψˆ(t) = Eexp(< t, X >) = exp(< t, x >)ψ(x)dx = (2π)−M/2

RM

∫ RM

exp(< t, A−1 y >)exp(−y T y/2)ΠM i=1 (1 +

= (2π)−M/2 ΠM i=1

∫

∑

c[n]Hn (yi ))dy

n≥1

exp((A−T t)i ξ)exp(−ξ 2 /2)(1 +

∑ n≥1

To calculate this integral, we first observe that ∫ (2π)−1/2 exp(tξ)exp(−ξ 2 /2)Hn (ξ)dξ R

c[n]Hn (ξ))dξ

466

General Relativity and Cosmology with Engineering Applications = (2π)

−1/2

∫ = tn

∫ (−1)

n

exp(tξ).Dξn exp(−ξ 2 /2)dξ

φ(ξ)exp(tξ)dξ = tn exp(t2 /2)

where integration by parts has been used. Thus for the above multivariate case, we get ∑ ψˆ(t) = exp(tT A−1 A−T t/2)ΠM c[n]((A−T t)i )n ) i=1 (1 + n≥1

= exp(tT (AT A)−1 t/2)ΠM i=1 (1 +

∑

c[n](A−T t)i )n )

n≥1

The approximate maximum likelihood estimator for an Edgeworth distribution: Suppose y = Ax + w where w has a multivariate Edgeworth expansion and x also has a multivariate Edgeworth expansion. We wish to estimate x based on y by maximizing p(x|y), ie, the MAP estimate. We have p(x|y)py (y) = p(y|x)px (x) = pw (y − Ax)px (x) Discrete time non-linear filtering theory applied to real time image param­ eter estimation. The parameter vector v[n] at time n satisfies the stochastic difference equation v[n + 1] = f (n, v[n]) + εv [n + 1] where εv [n] is an iid sequence. Thus, v[n] is a discrete time Markov process with transition density p(v[n + 1]|v[n]) = pεv (v[n + 1] − f (n, v[n])) The image vector x[n] is partitioned into patches Pi x[n] with each patch given by Pi x[n] = vi [n] + εi [n], i = 1, 2, ..., L or equivalently, P x[n] = v[n] + ε[n] where P is a non-singular square matrix and ε[n] is an iid sequence independent of the sequence εv [n], n ≥ 1. Finally, the measurement model for the image vector is given by y[n] = x[n] + w[n] where w[n] is again an iid sequence independent of both the sequences εv and ε. We assume that all the three random sequences εv , ε, w have multivariate Edgeworth probability densities with possibly different linear combination co­ efficients. The aim is to dynamically estimate v[n] based on Yn = {y[k] : k ≤ n}

General Relativity and Cosmology with Engineering Applications We have

467

∫

p(v[n+1]|Yn+1 ) = ∫

p(y[n + 1]|v[n + 1])p(v[n + 1]|v[n])p(v[n]|Yn )dv[n] p(y[n + 1]|v[n + 1])p(v[n + 1]|v[n])p(v[n]|Yn )dv[n]dv[n + 1]

We now observe that y[n] = P −1 (v[n] + ε[n]) + w[n] = P −1 v[n] + d[n] where

d[n] = P −1 ε[n] + w[n]

is again an iid vector valued noise. Thus, the MAP estimate of v[n + 1] given Yn+1 is given by ∫ vˆ[n + 1] = argmaxv' pd (y[n + 1] − Av ' )pεv (v ' − f (n, v))p(n, v|Yn )dv where A = P −1 . We assume that vˆ[n + 1] = f (n, vˆ[n]) + δv ' = vˆ0 [n + 1] + δv ' where vˆ0 [n + 1] = f (n, vˆ[n]), expand the above integral upto O((δv ' )2 ) and then maximize this w.r.t δv ' to get the extra correction. We have v0 [n + 1] + δv ' )) = pd (y[n + 1] − A(ˆ pd (y[n+1]−Avˆ0 [n+1])−p'd (y[n+1]−Avˆ0 [n+1])T Aδv ' 1 + δv 'T AT p''d (y[n+1]−Avˆ0 [n+1])Aδv ' 2 with neglect of O(|δv ' |3 ) terms. Likewise, pεv (ˆ v0 [n+1]+δv ' −f (n, v)) = pεv (ˆ v0 [n+1]−f (n, v))+p'εv (ˆ v0 [n+1]−f (n, v))T δv ' 1 + δv 'T p''εv (ˆ v0 [n + 1] − f (n, v))δv ' 2 with neglect of O(|δv ' |3 ). We thus obtain upto O(|δv ' |2 ), ∫ vˆ[n+1] = vˆ0 [n+1]+argmaxδv' (pd (y[n+1]−Avˆ0 [n+1]) −p'd (y[n+1]−Avˆ0 [n+1])T Aδv ' 1 + δv 'T AT p''d (y[n + 1] − Avˆ0 [n + 1])Aδv ' )(pεv (ˆ v0 [n + 1] − f (n, v))+ 2 1 v0 [n + 1] − f (n, v))T δv ' + δv 'T p''εv (ˆ v0 [n + 1] − f (n, v))δv ' )p(n, v|Yn )dv p'εv (ˆ 2 ∫ = vˆ0 [n + 1] + argmaxδv' [(

(pd (y[n + 1] − Avˆ0 [n + 1])×

p'εv (ˆ v0 [n+1]−f (n, v))−pεv (ˆ v0 [n+1]−f (n, v))AT p'd (y[n+1] −Avˆ0 [n+1]))p(n, v|Yn )dv)T δv '

468

General Relativity and Cosmology with Engineering Applications +δv 'T (

∫

(AT p'd (y[n + 1] − Avˆ0 [n + 1])p'εv (ˆ v0 [n + 1] − f (n, v))T +

1 '' (p (ˆ v0 [n + 1] − f (n, v)) + AT p''d (y[n + 1] − Avˆ0 [n + 1])A)p(n, v|Yn )dv)δv ' 2 εv This equation is of the form vˆ[n+1] = vˆ0 [n+1]+argmaxδv' [f (n, Yn+1 , vˆ0 [n+1])T δv ' 1 + δv 'T F (n, Yn+1 , vˆ0 [n+1])δv ' 2 = vˆ0 [n + 1] − F (n, Yn+1 , vˆ0 [n + 1])−1 (f (n, Yn+1 , vˆ0 [n + 1])) and provides the desired recursion. Remark: The following approximation provides an alternate technique for improving the speed of the recursion: ∫ ψ(n, y[n + 1], v)p(n, v|Yn )dv ≈ 1 ψ(n, y[n + 1], vˆ[n]) + T r(ψv'' (n, y[n + 1]ˆ v [n])Cov(v[n]|Yn )) 2 To apply this formula, we note that the vector and matrices f (n, Yn+1 , vˆ0 [n + 1]), F (n, Yn+1 , vˆ0 [n + 1]) can be expressed as integrals of the form f (n, Yn+1 , vˆ0 [n + 1]) = intψ1 (n, y[n + 1], vˆ0 [n + 1], v)p(n, v|Yn )dv ∫ F (n, Yn+1 , vˆ0 [n + 1]) = ψ2 (n, y[n + 1], vˆ0 [n + 1], v)p(n, v|Yn ) A.18.Cartan’s classification of the simple Lie algebras and the Weyl character formula for the irreducible representations of Compact semisimple Lie groups. A scheme S is a finite set linearly independent vectors (elements) α1 , ..., αn in a real vector space with an inner product (., .) such that a(α, β) = 2(α, β)/(α, α) is a non-positive integer for all α = β, α, β ∈ S. The Cauchy Schwarz inequality / then implies that 0 ≤ a(α, β)a(β, α) ≤ 3, α, β ∈ S, α = β / ie, the product a(α, β)a(β, α) assumes only the values 0, 1, 2, 3. It is known from the general theory of semisimple Lie algebras, that a set of simple positive roots of a semisimple Lie algebra forms a scheme. The numbers a(α, β) are called the Cartan integers. Obviously a(α, α) = 2. To pictorially display a scheme S having n elments, we arrange these elements as vertices with the weight of each vertex α marked by a number proportional to (α, α) = |α|2 .

General Relativity and Cosmology with Engineering Applications

469

Theorem 1: A connected scheme with n elements cannot have more than n − 1 links. For suppose that the elments of the scheme are αk , k = 1, 2, ..., n. Then consider n n ∑ ∑ 0 for all k /= m forms a dense subset of H. Hence, there exists a vector u such that < u, uk >, k = 1, 2, ..., p are all distinct. Hence the Vand-der-Monde matrix ((< u, uk >n ))1≤k,m≤p is non-singular implying that c(k) = 0, k = 1, 2, ..., p. Now, since ⊕ √ e(u) = 1 ⊕ tn u⊗n / n! n≥1

we get

√ ⊗n dn e(tu)| = n!u t=0 dtn and since any symmetric tensor can be expressed as a linear combination of tensors of the form u⊗n , it follows that the exponential vectors e(u), u ∈ H span a dense subspace of Γs (H). An adapted process Xt , t ≥ 0 is a family of operators ˜ t |e(ut] )) ⊗ |e(u(t ) > for all t ≥ 0 where we in Γs (H) such that Xt |e(u) >= (X have used the isomorphism that identifies |e(u ⊕ v) > with |e(u) > ⊗|e(v) >. . Ideally speaking, if T denotes the Here ut] = u ⊗ χ[0,t] and u(t = u ⊗ χ(t,∞) ⊕ ⊕ u ) in Γ ( H ) isomorphism that identifies the vector e( s i with ⊗i e(ui ) in i i i ⊗ Γ (H ), then we should write the definition of an adapted process as i i s ˜ t |e(ut] ) >) ⊗ e(u(t ) T (Xt |e(u) >) = (X Let P : 0 = t0 < t1 < ... < tn = T be a partition of [0, T ]. Its size is defined as |P | = max0≤k≤n−1 (tk+1 − tk ) and we define the partial sum I(X, A, P ) =

n−1 ∑

Xtk (Atk+1 (m) − Atk (m))

k=0

Note that X(t) is adapted so it acts in the Fock space Γs (Ht] ) while Atk+1 − Atk acts in the Fock space Γs (H(tk ,tk+1 ] since ∫ tk+1 ¯ (Atk+1 (m) − Atk (m))|e(u) >= ( dt)|e(u) > tk

Note that time unfolds in quantum stochastic calculus as a continuous tensor product of Hilbert spaces. This can be visualized also in the classical proba­ bilisitc setting by noting that if Ft , t ≥ 0 is a filtration on a probability space (Ω, F, P ) generated by a stochastic process X(t), t ≥ 0, then for any t1 < t2 < t3 , if F(t1 ,t2 ] denotes the σ field σ(X(t) : t1 < t ≤ t2 ), we can write L2 (F(t1 ,t3 ] ) = L2 (F(t1 ,t2 ] ) ⊗ L2 (F(t2 ,t3 ] ) in the sense that any measurable functional of X(t), t1 < t ≤ t3 can be expressed as a sum (poissbily infinite) of products of functions of {X(t) : t1 < t ≤ t2 } and of {X(t) : t2 < t ≤ t3 }. More generally, we can write 2 L2 (F(0,∞) = ⊗∞ i=1 L (F(ti ,ti+1 ] )

General Relativity and Cosmology with Engineering Applications

473

where 0 = t0 < t1 < ...., tn → ∞ We now have I(X, A, P )|e(u) >=

n−1 ∑

X(tk )|e(u) >> ((tk , tk+1 ])

k=0

where

∫

t

> ((s, t]) =

< m(t' ), u(t' ) > dt' , s ≤ t

s

Note that > can be extended to a complex measure on (R+ , B(R+ )). If Q is a partition finer that P , then it is clear that for each k = 0, 1, ..., n − 1, there exist integers a(k) < b(k) such that ∐

b(k)

(tk , tk+1 ] =

(sl , sl+1 ]

l=a(k)

and the points sl , l = 0, 1, ...m − 1 all form the partition Q. Thus, we have X(tk ) > ((tk , tk+1 ]) −

b(k) ∑

X(sl ) > ((sl , sl+1 ])

l=a(k)

=

b(k) ∑

(X(tk ) − X(sl )) > ((sl , sl+1 ])

l=a(k)

So || (I(X, A, P ) − I(X, A, Q))|e(u) >||≤ max|t−s|≤|P |,s,t∈[0,T ] || (X(t) − X(s))|e(u) >|| | > ([0, T ])| Assume that X(t) is strongly uniformly continuous on [0, T ]. Then limδ→0 max|t−s|≤δ,s,t∈[0,T ] || (X(t) − X(s))|e(u) >||= 0 and hence it follow that if Pn , n = 1, 2, ... is an increasing sequence of partitions, ie, Pn+1 > Pn ∀n and |Pn | → 0 as n → ∞, then |(I(X, A, Pn+m ) − I(X, A, Pn ))|e(u) >||→ 0, n → ∞, m = 1, 2, ... which implies that I(X, A, Pn )|e(u) >, n = 1, 2, ... is a Cauchy sequence in the Boson Fock space Γs (H ⊗ L2 (R+ )). and hence converges to an element of this space. Further, the limit is independent of the sequence of partitions Pn for if Qn , n = 1, 2, ... is another increasing sequence of partitions such that |Qn | → 0, then || (I(X, A, Pn )−I(X, A, Qn ))|e(u) >||≤|| (I(X, A, Pn )−I(X, A, Pn ∪Qn ))|e(u) >||

474

General Relativity and Cosmology with Engineering Applications + || (I(X, A, Pn ∪ Qn ), I(X, A, Qn )|e(u) >||

and by the above logic, both of the terms on the rhs converge to zero, proving that the lhs also converges to zero and hence the strong limits of I(X, A, Pn ) and of I(X, A, Qn ) are the same. A.20. Hartree-Fock equations: ψa (x), x ∈ R3 are Fermionic operator fields and they satisfy the standard anticommutation relations {ψa (x), ψb∗ (x' )} = δab δ 3 (x − x' ), {ψa (x), ψb (x' )} = 0, {ψa (x)∗ , ψb (x)∗ } = 0 Let

T (x) = −∇2x /2m

and let V (x, x' ) = V (x' , x) be a scalar potential field. The Hartree Fock seconed quantized Hamiltonian is defined by ∫

H = H0 + H1 ,

H0 =

ψa (x)∗ T (x)ψa (x)d3 x

with summation over the repeated index a being implied, ∫ H1 = V (x, x' )ψa (x)∗ ψa (x)ψb (x' )∗ ψb (x' )d3 xd3 x' Note that

H0∗ = H0

follows by integration by parts and H1∗ = H1 so that

H∗ = H

and hence H is a valid Hamiltonian. The Fermionic fields at time t are given by the rules of Heisenberg’s matrix mechanics: ψa (t, x) = exp(itH)ψa (x)exp(−itH), and its adjoint

ψa (t, x)∗ = exp(itH)ψa (x)∗ exp(−itH)

Now, ∂ψa (t, x) = iexp(itH)[H, ψa (x)]exp(−itH) ∂t [H, ψa (x)] = [H0 , ψa (x)] + [H1 , ψa (x)] ∫

We have [H0 , ψa (y)] =

[ψb (x)∗ T (x)ψb (x), ψa (y)]d3 x

General Relativity and Cosmology with Engineering Applications ∫ =− ∫ =

Now,

{ψb (x)∗ , ψa (y)}T (x)ψb (x)d3 x

δab δ 3 (x − y)T (x)ψb (x)d3 x = −T (x)ψa (x) ∫

[H1 , ψa (y)] =

475

V (x, x' )[ψb (x)∗ ψb (x)ψc (x' )∗ ψc (x' ), ψa (y)]d3 xd3 x' ψa (y)ψb (x)∗ ψb (x)ψc (x' )∗ ψc (x' ) =

(δab δ 3 (y − x) − ψb (x)∗ ψa (y))ψb (x)ψc (x' )∗ ψc (x' ) = δ 3 (y − x)ψa (y)ψc (x' )∗ ψc (x' ) + ψb (x)∗ ψb (x)ψa (y)ψc (x' )∗ ψc (x' ) So

[ψb (x)∗ ψb (x)ψc (x' )∗ ψc (x' ), ψa (y)] = −ψb (x)∗ ψb (x){ψc (x' )∗ , ψa (y)}ψc (x' ) − δ 3 (y − x)ψa (y)ψc (x' )∗ ψc (x' ) = −δac δ 3 (y − x' )ψb (x)∗ ψb (x)ψc (x' ) − δ 3 (y − x)ψa (y)ψc (x' )∗ ψc (x' ) = −δ 3 (y − x' )ψb (x)∗ ψb (x)ψa (y) − δ 3 (y − x)ψa (y)ψc (x' )∗ ψc (x' )

= −δ 3 (y − x' )ψb (x)∗ ψb (x)ψa (y) − δ 3 (y − x)(δac δ 3 (y − x' ) − ψc (x' )∗ ψa (y))ψc (x' ) = −δ 3 (y−x' )ψb (x)∗ ψb (x)ψa (y)−δ 3 (y−x)ψc (x' )∗ ψc (x' )ψa (y)−δ 3 (y−x)δ 3 (y−x' )ψa (y) Thus, we get ∫ = −iT (t, y)ψa (t, y) − 2i

∂ψa (t, y) = ∂t V (y, x)ψb (t, x)∗ ψb (t, x)d3 x − iV (y, y)ψa (t, y)

We write this equation as i

∂ψa (t, y) ˜ y)ψa (t, y) = H(t, ∂t

˜ (t, y) is the effective Hamiltonian operator defined by where H ∫ ˜ (t, y) = T (t, y) + 2 V (y, x)ψb (t, x)∗ ψb (t, x)d3 x + V (y, y) H and T (t, y) = exp(itH)T (y)exp(−itH) This is the Hartree-Fock equation. A.21. Quantum scattering theory. Explicit determination of the scattering operator. H0 = P 2 /2m, H = H0 + V . exp(−itH0 )ψ(Q) = ψt (Q)

476

General Relativity and Cosmology with Engineering Applications say. Then,

idψt (Q)/dt = H0 ψt (Q) = −∇2Q ψt (Q)/2m

ψt (Q) = exp(it∇2Q /2m)ψ(Q) ∫ / exp(it∇Q 2m) = (2πσ 2 )−3/2 exp(−|x|2 /2σ 2 )exp((x, ∇Q))d3 x Then,

2 /2m) exp(σ 2 ∇2Q /2) = exp(it∇Q

Hence,

σ 2 = it/m, σ =

So −3/2

∫

ψt (Q) = (2πit/m)

= (2πit/m)

−3/2

= (2πit/m)−3/2

∫ ∫

√ it/m

exp(−m|x|2 /2it)exp((x, ∇Q ))ψ(Q)dx exp(−m|x|2 /2it)ψ(Q + x)d3 x exp(−m|Q − x|2 /2it)ψ(x)d3 x

Define the Kernel function Kt (Q) = (2πit/m)−3/2 exp(−m|Q|2 /2it) Let W (t) = exp(itH)exp(−tH0 ) Then,

W ' (t) = iexp(itH).V (Q).exp(−itH0 )

(V is assumed to be a function of Q only. Thus, W ' (t) = iW (t)exp(itH0 )V (Q)exp(−itH0 ) Define Z(t) = exp(itH0 ).V (Q).exp(−itH0 ) Then,

Z ' (t) = iexp(itH0 )[H0 , V (Q)]exp(−itH0 )

and [H0 , V (Q)] = [P 2 , V (Q)]/2m = (2m)−1 ([Pa , V (Q)]Pa + Pa [Pa , V (Q)]) = −i(2m)−1 ((∇Q V (Q), P ) + (P, ∇Q V (Q))) (−i/m)(V ' (Q), P ) − (1/2m)∇2Q V (Q) Another way to evaluate this is to note that Z(t) = V (exp(itH0 )Q.exp(−itH0 ))

General Relativity and Cosmology with Engineering Applications

477

Now, exp(itH0 )Q.exp(−itH0 ) = exp(itad(H0 ))(Q) = Q+it[H0 , Q]+(it)2 [H0 , [H0 , Q]]+... =

Q + tP/m Thus, Z(t) = V (Q + tP/m) Thus,

W ' (t) = iW (t)V (Q + P t/m)

Now suppose |f >, |u >∈ Hac (H0 ). Then, by definition of the absolutely con­ tinuous spectrum of an operator, the Radon-Nikodym derivatives d < u|E0 (λ)|u > /dλ and d < f |E0 (λ)|f > /dλ exist and are finite. We have for V = |u >< u| with < u|u >= 1, ∫ W (t)|f >= exp(itH)exp(−itH0 )|f >= (i

t 0

exp(itH)V.exp(−itH0 )dt)|f >

and for Ω+ |f >= limt→∞ W (t)|f > to exist, it is sufficient that ∫ ∞ || V.exp(−itH0 )|f >|| dt < ∞ X= 0

Now, V.exp(−itH0 )|f >= |u >< u|exp(−itH0 )|f > so || V.exp(−itH0 )|f >||= | < u|exp(−itH0 )|f > | ∫ = | (d < u|E0 (λ)|f > /dλ)exp(−iλt)dλ| R

This is the magnitude of the Fourier transform of the function λ → d < u|E(λ)|f > /dλ. We note that | < u|dE(λ)|f > | ≤< u|dE) (λ)|u >1/2 < f |dE0 (λ)|f >1/2 Hence, |d < u|E0 (λ)|f > /dλ| ≤ (d || E0 (λ)|u >||2 /dλ)1/2 (d || E0 (λ)|f >||2 /dλ)1/2 A necessary condition for the magnitude of the Fourier transform of a function to be integrable is that the Fourier transform be finite. Thus, a necessary condition for X < ∞ is satisfied since the Radon-Nikodym derivatives d || E(λ)|u >||2 /dλ and d || E0 (λ)|f >||2 /dλ are finite because both |f > and |u > belong to Hac (H0 ). Suppose H0 = P (in one dimension) and V = V (Q). H = H0 + V = P + V (Q). Then, ∫ V.exp(itP )f (x) = V (x)f (x + t). So, Ω+ |f > will exist if the function t → ( R V (x)2 |f (x + t)|2 dx)1/2 is integrable on R+ .

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General Relativity and Cosmology with Engineering Applications

Consider now two Hamiltonians H0 , H = H0 + V and let φ : R → R. Consider now the Hamiltonians φ(H0 ), φ(H). Let Wφ (t) = exp(itφ(H))exp(−itφ(H0 )) Then, ∫ Wφ (t) = I + i

t 0

exp(isφ(H))(φ(H) − φ(H0 ))exp(−isφ(H0 ))ds

So Wφ (∞)|f > will exist if ∫ ∞ || (φ(H) − φ(H0 ))exp(−isφ(H0 ))|f >|| ds < ∞ 0

We note that

∫

< u|(φ(H)−φ(H0 ))exp(−isφ(H0 ))|f >= ∫ −

R

R

exp(−isφ(λ))d < u|φ(H)E0 (λ)|f >

exp(−isφ(λ))φ(λ)d < u|E0 (λ)|f >

In particular, if φ is an invertible function we can write < u|(φ(H) − φ(H0 ))exp(−isφ(H0 ))|f >= ∫

(exp(−isλ)d < u|φ(H)E0 (φ−1 (λ))|f > /dλ)dλ ∫

−

exp(−isλ)λ(d < u|E0 (φ−1 (λ))|f > /dλ)dλ

provided we assume that the concerned Radon-Nikodym derivatives exist. These will exist provided that all |u >, φ(H)|u > and |f > belong to the absolutely continuous parts of the spectral measure E0 oφ−1 ie they belong to Hac (φ(H0 )).

A.22.Hartree Fock approximation to the two electron problem of the Helium atom. The Hamiltonian is H = H1 + H2 + V12 where H1 = −∇21 /2m − 2e2 /r1 , H2 = −∇22 /2m − 2e2 /r2 , V12 = e2 /r12 Let us try the wave function (antisymmetric because the two electrons form a Fermionic pair) √ ψ = (ψ1 ⊗ ψ2 − ψ2 ⊗ ψ1 )/ 2

General Relativity and Cosmology with Engineering Applications

479

with the constraints < ψ1 |ψ1 >=< ψ2 |ψ2 >= 1, < ψ1 |ψ2 >= 0 As in all eigenvalue problems we extremize S =< ψ|H|ψ > −2E1 (< ψ1 |ψ1 > −1) − 2E2 (< ψ2 |ψ2 >) −2λ1 Re(< ψ1 |ψ2 >) − 2λ2 Im(< ψ2 |ψ1 >)) We first observe that taking into account the constraints, S =< ψ1 ⊗ ψ2 − ψ2 ⊗ ψ1 |(H1 + H2 + V12 )|ψ1 ⊗ ψ2 − ψ2 ⊗ ψ1 > −2E1 (< ψ1 |ψ1 > −1)−2E2 (< ψ2 |ψ2 >)−2λ1 Re(< ψ1 |ψ2 >)−2λ2 Im(< ψ2 |ψ1 >)) =< ψ1 |H1 |ψ1 > + < ψ2 |H1 |ψ2 > + < ψ1 |H2 |ψ1 > + < ψ2 |H2 |ψ2 > + < ψ1 ⊗ ψ2 |V12 |ψ1 ⊗ ψ2 > < ψ2 ⊗ ψ1 |V12 |ψ2 ⊗ ψ1 > − < ψ1 ⊗ ψ2 |V12 |ψ2 ⊗ ψ1 > − < ψ2 ⊗ ψ1 |V12 |ψ1 ⊗ ψ2 > −2E1 (< ψ1 |ψ1 > −1) −2E2 (< ψ2 |ψ2 >) − 2λ1 Re(< ψ1 |ψ2 >) − 2λ2 Im(< ψ2 |ψ1 >)) Now, δS/δψ¯1 = 0 gives 2H1 ψ1 (r1 ) + 2 < I ⊗ ψ2 |V12 |ψ1 ⊗ ψ2 > −2 < I ⊗ ψ2 |V12 |ψ2 ⊗ ψ1 > −2E1 ψ1 − λ1 |ψ1 > −iλ2 |ψ2 >= 0 and likewise another equation for δS/δψ¯2 = 0. Expanding the above, we get 2(−∇21 /2m−2e2 /r1 −E1 −λ1 )ψ1 (r1 )+2

∫

ψ¯2 (r2 )(e2 /r12 )(ψ1 (r1 )ψ2 (r2 )−ψ2 (r1 )ψ1 (r2 ))d3 r2

−iλ2 ψ2 (r1 ) = 0 We note that the second term can be expressed as ∫ ψ¯2 (r2 )(e2 /r12 )(ψ1 (r1 )ψ2 (r2 ) − ψ2 (r1 )ψ1 (r2 ))d3 r2 ∫ =[

2

2

3

(|ψ2 (r2 )| (e /r12 )d r2 ]ψ1 (r1 ) − (

∫

(e2 /r12 )ψ¯2 (r2 )ψ1 (r2 )d3 r2 )ψ2 (r1 )

The first term in this expression represents the potential energy produced by a smeared second electron charge on the first charge, the charge density of this smeared distribution being given by e|ψ2 (r2 )|2 . The second term in the above expression represents the effect of the spin interaction between the two electrons,

480

General Relativity and Cosmology with Engineering Applications

the interaction caused by the fact that both the electrons cannot occupy the same state. Remark: The constraint < ψ1 |ψ2 >= 0 need not be introduced. Neither do the constraints < ψ1 |ψ1 >=< ψ2 |ψ2 >= 1 need to be imposed. We only need to introduce the constraint that the overall wave function be normalized, ie, 1 =< ψ|ψ > which is equivalent to 2 =< ψ1 ⊗ ψ2 − ψ2 ⊗ ψ1 |ψ1 ⊗ ψ2 − ψ2 ⊗ ψ1 >= 2 < ψ1 |ψ1 >< ψ2 |ψ2 > −2| < ψ1 |ψ2 > |2 or equivalently, 1 =< ψ1 |ψ1 >< ψ2 |ψ2 > −| < ψ1 |ψ2 > |2 This results in the following version of the Hartree Fock equation ∫ 2(−∇21 /2m−2e2 /r1 )ψ1 (r1 )+ ψ¯2 (r2 )(e2 /r12 )(ψ1 (r1 )ψ2 (r2 )−ψ2 (r1 )ψ1 (r2 ))d3 r2 ∫ +

ψ¯2 (r1 )(e2 /r12 )(ψ2 (r1 )ψ1 (r2 ) − ψ1 (r1 )ψ2 (r2 ))d3 r2

−E(< ψ2 |ψ2 > ψ1 (r1 )− < ψ2 |ψ1 > ψ2 (r1 )) = 0 with another equation of the same type, ie, with the same value of E for ψ2 . We can generalize this to a system of N interacting Fermions. The Hamiltonian of such a system is given by H=

N ∑ i=1

∑

Hi +

Vij

1≤i= N !det((< ψa |ψb >)) = N !

∑

sgn(σ)ΠN k=1 < ψk |ψσk >= Λ[ψ]

σ

say. We have S1 [ψ] =< ψ|

N ∑ k=1

Hk |ψ >=

∑ σ,τ ∈SN

sgn(στ ) < ψσ1 ⊗ ...ψσN |Hk |ψτ 1 ⊗ ...ψτ N >

General Relativity and Cosmology with Engineering Applications =

∑

481

sgn(στ )(ΠN j=1,j/=k < ψσj , ψτ j >) < ψσk |Hk |ψτ k >

σ,τ,k

∑

=

sgn(στ )(ΠN j=1,j/=k < ψj , ψτ σ −1 j >) < ψk , |Hσ −1 k |ψτ σ −1 k >

σ,τ,k

∑

=

sgn(ρ)(ΠN j=1,j/=k < ψj , ψρj >) < ψk |Hσk |ψρk >

ρ,σ,k

and S2 [ψ] =< ψ|

∑

Vij |ψ >=

i) < ψσi ⊗ ψσj |Vij |ψρi ⊗ ψρj >

σ,τ

∑

=

sgn(ρ)(ΠN k=1,k/=i,j < ψk |ψρk >) < ψi ⊗ψj |Vσ −1 i,σ −1 j |ψρi ⊗ψρj >

σ,ρ,i,j:σ −1 i)Hσk |ψρk >

ρ,σ

+

∑

sgn(ρ)|ψρk > (ΠN l=1,l/=k,j < ψl |ψρl >) < ψj |Hσj |ψρj >

ρ,σ,j

+

∑

sgn(ρ)(ΠN m=1,m/=k,j < ψm |ψρm >)|ψρk >< I⊗ψj |Vσ −1 k,σ −1 j |I⊗ψρj >

σ,ρ,j:σ −1 k)|ψρk >< ψi ⊗I|Vσ −1 i,σ −1 k |ψρi ⊗I >

σ,ρ,i:σ −1 i)|ψρk >< ψi ⊗ψj |Vσ−1 i,σ−1 j |ψρi ⊗ψρj >

σ,ρ,σ −1 i (ΠN j=1,j /=k < ψj |ψσj >), k = 1, 2, ..., N

σ

A.23.Plasmonic waveguides.

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General Relativity and Cosmology with Engineering Applications

A rectangular waveguide is filled with a plasma. The charge per particle of the plasma is q and the particle distribution function is f (t, r, v). This distri­ bution function is assumed to vanish on the boundaries of the guide and hence its Fourier transform w.r.t time can be expanded as ∑ f f (ω, r, v) = fnm (ω, v)unm (x, y)exp(−γnm (ω)z) n,m≥1

where

√ unm (x, y) = (2/ ab)sin(nπx/a)sin(mπy/b), 0 ≤ x ≤ a, 0 ≤ y ≤ b

For a fixed n, m, we write f (ω, v) for fnm and γ for γnm . The Maxwell equations curlE = −jωμH, curlH = J + jωεE ∫

where J(ω, x, y) =

qvδf (ω, x, y, v)d3 v

where the f (ω, x, y, v)exp(−γz) is a component of the particle density corre­ sponding to propagation constant γ. In component form, we have Ez,y (x, y) + γEy (x, y) = −jωμHx (x, y), γEx + Ez,x = jωμHy , Ey,x − Ex,y = −jωμHz , Hz,y + γHy = Jx (x, y) + jωεEx , γHx + Hz,x = −Jy (x, y) − jωεEy , Hy,x − Hx,y = Jz (x, y) + jωεEz (x, y) where the frequency argument ω has been omitted. Here, ∫ ∫ 3 Jx (x, y) = qvx δf (x, y, v)d v, Jy (x, y) = qvy δf (x, y, v)d3 v, Jz (x, y) ∫ = qvz δf (x, y, v)d3 v−−−(a) These equations can be solved for Ex , Ey , Hx , Hy in terms of J(x, y) and the partial derivatives of Ez (x, y), Hz (x, y). Having done so, we substitute these into the Boltzmann kinetic transport equation with the unperturbed Maxwell distribution function f0 (v) being taken in place of f where multiplication with the em fields is concerned. Thus, we get the approximate equation jωδf (x, y, v)+vx δf,x (x, y, v)+vy δf,y (x, y, v)−γvz δf (x, y, v)+q(E+μv×H, ∇v )f0 (v) = −δf (x, y, v))/τ (v) − − − (b) where the relaxation time approximation for the collision term has been used. Here, δf (x, y, v) = f (x, y, v)−f0 (v). To proceed further, we first solve the above equations for Ex , Ey , Hx , Hy : ( ) γ −jωμ (Ex , Hy )T = (−Ez,x , Hz,y − Jx (x, y))T , jωε −γ

General Relativity and Cosmology with Engineering Applications (

γ jωε

jωμ γ

483

) (Ey , Hx )T = (−Ez,y , −Hz,x − Jy (x, y))T ,

Solving these gives Ex = −(γ/h2 )Ez,x − (jωμ/h2 )(Hz,y − Jx ) − − − (1) Ey = −(γ/h2 )Ez,y + (jωμ/h2 )(Hz,x + Jy ) − − − (2) Hx = (jωε/h2 )Ez,y − (γ/h2 )(Hz,x + Jy ) − − − (3) Hy = −(jωε/h2 )Ez,x − (γ/h2 )(Hz,y − Jx ) − − − (4) where

h2 = γ 2 + ω 2 εμ

Substituting these into the third components of the Maxwell curl equations gives (Hz,xx + Jy,x ) + (Hz,yy − Jx,y ) + h2 Hz = 0 (Ez,xx + Ez,yy ) − (γ/jωε)(Jx,x + Jy,y ) + h2 Ez = 0 or equivalently,

(∇2⊥ + h2 )Ez = (γ/jωε)(Jx,x + Jy,y ), (∇2⊥ + h2 )Hz = Jx,y − Jy,x

In accordance with the boundary conditions on the tangential components of E and the normal components of H, we have the expansions ∑ Enm unm (x, y)exp(−γnm z), Hz (x, y, z) Ez (x, y, z) = n,m

=

∑

Hnm wnm (x, y)exp(−γnm z)

n,m

where

√ wnm (x, y) = (2/ ab)cos(nπx/a)cos(mπy/b)

Formally, we can write Ez = (∇2⊥ + h2 )−1 (γ/jωε)(∇⊥ .J⊥ ) − − − (5) Hz = (∇2⊥ + h2 )−1 (Jx,y − Jy,x ) − − − (6) and express Ex , Ey , Hx , Hy in terms of J using eqns. (1)-(6). Finally, these expressions are substituted into the linearized Boltzmann eqn. (b) by mak­ ing use of (a). The result is a linear integro-partial differential equation for the Boltzmann particle distribution function f (ω, x, y, v) with γ as a parame­ ter. The solutions of this equation will generally lead to discrete values of the propagation constant γ. A.24. Winding number of planar Brownian motion. Let Zt = Xt + iYt be a complex Brownian motion, ie, X, Y are independent real standard Brownian motion processes. We write √ ρt = Xt2 + Yt2 = |Zt |, θt = T an−1 (Yt /Xt ) = Arg(Zt )

484

General Relativity and Cosmology with Engineering Applications Then, log(ρt ) = RelogZt , θt = Imlog(Zt ) Away from the origin, z → logz is an analytic function of a complex variable and hence its Laplacian vanishes. Thus, from Ito’s formula, dlog(Zt ) = dZt /Zt and hence log(Zt ) is a Martingale. Writing log(z) = u(x, y) + iv(x, y), z = x + iy it follows that u(Xt , Yt ) and v(Xt , Yt ) are both real Martingales. We have u(Xt , Yt ) = log(ρt ), v(Xt , Yt ) = θt Further, the quadratic variation of the Martingales u(Xt , Yt ) and v(Xt , Yt ) are the same processes. They are equal to ∫ t ∫ t ∫ t |∇u(Xs , Ys )|2 ds = |∇v(Xs , Ys )|2 ds = ds/|Zs |2 0

0

0

where we make use of the Cauchy-Riemann equations, u,x = v,y , u,y = −v,x and of course u,x + iu,y = u,x − iv,x , v,x + iv,y = −u,y + iv,y while on the other hand, writing f (z) = logz = u + iv, we have f ' (z) = u,x + iv,x , if ' (z) = u,y + iv,y Thus,

|f ' (z)|2 = |∇u|2 = |∇v|2 = 1/|z|2 = 1/(x2 + y 2 ) = 1/ρ2

Thus, writing

∫

t

C(t) = 0

ds/|Zs |2 =

∫ 0

t

ds/ρ2s

it follows that there exists a planar Brownian motion β(t) + iγ(t) such that log(ρt ) + iθt = logZt = β(C(t)) + iγ(C(t)) so that β(C(t)) = log(ρt ), γ(C(t)) = θt In fact, if τ () is the inverse function of C, then we have that u(Xτ (t) , Yτ (t) ) and v(Xτ (t) , Yτ (t) ) are independent Brownian motion processes which we denote by β(t) and γ(t) respectively. We have f (Zτ (t) ) = β(t) + iγ(t)

General Relativity and Cosmology with Engineering Applications and

485

df (Zτ (t) ) = f ' (Zτ (t) )dZτ (t)

(since (dZ)2 = 0). Thus, (df (Zτ (t) ))2 = f ' (Zτ (t) )2 (dZτ (t) )2 = 0 which proves independence of the Brownian motions β and γ. Note that we can write β(t) = log(ρτ (t) ), γ(t) = θτ (t) Remark: The real and imaginary parts β(t) and γ(t) ( of f (Zτ (t) ))are continuous dt 0 martingales with quadratic variation matrix equal to = dt.I2 which 0 dt proves by Levy’s theorem for vector valued Martingales that these two processes are independent standard Brownian motion processes. Now, the angle turned by the planar Brownian motion process Zt around the origin respectively inside and outside a circle of radius r are ∫ t ∫ t χ|Zs |r dlog(Zs ) θr− (t) = Im 0

0

and these can be expressed as ∫ θr− (t) = ∫ 0

∫

0

χρs r dZs /Zs

Using the above time change result, ∫ C(t) χβ(s)log(r) dγ(s)

Now define Ta = min(t ≥ 0 : β(t) = a) and σr = min(t ≥ 0 : ρt = r) Then, σr = min(t : log(ρt ) = log(r)) = min(t : β(C(t)) = log(r)) Thus, C(σr ) = min(C(t) : β(C(t)) = log(r)) = min(t : beta(t) = log(r)) = Tlog(r)

486

General Relativity and Cosmology with Engineering Applications

Now for a set E in C, consider ∫

t

I(t) = σ√t

χZ(s)∈E dZs /Zs

We write / 0 Zt = a + Zˆt , a = We assume that Z0 = a so that Zˆ0 = 0. We have ∫ t I(t) = χZˆs ∈E−1 dZˆs /(a + Zˆs ) σ√t

√ Now, the process Zˆts , s ≥ 0 has the same law as tZˆs , s ≥ 0 and hence I(t) has the same distribution as ∫ 1 √ √ ˜ I (t) = χZˆs ∈(E−1)/√t tdZˆs /(a + tZˆs ) ' t−1 σ√

t

√ where σr' is the same as σr but with the process a + tZˆs/t , s ≥ 0 used in place of Zˆs , s ≥ 0 (Note that these two processes have the same law). Now √ √ ' = min(s/t : |a + tZˆs/t | = t) t−1 σ√ t √ √ = min(s : |a + tZˆs | = t) √ = min(s : |a/ t + Zˆs | = 1) ' This gives the result that t−1 σ√ converges as t → ∞ to the random variable t ˆ min(s : |Zs | = 1) = σ say. It follows on taking lim t → ∞ that the limit in law of I(t) as t → ∞ is given by the random variable

∫

1

σ

where

χZˆs ∈E0 dZˆs /Zˆs

√ E0 = limt→∞ (E − 1)/ t

which is a finite random variable since dZˆs /Zˆs =√dlog(Zˆs ) and σ equals the limit as t → ∞ of the random variable min(s : |1/ t + Zˆs | = 1). In particular, we get that limt→∞ |θr− (t) − θr− (σ√t )|/logt = 0 and likewise for θr+ . Thus, it follows that if limt→∞ θr− (σ√t )/log(t) exists in law, then this limit coincides in law with the limit in law of θr− (t)/log(t) and likewise for θr+ (t). Now, ∫ θr−

(σ√

t) =

0

C(σ√t )

χβ(s) +θ(t' − t) < ψ(t' , r' )∗ ψ(t, r) >= Now we note that < ψ(t, r)ψ(t' , r' )∗ >=

∑

cn c¯m exp(−i(En t − Em t' ))un (r)¯ um (r' ) < an a∗m >

n,m

and since

< an a∗m >= δn,m

∗ ∗ an = δn,m and an |0 >= 0), it follows that (since an am + am

< ψ(t, r)ψ(t' , r' )∗ >=

∑

exp(−iEn (t − t' )un (r)¯ um (r' )

n

We note that

< ψ(t' , r' )∗ ψ(t, r) >= 0

since ψ(t, r)|0 >= 0. Thus, ∑ G(t, r, t' , r' ) = θ(t − t' ) exp(−iEn (t − t' ))un (r)¯ un (r' ) = G0 (t − t' , r, r' ) n

where

G0 (t, r, r' ) = G(t, r, 0, r' )

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General Relativity and Cosmology with Engineering Applications

We then get ∫ R

G0 (t, r, r' )exp(i(ω + iε)t)dt =

∑ un (r)¯ un (r' ) i(ω − En + iε) n

We have G,t (t, r, t' , r' ) = δ(t − t' ) < ψ(t, r)ψ(t, r' )∗ > +θ(t − t' ) < ψ,t (t, r)ψ(t' , r' )∗ > = δ(t − t' )δ 3 (r − r' ) − iθ(t − t' ) < H(r)ψ(t, r)ψ(t' , r' )∗ > = δ(t − t' )δ 3 (r − r' ) − iH(r)G(t, r, t' , r' )

A.28.Design of quantum gates using scattering theory. Let H0 , H be two self-adjoint operators in the same Hilbert space H. We define W (t) = exp(itH)exp(−itH0 ), t ∈ R The limits Ω+ = slimt→∞ W (t), Ω− = slimt→−∞ W (t) may exist on different domains. Let D+ be the domain of Ω+ and D− that of Ω− . Then Ω∗+ : H → D+ is defined and hence S = Ω∗+ Ω− : D− → D+ is defined. It is clear that W (t) is a unitary operator on H and hence ∗ Ω+ |g >=< Ω+ f |Ω+ g >= limt→∞ < W (t)f |W (t)g >=< f |g >, f, g ∈ D+ < f |Ω+

Thus Ω+ is unitary when restricted to D+ . Thus, Ω+ Ω∗+ : H → H is an orthogonal projection of H onto D+ . We write V = H − H0 and then find that ∫ W (t) − I =

t

W ' (s)ds == i

0

In particular,

t 0

∫ Ω+ = I + i

∫

exp(isH)V.exp(−isH0 )ds

∞ 0

exp(isH)V.exp(−isH0 )ds

Let E0 be the spectral measure of H0 and E that of H. We have ∫ ∞ ∫ ∞ d/dt(exp(itH0 ).exp(−itH))dt = I−i exp(itH0 )V.exp(−itH)dt Ω∗+ = I+i 0

0

Thus, I=

Ω∗+ Ω+

∫ = Ω+ − i

∞ 0

exp(itH0 )V.exp(−itH)Ω+ dt

General Relativity and Cosmology with Engineering Applications

499

Now, exp(−itH)Ω+ = lims→∞ exp(i(s−t)H)exp(−isH0 ) = lims→∞ exp(isH).exp(−i(s+t)H0 ) = Ω+ exp(−itH0 ) Thus,

∫ I = Ω+ − i

or equivalently,

∞

exp(itH0 )V Ω+ exp(−itH0 )dt

0

∫

Ω+ = I + i

[0,∞)×R

∫

=I−

R

exp(it(H0 − λ + iε))V Ω+ dtE0 (dΛ)

(H0 − λ + iε)−1 V Ω+ E0 (dλ)

The right side integral is to be interpreted as the limit of ε → 0+. This is the rigorous statement of the first Lippman-Schwinger equation in scattering theory. Roughly speaking, if |φ+ > is an output free particle state corresponding to an ”eigenfunction” of H0 with ”eigenvalue” λ and |ψ+ >= Ω+ |φ+ > is the corresponding output scattered state, which is an ”eigenfunction” of H with the same eigenvalue λ (energy is conserved during the scattering process), then we have |ψ+ >= |φ+ > −(H0 − λ + iε)−1 V |ψ+ > In a similar way, we get that if |φ− > is an input free particle state corresponding to an energy eigenfunction of H0 with eigenvalue λ and |ψ− >= Ω− |φ− > the corresponding input scattered state corresponding to an eigenfunction of H with eigenvalue λ, then we get the second Lippman-Schwinger equation |ψ− >= |φ− > −(H0 − λ − iε)−1 V |ψ− > This is seen as follows: Ω∗− = limt→−inf ty exp(itH0 )exp(−itH) = I + i ∫ 0

I = Ω∗− Ω− = Ω− + i

or equivalently, Ω− = I − i

∫ 0

−∞

exp(itH0 )V.exp(−itH)dt

exp(−itH0 )V.exp(itH)dt

Thus, ∫

0

∞

=I +i

= Ω− + i

∫

∫

∞ 0

exp(−itH0 )V.exp(itH)Ω− dt

∞ 0

∞

exp(−itH0 )V Ω− .exp(itH0 )dt

exp(−it(H0 − λ − iε))V Ω− dtE0 (dλ)

500

General Relativity and Cosmology with Engineering Applications ∫ =I−

(H0 − λ − iε)−1 V Ω− E0 (dλ)

R

which gives the second Lippman-Schwinger equation. Now, we derive an explicit form of the scattering matrix. We define R0 (λ) = (H0 −λ)−1 , R(λ) = (H −λ)−1 respectively for λ belonging to the resolvent set of H0 and of H. We have S = Ω∗+ Ω− =

∫ (I − i

∞

exp(itH0 )V.exp(−itH)dt)Ω−

0

∫

= Ω− − i

∞

exp(itH0 )V.Ω− exp(−itH0 )dt

0

∫ = Ω− +

R

(H0 − λ + iε)−1 V.Ω− E0 (dλ) ∫

= Ω− + Now,

R

∫ Ω− = I − i

∫ =I−

0

−∞ ∞

∫ =I −i

R0 (λ − iε)V Ω− E0 (dλ)

exp(itH).V.exp(−itH0 )dt

exp(−itH)V exp(itH0 )dt

0

×R

(H − λ − iδ)−1 V E0 (dλ)

Substituting this into the previous expression gives ∫ S−I = −

R

∫ R(λ+iδ)V E0 (dλ)+

∫ R

R0 (λ−iε)V E0 (dλ)−

R0 (λ−iε)V R(λ+iδ)V E0 (dλ)

A.29. Perturbed Einstein field equations with electromagnetic interactions. The unperturbed metric is g00 = 1, grs = −S(t)2 δrs , g0r = 0, A small change in the coordinate system can be made such that the first order perturbations in the metric δgμν satisfies δg0μ = 0. The unperturbed energymomentum tensor of the matter field is Tμν = (ρ(t) + p(t))Vμ Vν − p(t)gμν where Vr = 0, V0 = 1 (This is possible because the above unperturbed metric satisfies the comoving condition, ie, particles with Vr = 0 satisfy the geodesic equation). Thus, T00 = ρ(t), Trs = p(t)S 2 (t)δrs , g0r = 0

General Relativity and Cosmology with Engineering Applications

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We also define the tensor Sμν = Tμν − T gμν /2, T = g μν Tμν = ρ + p − 4p = ρ − 3p We have δSμν = δTμν − T δgμν /2 − gμν δT /2 so that since δg00 = 0, δS00 = δT00 − δT /2 = δρ − δρ/2 + 3δp/2 = (3δp + δρ)/2 Note that δρ, δp, δvμ , δgμν are in general functions of space and time, ie, of x = (t, r). δSrs = δTrs − T δgrs /2 − grs δT /2 = −δpgrs − (ρ − 3p)δgrs /2 − grs (δρ − 3δp)/2 = (3p − ρ)δgrs /2 + S 2 (δρ − δp)δrs /2 Note that we have the sequence of implications vμ v μ = 1, Vμ δv μ = 0, δv 0 = 0 More precisely, δ(gμν v μ v ν ) = 0 gives V μ V ν δgμν + gμν (V μ δv ν + V ν δv μ ) = 0 which implies

δg00 + 2δv 0 = 0

and since by the coordinate condition, δg00 = 0, it follows that δv 0 = 0. Hence, δv0 = (δg0μ v μ ) = δg0μ V μ + g0μ δv μ = 0 which gives δv0 = 0 since by the coordinate condition, δg0μ = 0 and g0r = 0. So δT00 = (p + ρ)(2V0 δv0 ) + V02 (δp + δρ) − pδg00 − g00 δp = δρ Further, δSr0 = δS0r = (ρ + p)δvr since δg0r = δgr0 = 0. Finally, we need to compute δRμν and equate this to K.δSμν . We have α ):ν − (δΓα δRμν = (δΓμα μν ):α (δΓα μα ):ν = α (δΓμν ):β = (δΓα μν ),β

502

General Relativity and Cosmology with Engineering Applications ρ ρ α +Γα ρβ δΓμν − Γμβ δΓρν ρ −Γνβ δΓα ρμ

Thus, (δΓα μν ):α = (δΓα μν ),α α α +Γρα δΓρμν − 2Γρμα δΓρν

Thus final form of the first order perturbation to the Ricci tensor is given by δRμν = ρ α (δΓα μα ),ν − Γμν δΓρα

−(δΓα μν ),α α α −Γρα δΓρμν + 2Γρμα δΓρν

The perturbed Einstein field equations have the general form A1 (μν, rs, t)δgrs (t, r) + A2 (μν, rsl, t)δgrs,l (t, r) + A3 (μν, rslp, t)δgrs,lp (t, r) +A4 (μν, rs, t)δgrs,0 (t, r) + A5 (μν, rsl, t)δgrs,l0 (t, r) + A6 (μν, rs, t)δgrs,00 (t, r) +A7 (μν, t)δρ(t, r) + A8 (μν, r, t)δvr (t, r) = KδSμν (t, r) where Sμν (t, r) is the energy-momentum tensor of the electromagnetic field. We have, Sμν = (−1/4)Fαβ F αβ gμν + Fμα Fνα √ Fμν = Aν,μ − Aμ,ν , (F μν −g),ν = 0 F0r = Ar,0 − A0,r = −(S 2 (t)Ar ),0 − A0,r s Frs = As,r − Ar,s = S 2 (t)(Ar,s − A,r )

The gauge condition is which reads

√ (Aμ −g),μ = 0 (A0 S 3 (t)),0 + S 3 (t)Ar,r = 0

The unperturbed Maxwell equations are F,r0r = 0, (F r0 S 3 ),0 + S 3 F,srs = 0 F 0r = g 00 g rr F0r = (1/S 2 )((S 2 Ar ),0 − A0,r ) and so the first Maxwell equation gives (S 2 Ar ),0r − ∇2 A0 = 0

General Relativity and Cosmology with Engineering Applications

503

or r − ∇2 A0 = 0 (S 2 )' Ar,r + S 2 A,0r

which reads on using the gauge condition, −(S 2 )' S −3 (A0 S 3 ),0 − S 2 (S −3 (A0 S 3 ),0 ),0 − ∇2 A0 = 0 and the second Maxwell equation gives 0 −S(As,r − Ar,s ),s − ((A,r + Ar,0 )S),0 = 0

which simplifies to S∇2 Ar − SAs,rs − (S(A0,r + Ar,0 )),0 = 0 and this becomes on using the gauge condition, S∇2 Ar + S −2 (A0 S 3 ),0r − (SA0,r ),0 − (SAr,0 ),0 = 0 or (∇2 Ar − Ar,00 ) + S −3 (A0 S 3 ),0r − S −1 (SA0,r ),0 − S −1 S ' Ar,0 = 0 Since the radiation field is homogeneous and isotropic, we assume that the vector potential has mean zero and statistical correlations of the form < Ar (t, r)As (t' , r' ) >= P (t, t' , r − r' )δrs We find that since (A0 S 3 (t)),0 + S 3 (t)Ar,r = 0 we have A0,0 (t, r)S 3 (t) + 3S 2 (t)S ' (t)A0 (t, r) + S 3 (t)Ar,r (t, r) = 0 and hence we can assume that A0 = 0 provided that we admit the Coulomb gauge constraint Ar,r (t, r) = 0 which gives

∂P (t, t' , r) =0 ∂xs

and so we can assume that P (t, t' , r) is independent of the spatial vector r = (xs ). Thus, we can write < Ar (t, r)As (t' , r' ) >= P (t, t' )δrs Now, the Maxwell equation (S 2 Ar ),0r − ∇2 A0 = 0

504

General Relativity and Cosmology with Engineering Applications is automatically satisfied since A0 = 0 and Ar,r = 0. The second Maxwell equation (∇2 Ar − Ar,00 ) + S −3 (A0 S 3 ),0r − S −1 (SA0,r ),0 − S −1 S ' Ar,0 = 0 gives using A0 = 0,

∇2 Ar − Ar,00 − S −1 S ' Ar,0 = 0

and hence taking correlations with As (t' , r' ), we get ∇2 P (t, t' ) − P,tt (t, t' ) − S −1 (t)S ' (t)P,t (t, t' ) = 0 and likewise, with t and t' interchanged. We assume that we have solved this equation to obtain P (t, t' ). Now, the average energy-momentum tensor of the radiation field before the metric has been perturbed by δgμν (t, r) is given by Sμν = (−1/4) < Fαβ F αβ > gμν + < Fμα Fνα > Now,

Fαβ F αβ = 2F0r F 0r + Frs F rs = −2S −2 (Ar,0 − A0,r )2 + S −4 (As,r − Ar,s )2 0 2 s r 2 = −2S −2 ((−S 2 Ar ),0 )2 − A,r ) + (A,r − A,s ) s r 2 = 2S −2 ((S 2 Ar ),0 )2 + (A,r − A,s ) s r 2 = 2S 2 (Ar,0 )2 + 2S −2 ((S 2 )' )2 (Ar )2 + (A,r − A,s )

and its average value is < Fαβ F αβ >= 6S 2 P,tt' (t, t' ) + 2S −2 ((S 2 )' )2 P (t, t' ) since P does not depend on the spatial coordinates. Note that by P,tt' (t, t), we mean that P,tt' (t, t) = limt' →t P,tt' (t, t' ) Thus,

< Fαβ F αβ >= 6S 2 (t)P,tt' (t, t) + 8(S ' )2 P (t, t' )

We next compute < Fμα Fνα >. For μ = ν = 0 this is < F0r F0r >= −S −2 < (Ar,0 )2 >= −S 2 < (Ar,0 )2 >= −3S 2 (t)P,tt' (t, t) For μ = r, ν = 0, we get < Frα F0α >= < Frs F0s >= −S −2 < (As,r − Ar,s )As,0 >= −S 2 < (As,r − Ar,s )As,0 >= 0 since P is independent of the spatial coordinates. Thus, < S00 >= (−3/2)S 2 (t)P,tt' (t, t)−2(S ' )2 P (t, t' )−3S 2 P,tt' (t, t) '

= (−9/2)S 2 (t)P,tt' (t, t)−2S 2 P (t, t' )

General Relativity and Cosmology with Engineering Applications

505

< Sr0 >=< S0r >= 0 Finally, < Srs >= (3/2)S 4 (t)P,tt' (t, t)δrs + < Fr0 Fs0 + Frm Fsm > < Fr0 Fs0 >=< Ar,0 As,0 >= S 4 (t) < Ar,0 As,0 >= S 4 (t)P,tt' (t, t)δrs < Frm Fsm >= (−1/S 2 ) < Frm Fsm >= (−1/S 2 ) < (Am,r −Ar,m )(Am,s −As,m ) >= 0 since P is independent of the spatial coordinates. Thus, < Srs >= (−1/2)S 4 (t)P,tt' (t, t)δrs The perturbed Maxwell equations and the perturbed energy-momentum tensor of the electromagnetic field: The perturbed Maxwell equations are √ (δ(F μν −g),ν = 0 The perturbed gauge condition is √ (δ(Aμ −g)),μ = 0 which gives

√ (S 3 (t)δAμ + Aμ δ −g),μ = 0

If we assume δA0 = 0, then this gauge condition gives √ S 3 (δAr ),r + (Ar δ −g),r = 0 Now,

δg = g(−S −2 )δgrr

(Note that our coordinate system is chosen so that δg0μ = 0. Thus, we get √ √ δ −g = −δg/2 −g = −Sδgrr /2 Thus the gauge condition with δA0 = 0 gives S 2 (δAr ),r − Ar δgss,r /2 = 0 where we have used Ar,r = 0 ,ie, the Coulomb gauge for the unperturbed em field. This gives us the following gauge condition for the perturbed em field (δAr ),r = S −2 Ar δgss,r /2 We now write down the first perturbed Maxwell equation using δA0 = 0 as S 2 δF,r0r − (F 0r δgss /2),r = 0 or

S 2 δAr,0r − (Ar,0 δgss /2),r = 0

506

General Relativity and Cosmology with Engineering Applications

which is since Ar,r = 0, the same as S 2 δAr,0r − Ar,0 δgss,r /2 = 0 Substituting for δAr,r from the perturbed gauge condition, we get S 2 (S −2 Ar δgss,r ),0 − Ar,0 δgss,r = 0 or equivalently

Ar (S −2 δgss,r ),0 = 0

and this condition is impossible to fulfill in general. So we cannot assume the Coulomb condition δA0 = 0 for the perturbed situation. Remark: We have assumed that A0 = 0 and hence that the gauge condition reads Ar,r = 0. We have also assumed that < Ar (t, r)As (t' , s) >= P (t, t' , r − r' )δrs and have hence derived the condition that P (t, t' , r) is independent of r. This is rather unrealistic since P being independent of r implies that the vector potentials between all the spatial points have the same correlation. A more realistic assumption would be to take < Ar (t, r)As (t' , s) >= P (t, t' )f r (r)f s (r' ) and hence derive from the gauge condition that r =0 divf = f,r

A more general approach: Assume that the unperturbed metric is gμν (x) and the unperturbed vector potential is Aμ (x). Their perturbed versions are gμν (x)+δgμν (x) and Aμ (x)+δAμ (x) respectively. The unperturbed velocity field of matter is v μ (x) and it gets perturbed by δv μ (x). Likewise the unperturbed density and pressure are ρ(x) and p(x) and they get perturbed by δρ(x) and δp(x) respectively. The unperturbed four vector potential correlations are Lμν (x, x' ) =< Aμ (x)Aν (x' ) > The Einstein field equations are Rμν = K(SM μν + SEμν )) where K = −8πG,

SM μν = TM μν − TM gμν /2

where TM μν = (ρ + p)vμ vν − pgμν , TM = g μν TM μν = ρ − 3p so that SM μν = (ρ + p)vμ vν − (p + ρ)gμν /2 and SEμν = (−1/4)Fαβ F αβ gμν + Fμα Fνβ g αβ

General Relativity and Cosmology with Engineering Applications

507

< SEμν >= gμν g αρ g βσ < Fαβ Fρσ > +g αβ < Fμα Fνβ > Now, < Fμα Fνβ >=< (Aα,μ − Aμ,α )(Aβ,nu − Aν,β ) > =< (gαρ Aρ ),μ − (gμρ Aρ ),α )((gβσ Aσ ),ν − (gνσ Aσ ),β ) > = (gαρ,μ − gμρ,α )(gβσ,ν − gνσ,β )Lρσ (x, x) ρ Aσ > +(gαρ gβσ < Aρ,μ Aσ,ν > +gαρ gβσ,ν < A,μ

+(gαρ,μ gβσ < Aρ Aσ,ν > −gαρ gνσ,β < Aρ,μ Aσ > −gαρ,μ gνσ < Aρ Aσ,β > −gαρ gνσ < Aρ,μ Aσ,β > −gμρ gβσ < Aρ,α Aσ,ν > −gμρ,α gβσ < Aρ Aσ,ν > −gμρ gβσ,ν < Aρ,α Aσ > σ > +gμρ gνσ,β < Aρ,α Aσ > +gμρ gνσ < Aρ,α Aσ,β > +gμρ,α gνσ < Aρ A,β

Using this formula, we can express < SEμν (x) >= C1 (μναβ, x) < Aα (x)Aβ (x) > +C2 (μναβρ, x) < Aα (x)Aβ,ρ (x) > β +C3 (μναβρσ, x) < Aα ,ρ (x)A,σ (x) >

where the functions Ck are constructed using the unperturbed metric gμν (x) and its first order partial derivatives gμν,α (x). The perturbation δ < SEμν (x) > of the average Maxwell energy momentum tensor is to be evaluated in terms of the metric perturbations. For evaluating this, we have to first express using the Maxwell equations, the perturbation δAμ (x) of the electromagnetic four vector potential in terms of the unperturbed potentials Aμ (x) and the metric perturbations δgμν (x). The Maxwell equations are √ (F μν −g),ν = 0 and its perturbation is √ √ ( −gδF μν + F μν δ −g),ν = 0 We have √ √ √ √ δ −g = −δg/2 −g = −gg μν δgμν /2 −g = ( −g/2)g μν δgμν δF μν = δ(g μα g νβ Fαβ ) = δ(g μα g νβ )Fαβ + g μα g νβ δFαβ Now, δ(g μα g νβ ) = −g μα g νρ g βσ δgρσ −g νβ g μρ g ασ δgρσ δFαβ = δ(Aβ,α − Aα,β ) =

508

General Relativity and Cosmology with Engineering Applications (δAβ ),α − (δAα ),β = δAβ = δ(gβmu Aμ ) = Aμ δgβμ + gβμ δAμ

The perturbation of the gauge condition √ (Aμ −g),μ = 0 is given by Now

√ √ ( −gδAμ ),μ + (Aμ δ −g),μ = 0

√ √ √ √ δ −g = −δg/2 −g = −gg μν δgμν /2 −g = −gg μν δgμν /2

Combining all these equations, it follows that δAμ satisfies an equation of the form ν ν (x) + D2 (μνρ, x)δA,ρ (x) + D3 (μνραβ, x)Anu D1 (μνρσ, x)δA,ρσ ,ρ (x)δgαβ (x)

+D4 (μναβ, x)Aν (x)δgαβ (x)+ D5 (μνραβσ, x)Aν,ρ (x)δgαβ,σ (x) +D6 (μναβσ, x)Aν (x)δgαβ,σ (x) = 0 This equation can formally be solved to give ∫ δAμ (x) = M (μν, αβ, x, x' , x'' )Aν (x' )δgαβ (x'' )d4 x' d4 x'' The perturbation to δ < SEμν (x) >=< δSEμν (x) > to the average energy momentum tensor of the electromagnetic field can be expressed as follows: δSEμν (x) = (−1/4)δ(Fαβ F αβ gμν ) + δ(Fμα Fνβ g αβ ) = (E1 (μνρσαβ)(x)Aρ,σ (x) + E2 (μνραβ, x)Aρ (x))δAα ,β (x) +(E3 (μνρσα)(x)Aρ,σ (x) + E4 (μνρα, x)Aρ (x))δAα (x) +(E5 (μνρδαβσ, x)Aδ (x)Aρ (x) + E6 (μνργδαβσ, x)Aδ (x)Aρ,γ (x))δgαβ,σ (x) δ +(E7 (μνρδχαβ, x)Aδ,χ Aρ (x) + E8 (μνρδχγαβ, x)A,χ (x)Aρ,γ (x))δgαβ (x)

A.30. BCS theory of superconductivity. The second quantized Hamiltonian is given by ∫ ∫ ∗ 2 3 K = ψa (x) (−∇ /2m+μ)ψa (x)d x+ V (x, y)ψa (x)∗ ψa (x)ψb (y)∗ ψb (y)d3 xd3 y with the Fermionic fields ψa satisfying the anticommutation relations {ψa (x), ψb (y)∗ } = δab δ 3 (x − y),

General Relativity and Cosmology with Engineering Applications

509

{ψa (x), ψb (y)} = 0 We define ψa (tx) = exp(itK).ψa (x).exp(−itK) Then, ψa,t (tx) = iexp(itK)[K, ψa (x)]exp(−itK) The Green’s function is defined as Gab (t, x|s, y) =< T (ψa (tx)ψb (s, y)∗ ) >= T r(ρG T (ψa (tx)ψb (sy)∗ )) where ρG = exp(−βK)/T r(exp(−βK)) We observe that for t > s, Gab (tx|sy) = T r(exp(−βK).exp(itK)ψa (x)exp(i(s − t)K)ψb (y)∗ .exp(−isK)) = T r(exp((i(t − s) − β)K)ψa (x)exp(i(s − t)K))ψb (y)∗ ) We make the approximation of replacing ∫ V (x, y)ψa (x)∗ ψa (x)ψb (y)∗ ψb (y)d3 xd3 y by ∫

V (x, y)(2 < ψa (y)∗ ψa (y) > ψb (x)∗ ψb (x)d3 xd3 y+

∫ +

V (x, x)ψa (x)∗ ψa (x)d3 x +

∫

∫

V (x, y) < ψa (x)ψb (y)∗ > ψa (x)∗ ψb (y)d3 y

V (x, y) < ψa (x)ψb (y) > ψa (x)∗ ψb (y)∗ d3 xd3 y

plus other quadratic terms. We can write down the general form of the above approximation to the potential energy integral as ∫ F1 (x, y) < ψb (y)∗ ψb (y) > ψa (x)∗ ψa (x)d3 xd3 y ∫

F2 (x, y) < ψb (x)∗ ψa (y) > ψa (y)∗ ψb (x)d3 xd3 y

+ ∫

F3 (x, y) < ψb (y)ψa (x) > ψa (x)∗ ψb (y)∗ d3 xd3 y

+ ∫

F4 (x, y) < ψb (y)∗ ψa (x)∗ > ψa (x)ψb (y)d3 xd3 y

+

= V1 + V2 + V3 + V4 say. We have [V1 , ψc (x' )] =

∫

F1 (x, y) < ψb (y)∗ ψb (y) > [ψa (x)∗ ψa (x), ψc (x' )]d3 xd3 y '

510

General Relativity and Cosmology with Engineering Applications ∫ =−

F1 (x, y) < ψb (y)∗ ψb (y) > δac δ 3 (x − x' )ψa (x)d3 xd3 y ∫ = −(

[V2 , ψc (x' )] = ∫ =−

∫

F2 (x, y) < ψb (x)∗ ψa (y) > [ψa (y)∗ ψb (x), ψc (x' )]d3 xd3 y

F2 (x, y) < ψb (x)∗ ψa (y) > δac δ 3 (y − x' )ψb (x)d3 xd3 y ∫ = −(

F2 (x, x' ) < ψb (x)∗ ψa (x' ) > ψb (x)d3 x [V3 , ψc (x' )] =

∫

Now,

F1 (x' , y) < ψb (y)∗ ψb (y) > d3 y)ψc (x' )

F3 (x, y) < ψb (y)ψa (x) > [ψa (x)∗ ψb (y)∗ , ψc (x' )]d3 xd3 y

ψa (x)∗ ψb (y)∗ ψc (x' ) = ψa (x)∗ (δbc δ 3 (y − x' ) − ψc (x' )ψb (y)∗ ) = δbc δ 3 (y − x' )ψa (x)∗ − (δac δ 3 (x − x' ) − ψc (x' )ψa (x)∗ )ψb (y)∗ = δbc δ 3 (y − x' )ψa (x)∗ − δac δ 3 (x − x' )ψb (y)∗ + ψc (x' )ψa (x)∗ ψb (y)∗

Thus, [V3 , ψc (x' )] =

∫ =

∫

F3 (x, y) < ψb (y)ψa (x) > (δbc δ 3 (y−x' )ψa (x)∗ −δac δ 3 (x−x' )ψb (y)∗ )d3 xd3 y

F3 (x, x' ) < ψc (x' )ψa (x) > ψa (x)∗ d3 x−

∫

F3 (x' , y) < ψb (y)ψc (x' ) > ψb (y)∗ d3 y

Finally, '

∫

[V4 , ψc (x )] =

F4 (x, y) < ψb (y)∗ ψa (x)∗ > [ψa (x)ψb (y), ψc (x' )]d3 xd3 y = 0

A.31. Maxwell’s equations in the closed isotropic model of the universe. The metric is dτ 2 = dt2 − f 2 (r)S 2 (t)dr2 − r2 S 2 (t)(dθ2 + sin2 (θ)dφ2 ) where

f (r) = (1 − kr2 )−1/2

Thus, g00 = 1, g11 = −f 2 (r)S 2 (t), g22 = −r2 S 2 (t), g33 = −r2 S 2 (t)sin2 (θ) √ −g = r2 f (r)S 3 (t)sin(θ)

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511

Assume A0 = 0. The gauge condition √ (Aμ −g),μ = 0 then gives sin(θ)(A1 r2 f ),1 + r2 f (A2 sin(θ)),2 + r2 f sin(θ)A3,3 = 0 or equivalently, r−2 (r2 f A1 ),1 + (sin(θ))−1 (A2 sin(θ)),2 + A3,3 = 0 The Maxwell equation

√ (F 0k −g),k = 0

under the assumption A0 = 0 gives √ (g kk Ak,0 −g),k = 0 or equivalently,

√ (g kk −g(gkk Ak ),0 ),k = 0

or equivalently,

(r2 f sin(θ)(S 2 Ak ),0 ),k = 0

The above two equations are not compatible. So we assume that A0 /= 0 and then the above two equations, ie, the gauge condition and Gauss’ law respec­ tively become sin(θ)r2 f (S 3 A0 ),0 + sin(θ)(A1 r2 f ),1 + r2 f (A2 sin(θ)),2 + r2 f sin(θ)A3,3 = 0 and

√ 0 = (F 0k −g),k √ = (g kk F0k −g),k = √ (g kk (Ak,0 − A0,k ) −g),k = √ √ (g kk (gkk Ak ),0 −g),k − (g kk A0,k −g),k

This further simplifies to ((S 2 Ak ),0 r2 f sin(θ)),k − S 2 (g kk A0,k r2 f sin(θ)),k = 0 The Maxwell equation √ √ (F rs −g),s + (F r0 −g),0 = 0 gives taking r = 1, √ √ √ (F 12 −g),2 + (F 13 −g),3 + (F 10 −g),0 = 0

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and likewise for r = 2, 3. This equation can be expressed as √ √ √ 0 = (F 12 −g),2 + (F 13 −g),3 + (F 10 −g),0 √ √ = (g 11 g 22 (A2,1 − A1,2 ) −g),2 + (g 11 g 33 (A3,1 − A1,3 ) −g),3 √ +(g 11 (A0,1 − A1,0 ) −g),0 = (g 11 g 22 ((g22 A2 ),1 − g 11 A1,2 ),2 + √ (g 11 g 33 (g33 A3 ),1 − (g11 A1 ),3 ) −g),3 √ +(g 11 (A0,1 − (g11 A1 ),0 ) −g),0

A.32. Intuitive proof of Cramer’s theorem for large deviations for iid random variables. Let Xi , i = 1, 2, ... be iid random variables with Eexp(λX1 ) = exp(Λ(λ)) Define I(x) = supλ∈R (λx − Λ(λ)) Also define Sn =

n ∑

Xi

i=1

Then, we have for any Borel set E and any λ ∈ R, the following: Let x ∈ E be arbitrary. Then, 1 = E[exp(λSn − nΛ(λ))] ≥ E[exp(λSn − nΛ(λ))χSn /n∈E ] ≥ E[infx∈E exp(nλx − nΛ(λ))χSn /n∈E ] = infx∈E exp(n(λx − Λ(λ)))P (Sn /n ∈ E) and therefore, n−1 .log(P (Sn /n ∈ E) + infx∈E (λx − Λ(λ)) ≤ 0 or equivalently, n−1 .log(P (Sn /n ∈ E) ≤ −infx∈E (λx − Λ(λ)) If we assume that for each x ∈ E the function λ → (λx − Λ(λ)) attains it supremum at some λ(x) ∈ R, then we can conclude that n−1 .log(P (Sn /n ∈ E) ≤ −infx∈E I(x) where I(x) = supλ∈R (λx − Λ(λ)) = λ(x)x − Λ(λ(x))

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This is called the Large deviations upper bound. To get the lower bound, we define a new probability measure P˜n by the prescription, P˜n = exp(λSn − nΛ(λ)).Pn where Pn is the distribution of (X1 , ..., Xn ), then we have ∫ n ˜ 1 = Pn (R ) = exp(λξ − nΛ(λ))dPn (ξ) and we obtain on differentiating both sides of this equation w.r.t. λ, ∫ 0 = (ξ − nΛ' (λ))dP˜n (ξ) and hence if η denotes the mean of X1 under P˜1 , then we have 0 = η − Λ' (λ) ie,

η = Λ' (λ)

˜ 1 (x) = Note that X1 , ..., Xn under P˜n are iid random variables with distribution dP exp(xλ − Λ(λ))dP1 (x). Thus, we get for λ > 0 and η > 0, ∫ Pn (E) = exp(−λξ + nΛ(λ))dP˜n (ξ) = E

EP˜ [exp(−n(λSn /n − Λ(λ)))χSn /n∈E ] ≥ EP˜ [exp(−n(λSn /n − Λ(λ)))χSn /n∈E χ|Sn /n−η|≤ε ] We are assuming that λ is a function of η determined by the condition Λ' (λ) = η. It follows that ∩ Pn (E) ≥ supη+ε∈E exp(−n(λ(η+ε)−Λ(λ)))P˜n (|{Sn /n ∈ E} {|Sn /n−η| ≤ ε}) Now assume that E = (−∞, a]. By the law of large numbers for iid random variables, we have limn→∞ P˜n (|Sn /n − η| ≤ ε) = 1 ∀ε > 0. Note that P˜ is the probability measure on RZ+ having marginals P˜n , n = 1, 2, .... If we further assume that η < a, the it follows by the law of large numbers that P˜n ({Sn /n ∈ E} ∩ {|Sn /n − η| < ε) → 1 and hence liminfn→∞ n−1 .log(Pn (E)) ≥ supη∈E (−λη + Λ(λ))

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We can write the above as liminfn→∞ n−1 .log(Pn (E)) ≥ supη∈E (−λη + Λ(λ)) = −infη∈E I(η) A similar argument works for η > a. A.33. Current in the BCS theory of superconductivity: ψα (x) are the Fermionic field operators. They satisfy the anticommutation rules {ψα (x), ψβ (y)∗ } = δαβ δ 3 (x − y), {ψα (x), ψβ (y)} = 0, {ψα (x)∗ , ψβ (y)∗ } = 0 The BCS Hamiltonian has the form ∫ ∫ ∗ 2 3 H = (−1/2m) ψα (x) (∇ + ieA(x)) ψα (x)d x + μ ψα (x)∗ ψα (x)dx ∫ + ∫ + ∫ + ∫ +

V1 (x, y) < ψα (x)ψβ (y) > ψα (x)∗ ψβ (y)∗ dxdy V¯1 (x, y) < ψβ (y)∗ ψα (x)∗ > ψβ (y)ψα (x)dxdy V2 (x, y) < ψα (x)ψβ (x)∗ > ψα (y)∗ ψβ (y)dxdy V3 (x, y) < ψα (x)ψβ (y)∗ > ψα (x)∗ ψβ (y)dxdy

where for Hermitianity of H, we require that V¯2 (x, y) = V2 (x, y), V¯3 (x, y) = V3 (y, x) The Gibbs density operator for this Hamiltonian is ρG = exp(−βH)/T r(exp(−βH)) and averages are defined w.r.t this density, ie for any operator X constructed out of the quantum fields ψα (x), ψα (x)∗ , α = 1, 2, x ∈ R3 , we define < X >= T r(ρG X) ∫

We write T = (−1/2m) Then,

ψα (x)∗ (∇ + ieA(x))2 ψα (x)dx

[T, ψα (x)] = (1/2m)(∇ + ieA(x))2 ψα (x) ∫

Also [

psiβ (y)∗ psiβ (y)d3 y, ψα (x)] =

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−ψα (x) ∫ ∫ + ∫ +

[V, ψγ (z)] = V1 (x, y) < ψα (x)ψβ (y) > [ψα (x)∗ ψβ (y)∗ , ψγ (z)]dxdy V¯1 (x, y) < ψβ (y)∗ ψα (x)∗ > [ψβ (y)ψα (x), ψγ (z)]dxdy V2 (x, y) < ψα (x)ψβ (x)∗ > [ψα (y)∗ ψβ (y), ψγ (z)]dxdy

∫

V3 (x, y) < ψα (x)ψβ (y)∗ > [ψα (x)∗ ψβ (y), ψγ (z)]dxdy ∫ ∫ = δ1αγ (x, z)V1 (x, z)ψα (x)∗ dx − Δ1γα (z, x)V1 (z, x)ψα (x)∗ dx ∫ + V2 (x, z)Δ2γα (x, x)ψα (z)dx +

A.34. Some aspects of nonlinear filtering theory. Let Xt be a Markov process with transition generator kernel Kt (x, y), ie, ∫ E(f (Xt+dt )|Xt = x) − f (x) = dt Kt (x, y)f (y)dy + o(dt) = (Kt f )(x)dt + o(dt) Assume that the measurement process is zt defined by dzt = ht (Xt )dt + dvt where vt is a Levy process, ie, an independent increment process with moment generating function given by Eexp(svt ) = exp(tψ(s)) Let Zt = σ(zs , s ≤ t), ie, the measurement process upto time t and for any function φ(x) defined on the state space of the Markov process Xt , we define πt (φ) = E(φ(Xt )|Zt ) We can write a stochastic differential equation for the process πt (φ) as dπt (φ) = Ft (φ)dt +

∞ ∑

Gkt (φ)(dzt )k

k=1

where Ft (φ), Gkt (π) are Zt measurable functions. The processes Ft (φ) and Gkt (φ) need to be calculated. We define a process yt via the sde ∑ fk (t)(dzt )k yt , y0 = 1 dyt = k≥1

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where the fk' s are arbitrary non-random functions. Then yt is a process adapted to Zt and we have the relation E[(φ(Xt ) − πt (φ))yt ] = 0 by the definition of conditional expectation. We write φt = πt (φ). It follows that E[(dφt − dπt (φ))yt ] + E[(φt − pit (φ))dyt ] + E[(dφt − dπt (φ))dyt ] = 0 by Ito’s formula. This equation can be expressed as ∑ ∑ fk (t)E[(φt −πt (φ))(dzt )k yt ]+ E[(dφt −dπt (φ))(dzt )k yt ] = 0 E[(dφt −dπt (φ))yt ]+ k≥1

k

and from the arbitrariness of the functions fk , we get E[(dφt − dπt (φ))|Zt ] = 0 − − − (1) E[(φt − πt (φ))(dzt )k |Zt ] + E[(dφt − dπt (φ))(dzt )k |Zt ] = 0, k ≥ 1 − − − (2) We note that (dzt )k = (dvt )k , k ≥ 2 and hence writing E[(dvt )k ] = μk dt we get from (1) and (2), assuming μ1 = 0, ∑ μk Gkt (φ) = πt (Kt φ) − − − (3), Ft (φ) + k≥1

(πt (ht φ) − πt (φ)πt (ht )) + πt ((Kt φ)ht ) − Ft (φ)πt (ht ) − μk πt (Kt (φ)) −

∑

∑

μr+1 Grt (φ)πt (ht ) = 0,

r≥1

μr+k Grt (φ) = 0, k ≥ 2

r≥1

A.35. Some aspects of supersymmetry. Assume that xμ , μ = 0, 1, 2, 3 are the Bosonic spatial variables and θμ , μ = 0, 1, 2, 3 are the Fermionic anticommuting variables. A super field S(x, θ) can be expanded as S(x, θ) = S0 (x) + S1μ (x)θμ + S2μν (x)θμ θν + S3μνρ (x)θμ θν θρ +S4 (x)θ1 θ2 θ3 θ4 Note that a product of more than four θ' s is zero. The sum in each term is over the repeated variables and we may assume that the summation is over μ < ν < ρ < σ. We now need to introduce an infinitesimal supersymmetry transformation as a super vector field and describe the transformation laws of

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the component fields S0 , S1μ , S2μν , S3μνρ , S4 under such a supersymmetry trans­ formation. The infinitesimal super-symmetry transformation may be derived by assuming the following group conservation law for Bosonic and Fermionic vari­ ables (x, θ), x = (xμ ), θ = (θμ ): '

'

'

(xμ , θα ).(x' μ, θ α ) = (xμ + x μ + θT γ μ θ' , θα + θ alpha ) We express this equation as (x, θ).(x' , θ' ) = m((x, θ), (x' , θ' )) = (m1 ((x, θ), (x' , θ' )), m2 ((x, θ), (x' , θ' ))) where

'

m1 ((x, θ), (x' , θ' )) = (xμ + x μ + θT γ μ θ) '

m2 ((x, θ), (x' , θ' )) = (θα + θ α ) Then to get left and right-invariant vector fields on the supermanifold described by Bosonic and Fermionic coordinates (x, θ), we define Dxμ =

∂mν2 ∂ ∂mν1 ∂ + 'μ ∂x' μ ∂θν ∂x ∂xν

∂ = ∂xμ ∂xμ ∂mν1 ∂ ∂mν2 ∂ = + 'α ∂θ' α ∂θν ∂θ ∂xν =

Dθα

= (θT γ ν )α ∂xμ + ∂θα = (θT γ μ )α

∂ ∂ + α ∂xμ ∂θ

A.36. Comparison between the classical and quantum motions of a simple pendulum perturbed by white Gaussian noise. Classical case: θ'' (t) = −a.sin(θ(t)) + σw(t) w(t) = B ' (t) B(t) is standard Brownian motion. We solve this by perturbation theory: Let θ(t) = θ0 (t) + σθ1 (t) + σ 2 θ2 (t) + ... Then substituting this expression into the equation of motion and equating coefficients of σ m , m = 0, 1, 2 successively gives us θ0'' (t) = −a.sin(θ0 (t)), θ1'' (t) = −a.cos(θ0 (t))θ1 (t) + B ' (t), θ2'' (t) = −a.cos(θ0 (t))θ2 (t) + (a/2)sin(θ0 (t))θ1 (t)2

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General Relativity and Cosmology with Engineering Applications

(

Define the matrix A(t) =

0 −a.cos(θ0 (t))

1 0

) ,

and the 2 × 2 state transition matrix Φ(t, τ ) by ∂Φ(t, τ ) = A(t)Φ(t, τ ), t ≥ τ, Φ(τ, τ ) = I2 ∂t Then,

∫ θ1 (t) = ∫ θ2 (t) = (a/2)

t 0

Φ12 (t, τ )dB(τ ),

Φ12 (t, τ )sin(θ0 (τ ))θ1 (τ )2 dτ

Using these formulas, the statistics of θ(t) can be computed upto O(σ 2 ). We leave the problem of calculating the mean and autorcorrelation of θ(t) upto O(σ 2 ) as an exercise to the reader. The quantum case: Unitarity of the Schrodinger evolution operator U (t) is guaranteed if we take into account an Ito correction term in the Hamiltonian. Thus, the Schrodinger evolution is given by dU (t) = [−(iH0 + σ 2 V 2 /2)dt − iσV dB(t)]U (t) where

H0 = −∂θ2 /2ml2 − mgl.cos(θ)

We note that in this formalism, U (t) should be regarded as an integral kernel U (t, θ, θ' ) so that ∫ (U (t)ψ)(θ) = U (t, θ, θ' )ψ(θ' )dθ' Heisenberg matrix mechanics: Let X be an observable and define X(t) = U (t)∗ XU (t) Then, assuming that V = V (θ) is a multiplication operator, dX(t) = dU (t)∗ XU (t) + U (t)∗ XdU (t) + dU (t)∗ XdU (t) = U (t)∗ (i[H0 , X]dt − (σ 2 /2)(V 2 X + XV 2 − V XV )dt + iσ[V, X]dB(t))U (t) For example, taking X = θ, pθ respectively gives dθ(t) = U (t)∗ (pθ /ml2 − (σ 2 /2)(V 2 θ + θV 2 − V θV ) + iσ[V, θ]dB(t))U (t) = pθ (t)/ml2 , dpθ (t) = U (t)∗ (i[−mglcos(θ), pθ ]−(σ 2 /2)(V 2 pθ +pθ V 2 −V pθ V )+iσ[V, pθ ]dB(t))U (t) = U (t)∗ (−mglsin(θ) − (σ 2 /2)(V [V, pθ ] + [pθ , V ]V ) + iσ[V, pθ ]dB(t))U (t)

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= −mglsin(θ(t)) − σV ' (θ(t))dB(t) Thus, we get the quantum analogue of the classical sde. However, to be more precise and obtain further generalizations, we must take our noise processes as the creation, annihilation and conservation processes of the Hudson-Parthasarathy quantum stochastic calculus. HP calculus generalizations: Consider the qsde dU (t) = (−(iH0 + P )dt + L1 dA(t) − L2 dA(t)∗ + SdΛ(t))U (t) where L1 , L2 , P, S are system operators chosen to make U (t) unitary for all t. Here, H0 = −∇2 /2m + U (r) acts in the system Hilbert space h = L2 (R3 ). Let X be a system observable. At time t it evolves under HP noisy Heisenberg dynamics to X(t) = U (t)∗ XU (t) X(t) is defined on h⊗Γs (L2 (R+ )) and A(t), A∗ (t), Λ(t) are the HP noise operator processes satisfying the quantum Ito formula dA(t)dA(t)∗ = dt, dA∗ dA = 0, (dA)2 = 0, (dA∗ )2 = 0, (dΛ)2 = dΛ, dA.dΛ = dA, dΛ.dA∗ = dA∗ , dΛ.dA = 0, dA∗ dΛ = 0 Remark: Formally, we can write dΛ = dA∗ .dA/dt and hence using dA.dA∗ = dt, we get dΛ.dA∗ = dA∗ , dA.dΛ = dA We get dX(t) = dU (t)∗ XU (t) + U (t)∗ XdU (t) + dU (t)∗ XdU (t) = U (t)∗ (i[H0 , X]dt − (P X + XP )dt + (L∗1 X − XL2 + S ∗ XL∗1 )dA∗ +(−L∗2 X + XL∗1 − L∗2 XS)dA + (S ∗ X + XS ∗ + S ∗ XS)dΛ)U (t) Exercise: Evaluate the Heisenberg equations of motion taking X = qk , X = pk , k = 1, 2, 3 where r = (q1 , q2 , q3 ), p = (p1 , p2 , p3 ). Assume that the system operators L1 , L2 , S, P are arbitrary functions of q, p subject to the constraint that makes U (t) unitary for all t. Specialize to the case when L1 , L2 , S, P are functions of q alone. A.37. The magnetic field produced by a transmission line current when hysteresis effects are taken into account. The line equations are ∫ V ' (ω, z) + Z(ω, z)I(ω, z) + δ H1 (ω1 , ω − ω1 , z)I(ω1 , z)I(ω − ω1 , z)dω1 = 0

520

General Relativity and Cosmology with Engineering Applications ∫

I ' (ω, z) + Y (ω, z)V (ω, z) + δ

H2 (ω1 , ω − ω1 , z)V (ω1 , z)V (ω − ω1 , z)dω1 = 0

Here, Z(ω, z) = R(z) + jωL(z), Y (ω, z) = G(z) + jωC(z) We can expand in a Fourier series ∑ Zn (ω)exp(jnβz), β = 2π/d Z(ω, z) = n∈Z

Y (ω, z) = V (0) (ω, z) =

∑

∑

Yn (ω)exp(jnβz)

n

Vn(0) (ω)exp((γ + jnβ)z),

n

I

(0)

(ω, z) =

∑

In(0) (ω)exp((γ + jnβ)z)

n

V (ω, z) = V

(0)

(ω, z) + δV (1) (ω, z) + O(δ 2 )

I(ω, z) = I (0) (ω, z) + δI (1) (ω, z) + O(δ 2 ) Then, substituting these expressions into the above line differential equations and equating coefficients of δ 0 and δ 1 respectively gives us ∑ (0) (γ + jnβ)Vn(0) (ω) + Zn−m (ω)Im (ω) = 0, m

(γ + jnβ)In(0) (ω) +

∑

Yn−m (ω)Vm(0) (ω) = 0

m

∂V

(1)

(ω, z) + Z(ω, z)I (1) (ω, z) + H1 .(I (0 ⊗ I (0) )(ω, z) = 0, ∂z

∂I (1) (ω, z) + Y (ω, z)I (1) (ω, z) + H2 (V (0) ⊗ V (0) )(ω, z) = 0 ∂z Let Φ(ω, z, z ' ) ∈ C2×2 denote the state transition matrix corresponding to the forcing matrix ( ) 0 −Z(ω, z) A(ω, z) = −Y (ω, z) 0 Then, the solutions to the first order line voltage and current perturbations are given by ∫ z Φ11 (ω, z, z ' )H1 .(I (0 ⊗ I (0) )(ω, z ' )dz ' V (1) (ω, z) = − 0

∫ − I

(1)

z 0

Φ12 (ω, z, z ' )H2 1.(V (0 ⊗ V (0) )(ω, z ' )dz '

(ω, z) = −

∫

z 0

Φ21 (ω, z, z ' )H1 .(I (0) ⊗ I (0) )(ω, z ' )dz '

General Relativity and Cosmology with Engineering Applications ∫ −

z 0

521

Φ22 (ω, z, z ' )H2 .(V (0) ⊗ V (0) )(ω, z ' )dz '

Denoting the infinite dimensional matrix ((Zn−m (ω))) by Z(ω), ((Yn−m (ω))) by Y(ω), the diagonal matrix diag[n : n ∈ Z] by D, the unperturbed voltage (0) (0) and current Fourier coefficient vectors ((Vn (ω))) and ((In (ω))) by V(0) (ω) (0) and I (ω) respectively, the above unperturbed eigenvalue equations can be expressed as T(ω)u(ω) = −γ(ω)u(ω) − − − (1) (

where

jβD Z(ω) T(ω) = Y(ω) jβD ( (0) ) V (ω) u(ω) = I(0) (ω)

) ,

Let −γn (ω), un (ω), n = 1, 2, ... be the complete set of eigenvalues and corre­ sponding eigenvectors of T(ω) as in (1). Then we can write ∑ u(ω) = cn (ω)un (ω) n∈Z

and we have on defining the infinite dimensional column vector en (ω, z) = ((exp((γn (ω) + jkβ)z)))k∈Z , the following expression for the unperturbed line voltage and current: ∑ V (0) (ω, z) = cn (omega)en (ω, z)T vn (ω), n

I (0) (ω, z) =

∑

cn (ω)en (ω, z)T wn (ω)

n

(

where un (ω) =

vn (ω) wn (ω)

)

The far field magnetic vector potential produced by this unperturbed line cur­ rent is given by ∫ d I (0) (ω, ξ)exp(jKξ.cos(θ))dξ Az (ω, r) = (μ.exp(−jKr)/r) 0

= (μ.exp(−jKr)/r)P (ω, θ) where K = ω/c and ∫

d

P (ω, θ) = 0

I (0) (ω, ξ)exp(jkξ.cos(θ))dξ

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A.38. Problems in quantum mechanics with solutions: [1] Obtain the supersymmetry generators in the form of super vector fields for four Boson and four Fermionic variables (xμ ), (θμ ). hint: Consider the generators of the form Dα = (Aμ θ)α ∂/partialxμ + Bαβ ∂/∂θβ and ¯ α = (C μ θ)α ∂/partialxμ + F β ∂/∂θα D α The matrices Aμ , C μ and the complex numbers Bαβ and Fαbeta are to be chosen so that the supersymmetric commutation relations hold in the form: ¯ β ] = K μ ∂/∂xμ [Dα , D αβ [2] Consider a periodic potential V (r), r ∈ R3 with three linearly independent period vectors, a1 , a2 , a3 ∈ R3 so that V (r + n1 a1 + n2 a2 + n3 a3 ) = V (r), n1 , n2 , n3 ∈ Z Expand V as a 3-D Fourier series using the reciprocal lattice vectors and hence formulate the stationary state Schrodinger equation −∇2 ψ(r)/2m + V (r)ψ(r) = Eψ(r), r ∈ R3 so that in view of the periodicity of V , we have ψ(r + n1 a1 + n2 a2 + n3 a3 ) = C1n1 C2n2 C3n3 ψ(r), n1 , n2 , n3 ∈ Z where |Ck | = 1. If there are Nk atoms along ak k − 1, 2, 3, then we may apply the periodic boundary conditions on ψ without loss of generality in the form ψ(r + Nk ak ) = ψ(r), k = 1, 2, 3 This leads to CkNk = 1, k = −1, 2, 3 so that Ck = exp(2πilk /Nk ), k = 1, 2, 3 for some lk ∈ {0, 1, ..., Nk − 1}. Now define the Bloch wave functions ul1 l2 l3 (r) by ψ(r) = exp(2πi(l1 b1 /N1 + l2 b2 /N2 + l3 b3 /N3 , r))ul1 l2 l3 (r) where {b1 , b2 , b3 } are the reciprocal lattice vectors corresponding to {a1 , a2 , a3 }. In other words, (bi , aj ) = δij Then, we have ul1 l2 l3 (r + m1 a1 + m2 a2 + m3 a3 ) = ul1 l2 l3 (r), m1 m2 , m3 ∈ Z

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ie, ul1 l2 l3 is also periodic with periods a1 , a2 , a3 like V (r) and hence it can also be developed into a Fourier series: ∑ Ul1 l2 l3 [n1 n2 1n3 ]exp(2πi(n1 b1 + n2 b2 + n3 b3 , r)) ul1 l2 l3 (r) = n1 n2 n3 ∈Z

V (r) =

∑

V [n1 n2 n3 ]exp(2πi(n1 b1 + n2 b2 + n3 b3 , r))

n1 n2 n3 ∈Z

The problem is to subsitute these two Fourier series expansions for the wave function and the potential into the stationary state Schrodinger wave equation and derive a difference equation for Ul1 l2 l3 [n1 n2 n3 ]. [3] Quantum control of the Hudson-Parthasarathy equation based on the Belavkin filter observer. HP equn: dU (t) = (−(iH + P )dt + L1 dA + L2 dA∗ + SdΛ)U (t) For unitarity of U (t), we require 0 = d(U ∗ U ) = dU ∗ .U + U ∗ .dU + dU ∗ .dU This gives using the quantum Ito formula dAdA∗ = dt, dΛ.dA∗ = dA∗ , dA.dΛ = dA, (dΛ)2 = dΛ, (all the other products of differentials are zero), P = L∗2 L2 /2, L∗2 + L1 + L∗2 S = 0, L∗1 + L2 + S ∗ L2 = 0, S + S∗ + S∗S = 0 L1 , L2 , S, H, P are system operators with H, P self-adjoint. We have (I + S)∗ (I + S) = I + S ∗ + S + S ∗ S = I so we can write I + S = exp(iλ(t)Z) where Z is selfadjoint and λ(t) is a real valued controllable function of time. Then, we have L1 = −L∗2 (I + S) = −L∗2 exp(iλ(t)Z) We write L2 = L =

p ∑ k=1

ck (t)Nk = L({ck (t)})

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General Relativity and Cosmology with Engineering Applications Then L1 = −(

∑

c¯k (t)Nk∗ ))exp(iλ(t)Z) = L1 ({ck (t)}, λ(t)),

k

S = exp(iλ(t)Z) − 1 = S(λ(t)) ck (t) are complex valued controllable functions of time. The real time control algorithms for controlling the ck (t)' s and λk (t) are based on minimizing the expected value of the Belavkin observer error E(t) = Xd (t) − πt (X) where Xd (t) is the desired system state at time t. This is a system observable and πt (X) is a measurable function of the noise algebra σ(Y )t upto time t. The noise is taken as Y (t) = U (t)∗ Yi (t)U (t), Yi (t) = a1 B(t) + a2 Λ(t), B(t) = A(t) + A(t)∗ , a1 , a2 ∈ R It is easily verified that Y satisfies the non-demolition Abelian property, ie, [Y (t), Y (s)] = 0∀t, s ≥ 0 and [Y (s), jt (X)] = 0, t ≥ s, jt (X) = U (t)∗ XU (t) = U (t)∗ (X ⊗ I)U (t) We have

Y (t) = jt (Yi (t)) = U (t)∗ Yi (t)U (t), dY (t) = dYi (t) + dU (t)∗ dYi (t).U (t) + U (t)∗ dYi (t)dU (t) =

dYi (t) + U (t)∗ (L∗1 dA∗ + L∗2 dA + S ∗ dΛ)dYi + dYi (L1 dA + L2 dA∗ + SdΛ))U (t) Now,

dYi (L1 dA + L2 dA∗ + SdΛ) = (a1 (dA + dA∗ ) + a2 dΛ)(L1 dA + L2 dA∗ + SdΛ) = a1 L2 dt + a2 SdΛ + a2 L2 dA∗ + a1 SdA

We thus get dY (t) = dYi (t)+jt (a1 (L2 +L∗2 ))dt+jt (a2 (S+S ∗ ))dΛ+jt (a2 L2 +a1 S ∗ )dA∗ +jt (a2 L∗2 +a1 S)dA write πt (X) = E(jt (X)|σ(Y )t ) where the expectation is taken in the state |f φ(u) > with f ∈ h, the system Hilbert space and |φ(u) >= exp(− || u ||2 /2)|e(u) > the normalized exponential vector in the Boson Fock space Γs (L2 (R+ )). We assume that the optimal filter is described by the following qsde: dπt (X) = Ft (X)dt + Gt (X)dY (t)

General Relativity and Cosmology with Engineering Applications

525

where Ft (X), Gt (X) ∈ σ(Y )t . Then applying the orthogonality principle E[(jt (X) − πt (X))C(t)] = 0 for C(t) defined via the qsde dC(t) = f (t)C(t)dY (t), C(0) = 1 and using the arbitrariness of f (t) gives us E[(djt (X) − dπt (X))|σ(Y )t ] = 0, E[(djt (X) − dπt (X))dY (t)|σ(Y )t ] + E[(jt (X) − πt (X))dY (t)|σ(Y )t ] = 0 These two equations give us two equations for the observables Ft (X) and Gt (X), solving which the Belavkin filter is obtained. To derive these two equations, we compute using quantum Ito’s formula, djt (X) = dU ∗ XU + U ∗ XdU + dU ∗ XdU = = U ∗ (i[H, X] − P X − XP + L∗2 XL2 )dt+ (L∗2 X+XL1 +L∗2 XS)dA+(L∗1 X+XL2 +S ∗ XL2 )dA∗ +(S ∗ X+XS+S ∗ XS)dΛ)U = jt (θ0 (X))dt + jt (θ1 (X))dAt + jt (θ2 (X))dA∗t + jt (θ3 (X))dΛt where

θ0 (X) = i[H, X] − L∗ LX/2 − XL∗ L/2 + L∗ XL∗ θ1 (X) = L∗ X − XL∗ (I + S) + L∗ XS, θ2 (X) = −(1 + S)∗ LX + XL − S ∗ XL, θ3 (X) = S ∗ X + XS + S ∗ XS

Note that L2 = L = L({ck (t)}), S = S(λ(t)) and

L1 = −L∗ (1 + S) = L1 ({ck (t), λ(t))

The equation E[(djt (X) − dπt (X))|σ(Y )t ] = 0, thus gives u(t) + πt (θ3 (X))|u(t)|2 πt (θ0 (x)) + πt (θ1 (X))u(t) + πt (θ2 (x))¯ −Gt (X)[a1 (u(t) + u ¯(t)) + a2 |u(t)|2 + πt (a1 (L2 + L∗2 )) + πt (a2 (S + S ∗ ))|u(t)|2 + u(t) + πt (a2 L∗2 + a1 S)u(t)] = 0 πt (a2 L2 + a1 S ∗ )¯ or equivalently, Ft (X) =

526

General Relativity and Cosmology with Engineering Applications πt (θ0 (x)) + πt (θ1 (X))u(t) + πt (θ2 (x))¯ u(t) + πt (θ3 (X))|u(t)|2 −Gt (X)[a1 (u(t) + u ¯(t)) + πt (a1 (L2 + L∗2 )) + πt (a2 (S + S ∗ ))|u(t)|2 +pit (a2 L2 + a1 S ∗ )¯ u(t) + πt (a2 L∗2 + a1 S)u(t)]

The equation E[(djt (X) − dπt (X))dY (t)|σ(Y )t ] + E[(jt (X) − πt (X))dY (t)|σ(Y )t ] = 0 gives u(t) + πt (θ3 (X))|u(t)|2 πt (θ0 (X)) + πt (θ1 (X))u(t) + πt (θ2 (X))¯ −Ft (X) − Gt (X)E[(dY (t))2 |σ(Y )t ] + πt (X)(a1 (u(t) + u ¯(t)) + a2 |u(t)|2 + πt (a1 (L2 + L∗2 )) + πt (a2 (S + S ∗ ))|u(t)|2 +πt (a2 L2 + a1 S ∗ )¯ u(t) + πt (a2 L2 + a1 S)u(t) = 0 Now, from the equation dY (t) = dYi (t)+jt (a1 (L2 +L∗2 ))dt+jt (a2 (S+S ∗ ))dΛ+jt (a2 L2 +a1 S ∗ )dA∗ +jt (a2 L∗2 +a1 S)dA = jt (a1 (L2 +L∗2 ))dt+jt (a2 (S+S ∗ +1))dΛ+jt (a2 L2 +a1 (S ∗ +1))dA∗ +jt (a2 L∗2 +a1 (S+1))dA we get using quantum Ito’s formula and the homomorphism property of jt , dt−1 E[(dY (t))2 |σ(Y )t ] = πt (a22 (S + S ∗ + 1)2 )|u(t)|2 + πt ((a2 L∗2 + a1 (S + 1))(a2 L2 + a1 (S ∗ + 1)))+ u(t)+πt (a2 (a2 L∗2 +a1 (S+1))(S+S ∗ +1))u(t) πt ((a2 (S+S ∗ +1)(a2 L2 +a1 (S ∗ +1)))¯ [4](Reference: S.Wienberg, The quantum theory of fields, vol.II, Cambridge University Press)Evaluate approximately the path integral ∫ ∫ Z(J) = exp(iI(φ) − i J(x)φ(x)d4 x)Dφ where J(x) is an external current source and ∫ I(φ) = (∂μ φ)(∂ μ φ)/2 − m2 φ2 /2 − εV (φ)d4 x ie, the Klein-Gordon action functional with a small perturbative correction. Calculate using this expression the propagators ∫ exp(iI(φ)φ(x1 )...φ(xn )Dφ/Z(0)

General Relativity and Cosmology with Engineering Applications by using the formula ∫ = in

527

∂ n bZ(J) |J=0 ∂J(x1 )...∂J(xn ) exp(iI(φ))φ(x1 )...φ(xn )dφ

∫ Now derive the equation for J at which log(Z(J)) − J(x)φ0 (x)d4 x becomes stationary for a fixed field φ0 . Denoting this stationary solution by J0 (x), show that −φ0 (x) + δlogZ(J0 )/δJ(x) = 0 or equivalently, φ0 (x) + iZ(J0 )

−1

∫

∫ exp(i

(I(φ) − J0 (x)φ(x))d4 x)φ(x).Dφ = 0

We write J0 (x) = J0 (φ0 )(x) or inverting this, φ0 (x) = φ0 (J0 )(x) It follows that (δφ0 (x)/δJ0 (y) = δ 2 log(Z(J0 ))/δJ(x)δJ(y) ∫

We define Γ(φ0 ) = log(Z(J0 )) −

J0 (x)φ0 (x)d4 x

Thus, ∫ δΓ(φ0 )/δφ0 (y) =

∫ φ0 (x)(δJ0 (x)/δφ0 (y))d x− (δJ0 (x)/δφ0 (y))φ0 (x)d4 x−J0 (y) 4

= −J0 (y) Γ(φ0 ) is called the quantum effective action and the above equation is called the equation of motion for the quantum effective ation. [5] Calculation of path integrals for gauge invariant theories: Let φ(x) be the set of fields and f [φ] a gauge fixing functional. We consider a path integral of the form ∫ X = G[φ]B[f [φ]]F [φ]Dφ where F [φ] is the Jacobian determinant: F [φ] = det(df [φΛ ]/dΛ)|Λ=id where φ → φΛ is the gauge transformed field. Λ is the gauge transorming func­ tion. We wish to show that in a certain sense, X does not depend on the gauge

528

General Relativity and Cosmology with Engineering Applications

fixing functional f . We assume that the combined action functional G[φ] along with the measure Dφ, ie, G[φ]Dφ is invariant under the gauge transformation, ie, for all gauge transformations Λ, we have G[φΛ ]DφΛ = G[φ]Dφ Then, it follows that

∫ G[φΛ ]B[f [φΛ ]]F [φΛ ]DφΛ

X= ∫ =

G[φ]B[f [φΛ ]]F [φΛ ]Dφ

Now, if Λ and λ are two gauge transformations, then det(df [φΛoλ ]/dλ)|λ=id = F [φΛ ]/ρ(Λ) where

ρ(Λ)−1 = det(d(Λoλ)/dλ)λ=id

So,

∫ X=

G[φ]B[f [φΛ ]]ρ(Λ)det(df [φΛ ]/dΛ)Dφ

Integrating this equation w.r.t Λ over the gauge group gives us after appropri­ ately normalizing the Haar measure, ∫ ∫ X = G[φ].B[f ]]df.Dφ = C G[φ]Dφ where C is a constant defined by ∫ ∫ C = B[f ]Df = B[f [φΛ ]]det(df [φΛ ]/dΛ)ρ(Λ)dΛ where we use the fact that ρ(Λ) as defined above is the left invariant Haar density on the gauge group and also the assumption that the gauge group acts transitively on the matter fields φ. [6] Quantum teleportation: The simplest version of this idea involves trans­ mitting one qubit of information by transmitting just two classical bits of in­ formation. Alice and Bob share an entangled state |00 > +|11 > apart from normalization factor of 2−1/2 . Alice prepares another state |ψ >= c1 |1 > +c2 |0 > where c1 , c2 ∈ C and |c1 |2 + |c2 |2 = 1 which she wishes to transmit to Bob by making use of the entangled state that she shares with Bob. The overall state of Alice and Bob is thus |ψ > (|00 > +|11 >) = (c1 |1 > +c2 |0 >)(|11 > +|00 >) =

General Relativity and Cosmology with Engineering Applications

529

c1 |111 > +c1 |100 > +c2 |011 > +c2 |000 > The first two qubits of this state can be controlled only by Alice and the last only by Bob. Alice applies a phase gate to her two qubits thus obtaining the overall state as |φ >= c1 |111 > +c1 |100 > +c2 |011 > −c2 |000 > We express this state in terms of the orthogonal one qubit state |+ >= |1 > +|0 >, |− >= |1 > −|0 > or equivalently, |1 >= |+ > +|− >, |0 >= |+ > −|− > Thus the overall state of Alice and Bob is given by |φ >= c1 (|+ > +|− >)(|+ > +|− >)|1 > +c1 (|+ > +|− >)(|+ > −|− >)|0 > +c2 (|+ > −|− >)(|+ > +|− >)|1 > −c2 (|+ > −|− >)(|+ > −|− >)|0 > Alice now performs a measurement on her two qubits in this shared state using the orthonormal basis | + + >, | + − >, | − + >, | − − >. If she measures | + + >, then clearly from the above expression for |φ >, Bob’s state collapses to c1 |1 > +c1 |0 > +c2 |1 > −c2 |0 >= c1 (|1 > +|0 >) + c2 (|1 > −|0 >) If Alice measures | + − >, then Bob’s state collapses to c1 |1 > −c1 |0 > +c2 |1 > +c2 |0 >= c1 (|1 > −|0 >) + c2 (|1 > +|0 >) If Alice measures | − + >, then Bob’s state collapses to c1 |1 > +c1 |0 > −c2 |1 > +c2 |0 >= c1 (|1 > +|0 >) + c2 (|0 > −|1 >) Finally, if Alice measures | − − >, then Bob’s state collapses to c1 |1 > −c1 |0 > −c2 |1 > −c2 |0 >= c1 (|1 > −|0 >) − c2 (|1 > +|0 >) Using two classical bits, Alice reports to Bob the outcome of her measurements. If she reports | + + >, then Bob applies the unitarty gate U1 to his state defined by √ √ U1 (|1 > +|0 >) 2 = |1 >, U1 (|1 > −|0 >)/ 2 = |0 >, to recover |ψ >. If Alice reports | + − >, then Bob applies the unitary gate U2 to his state defined by √ √ U2 (|1 > −|0 >)/ 2 = |1 >, U2 (|1 > +|0 >)/ 2 = |0 > to recover |ψ >. If Alice reports | − + >, then Bob applies the unitary gate U3 to his state defined by √ √ U3 (|1 > +|0 >)/ 2 = |1 >, U3 (|1 > −|0 >)/ 2 = −|0 >

530

General Relativity and Cosmology with Engineering Applications

to recover |ψ >. Finally, if Alice reports | − − >, then Bob applies the unitary gate U4 to his state defined by √ √ U4 (|1 > −|0 >)/ 2 = |1 >, U4 (|1 > +|0 >)/ 2 = −|0 > to recover the state |ψ >. [7] Quantum Boltzmann equation. N identical particles. ρ(t) is the joint density matrix of the particles. The k th particle acts in the Hilbert space Hk . The Hilbert space of the whole system is H=

N ⊗

Hk

k=1

Each Hk is an identical copy of a fixed Hilbert space H0 . Schrodinger-Von-Neumann-Liouville equation

ρ satisfies the

iρ' (t) = [H, ρ(t)] where H=

N ∑

Hk +

k=1

∑

Vkj

1≤k< i1 , ..., in | where |i1 , ..., , in > are eigenvectors of ρ¯⊗n obtained by tensoring the orthonor­ mal basis of eigenvectors of ρ¯. Then, we have ∑ ˜ (n, ui , δ)) = ˜ (n, ui , δ)π) T r(ρ(ui )πE T r(ρ(ui )E π

=

∑

˜ (n, ui , δ)) T r(πρ(ui )πE

π

˜ (n, ui , δ)) = T r(˜ ρ(ui )E ≥ 1 − N/δ 2 Thus,

˜ (n, ui , δ)) ≤ N/δ 2 T r(ρ(ui )E

and also

T r(ρ(ui )E(n, ui , δ)) ≤ N/δ 2

Thus, we get T r(Di' ) ≥ 2n = 2n

∑

∑ x

x

√ P (x)S(ρ(x))−K6 δ n

√ P (x)S(ρ(x))−K6 δ n

(1 − ε − βN/δ 2 − N/δ 2 )

(1 − ε − (β + 1)N/δ 2 )

563

General Relativity and Cosmology with Engineering Applications It follows that T r(

M ∑

Di' ) ≥ M.2n

∑

√ P (x)S(ρ(x))−K6 δ n

x

(1 − ε − (β + 1)N/δ 2 )

i=1

on the one hand and on the other, T r(

M ∑

Di' ) =

i=1

M ∑

˜ (n, ui , δ)Di E ˜ (n, ui , δ)) T r(E

i=1

We have already noted that if P = Pu , then √ ˜ (n, u, δ) ≤ E(¯ E ρ⊗n , δ a) so if we assume that Pui = P ∀i, then we get T r(

M ∑

√ √ Di' ) ≤ T r(E(¯ ρ⊗n , δ a)DE(¯ ρ⊗n , δ a)

i=1

√ ≤ T r(E(¯ ρ⊗n , δ a)) ¯ 7δ ≤ 2nS(ρ)+K

√

an

and hence we get ¯ 7 2nS(ρ)+K

√

an

≥ M.2n

∑ x

√ P (x)S(ρ(x))−K6 δ n

(1 − ε − (β + 1)N/δ 2 )

from which we get the upper bound on M in the special case when Pui = P ∀i: ¯ M ≤ (1 − ε − (β + 1)N/δ 2 )−1 2n(S(ρ)−

∑ x

P (x)S(ρ(x)))+(K6 +K7

√

√ a)δ n

Suppose now that u1 , ..., uM , D1 , ..., DM are as in the greedy algorithm. Let Q be any empirical probability distribution on A corresponding to the integer n. This means that there is a sequence u ∈ An such that Q(x) = N (x|u)/n, x ∈ A. It is clear that the number of empirical probability distributions on A corresponding to n cannot exceed (n + 1)a . (Each symbol x ∈ A can occur in a sequence of length n, k times where k = 0, 1, ..., n). We now consider the subset FQ of all integers 1, 2, ..., M such that ui is of empirical type Q, ie, N (x|ui )/n = Q(x), ∀x ∈ A. Let MQ denote the cardinality of FQ . Then it is clear that ∑ M= MQ Q

where the summation is over all empirical distributions Q, there being atmost (n + 1)a of them. By the above∑inequality, we have since T r(ρ(ui )Di ) ≥ 1 − ε, Di ≤ E(n, ui , δ), ∀i ∈ FQ and i∈FQ Di ≤ I, MQ ≤ (1 − ε − (β + 1)N/δ 2 )−1 2n(S(ρ¯Q )−

∑ x

Q(x)S(ρ(x)))+(K6 +K7

√

√ a)δ n

564

General Relativity and Cosmology with Engineering Applications ≤ 1 − ε − (β + 1)N/δ 2 )−1 2nC+(K6 +K7

where

∑

ρQ =

√

√ a)δ n

Q(x)ρ(x)

x

and C = maxP (S(

∑

P (x)ρ(x)) −

x

∑

P (x)S(ρ(x)))

x

the maximum being taken over all probability distributions P on A. Summing this over all empirical distributions Q of length n on A gives us ∑ √ √ M= MQ ≤ (n + 1)a .(1 − ε − (β + 1)N/δ 2 )−1 2nC+(K6 +K7 a)δ n Q

which is the desired upper bound on M . A remark: Now suppose we √ do not assume Pui = P ∀i. Since ui ∈ T (P, n, δ), we have |Pui (x) − P (x)| ≤ δ/ 2n∀i, x. ˜ (n, u, δ) in the same way as above except that we For any u, we define E replace P by Pu . Then, by the same arguments as above, we have √ n ˜ (n, u, δ) ≤ E(¯ E ρ⊗ u , δ a) where ρ¯u =

∑

Pu (x)ρ(x)

x∈A

˜ We proceed in the same way as above by defining Di' in terms of the new E leading to the results: ∑ ∑ ˜ (n, ui , δ)Di E ˜ (n, ui , δ) Di' = E D' = i

i

√

√ = E(¯ ρ⊗n , δ a)DE(¯ ρ⊗n , δ a)+ ∑ ∑ ¯ )Di (Ei − E ¯) + ¯ )Di E ¯) ¯ + ED ¯ i (Ei − E (Ei − E (Ei − E i

where

i

√ ˜ (n, ui , δ), E ¯ = E(¯ ρ⊗n , δ a) Ei = E

Thus, in terms of norms, ¯ ) + 2.maxi || Ei − E ¯ ||2 +maxi || Ei − E ||1 T r(D' ) ≤ T r(E 1 Note that ui ∈ T (P, n, δ). Now if u ∈ T (P, n, δ), then Pu is close to P , hence ˜ (n, u, δ) will be close E ¯ . Thus, Ei will be close ρ¯u will be close to ρ¯ and so E ' ¯ ) by an order of an ¯ to E and we would get that T r(D ) cannot exceed T r(E ¯ ||2 cannot exponential of n provided that we are able to show that || Ei − E

General Relativity and Cosmology with Engineering Applications

565

grow faster than an exponential of∑np where p < 1). √ Hence, the desired lower ¯ x P (x)S(ρ(x))+Kδ n would follow. To make bound of the form M ≤ 2n(S(ρ)− this more precise, we observe that u ∈ T (P, n, δ) implies |Pu⊗n (v) − P ⊗n (v)| = |Πx∈A Pu (x)N (x|v) − Πx∈A P (x)N (x|v) | ∑

= |e

x

=≤ |exp(

N (x|v)log(Pu (x))

∑

−e

∑ x

N (x|v)log(P (x))

|

N (x|v)(log(Pu (x)) − log(P (x))) − 1|

x

We can write ∑ √ Pu (x) = P (x) + f (x)/ n, x ∈ A, |f (x)| ≤ 1, f (x) = 0 x

Hence, |exp( |exp(

∑

∑

|Pu⊗n (v) − P ⊗n (v)| ≤

√ N (x|v)log(1 + f (x)/P (x) n)) − 1| =

x

√ N (x|v)(f (x)/P (x) n − f (x)2 /2P (x)2 n + ...)) − 1|

x

√ If v ∈ / T (P, n, δ), then N (x|v) < nP (x) + O( n) in which case we see that ∑ √ √ N (x|v)(f (x)/P (x) n − f (x)2 /2P (x)2 n + ...) < O(1/ n) x

∑

√ since f (x) = 0. If v ∈ T (P, n, δ), then N (x|v) = nP (x) + g(x) n where |g(x)| < 1. Hence, in this case, ∑ √ N (x|v)(f (x)/P (x) n−f (x)2 /2P (x)2 n+..) x

=

∑

√ (g(x)f (x)/P (x)−f (x)2 /2P (x))+O(1/ n)

x

√ = c0 + O(1/ n) Other useful inequalities are |log(Pu⊗n (v)) − log(P ⊗n (v))| = ∑ ∑ N (x|v)log(P (x))| = =| N (x|v)log(Pu (x)) − x

|

∑

x

√ N (x|v)(log(1 + f (x)/P (x) n)|

x

[20] Restricted quantum gravity in one spatial dimension and one time di­ mension. The metric is dτ 2 = (1 + 2U (t, x))dt2 − (1 + 2V (t, x))dx2

566

General Relativity and Cosmology with Engineering Applications

The position fields are U (t, x) and V (t, x) and to find the momentum fields, we must first evaluate the Lagrangian density √ β β α L = g μν −g(Γα μν Γαβ − Γμβ Γαβ ) This Lagrangian density is a function of U, V, U,μ , V,μ . Define the position fields as U, V and the canonical momentum fields as πU =

∂L ∂L , πV = ∂V,0 ∂U,0

Then, apply the Legendre transformation after solving for U,0 , V,0 in terms of U, V, ∇U, ∇V to get the Hamiltonian density as H(U, V, ∇U, ∇V, πU , πV ) = πU U,0 + πV V,0 − L Then, set up the Schrodinger wave equation ∫ ( H(U (x), V (x), ∇U (x), ∇V (x), −iδ/δU (x), −iδ/δV (x))dx)ψt ({U (x), V (x) : x ∈ R) =i

∂ ψt ({U (x), V (x) : x ∈ R) ∂t

[21] Shannon’s noisy coding theorem for classical channels. The input al­ phabet is X and the output alphabet is Y . The channel is described by the transition probability P r(y|x) = px (y), x ∈ X, y ∈ Y The input probability distribution (source) is p(x), x ∈ X and the output prob­ ability distribution is ∑ q(y) = p(x)px (y), y ∈ Y x∈X

For δ > 0 define V ⊂ X × Y by V = {(x, y) : ||log(px (y)/q(y)) − I(X, Y )| < δ} where I(X, Y ) = E[logpx (y)/q(y)] = −

∑ y

q(y)log(q(y))+

∑

p(x)px (y)log(px (y))

x,y

= H(Y )−H(Y |X) We have with ω denoting the probability distribution of (x, y) on X × Y , ie, ω(x, y) = p(x)px (y), x ∈ X, y ∈ Y

567

General Relativity and Cosmology with Engineering Applications by Chebyshev’s inequality, ω(V ) ≥ 1 − V ar(log(px (y)/q(y))/δ 2 = 1 − α/δ 2 where α = V ar(log(px (y)/q(y))) = V ar(ω(x, y)/p(x)q(y)) We also have by definition, 2I−δ q(y) ≤ px (y) ≤ 2I+δ q(y), x ∈ X, y ∈ Y and hence 2I−δ q(Vx ) ≤ px (Vx ) ≤ 2I+δ q(Vx ), x ∈ X where Vx = {y ∈ Y : (x, y) ∈ V }, x ∈ X

Now let ε > 0 and choose x1 ∈ X such that px1 (V1 ) > 1−ε where V1 = Vx1 . Then / x1 such that px2 (V2 ) > 1 − epsilon where V2 = Vx2 ∩ V1c . choose x2 ∈ X, x2 = In this way, choose distinct x1 , x2 , ..., xN such that pxk (Vk ) > 1 − ε, where ∪k−1 Vk = Vxk ∩ ( i=1 Vi )c , k = 1, 2, ..., N and N is maximal, ie, for any x ∈ X ∪N ∪N px ( k=1 Vk )c ) ≤ 1−ε. Note that for k = 1, 2, ...N , pxk (( i=1 Vi )c ) ≤ pxk (Vkc ) ≤ ε Note also that for any x ∈ X, 2−δ−I px (Vx ) ≤ q(Vx ) and also

2−δ−I px (Vk ) ≤ q(Vk ), k = 1, 2, ..., N

and hence choosing x = xk , we get 2−δ−I (1 − ε) ≤ q(Vk ), k = 1, 2, ..., N so that by the disjointness of the Vk' s, N ∐

N 2−δ−I (1 − ε) ≤ q(

Vk ) ≤ 1

k=1

ie, N ≤ 2δ+I /(1 − ε) Further, 1 − α/δ 2 ≤ ω(V ) =

∑

p(x)px (y) =

x,y

=

∑

N ∐

p(x)px (Vx ∩

x

∑

Vk ) +

x

p(x)px (

p(x)px (Vx )

x

∑

p(x)px (Vx ∩ (

x

k=1

≤

∑

N ∐ k=1

Vk ) + 1 − ε

N ∐ k=1

Vk ) c )

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General Relativity and Cosmology with Engineering Applications

so that ε − α/δ 2 ≤ q(

N ∐

Vk )

k=1

=

N ∑

q(Vk )

k=1

Now, the equation 2I−δ q(Vk ) ≤ px (Vk ) ≤ 1 gives us and so,

ε − α/δ 2 ≤ N.2−I+δ N ≥ 2I−δ (ε − α/δ 2 )

This result is known as the ”Feinstein-Khinchin fundamental lemma. If we apply this to a direct product of n channels, we get (ε − α/nδ 2 )2n(I−δ) ≤ Nn and hence liminfn→∞ log(Nn )/n ≥ I = I(X, Y ) and likewise from one of the above inequalities, Nn ≤ 2n(I+δ) /(1 − ε) so that limsupn→∞ log(Nn )/n ≤ I = I(X, Y ) Hence, we arrive at the relation that the maximum rate of information trans­ mission for recovery with arbitrary small error probability is given by limn→∞ log(Nn )/n = I(X, Y ) This result is known as ”Shannon’s noisy coding theorem”. [22] General relativistic version of the Vlasov equations. Let f (t, r, v) denote the Boltzmann distribution function for one species of charge with a charge of q Coulombs per particle. Here, v = (v r , r = 1, 2, 3) are the spatial components of the four velocity vector. We have gμν v μ v ν = g00 v 02 + 2g0r v 0 v r + grs v r v s = 1 and so v 0 can be solved for. We can express the equation of motion of a single particle taking into account electromagnetic interactions can be expressed as dv r /dτ = (q/m)(F0r v 0 + Fsr v s ) − Γr00 v 02 − 2Γr0s v 0 v s − Γrkm v k v m

General Relativity and Cosmology with Engineering Applications

569

which can be rewritten as dv r /dt = (q/m)(dτ /dt)(F0r v 0 + Fsr v s ) − (dτ /dt)(Γr00 v 02 + 2Γr0s v 0 v s + Γrkm v k v m ) where so that

dτ 2 = g00 dt2 + 2g0r dtdxr + grs dxr dxs g00 (dt/dτ )2 + 2g0r v r (dt/dτ ) + grs v r v s − 1 = 0

This is a quadratic equation for dt/dτ and can be solved to give √ 2 = γ(x, v), x = (t, r) dt/dτ = −g0r v r /g00 + γrs v r v s + 1/g00 and its reciprocal gives dτ /dt in terms of gμν (x) and v r , r = 1, 2, 3. Thus, the above equation of motion of a charged particle can be expressed as r r (x)v 02 +2Γr0s (x)v 0 v s +Γkm (x)v k v m ) dv r /dt = (q/mγ)(F0r (x)+Fsr (x)v s )−γ −1 (Γ00

= F r (t, r, v)

say, where

x = (t, r), r = (x1 , x2 , x3 )

Thus, our kinetic transport equation becomes f,t (t, r, v) + (v, ∇r )f (t, r, v) + (F (t, r, v), ∇v )f (t, r, v) = (f,t )coll The energy momentum tensor of the plasma has components T 00 = (ρ + p) < v 02 > −pg 00 T 0r = (ρ + p)v 0 v r − pg 0r , where

T rs = (ρ + p) < v r v s > −pg rs ∫ ∫ 3 r s ρ = f (t, r, v)d v, < v v >= v r v s f (t, r, v)d3 v/ρ, ∫ p=

f (t, r, v)

3 ∑

(v m − < v m >)2 d3 v

m=1

where ρ < v 02 >=

∫

v 02 f (t, r, v)d3 v,

In short, T μν (t, r) is a linear functional of the Boltzmann distribution function f . We can then formulate the Einstein field equations as Rμν − (1/2)Rg μν = −8πGT μν

570

General Relativity and Cosmology with Engineering Applications

We however need to verify for consistency that T:νμν = 0 This should in fact appear as a consequence of the Boltzmann kinetic transport equation. If not, then the kinetic transport equation should be modified such that this is valid. [23] Quasi-classical quantum mechanics in a curved space-time based on the KG equation. The KG equation reads [pμ + eAμ )g μν (pν + eAν ) + m2 ]ψ = 0 where pμ = i∂μ Equivalently, μν g,μ (−ψ,ν + ieAν ψ) + g μν (−ψ,μν + ieAν,μ ψ + ieAν ψ,μ +

ieAμ ψ,ν + e2 Aμ Aν ψ) + m2 ψ = 0 We substitute ψ(t, r) = F (t, r)exp(2πiS(t, r)/h) into this equation and equate real and imaginary parts. Observe that ∂μ ψ = (F,μ + 2πiF S,μ /h)exp(2πiS/h), ∂ν ∂μ ψ = (F,μν + 2πi(F,μ S,ν + F,ν S,μ )/h + 2πiF S,μν /h)exp(2πiS/h) [24] Schumacher’s noiseless quantum coding theorem or quantum compres­ sion theorem. Let ρ be a state in a finite dimensional Hilbert space H and write its spectral decomposition as ∑ p(a)|a >< a| ρ= a∈A

where μ(A) = dimH = N < ∞. Here, μ(E) denotes the cardinality of a set E. Define for δ > 0 and n = 1, 2, 3, ... T (n, p, δ) = {(a1 , ..., an ) : |n−1 log(p(a1 )...p(an )) + S(ρ)| < δ} = {x ∈ An : |n−1 p⊗n (x) + S(ρ)| < δ} Here, S(ρ) = −

∑ a∈A

p(a)log(p(a)) = −T r(ρ.log(ρ))

General Relativity and Cosmology with Engineering Applications We have

571

2−n(S+δ) ≤ p⊗n (x) ≤ 2−n(S−δ) , x ∈ T (n, p, δ)

and hence, 2−n(S+δ) μ(T (n, p, δ)) ≤ p⊗n (T (n, p, δ)) ≤ 2−n(S−δ) μ(T (n, p, δ)) from which, it follows in particular that μ(T (n, p, δ)) ≤ 2n(S+δ) We also have from Chebyshev’s inequality, p⊗n (T (n, p, δ)) ≥ 1 − V arlogp/nδ 2 = 1 − α/nδ 2 We also have

μ(T (n, p, δ)) ≥ 2n(S−δ) p⊗n (T (n, p, δ)) ≥ 2n(S−δ) (1 − α/nδ 2 )

from which we get S−δ ≤ liminfn→∞ log(μ(T (n, p, δ))/n ≤ limsupn→∞ log(μ(T (n, p, δ))/n ≤ S+δ and simultaneously

limn→∞ p⊗n (T (n, p, δ)) = 1

This is the Shannon noiseless coding theorem (classical probability). Now we prove the quantum version of this result due to Schumacher. Define the orthog­ onal projection ∑ E(n, ρ, δ) = |x >< x| x∈T (n,p,δ)

in H⊗n . We have ρ⊗n E(n, ρ, δ) =

∑

p⊗n (x)|x >< x|

x∈T (n,p,δ)

and hence 2−n(S+δ) E(n, ρ, δ) ≤ ρ⊗n E(n, ρ, δ) ≤ 2−n(S−δ) E(n, ρ, δ) Define for given ε > 0, ν(n, ρ, ε) = min{T r(F ) : T r(ρ⊗n F ) > 1 − ε} where F varies over all orthogonal projections in H⊗n . Then, since T r(E(n, ρ, δ)) = μ(T (n, p, δ)) ≥ 1 − α/nδ 2 So for all sufficiently large n, T r(E(n, ρ, δ)) ≥ 1 − ε

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General Relativity and Cosmology with Engineering Applications

and hence for all sufficiently large n. ν(n, ρ, ε) ≤ T r(E(n, ρ, delta)) Now have T r(E(n, ρ, δ)) ≤ 2n(S+δ) p(T (n, p, δ)) ≤ 2n(S+δ) so we get for all sufficiently large n n−1 log(ν(n, ρ, ε)) ≤ S + δ and thus, limsupn→∞ n−1 log(ν(n, ρ, ε)) ≤ S + δ from which it follows that limε→0 limsupn→∞ n−1 log(ν(n, ρ, ε)) ≤ S This is the first part of Schumacher’s theorem. For the converse part, suppose for some η > 0, liminfn→∞ n−1 log(ν(n, ρ, ε)) ≤ S − η Then, it follows that there exists a sequence of integers nj → ∞ and orthogonal projections Fnj in H⊗nj such that T r(ρ⊗nj Fnj ) > 1 − ε and T r(Fnj ) < 2nj (S−η) for all j. It follows that 1 − ε < T r(ρ⊗nj Fnj ) = T r(ρ⊗nj (1 − E(nj , ρ, δ))) + T r(ρ⊗nj E(nj , ρ, δ)Fnj ) ≤ α/nj δ 2 + 2−nj (S−δ) T r(Fnj ) ≤ α/nj δ 2 + 2−nj (S−δ)+nj (S−η) = 2−nj (η−δ) It follows that if δ < η, the rhs converges to zero as j → ∞ leading to a contradiction. This proves that for all η, ε > 0, we have S − η ≤ liminfn→∞ n−1 log(ν(n, ρ, ε)) and hence we get Schumacher’s noiseless quantum coding theorem: For any δ, ε > 0, S − δ ≤ liminfn n−1 log(ν(n, ρ, ε)) ≤ limsupn n−1 log(ν(n, ρ, ε)) ≤ S + δ

General Relativity and Cosmology with Engineering Applications

573

th [25] Training quantum neural ⊕p networks: The state of the i layer is a vector in the Hilbert space Hi = j=1 Hij where i = 1, 2, ..., L is the total number of layers and p is the number of nodes in any given layer. We write

|ψi >= (|ψi1 >, ..., |ψip >), |ψij >∈ Hij , i = 1, 2, ..., L The weight matrix of this quantum feed-forward neural network is defined by a set of unitary operators U1 , ..., Up−1 where each unitary operator Ui is a function of |ψi > and some finite dimensional complex vector valued weight wi . Thus, we can write Ui = Ui (wi , |ψi >) It should be noted that Ui maps Hi onto Hi+1 which means that all the Hilbert spaces Hi , i = 1, 2, ..., L must be isomorphic. We shall be assuming without loss of generality that the Hi' s are all the same. One way to define Uk would be to express it as Uk = exp(i(

s ∑

pkr (|ψk >)Akr + p¯kr (|ψk >)A∗kr ))

r=1

where Akr are linear operators in Hk . and pkr : Hk → C is a polynomial for each r with coefficients being elements of the weight vector wk . The i/o transfer characteristic of such a network is given by |ψk+1 >= Uk (wk , |ψk >)|ψk >, k = 1, 2, ..., L − 1 |psi1 > is the input state and |ψL > is the output state. The process of acting on the k th layer state |psik > by Uk (wk , |ψk >) to produce the (k + 1)th layer state |ψk+1 > should be compared to the classical feed-forward neural network in which the k th layer state xk [n] and the (k +1)th layer state xk+1 [n] are related by a linear combination followed by the actio of a nonlinear sigmoidal function fk : xk+1 [n] = fk (WkT xk [n]), k = 1, 2, ..., L − 1 where x1 [n] is the input signal vector and y[n] = xL [n] is the output signal vector. The only difference is that in the quantum case, unitarity of the trans­ formation must be imposed to guarantee conservation of total probability from layer to layer. The final output state can thus be expressed as a cascade of state dependent unitaries: |ψL >= UL−1 UL−2 ...U1 |ψ1 > We have to select the weight vectors wk such that |ψL > is as close to a desired output state |ψd >, ie, we must minimize | || |ψL > −|ψd >||2 For example, if L = 2, then we have to minimize F (w1 , w2 ) = | || U2 (w2 , U1 (w1 , |ψ1 >)U1 (w1 , |ψ1 >)|ψ1 > −|ψd >||2

574

General Relativity and Cosmology with Engineering Applications

All this is about static quantum neural networks. To obtain a formulation of dynamic neural networks, we model the state evolution at each layer as contin­ uously evolving unitary operator dependent on the state of the previous layer using a quantum stochastic differential equation in the sense of Hudson and Parthasarathy and if the output state is |ψL (t) > while the desired state is |ψd (t) >, then we must minimize ∫ T || |ψd (t) > −|ψL (t) >||2 dt 0

with respect to the weights. The HP equation for the k th layer is given by dUk (t) = (−i(Hk + Pk )dt + L1k dAk (t) + L2k dAk (t)∗ + Sk dΛk (t))Uk (t) where the operators Hk , Pk , L1k , L2k , Sk act in the Hilbert space Hk = H1 and that these operators are functions of ρk (t) = T r2 (|ψk (t) >< ψk (t)|) and certain complex weight vectors wk (t) and that |ψk (t) >= Uk−1 (t)|ψk−1 (t) >. We thus have for the output layer, |ψL (t) >= UL−1 (t)...U1 (t)|ψ0 (t) > Note that the Uk (t)' s satisfy nonlinear Schrodinger equations since the coeffi­ cients appearing in the associated qsde’s are functions of the current system state. An example of a nonlinear Schrodinger equation: dU (t) = (−(iH + P )dt + L1 dA + L2 dA∗ + SdΛ)U (t) where H ∗ = H and P, L1 , L2 , S are system operators chosen to make U (t) unitary. Let |ψ(t) >= U (t)|f ⊗ φ(u) >, |φ(u) >= |e(u) > Then, we have dA|φ(u) >= u(t)dt|φ(u) >, dA∗ |φ(u) >= (dB(t) − u(t)dt)|φ(u) > −, dΛ|φ(u) >= (dA∗ dA/dt)|φ(u) >= u(t)dA∗ |φ(u) >= u(t)(dB(t)−u(t)dt)|φ(u) > where B(t) = A(t) + A(t)∗ is a classical Brownian motion. Thus, |ψ(t) > satisfies a classical sde d|ψ(t) >= (−(iH+P )dt+u(t)L1 dt+(dB(t)−u(t)dt)L2 +u(t)(dB(t)−u(t)dt)S)|ψ(t) >

Now define the collapsed state ρ(t) = T r(t,∞) (|ψ(t) >< ψ(t)|)

General Relativity and Cosmology with Engineering Applications

575

Note that ∫ T r(t,∞) (|ψ(0) >< ψ(0)|) = exp(−

0

t

|u(s)|2 ds)|f ⊗ e(ut] ) >< f ⊗ e(u(t )|

= |f ⊗ φ(u(t ) >< f ⊗ φ(u(t )| It is also clear that dρ(t) = dT r(t,∞) (|ψ(t) >< ψ(t)|) = T r(t,∞) d(|ψ(t) >< ψ(t)|) Now, d(|ψ(t) >< ψ(t)|) = (d|ψ(t) >) < ψ(t)|+|ψ(t) > d < ψ(t)|+(d|ψ(t) >)(d < ψ(t)|) = [−(iH+P )dt+u(t)L1 dt+(dB(t)−u(t)dt)L2 +u(t)(dB(t)−u(t)dt)S)]|ψ(t) >< ψ(t)| ¯(t)dt)L∗2 +¯ u(t)(dB(t)−u ¯(t)dt)S ∗ ] +|ψ(t) >< ψ(t)|[(iH−P )dt+¯ u(t)L∗1 dt+(dB(t)−u ¯(t)S ∗ )dt +(L2 + u(t)S)|ψ(t) >< ψ(t)|(L∗2 + u and hence dρ(t) = [−i[H, ρ(t)] + (u(t)(L1 − L2 ) − u(t)2 S − P )ρ(t)+ ρ(t)(¯ u(t)(L∗1 − L∗2 ) − u ¯(t)2 S ∗ − P ) + (L2 + u(t)S)ρ(t)(L∗2 + u ¯(t)S ∗ )]dt+ ¯(t))]dB(t) [(L2 + u(t))ρ(t) + ρ(t)(L∗2 + u Note that P ∗ = P . Let E denote expectation w.r.t the Brownian motion B(.). Then ρ0 (t)E(ρ(t)) = T r2 (|ψ(t) >< ψ(t)|) satisfies the GKSL equation ρ'0 (t) = −i[H, ρ(t)] + (u(t)(L1 − L2 ) − u(t)2 S − P )ρ(t) + ρ(t)(¯ u(t)(L∗1 − L∗2 )− u ¯(t)2 S ∗ − P ) + (L2 + u(t)S)ρ(t)(L∗2 + u ¯(t)S ∗ ) It is also easy to see that ρ0 (t) = E(|ψt (B) >< ψt (B)|) and in particular, ∫ 1 = T r(ρ(t)) = E[< ψt (B)|ψt (B) >] =

< ψt (B)|ψt (B) > dμ(B)

where μ denotes the Wiener probability measure and |ψt (B) > satisfies the the same sde as |ψ(t) >, ie, d|ψt (B) >= (−(iH + P )dt + u(t)L1 dt + (dB(t) − u(t)dt)L2 + u(t)(dB(t) − u(t)dt)S)|ψt (B) >

576

General Relativity and Cosmology with Engineering Applications

= [(−(iH + P ) + u(t)(L1 − L2 ) − u(t)2 S)dt + (L2 + u(t)S]dB(t))|ψt (B) > Note that < ψt (B)|ψt (B) >/= 1 in general. So if we take measurements of the classical Brownian path B upto time t, then the state of the system collapses to the pure state (after averaging over the environment from (t, ∞)) |χ(t) >=

|ψt (B ) > < ψt (B )|ψt (B ) >1/2

We now show that |χ(t) > satisfies a nonlinear Stochastic Schrodinger equation. We have by Ito’s formula, d(< ψt (B)|ψt (B) >−1/2 ) = (−1/2) < ψt (B)|ψt (B) >−3/2 d < ψt (B)|ψt (B) > +(3/8) < ψt (B)|ψt (B) >−5/2 (d < ψt (B)|ψt (B) >)2 Now, d < ψt (B)|ψt (B) >=< dψt (B)|ψt (B) > + < ψt (B)|dψt (B) > + < dψt (B)|dψt (B) >

The classical and quantum back-propagation scheme: In the classical case, we have at the k th layer, the state vector xk (t) ∈ Rp and the k th layer output sig­ nal yk (t) = gk (wkT xk (t)) where k = 1, 2, ..., L. The state transfer characteristic of the feed-forward network is described by the recursive relations xk (t) = fk (ηkT xk−1 (t)), k = 1, 2, ..., L where x0 (t) is the input vector signal. ηk are the weight vectors which define the transfer characteristics between layers and the fk' s are the sigmoidal functions. The gk' s are also certain nonlinear functions. The desired output at the different layers is dk (t), k = 1, 2, ..., d and the weight vectors wk , ηk , k = 1, 2, ..., L are to be selected so that the total error energy of the network over the time duration t = 1, 2, ..., T is a minimum. This error energy is given by E(T ) =

T ∑

2

|| y(n) − d(n) || =

n=1

L ∑ T ∑

(yk (n) − dk (n))2

k=1 n=1

So far, everything is feed-forward, ie, no back-propagation terms are included. In order to minimize E(T ), we have have to compute its gradient w.r.t. the weights ηk , wk , k = 1, 2, ..., L. Thus, ∂E(T )/∂wk = 2

T ∑ n=1

(yk (n) − dk (n))∂yk (n)/∂wk

General Relativity and Cosmology with Engineering Applications

=2

T ∑

577

(yk (n) − dk (n))gk' (wkT xk (n))xk (n)

n=1

Further, ∂E(T )/∂ηj = 2

∑

(yk (n) − dk (n))∂yk (n)/∂ηj

k,n

Clearly, since the network is feed-forward, ∂yk (n)/∂ηj = 0 for j < k and for j ≥ k, ∂yk (n)/∂ηjm = gk' (wkT xk (n))wkT ∂xk (n)/∂ηjm where ηj = [ηj1 , ..., ηjp ]T with p denoting the number of nodes in each layer. We have further, ∂xk (n)/∂ηk fk' (ηkT xk−1 (n))xk−1 (n), while for j < k, ∂xk (n)/∂ηjm = fk' (ηkT xk−1 (n))ηkT ∂xk−1 (n)/∂ηjm In other words, defining δk,j,m (n) = ∂xk (n)/∂ηjm we get and for j < k,

δk,k,m (n) = fk' (ηkT xk−1 (n))xk−1 (n), δk,j,m (n) = fk' (ηkT xk−1 (n))ηkT δk−1,j,m (n)

[26] Interaction between a gravitational wave and an electromagnetic wave. Four cases need to be studied here. (1) classical gravitational and classical em wave, (2) classical gravitational and quantum em wave, (3) quantum gravita­ tional and classical em wave and (4) quantum gravitational and quantum em wave. Weak gravitational waves are described after an appropriate coordinate change by ηαβ hμν,αβ = 0 ie the classical wave equation. The solution is ∫ ¯μν (K)exp(−i(|K|ct − K.r))]d3 K hμν (t, r) = [aμν (K)exp(i(|K|ct − K.r)) + a

578

General Relativity and Cosmology with Engineering Applications

where the coordinate condition is hμν,μ − (1/2)h,ν = 0 To check this, we only need to linearize the Einstein field equations Rμν = 0 around the flat space-time metric. Thus writing gμν = ημν + hμν (x) we get on neglecting second and higher degree terms in hμν and its partial derivatives, α δRμν = Γα μα,ν − Γμν,α = ηαβ (Γβμα,ν − Γβμν,α ) = 0 or ηαβ (hβμ,αν + hβα,μν − hαμ,βν ) −etaαβ (hβμ,να + hβν,μα − hνμ,βα ) = 0 or α h,μν − hα μ,αν − hν,μα + ηαβ hμν,αβ = 0

If we now impose the coordinate condition hμν,μ − (1/2)h,ν = 0 then we get the flat space-time wave equation for hμν : ηαβ hμν,αβ = 0 The Lagrangian density for the electromagnetic field in this gravitational field is √ LEM = K.Fμν F μν −g from which the terms that are linear in hμν correspond to interaction Lagrangian density between the gravitational field and the em field: LEM = K(1 + h/2)Fμν Fαβ (ηαμ − hαμ (ηβν − hβν ) +O(h2 ) = Kηαμ ηβν Fμν Fαβ + K.Fμν Fαβ [(h/2)ημα ηνβ − hμα ηνβ − ημα hνβ ] The first term on the rhs is the Lagrangian density of the em field and the second term represents the interaction Lagrangian between the gravitational field and the em field: Lint = K.Fμν Fαβ [(h/2)ημα ηνβ − hμα ηνβ −ημα hνβ ]

579

General Relativity and Cosmology with Engineering Applications The Maxwell equations in the absence of external charges and currents are √ (F μν −g),ν = 0 and the gauge condition on the em four potential is √ (Aμ −g),μ = 0

Retaining terms upto linear orders in hμν , these equations can be expressed as [Fαβ (ημα − hμα )(ηνβ − hνβ )(1 + h/2)],ν = 0 or (ημα ηνβ Fαβ ),ν + (−ημα h

νβ

− ηνβ hμα + (ημα ηνβ h/2)Fαβ ),ν = 0

The em potential gauge condition when expressed upto linear orders in h be­ comes [(ημν − hμν )(1 + h/2)Aν ],μ = 0 or (ημν Aν ),μ + [Aν (−hμν + hημν /2)],μ = 0 Exercise: Now simplify the Maxwell equations using this gauge condition and bring it into the final form ηαβ Aμ,αβ + C(μναβρσ)(Aα,β hρσ ),ν = 0 where C(μναβρσ) are constants. Hence using first order perturbation theory, express the solution to the em four potential in the form Aμ = Aμ(0) + Aμ(1) , (0)

ηαβ Aμ,αβ = 0, (1)

(0)

ηαβ Aμ,αβ + C(μναβρσ)(Aα,β hρσ ),ν = 0 (0)

(1)

Thus, Aμ can be expressed as a superposition of plane waves and Aμ can be expressed in terms of this plane wave and the gravitational potential hμν using retarded potential theory. [27] Statement of the spectral theorem for bounded self-adjoint operators in a Hilbert space. Let H be a Hilbert space and T a bounded Hermitian(self­ adjoint) operator in H. We wish to express T as a spectral integral ∫ T = λdP (λ) R

The integral may be taken over [− || T ||, || T ||].

580

General Relativity and Cosmology with Engineering Applications

[28] Feinstein-Khinchin fundamental lemma and the proof of Shannon’s noisy coding theorem for classical probability. The channel is defined by the transition probabilities: νx (y) = p(y|x), x ∈ X, y ∈ Y Input probability distribution is p(x), x ∈ X. Output probability distribution is ∑ p(x)νx (y), y ∈ Y q(y) = x∈X

Let Vδ = {(x, y) : |log(νx (y)/q(y)) − I| < δ} where I = I(X, Y ) = E[log(νx (y)/q(y))] =

∑

p(x)νx (y)log(νx (y)/q(y)) = H(Y )−H(Y |X)

x,y

where H(Y ) = −

∑

q(y)log(q(y)), H(Y |X) = −

y

∑

p(x)νx (y)log(νx (y)) = −E[log(νx (y))]

x,y

Thus, we can write Vδ =

∐

({x} × Vx )

x∈X

where Vx {y : (x, y) ∈ Vδ }, x ∈ X Then we get from the definition of Vδ 2I−δ q(y) ≤ νx (y) ≤ 2I+δ q(y), y ∈ Vx , x ∈ X Thus 2I−δ q(Vx ) ≤ νx (Vx ) ≤ 2I+δ q(Vx ), x ∈ X In particular, q(Vx ) ≤ 2−(I−δ) , x ∈ X Now choose x1 ∈ X so that νx1 (V1 ) > 1 − ε where V1 = Vx1 . Then choose x2 /= x1 such that νx2 (V2 ) > 1 − ε where V2 = Vx1 ∩ V1c . In this way, choose distinct x1 , x2 , ..., xM such that M is maximal subject to νxk (Vk ) > 1 − ε, k = ∪k−1 1, 2, ..., M where Vk = Vxk ∩ ( i=‘ Vi )c , k = 2, 3, ..., M . This means that for ∪M any x ∈ / {x1 , ..., xM } we have νx (( i=1 Vi )c ) ≤ 1 − ε. We also note that for i = 1, 2, ..., M , V1 , ..., VM are pairwise disjoint. νxi ((

M ∐ i=1

Vi )c ) ≤ νxi (Vic ) ≤ ε

General Relativity and Cosmology with Engineering Applications

581

and hence if we assume that ε < 1/2, we have that νx ((

M ∐

Vi )c ) < 1 − ε, x ∈ X

i=1

We also have by Chebyshev inequality ∑ p(Vδ ) = p(x)νx (Vx ) > 1 − α/δ 2 x∈X

where α = V ar(log(νx (y)/q(y))) Thus, 1 − α/δ 2 ≤ p(Vδ ) =

∑

p(x)νx (Vx )

x∈X

∑

p(x)νx (Vx ∩

∑

p(x)νx (Vx ∩ (

∑

=

M ∐

p(x)ν(

Vi ) +

i=1

x∈X M ∑

M ∐

Vi ) c )

i=1

x∈X

≤

Vi )

i=1

x∈X

+

M ∐

q(Vi ) +

i=1

≤

p(x)νx (

∑

p(x)νx ((

M ∐

Vi ) c )

i=1

x∈X

x∈X M ∑

∑

M ∐

Vi ) c )

i=1

q(Vxi ) + 1 − ε

i=1

(since Vi ⊂ Vxi ). It follows that ε − α/δ 2 ≤ M.2−(I−δ) so that

M ≥ (ε − α/δ 2 ).2I−δ

It follows that if we use an iid source and an iid product channel, then with X replaced by X n and Y by Y n so that X ×Y is replaced by X n ×Y n = (X ×Y )n , we get the result that for all sufficiently large n there exists a code having error probability smaller than ε and a size of Mn such that Mn ≥ (ε − α/nδ 2 )2(n(I−δ) which implies that liminfn→∞ log(Mn )/n ≥ I − δ

582

General Relativity and Cosmology with Engineering Applications

for any δ > 0. This proves the direct part of Shannon’s noisy coding theorem. Proof of the converse part in Shannon’s noisy coding theorem. X is the input alphabet, Y is the output alphabet. The channel transition probabilities are νx (y). For u = (x1 , ..., xn ) ∈ X n and v = (y1 , ..., yn ) ∈ Y n , we define νu (v) = Πni=1 νxi (yi ) Thus, νu (v) is a transition probability from X n ∑ to Y n . We write p(x), x ∈ X for the input probability distribution and q(y) = x∈X p(x)νx (y), y ∈ Y for the output probability distribution. We write p(u) = Πni=1 p(xi ), q(v) = Πni=1 q(yi ). Thus, ∑ q(v) = p(u)νu (v), v ∈ X n u∈X n

∑n Write T (n, p, δ) for the set of all u = (x1 , ..., xn ) ∈ X n for which |n−1 i=1 log(p(xi ))+ H(X)| < δ, ie, |n−1 log(p(u)) + H(X)| < δ. Likewise for T (n, q, δ). Let N (x|u) denote the number of times x ∈ X appears in the sequence u. We define Pu (x) = N (x|u)/n. Then, Pu is a probability distribution on X. T (N (x|u), νx , δ) is the ⊗N (x|u) set of all vx ∈ Y N (x|u) for which |log(νx (vx ))/N (x|u) + H(Y |x)| < δ. In other words, vx ∈ T (N (x|u), νx , δ) iff 2−N (x|u)(H(Y |x)+δ) < νx⊗N (x|u) (vx ) < 2−N (x|u)(H(Y |x)−δ) Writing v = ∪x∈X vx , it follows that if vx ∈ T (N (x|u), νx , δ) for all x ∈ X, then ∑ ∑ Pu (x)H(Y |x) − δ) 2−n( x∈X Pu (x)H(Y |x)+δ) < νu (v) < 2−n( x∈X

We say that v ∈ T (n, νu , δ) iff vx ∈ T (N (x|u), νx , δ)∀x ∈ X. Therefore, if Pu = p and p is the input probability distribution, we can write 2−n(H(Y |X)+δ) < νu (v) < 2−n(H(Y |X)−δ) , v ∈ T (n, νu , δ) Suppose u1 , ..., uM are M distinct sequences each of length n with Puk = p for all k = 1, 2, ..., M and D1 , ..., DM are pairwise disjoint sequences in Y n such that νuk (Dk ) > 1 − ε, k = 1, 2, ..., M . Define ˜ k = Dk ∩ T˜(n, νu , δ) D k ˜ where √ T (n, νu , δ) is the set of all n length v such that |N (y|vx )−N (x|u)νx (y)| < δ N (x|u)νx (y)(1 − νx (y)) for all y ∈ Y and all x appearing in u. Remark: In general, for any probability distribution μ on a finite set A, n ˜ we √ define T (n, μ, δ) to be the set of all u ∈ A such that |N (x|u) − nμ(x)| < δ nμ(x)(1 − μ(x)), ∀x ∈ A. We then see that Chebyshev’s inequality and the union bound give μ(T˜(n, μ, δ)) ≥ 1 − a/δ 2 where a is the number of elements in A and further, u ∈ T˜(n, μ, δ) implies √ √ nμ(x) − δ nμ(x)(1 − μ(x)) ≤ N (x|u) ≤ nμ(x) + δ nμ(x)(1 − μ(x)), x ∈ A

General Relativity and Cosmology with Engineering Applications

583

This implies ∑ ∑√ μ(x)log(μ(x)) + (δ/sqrtn) μ(x)(1 − μ(x))log(μ(x)) ≤ x

x

≤ n−1 ∑

∑

N (x|u)log(μ(x)) ≤

x∈A

μ(x)log(μ(x)) − (δ/sqrtn)

∑√ μ(x)(1 − μ(x))log(μ(x))

x

x

or ∑ equivalently, with Pu (x) = N (x|u)/n, we have that for all n, and K = √ − x μ(x)(1 − μ(x))log(μ(x)) ∑ √ √ (H(μ) − Kδ/ n) ≤ − Pu (x)log(μ(x)) ≤ (H(μ) + Kδ/ n) x

This equation is the same as √ √ H(μ) − Kδ/ n ≤ n−1 log(μn (u)) ≤ H(μ) + Kδ/ n since

μn (u) = Πx∈u μ(x) = Πx∈A mu(x)N (x|u)

and so log(μn (u)) =

∑

N (x|u)log(μ(x))

x∈A

In other words, we have established that √ T˜(n, μ, δ) ⊂ T (n, μ, Kδ/ n) Now, note that N (y|vx ) is the number of occurrences of y in the sequence vx whose entries are those in v occurring precisely at those positions where x occurs in u. We have Then assume Pu = p and v ∈ T˜(n, νu , δ). Then for any y ∈Y, ∑ ∑ |N (y|v) − nq(y)| = | N (y|vx ) − np(x)νx (y)| =|

∑

x∈X

N (y|vx ) −

x∈X

≤

∑

∑

x∈X

N (x|u)νx (y)|

x∈X

|N (y|vx ) − N (x|u)νx (y)| ≤

x∈X

≤

∑ √ δ N (x|u)νx (y)(1 − νx (y))

x∈X

√ ∑√ =δ n p(x)νx (y)(1 − νx (y)) x∈X

584

General Relativity and Cosmology with Engineering Applications ∑ √ ≤ δ an.( p(x)νx (y)(1 − νx (y)))1/2 x∈X

√

= δ an.(q(y) − q(y)2 )1/2 = δ

√

na(q(y)(1 − q(y))

since by the Cauchy-Schwarz inequality, ∑ q(y) = p(x)νx (y) x∈X

implies q(y)2 ≤

∑

p(x)νx (y)2

x∈X

Here, a denotes the number of elements in X. We have thus proved that if Pu = p, then √ T˜(n, νu , δ) ⊂ T˜(n, q, δ a) We also note that as seen above, by Chebyshev’s inequality and the union bound, μn (T˜(n, μ, δ)) ≥ 1 − a/δ 2 and hence,

νu (T˜(n, νu , δ)) = Πx∈X νxN (x|u) (T˜(N (x|u), νx , δ)) ≥ Πx∈X (1 − b/δ 2 ) = (1 − b/δ 2 )a ≥ 1 − ab/δ 2

where now a is the number of elements in X and b is the number of elements in Y. Now since Puk = p, k = 1, 2, ..., M , we get ˜ k ), k = 1, 2, ..., M 1 − a/δ 2 − ε ≤ νuk (D ˜ k implies v ∈ T˜(n, νu , δ) implies v ∈ T˜(n, q, δ √a) and we also get that v ∈ D k √ which implies v ∈ T (n, q, Kδ a/n) which implies 2−(nH(Y )+δ1

√

n)

≤ q(v) ≤ 2−(nH(Y )−δ1

This in turn implies (on summing over v ∈ μ0 (

M ∐

∪M k=1

√

n)

˜ k ) that D

˜ k ) ≤ 2nH(Y )+δ1 D

√

n

k=1

where μ0 (E) denotes the number of elements in a set E. We now recall that 2−n(H(Y |X)+δ) < νuk (v) < 2−n(H(Y |X)−δ) , v ∈ T (n, νuk , δ) and since

√ ˜ k ⊂ T˜(n, νu , δ) ⊂ T (n, νu , Kδ/ n) D k k

585

General Relativity and Cosmology with Engineering Applications ˜ k , we have it follows that for all v ∈ D 2−nH(Y |X)−Kδ

√

n

≤ νuk (v) ≤ 2−nH(Y |X)+Kδ

√

n

˜ k and using the above upper bound on which on summing over v ∈ D ∪Mgives ˜ ' s, the result ˜ k ) and the disjointness of the D μ0 ( k=1 D k ˜ k ) ≤ 2−nH(Y |X)+Kδ (1 − ab/δ 2 − ε) ≤ νuk (D

√

so that M (1 − ab/δ 2 − ε) ≤ 2−n(H(Y |X)+Kδ

√

n

μ0 (

n

˜ k) μ0 (D

M ∐

˜ k) D

k=1

≤ 2n(H(Y )−H(Y |X))+K1 δ

√

n

and hence the required upper bound on M . This proof of the converse of Shannon’s noisy coding theorem is due to Wolfowitz. [29] Problem: Consider a rotating blackhole with metric dτ 2 = A(r, θ)dt2 − B(r, θ)dr2 − C(r, θ)dθ2 − D(r, θ)(dφ − ω(r, θ)dt)2 Write down the Einstein field equations Rμν = 0 for this metric and derive the Kerr solution for A, B, C, ω. Equivalently, using Cartan’s equations of structure, write down the Einstein field equations in the tetrad basis √ √ √ √ e0 = Adt, de1 = Bdθ, e2 = Cdθ, e3 = D(dφ − ωdt) and solve for A, B, C, D, ω. Note that the metric is diagonal in this tetrad basis: g = e0 ⊗ e0 −

3 ∑

er ⊗ e r

r=1

[30] Let R(t) be a 3 × 3 rotation matrix dependent on time t. Let B denote the region of space occupied by a rigid body at time t = 0. Then if ρ0 denotes the rest mass density of the body, its general relativistic Lagrangian can be expressed as ∫ L(R(t), R' (t), t) = −ρ0 (g00 (t, R(t)ξ) + 2g0m (t, R(t)ξ)(R' (t)ξ)m B

+gmk (t, R(t)ξ)(R' (t)ξ)m (R' (t)ξ)k )1/2 d3 ξ

586

General Relativity and Cosmology with Engineering Applications

Note that (R(t)ξ)m = Rms (t)ξ s with summation over the repeated index s assumed. Now write R(t) = Rz (φ(t))Rx (θ(t))Rz (ψ(t)) with φ, θ, ψ the Euler angles and set up the general relativistic equations of mo­ tion of the rigid body, ie, rigid body in a gravitational field. [31] The energy-momentum tensor of a system of N particles with rest masses {mi } is given by T μν (x) =

N ∑

mi δ 3 (x − xi )(dxμi /dt)(dxμi /dτi )(−g(x))−1/2

i=1

Justify this by considering the special relativistic case and then showing that ∫ ∫ ∑ √ μν 4 T (x) −g(x)d x = mi viμ dxμi i

is a tensor where with dτi = particle.

√

viμ = dxμi /dτi gμν (xi )dxμi dνi being the proper time along the world line of the ith

[32] Quantum Boltzmann equation: The Hamiltonian of a system of N iden­ tical particles is given by H=

N ∑ a=1

Ha +

∑

Vab

1≤a 0. ρ(t) satisfies iρ' (t) = [H, ρ(t)] So iρ'1 (t) = iT r23...N ρ' (t) = [H1 , ρ1 (t)] + (N − 1)T r2 [V12 , ρ12 (t)] − − − − − (1) iρ'12 (t) = iT r34...N ρ' (t) = [H1 +H2 +V12 , ρ12 (t)]+(N −2)T r3 [V13 +V23 , ρ123 (t)]−−−−−(2)

General Relativity and Cosmology with Engineering Applications

587

Neglecting ρ123 (or equivalently assuming that ρ123 can be separated as ρ1 ⊗ ρ2 ⊗ ρ3 plus negligible terms, or more generally as g12 ⊗ g3 which ensures that the last commutator vanishes), we get approximately, iρ'12 (t) = [H1 + H2 + V12 , ρ12 (t)] an hence ρ12 (t) = exp(−itad(H1 + H2 + V12 ))(ρ12 (0)) Then substituting this into the equation for ρ1 gives us iρ'1 (t) = [H1 , ρ1 (t)] + (N − 1)T r2 [ad(V12 )(exp(−itad(H1 + H2 + V12 ))(ρ12 (0)))] This may be termed as the quantum Boltzmann equation with the second term on the rhs being the collision term. The solution is ρ1 (t) = exp(−itad(H1 ))(ρ1 (0))− ∫ i(N − 1)

t

0

exp(−i(t − s)ad(H1 ))(T r2 [ad(V12 )(exp(−is.ad(H1 + H2 + V12 ))])ds

There are other versions of the quantum Boltzmann equation derivable from the exact equations (1) and (2). One of these versions is as follows. Let ρ12 (t) = ρ1 (t) ⊗ ρ2 (t) + g12 (t) where ρ2 (t) = ρ1 (t) and g12 is small. Then assuming that Vab are also small, we can neglect the products V12 ⊗ g12 and obtain approximately from (1) and (2), ' (t) = [H1 + H2 , g12 (t)] + [V12 , ρ1 (t) ⊗ ρ2 (t)] ig12

so that ∫ g12 (t) = −i

t 0

exp(−i(t − s)ad(H1 + H2 ))([V12 , ρ1 (s) ⊗ ρ2 (s)]ds − − − (3)

Again from (1) and (3) we then get iρ'1 (t) = [H1 , ρ1 (t)]− ∫ i(N −1)

t 0

T r2 [ad(V12 )exp(−i(t−s)ad(H1 +H2 ))ad(V12 )(ρ1 (s)⊗ρ1 (s))]ds−−−(4)

where we have used ρ2 (t) = ρ1 (t). This equation is closer in form to the classical Boltzmann equation since it is also an integro differential equation with the collision term being a quadratic function of the single particle density matrices as is the classical case when we consider only binary collisions. [33] Interacting Dirac particles. Let αk = (αkx , αky , αkz ), betak , k = 1, 2, ..., N denote N copies of the Dirac matrices acting on different components of the tensor product of N Hilbert spaces, each Hilbert space being L2 (R3 )⊗4 . pk =

588

General Relativity and Cosmology with Engineering Applications

(pkx , pky , pkz ), k = 1, 2, ..., N are correspondingly the N momentum operators acting on the different component Hilbert spaces and qk = (qkx , qky , qkz ), k = 1, 2, ..., N are correspondingly the N position vectors acting on the different component Hilbert spaces. Let vk (t) denote the three velocity of the k th par­ ticle. Then from standard Hamiltonian mechanics, the operator vk (t) is to be replaced by pk − ek Ak (t, qk ) and Ek , the energy of the k th particle is to be replaced by Ek − ek V (t, qk ) where A(t, qk ) and V (t, qk ) are respectively the magnetic vector potential and electric scalar potential produced by the other N − 1 particles at the site of the k th particle. If we make the non-relativistic approximation for computing the electromagnetic potentials, then ∑ ∑ ej vj (t)/|qk − qj | = ej (pj + ej Aj (t, qj ))/|qk − qj |, k = 1, 2, ..., N A(t, qk ) = j/=k

j=k /

V (qk ) =

∑

ej /|qk − qj |

j/=k

and our approximate Dirac Hamiltonian for the system of N particles is given by N ∑ [(αk , pk − ek A(t, qk )) + βk mk + ek V (qk )] H= k=1

Problem: Solve for A(t, qk ) in terms of pj , qk , ej , j = 1, 2, ..., N and wherever cross terms appear between the j th and k th particles, divide by two to take care of the fact that inter-particle interactions are not counted twice. Let ∑ e2j Aj (t, qj )/|qk − qj | Fk = j/=k

Then we can write A(t, qk ) =

∑

ej /|qk − qj | + Fk = V (qk ) + Fk

j/=k

An alternate approach: Let A(t, q), V (t, q) denote respectively the magnetic vector potential and electric scalar potential in space produced by the N Dirac particles having mass mk and charge ek , k = 1, 2, ..., N . The N particle four component Dirac wave function ψ(t, q1 , ..., qN ) satisfies iψ,t = Hψ where H=

N ∑

((αk , pk − ek A(t, qk )) + βk mk + ek V (t, qk )), pk = −i∂/∂qk

k=1

The current and charge density fields can be taken as J m (t, q) =

N ∫ ∑ k=1

∫ ek

ψk (t, q)∗ αm ψk (t, q), m = 1, 2, 3,

General Relativity and Cosmology with Engineering Applications ρ(t, q) =

N ∑

589

ek ψk (t, q)∗ ψk (t, q), q ∈ R3

k=1

where ψ(t, q1 , ..., qN )T = [ψ1 (t, q1 )T , ..., ψN (t, qN )T ] ∈ C4N , qk ∈ R3 , ψj (t, q) ∈ C4 We have for A(t, q) and V (t, q), the retarded potentials ∫ ∫ A(t, q) = J(t, |q−q ' |/c, q ' )d3 q ' /|q−q ' |, V (t, q) = ρ(t−|q−q ' |/c, q ' )d3 q ' /|q−q ' | Remark: While calculating A(t, qk ) and V (t, qk ) at the site of the k th particle, the k th terms in the summation expressions for J and ρ are to be omitted. In­ deed, this means that the potentials at the site of any given particle is generated by all the other particles only, not by the given particle. We have thus expressed the electromagnetic four potential in terms of the individual wave functions of the particles. We substitute this expression into the Dirac equation i∂ψk (t, qk )/∂t = ((αk , pk − ek qk ) + βk mk + ek V (t, qk ))ψk (t, qk ), k = 1, 2, ..., N or equivalently, i∂ψk (t, q)/∂t = ((α, p − ek q) + βmk + ek V (t, q))ψk (t, q), k = 1, 2, ..., N and solve this nonlinear coupled integro-differential equations for the individual wave functions ψk . [34] Dirac’s equation for a rigid body, an approximation. Let B denote the volume of the rigid body at time t = 0. After time t each point r ∈ B goes over to R(t)r where R(t) ∈ SO(3). The three velocity of this point is R' (t)r and we can express the total kinetic energy plus rest mass energy of this rigid body as ∫ K(t) = ρ (1 − |R' (t)r|2 /c2 )−1/2 d3 r B

where ρ is the volume mass density of the body. Using the binomial theorem, we can make the approximation ∫ K(t) = ρ (1 + |R' (t)r|2 /2c2 + 3|R' (t)r|4 /8c4 )d3 r B

Exercise: Now apply the Legendre transformation to obtain the Hamiltonian as a function of canonical angular position and momentum coordiates and perform an approximate factorization as in the derivation of the Dirac equation from the Klein-Gordon equation. [35] Estimating the metric of space-time from measurements of the scattered electromagnetic field.

590

General Relativity and Cosmology with Engineering Applications The em field equations in a background metric gμν are √ (F μν −g),ν = 0 assuming no charge and current sources, or equivalently, √ (g μα g νβ −gFαβ ),ν = 0 (0)

We are given an incident em field Fμν and we measure the scattered em field (1) Fμν . The total em field is given by (0) (1) Fμν = Fμν + Fμν

The scattered field is small compared to the incident field and arises owing to the small perturbation of the metric from flat space-time. The incident em field satisfies the unperturbed wave equation, ie, the Minkowskian wave equation and hence is representable as superposition of plane waves. We write the total metric as gμν = ημν + hμν (x) where ((ημν )) = diag(1, −1, −1, −1) is the Minkowskian metric and hμν (x) is a small perturbation of the Minkowskian metric. We have upto linear orders in hμν , √ g μα g νβ −g = (ημα − hμα )(ηνβ − hνβ )(1 + h/2) = ημα ηνβ + [ημα ηνβ h/2 − ημα hνβ − ηνβ hμα ] We then get using first order perturbation theory, (0)

ημα ηνβ Fαβ,ν = 0 or equivalently,

(0)

ηνβ Fαβ,ν = 0 and

(1)

(0)

ημα ηνβ Fαβ,ν = −(fμναβ Fαβ ),ν or equivalently,

(1)

(0)

ηνβ Fμβ,ν = −ημα (fανρβ Fρβ ),ν where fμναβ = [ημα ηνβ h/2 − ημα hνβ − ηνβ hμα ] We write the perturbation expansion of the covariant em four potential as Aμ = Aμ(0) + Aμ(1) so that

(0) (0) = A(0) Fμν ν,μ − Aμ,ν ,

General Relativity and Cosmology with Engineering Applications

591

(1) (1) Fμν = A(1) ν,μ − Aμ,ν

The gauge condition on the em potentials introduced is √ (Aμ −g),μ = 0 or equivalently,

√ (g μν −gAν ),μ = 0

Retaining terms only upto first order we can write this equation as (0) (1) (ημν A(0) ν + aμν Aν + ημν Aν ),μ = 0

Equating zeroth order terms gives us ημν A(0) ν,μ = 0 which is the standard Lorentz gauge condition in special relativity: (0)

(0)

(0)

(0)

A0,0 − A1,1 − A2,2 − A3,3 = 0 Equating first order terms gives us (1) ημν Aν,μ = −(aμν A(0) ν ),μ

√ In these expressions, aμν is the first order component in g μν −g, ie, aμν = ημν h/2 − hμν Note that the indices of hαβ are raised using the Minkowskian metric ημν . We now find using this gauge condition that □A(0) μ = 0, where □ = ημν ∂μ ∂ν is the wave operator of special relativity and (1)

ημα ηνβ Fαβ,ν = (1)

(1)

ημα ηνβ (Aβ,αν − Aα,βν ) (0)

= −ημα (aνβ Aβ ),να − ημα □A(1) α so that [36] When the parameters of a quantum system change slowly with time, then based on the second order truncated Dyson series for the unitary evolution operator of a quantum system, we derive by an application of the RLS lattice algorithm for Volterra filters, a recursive/real time method for updating the

592

General Relativity and Cosmology with Engineering Applications

parameter estimates. The quantum system is specified by the time varying Hamiltonian H(t) = H0 + f (t)V where f (t) is the input signal applied to the quantum system which modulates a potential V as a perturbation to the system Hamiltonian H0 . The Hamiltonian H0 as well as the perturbing potential V may depend on a parameter vector θ ∈ Rp which needs to be estimated by continuously measuring an observable X on the system without incorporating the collapse postulate. To show the explicit dependence of H0 , V on the parameter vector θ, we write H0 (θ) and V (θ). Examples are V (θ) = V0 +

p ∑

θk Vk , H0 = H00 +

k=1

p ∑

θ k Hk

k=1

Writing V˜ (t) = V˜ (t, θ) = exp(itH0 ).V.exp(−itH0 ) we know that the solution to the Schrodinger unitary evolution operator U (t) which satisfies Schrodinger’s equation iU ' (t) = H(t)U (t), t ≥ 0, U (0) = I can be expressed as a Dyson series: U (t) = U0 (t)W (t), U0 (t) = exp(−itH0 ), W (t) = I +

∞ ∑

∫ (−i)

n 0 (t) = T r(ρ0 X0 (t)) − i

t 0

˜ I (t' ), ρ0 ]X0 (t))dt' T r([H

603

General Relativity and Cosmology with Engineering Applications where H0 is the unperturbed system Hamiltonian and ˜ I (t) = exp(itH0 ).HI (t).exp(−itH0 ), X0 (t) = exp(itH0 )X.exp(−itH0 ) H We write ξ2 (t) =

∑

ξ1 (t) =< qφ(u)|HI (t)|pφ(u) >, [uk < q|(χkn (r), ∇r )|p > Cn (θ(t), θ' (t))

kn

+

−u ¯k < q|(χ ¯kn (r), ∇r )|p > C¯n (θ, θ' (t))]

∑ ¯ n (θ(t), θ' (t))] ¯k < q|η¯kn (r)|p > D [uk < q|ηkn (r)|p > Dn (θ(t), θ' (t)) + u kn

= F1 (θ(t), θ' (t)) say. Also let

+

∑

ξ3 (t) =

∫ [uk < q|ρkn (r)|p >

kn

t 0

En (θ(s), θ' (s))ds+¯ uk < q|ρ¯kn (r)|p >

∫

t 0

¯n (θ(s), θ' (s))ds] E

Then, we have ξ3' (t) = ξ4 (t) =

∑

¯n (θ(t), θ' (t))] [uk < q|ρkn (r)|p > En (θ(t), θ' (t))+¯ uk < q|ρ¯kn (r)|p > E

kn

= F2 (θ(t), θ' (t)) say. We have ∫ PT (p → q|u) = |

T

0

so defining

exp(−iEt)ξ1 (t)dt|2 , E = Eq − Ep ∫

ξ5 (t) =

t 0

exp(−iEs)ξ1 (s)ds

we have Pt = Pt (p → q|u) = |ξ5 (t)|2 = ξ5 (t)ξ¯5 (t) and we have dPt = ξ5 dξ¯5 + ξ¯5 dξ5 = = 2Re(ξ5 (t)ξ¯1 (t)exp(iEt))dt, dξ5 (t) = exp(−iEt)ξ1 (t)dt, ξ1 (t) = ξ2 (t) + ξ3 (t), ξ2 (t) = F1 (θ(t), θ' (t)) dξ3 (t) = F2 (θ(t), θ' (t))dt

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From these equations, we can construct an EKF for estimating the fan angle using quantum mechanical measurements. The above equations in the presence of measurement noise can be expressed as dPt = 2Re(ξ5 (t)ξ1 (t)exp(iEt))dt + dV (t) = [2(ξ5R (t)(ξ3R (t) + F1R (θ(t), θ' (t))) − ξ5I (t)(ξ3I (t)) + F1I (θ(t), θ' (t)))cos(Et)dt −2(ξ5R (t)(ξ3I (t)+F1I (θ(t), θ' (t)))+ξ5I (t)(ξ3R (t)+F1R (θ(t), θ' (t)))sin(Et)]dt+dV (t) dξ5R (t) = [(cos(Et)(F1R (θ(t), θ' (t))+ξ3R (t))+sin(Et)(F1I (θ(t), θ' (t))+ξ3I (t))]dt+dW5R (t), dξ5I (t) = [(−sin(Et)(F1R (θ(t), θ' (t))+ξ3R (t))+cos(Et)(F1I (θ(t), θ' (t))+ξ3I (t))]dt+dW5I (t),

dξ3R (t) = F2R (θ(t), θ' (t))dt + dW3R (t), dξ3I (t) == F2I (θ(t), θ' (t))dt + dW3I (t) dθ(t) = θ' (t)dt, dθ' (t) = −(γ/J)θ' (t) − (K/J)θ(t) − A.sin(θ(t))/J + dWθ (t) Note that the state variables are ξ3R , ξ3I , ξ5R , ξ5I , θ, θ' while Pt is the measure­ ment process. The EKF can now be directly applied to this to obtain estimates of the states on a real time basis based on our measurement process. Reference: Rohit Singh, Naman Garg and H.Parthasarathy, ”Estimating the angular dynamics of a fan window stroboscope from noisy quantum image measurements”, Technical report, NSIT, 2017. [39] Two dimensional nonlinear difference equations in general relativity. Suppose we have only one spatial dimension and one time dimension. Then, a space time point is specified by an ordered pair (t, x) ∈ R2 and the metric of space-time has the form dτ 2 = g00 (t, x)dt2 + g11 (t, x)dx2 + 2g12 (t, x)dtdx We set up the Einstein field equations in the presence of a source field having density ρ(t, x) and velocity (v 0 (t, x), x1 (t, x)) where g00 v 02 + g11 v 12 + 2g01 v 0 v 1 = 1 The energy-momentum tensor of the source is T 00 (t, x) = ρ(t, x)v 02 (t, x), T 11 (t, x) = ρ(t, x)v 12 (t, x), T 01 (t, x) = T 10 (t, x) = ρ(t, x)v 0 (t, x)v 1 (t, x) Writing

T = g00 T 00 + g11 T 11 + 2g01 T 01

the field equations can be written as R00 = KS00 , R11 = KS11 , R22 = KS22

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where Sab = Tab − T gab /2, a, b = 0, 1 and

2 2 T 00 + g01 T 11 + 2g00 g01 T 01 T00 = g00 2 2 T11 = g11 T 11 + g01 T 00 + 2g01 g11 T 01 , 2 T 01 + g01 g11 T 11 T01 = T10 = g00 g11 T 01 + 2g01

Two coordinate conditions give us two additional relations for the metric coef­ ficients. Thus, in all we have 5 pde’s for the five functions (g00 , g01 , g11 , ρ, v 1 ). These five equations are nonlinear second order pde’s in the variables (t, x). Af­ ter space-time discretization, these become five nonlinear coupled second order two dimensional difference equations and if we retain only upto second degree nonlinear terms, they are of the form [40] Quantization of the gravitational field: g μν = q μν + nμ nν q μν is the spatial part of the metric and nμ nν is the time part of the metric. nμ q μν = 0. Let Kμν = qμα qνβ nα:β We have R = Rμνρσ g νρ g μσ = Rμνρσ q νρ q μσ + 2Rμνρσ q νρ nμ nσ ˜ + 2q νρ nσ (nν:ρ:σ − nν:σ:ρ ) =R (Note: Rμνρσ = −Rνμρσ = −Rμνσρ = Rρσμν . The action integral is ∫ ∫ ∫ √ ˜ √−gd4 x + 2 (q νρ nσ (nν:ρ:σ − nν:σ:ρ )√−gd4 X S = R −gd4 x = R With neglect of a perfect divergence (which does not contribute to the action integral), we have q νρ nσ (nν:ρ:σ − nν:σ:ρ ) = −(q νρ nσ ):σ nν:ρ + (q νρ nσ ):ρ nν:σ = −g νρ nσ:σ nν:ρ + (nν nρ nσ ):σ nν:ρ +g νρ nσ:ρ nν:σ − (nν nρ nσ ):ρ nν:σ = −g νρ nσ:σ nν:ρ + g νρ nσ:ρ nν:σ = −nσ:σ nρ:ρ + nσ:ρ nρ:σ Now K = g μν Kμν = q μν nμ:ν = g μν nμ:ν = nμ:μ

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General Relativity and Cosmology with Engineering Applications

since nμ nμ:ν = 0 Also, Kμν K νμ = qμα qνβ nα:β q ρν q σμ nρ:σ = q ασ q βρ nα:β nρ:σ = g ασ g βρ nα:β nρ:σ = nσ:β nβ:σ since nα nα:β = 0 Thus,

∫ S=

˜ + 2(Kμν K νμ − K 2 ))√−gd4 X (R

Note that Kμν = Kνμ . To prove this, we observe that owing to the symmetry of the connection, nμ:ν − nν:μ = nμ,ν − nν,μ and further, since nμ is the normal to a 3-D surface of the form F (x) = 0, we can write nμ (x) = G(x)F,μ (x) and hence nμ,ν − nν,μ = G,ν F,μ − G,μ F,ν = (logG),ν nμ − (logG),μ nν so that Kαβ − Kβα = qαμ qβν ((logG),ν nμ − (logG),μ nν ) = 0 since qαμ nμ = 0

Spatial covariant derivative: Let uμ be a spatial vector field, ie, uμ nμ = 0. Then define Dμ uν = qμα qνβ uα:β Note that we are embedding a 3-D surface Σt at time t = x0 described by coordinates (xa , a = 1, 2, 3) inside the four dimensional space-time R4 described by coordinates (X μ (x), μ = 0, 1, 2, 3). The unit normal to this 3-D surface is denoted by nμ . Tangent vectors to Σt relative to the X-system are given by μ (X,a )μ , a = 1, 2, 3

General Relativity and Cosmology with Engineering Applications

607

Further, X,tμ can be decomposed into a component tangential to Σt and a com­ ponent normal to Σt as follows: X,tμ = N nμ + N μ where N μ is a spatial vector field, ie, expressible as μ N μ = N a X,a '

(summation over a = 1, 2, 3). The N a s are determined from the orthogonality condition ν gμν nμ X,b = 0, b = 1, 2, 3 or equivalently, μ ν gμν (X,tμ − N a X,a )X,b = 0,

or equivalently, g˜ba N a = g˜0b , b = 1, 2, 3 where g˜μν is the metric in the x-system. These are three linear equations for the three functions N a , a = 1, 2, 3. It is usual to denote g˜ab by qab can call it as the spatial metric. We next show that if ((q ab )) is the inverse of the matrix ((qab )), then μ ν q μν = q ab X,a X,b Indeed, we have μ ν q μν + nμ nν = g μν = g˜αβ X,α X,β μ ν = g˜ab X,a X,b + g˜0b X,tμ Xbν + μ X,tν + g˜00 X,tμ X,tν +˜ g a0 X,a

Equating the spatial components of the metric on both the sides gives μ ν ν q μν = g˜ab X,a X,b + g˜0b N μ X,b

+˜ g a0 X,tμ N ν + g˜00 N μ N ν μ ν = X,a X,b (˜ g ab + g˜0b N a + g˜a0 N b + g˜00 N a N b )

Thus, we have to verify that the inverse of ((qab )) = ((˜ gab )) is given by q ab = g˜ab + g˜0b N a + g˜a0 N b + g˜00 N a N b where (N a ) satisfies g˜ba N a = g˜0b , b = 1, 2, 3 This is an elementary matrix identity and we leave it as an exercise to the reader. To prove it, we must simply use the identity g˜μν g˜να = δμα

608

General Relativity and Cosmology with Engineering Applications so that

g˜ab g˜bc = δac − g˜a0 g˜0c

and

g˜ab g˜b0 = −g˜a0 g˜00 , g˜0b g˜bc = −g˜00 g˜0c , g˜0b g˜b0 = 1 − g˜00 g˜00

Now consider a spatial vector uμ , ie, uμ nμ = 0. Then, we have (writing ∇μ fν = fν:μ ) Dμ uν = qμα qνβ ∇α uβ and D ρ D μ uν = Dρ qμα qνβ ∇α uβ '

'

= qρσ qμμ qνν ∇σ qμα' qνβ' ∇α uβ Remark: The spatial covariant derivative Dμ annihilates the spatial metric tensor qμν just as the covariant derivative ∇μ annihilates the metric tensor gαβ . In fact, we have since 0 = ∇μ g αβ = ∇μ (q αβ + nα nβ ) the following: Dμ q αβ = qμρ qσα qγβ ∇ρ q σγ = −qμρ qσα qγβ ∇ρ (nσ nγ ) = 0 since ∇ρ is a derivation and qσα nσ = 0. ˜ defined above to the curvature We now wish to relate the spatial curvature R of the spatial covariant derivative Dμ . We have ' ' '' '' ' Dρ Dμ uν = qρρ qμμ qνν ∇ρ' qμμ'' qνν'' ∇μ' uν ' '

''

'

''

'

= qρρ qμμ qνν qμμ'' qνν'' ∇ρ' ∇μ' uν ' '

''

'

''

'

+qρρ qμμ qνν ∇ρ' (qμμ'' qνν'' )∇μ' uν ' '

'

'

= qρρ qμμ qνν ∇ρ' ∇μ' uν ' '

''

'

''

'

'

'

−qρρ qμμ qνν ∇ρ' (δμμ'' nν nν '' + δνν'' nμ nμ'' )∇μ' uν ' '

'

'

= qρρ qμμ qνν ∇ρ' ∇μ' uν ' '

'

''

'

'

''

'

'

−qρρ qμμ qνν ∇ρ' (nν nν '' )∇μ' uν ' −qρρ qμμ qνν ∇ρ' (nμ nμ'' )∇μ' uν ' '

'

'

= qρρ qμμ qνν ∇ρ' ∇μ' uν '

General Relativity and Cosmology with Engineering Applications '

'

''

'

'

''

'

'

609

−qρρ qμμ qνν ∇ρ' (nν '' )nν ∇μ' uν ' −qρρ qμμ qνν ∇ρ' (nμ'' )nμ ∇μ' uν ' '

'

'

= qρρ qμμ qνν ∇ρ' ∇μ' uν ' '

'

'

'

'

'

−qμμ Kρν nν ∇μ' uν ' −qνν Kρμ nμ ∇μ' uν ' Now observe that

−qμμ Kρν nν ∇μ' uν ' '

= qμμ Kρν uν ' ∇μ' nν

'

'

= qμμ uα qνα' Kρν ∇μ' nν

'

= Kμα Kρν uα It follows therefore by interchanging ρ and μ and taking the difference that [Dρ , Dμ ]uν = '

'

'

qρρ qμμ qνν [∇ρ' , ∇μ' ]uν ' +(Kμα Kρν − Kνα Kρμ )uα '

'

'

= [qρρ qμμ qνν Rρα' μ' ν ' +(Kμα Kρν − Kνα Kρμ )]uα ˆ denotes the curvature (spatial) associated with the spatial co­ Therefore, if R variant derivative Dμ , then we get ˆ=R ˜ + Kμρ K μρ − K 2 R ∫

and hence S=

∫ =

˜ + 2(Kμν K νμ − K 2 ))√−gd4 X (R ˆ + (Kμν K μν − K 2 ))√−gd4 X (R

(Note that K μν = K νμ ). μ . Note that this is a vector For each a = 1, 2, 3, consider the vector field X,a field since ¯μ ¯ μ = ∂X X ν X ,a ∂X ν ,a ¯ are three different coordinate systems and X μ and X ¯ μ stand Here x, X, X ,a ,a μ μ μ ¯ μ ∂X ∂X μ and . Note that T = X = is also a vector field respectively for ∂X a a 0 ,t ∂x ∂x ∂x and as mentioned above, it is decomposed as

T μ = N μ + N nμ

610

General Relativity and Cosmology with Engineering Applications μ . where gμν N μ nν = 0 with the condition that N μ is spatial, ie, N μ = N a X,a We now compute the Lie derivative μ ν X,b (Ln q)μν (Ln q)ab = X,a

Now, (Ln q)μν = nρ qμν,ρ + qρν nρ,μ + qρμ nρ,ν since qρν nρ = 0 we get (Ln q)μν = (T ρ − N ρ )qμν,ρ /N + qρν (T ρ − N ρ ),μ /N + qρμ (T ρ − N ρ ),ν /N = N −1 (LT −N q)μν = N −1 (LT q)μν − N −1 (LN q)μν Now μ ν (LT q)ab = X,a X,b (LT q)μν μ ν α = X,a X,b (T α qμν,α + qαν T,μ + qαμ T,να ) ν α ν μ μ X,b qμν,t + qαν T,a X,b + qαμ T,bα X,a = X,a

Now, μ ν X,b ),t = qab,t = (qμν X,a μ μ μ ν ν ν + qμν X,a X,bt X,b + qμν X,at X,b qμν,t X,a

and noting that α T,bα = X,bt

it follows that qab,t = (LT q)ab Now define μ ν X,b Kμν Kab = X,a

It is elementary that μ ν X,b ∇ μ nν Kab = Kba = X,a

(Ln q)μν = nα ∇α qμν + qαν ∇μ nα + qαμ ∇ν nα Now, nα ∇α qμν = −nα ∇α (nμ nν ) Thus, μ ν X,b nα ∇α qμν = X,a μ ν X,b ∇α (nμ nν ) = 0 −nα X,a

since μ nμ = 0 X,a

Further, μ ν X,b qαν ∇μ nα X,a

General Relativity and Cosmology with Engineering Applications μ ν = X,a X,b gαν ∇μ nα μ ν = X,a X,b ∇μ nν = Kab

Thus, μ ν X,b (Ln q)μν = 2Kab (Ln q)ab = X,a

Thus, we obtain the fundamental identity 2N Kab = qab,t − (LN q)ab We note that (LN q)ab is a purely spatial tensor. In fact, (LN q)ab = X,a μ X,b ν (LN q)μν μ ν α α = X,a X,b (N α qμν,α + qαν N,μ + qαμ N,ν )

Now, μ ν α X,b N c X,c qμν,α = X,a μ ν μ ν μ ν X,b = N c ((qμν X,a X,b ),c − qμν (X,a X,b ),c ) N c qμν,c X,a ν μ X,b ),c ) = N c (qab,c − qμν (X,a ν μ − N c qμb X,ac = qab,c N c − N c qaν X,bc

Further, μ ν α X,b N,μ qαν = X,a α α qαb = (N c X,c ),a qαb = = N,a c α qca + N c X,ca qαb N,a

Likewise for the last term. Combining all this we get c qac + N,bc qbc (LN q)ab = N c qab,c + N,a

proving that the lhs is purely spatial. We can now write μ ν X,b = K μν Kμν Kab K ab = K ab Kμν X,a

and μ ν X,b = q μν Kμν = K q ab Kab = q ab Kμν X,a

Thus, we can write ∫ S=

ˆ + Kab K ab − K 2 )√−gd4 X = (R

where '

'

Kab K ab = (4N 2 )−1 (q aa q bb (qab,t − (LN q)ab )(qa' b' ,t − (LN q)a' b' ) and

K = q ab Kab = (2N )−1 q ab (qab,t − (LN q)ab )

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General Relativity and Cosmology with Engineering Applications

This gives us an explicit representation of S in terms of the space and time derivatives of the position fields (qab , N, N a ) provided we also decompose the √ volume element −g into similar terms. We have μ ν gμν X,a X,b = qab = g˜ab

qab N b = qa0 = g˜a0 g˜00 = gμν T μ T ν = gμν (N μ + N nμ )(N ν + N nν ) = qab N a N b + N 2 ˜ = ((˜ Thus, we can write in matrix notation using G = ((gμν )), G gμν )), Q = a ((qab )), Z = (N ), ( ) Q QZ ˜ G= (QZ)T Z T QZ + N 2 It is easy to verify that

˜ = g˜ = qN 2 detG

where q = detQ = det((qab )) Thus, we get using √

−gd4 X =

√ √ −˜ gd4 x = N −qd4 x

and hence the final form of our action integral is ∫ ˆ + (4N 2 )−1 (q aa' q bb' (qab,t − (LN q)ab )(qa' b' ,t − (LN q)a' b' ) S = (R √ −(4N 2 )−1 [q ab (qab,t − (LN q)ab )]2 )N −qd4 x

Some remarks on quantum gravity qab is the spatial metric and eia is a triad corresponding to it, ie, i, a = 1, 2, 3 and eia eib = qab with summation over the repeated index i being assumed. Thus, the metric after transforming spatial vectors by such a triad, is the Euclidean metric in R3 which is invariant under SO(3) or equivalently, under SU (2). Thus, if τi , i = 1, 2, 3 are the Pauli spin matrices, we can regard the triad components √ eia as the su(2) matrices eai τi . It is customary to replace the τj' s by i = −1 times the corresponding Pauli spin matrices. From the triads eia , Ashtekar constructs Eai which will be the quantized momentum field operators and also ' ' s and Eai s, Ashtekar constructs the quantized position field op­ from the Kab i erators Aa . Earlier, we had expressed the action S in ADM form, ie, as the sum of the spatial part of the curvature scalar and a component involving the qab , qab,t and Kab . This form of the action integral immediately enables us to construct the momentum field operators P ab = δS/δqab,t and express these in terms of {Kab , qab , qab,t }. There are other components of the position field,

General Relativity and Cosmology with Engineering Applications

613

namely N, N a , a = 1, 2, 3 apart from qab but although their spatial derivatives appear in the action integral, their time derivatives do not and hence the canon­ ical momentum operators corresponding to these are zero. This means that we are now dealing with a Lagrangian field problem with constraints and hence the Poisson bracket has to be replaced by the Dirac bracket. Without going into this in too much detail, we emphasize that Ashtekar’s position and momentum variables Eai , Aia can be regarded as our new position and momentum variables in place of the previous ones {qab , N, N a }, P ab . Ashtekar proved that if these new variables satisfy the canonical commutation relations, then the previous ones will also satisfy the same and further that the Hamiltonian of the gravita­ tional field can be expressed as integrals of elementary polynomials of the new variables. In loop quantum gravity, one normally constructs the one parameter subgroup of SU (2) generated by the su(2) valued vector field Aa = Aia τi and discretizes the action integral into sum’s over the edges of a graph of the these SU (2) elements. Specifically, the continuous spatial manifold is replaced by a discrete graph consisting of edges connecting points in the spatial manifold and the Hamiltonian (the integral of the Hamiltonian density over the spatial variables) is approximated by replacing the spatial integrals involving Aia with sums of integrals of the same over the edges of a small box with the integrals of ∫ u(2) Lie algebra elements replaced by the corresponding SU (2) group ele­ ments obtained by translating the Lie algebra valued field along edges of the box. Specifically, we solve the differential equation '

dh(t' )/dt' = pa (t' )h(t' )Aa (t, p(t' )) for 0 ≤ t' ≤ T where t' ∈ [0, T ] → p(t) is the concerned edge of the graph. If h(0) ∈ SU (2), then it is clear that h(T ) ∈ SU (2) and one can use this translation formula to approximate the su(2) valued connection Aa (t, r) by SU (2) elements of the form (h(0)−1 h(T )−I2 )/T . After this, the Hamiltonian becomes a function of position and momentum variables where now the position variables are SU (2) elements and the momenta are the same, ie, Ea = Eai τi which can be regarded as vector fields on a differentiable manifold whose points are SU (2) elements. A wave function for the gravitational field in the position representation is thus simply a function of the SU (2) group elements he where e ranges over all the edges of the graph. Such a function can be expressed using the PeterWeyl theorem as a linear combination of the product Πe∈Γ [πe (he )]me ,ne where Γ is the set of the edges of the graph and πe is an irreducible representation of SU (2) with me , ne positive integers so that Xme ,ne denotes the (me , ne )th element of the matrix X. Now fix an edge e of the graph Γ and consider the corresponding position variable he ∈ SU (2). We wish to construct the associated momentum operator. From quantum mechanics, this momentum operator must be something like −i∂/∂he , ie a left/right invariant vector field on SU (2). The position space is the set of all maps SU (2)Γ of all maps from the set of edges Γ of the graph into SU (2), or equivalently, the ordered tuple {he : e ∈ Γ} where he ∈ SU (2). Consider the left invariant vector field Lj on

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General Relativity and Cosmology with Engineering Applications

SU (2) defined by d f (g.exp(tτj ))|t=0 dt and the corresponding right invariant vector field Rj defined by Lj f (g) =

Rj f (g) =

d f (exp(tτj )g)|t=0 dt

We can write these operators as Lj =

∑

(gτj )AB

∂ ∂gAB

(τj g)AB

∂ ∂gAB

A,B=1,2

Rj =

∑ A,B=1,2

Thus, Lj gCD f (g) = (gτj )CD f (g) + gCD Lj f (g) Thus, [Lj , gCD ] = (gτj )CD and likewise, [Rj , gCD ] = (τj g)CD In the limit when g → I2 so that g ≈ I2 + xi τi , ie g is parametrized by its Lie algebra coordinates xi , i = 1, 2, 3, we get writing f1 (x1 , x2 , x3 ) = f (g) = f (I + xi τi ), d Lj f1 (x1 , x2 , x3 ) = f ((I + xi τi )(I + tτj ))|t=0 dt d = f (I + tτj + xi τi + txi τi τj )|t=0 dt d = f (I + tτj + xi τi + tε(ijk)xi τk )|t=0 dt d = f1 (x1 + t(δ1j + ε(ij1)xi ), x2 + t(δ2j + ε(ij2)xi ), x3 + t(δ3j + ε(ij3)xi ))|t=0 dt Taking j = 1, 2, 3 successively gives us L1 f1 (x) =

d f1 (x1 + t, x2 + tx3 , x3 − tx2 )|t=0 = dt

= (∂/∂x1 + (x3 ∂/∂x2 − x2 ∂/∂x3 ))f1 (x) More precisely, we can define the position and momentum operators correspond­ ing to an edge e of the graph by regarding a function f : SU (2) → C as the function f1 : su(2) → C defined by f1 (X) = f (exp(X))

General Relativity and Cosmology with Engineering Applications

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Then the position operators at e are multiplication by X 1 , X 2 , X 3 where X = X i τi and the momentum operators are Pk = −i∂/∂X k , k = 1, 2, 3. We require to evaluate the action of the momentum operators on f : iPk f1 (X) = ∂f (exp(X))/∂X k To evaluate this, we require the differential of the exponential map. From basic Lie group theory, dexp(X) = exp(X).[(I − exp(−ad(X))/ad(X)](dX) so exp(−X)(∂/∂X k )exp(X) = [(I − exp(−ad(X)))/ad(X)](τk ) Let k = 1. Then, ((I − exp(−ad(X))/ad(X))(τ1 ) =

∞ ∑

(−1)n−1 (n!)−1 (ad(X))n−1 (τ1 )

n=1

Now,

ad(X)(τ1 ) = X 2 [τ2 , τ1 ] + X 3 [τ3 , τ1 ] = 2τ2 X 3 − 2τ3 X 2 (ad(X))2 (τ1 ) = 2[X 1 τ1 + X 3 τ3 , τ2 ]X 3 − 2[X 1 τ1 + X 2 τ2 , τ3 ]X 2 = 4(X 1 τ3 − X 3 τ1 )X 3 + 4(X 1 τ2 − X 2 τ1 )X 2 = 4(−(X 22 + X 32 )τ1 + X 1 X 2 τ2 + X 1 X 3 τ3 )

etc. Another way to specify the momentum operators on SU (2) would be by expressing any g ∈ SU (2) as g = exp(x1 τ1 ).exp(x2 τ2 ).exp(x3 τ3 ) and then write f (g) = f1 (x1 , x2 , x3 ) Then,

f (g.exp(tτ3 )) = f1 (x1 , x2 , x3 + t)

so that the left invariant vector field corresponding to τ3 is ∂/∂x3 . Likewise, ∂f (g)/∂x2 = (d/dt)f (exp(x1 τ1 )exp(x2 τ2 )exp(tτ2 )exp(x3 τ3 ))|t=0 = (d/dt)f (g.exp(−x3 τ3 )exp(tτ2 )exp(x3 τ3 ))|t=0 Now,

exp(−x3 τ3 )τ2 exp(x3 τ3 ) = exp(−x3 ad(τ3 ))(τ2 ) = τ2 − 2x3 τ1 + (−2x3 )2 τ2 /2! + ... = cosh(2x3 )τ2 − sinh(2x3 )τ1

Thus, [41] Plasmonic waveguides

616

General Relativity and Cosmology with Engineering Applications Plasma equations inside a cavity resonator. f (t, r, v) = f0 (r, v) + f1 (t, r, v) f0 (r, v) = C.exp(−βm(v 2 /2 + U (r))

f0 is the equilibrium distribution function. It satisfies (v, ∇r )f0 + (−∇U (r), ∇v )f0 = 0 ie the equilibrium collisionless Boltzmann equation. f1 is the perturbation to the distribution function caused by the interaction of the plasma with the em fields in the guide. The charge density within the guide is ∫ ρ(t, r) = q f (t, r, v)d3 v and the current density is ∫

vf1 (t, r, v)d3 v

J(t, r) = q

f1 satisfies the perturbed Boltzmann equation: f1,t + (v, ∇r )f1 + (q/m)(E(t, r) + v × B(t, r), ∇v )f0 (t, r) = −f1 (t, r, v)/τ (v) or equivalently, f1,t + (v, ∇r )f1 − βq(E, v)f0 + f1 /τ = 0 In the frequency domain, this translates to jωf1 (ω, r, v) + (v, ∇r )f1 (ω, r, v) − βq(E(ω, r), v)f0 (r, v) + f1 (ω, r, v)/τ = 0 where E(t, r) and B(t,r) are calculated from ∫ ∇ × E = −B,t , ∇ × B = μq vf1 (t, r, v)d3 v + μεE,t , ∫ ∇.E = (q/ε)

f1 (t, r, v)d3 v,

∇.B = 0 Thus,

∇2 E − μεE,tt = ∇ρ/ε + μJ,t ∇2 B − μεB,tt = −μ∇ × J

or equivalently, in the frequency domain, (∇2 + k 2 )E(ω, r) = ∇ρ(ω, r)/ε + jωμJ(ω, r)

General Relativity and Cosmology with Engineering Applications

617

(∇2 + k 2 )B(ω, r) = −μ∇ × J(ω, r) We assume that the resonator walls are perfect electric conductors. Ex vanishes when y = 0, b, z = 0, d, Ey vanishes when x = 0, a, z = 0, d, Ez vanishes when x = 0, a, y = 0, b. Hx vanishes when x = 0, a, Hy vanishes when y = 0, b and finally Hz vanishes when z = 0, d. So these fields admit the expansions ∑ Ex (x, y, z, t) = Ex [n, m, p, t]cos(nπx/a)sin(mπy/b)sin(pπz/d) n,m,p≥1

∑

Ey (x, y, z, t) =

Ey [n, m, p, t]sin(nπx/a)cos(mπy/b)sin(pπz/d)

n,m,p≥1

∑

Ez (x, y, z, t) =

Ez [n, m, p, t]sin(nπx/a)sin(mπy/b)cos(pπz/d)

n,m,p

∑

Hx (x, y, z, t) =

Hx [n, m, p, t]sin(nπx/a)cos(mπy/b)cos(pπz/d)

n,m,p

∑

Hy (x, y, z, t) =

Hy [n, m, p, t]cos(nπx/a)sin(mπy/b)cos(pπz/d)

n,m,p

Hz (x, y, z, t) =

∑

Hz [n, m, p, t]cos(nπx/a)cos(mπy/b)sin(pπ/d)

n,m,p

These expansions can be derived from the standard Maxwell curl relations along with the stated boundary conditions in a region free of sources (we are assuming that in a neighbourhood of the resonator walls, there is no plasma): ˆ E⊥ = (−γ/h2 )∇⊥ Ez − (jωμ/h2 )∇Hz × z, H⊥ = (−γ/h2 )∇⊥ Hz + (jωε/h2 )∇Ez × zˆ where γ = −∂/∂z and further, (∇2⊥ + h2 )Ez = 0, (∇2⊥ + h2 )Hz = 0 Now, consider the vector source terms: s(r, t) = ∇ρ/ε + μJ,t g(r, t) = −μ∇ × J appearing on the right side of the above wave equations for the electric and magnetic fields. We can express these in terms of the perturbation f1 to the Boltzmann distribution function: ∫ ∫ s(r, t) = (q/ε) ∇r f (t, r, v)d3 v + μq vf1,t (t, r, v)d3 v ∫ g(r, t) = −qμ

(∇r f1 (t, r, v) × v)d3 v

618

General Relativity and Cosmology with Engineering Applications In accordance with the boundary conditions on E, we expand ∑ sx (t, r) = sx [n, m, p, t]cos(nπx/a)sin(mπy/b)sin(pπ/d) n,m,p

sy (t, r) =

∑

sy [n, m, p, t]sin(nπx/a)cos(mπy/b)sin(pπz/d)

n,m,p

sz (t, r) =

∑

sz [n, m, p, t]sin(nπx/a)sin(mπy/b)cos(pπz/d)

nmp

Then we get in the frequency domain Ex (ω, x, y, z) =

∑

sx [n, m, p, ω] 2 − π 2 (n2 /a2 + m2 /b2 + p2 /d2 ) k nmp

and likewise for Ey , Ez . Our aim is to express E as a linear functional of f1 and then substitute this expression into the Boltzmann equation to obtain a homo­ geneous linear integro-differential equation for f1 from which the characteristic freqyencies of plasma oscilations can be computed by setting the determinant of the linear operator acting on f1 to zero after appropriately approximating this operator by a finite matrix. We note that ∫ ∫ s(ω, r) = q ∇r f1 (ω, r, v)d3 v + μqjω vf1 (ω, r, v)d3 v so that on using integration by parts with the assumption that f1 vanishes on the boundary of the resonator, ∫ ∫ sx (ω, r) = (q/ε) f1,x (ω, r, v)d3 v + jμqω vx f1 (ω, r, v)d3 v = −(8q/εabd) ∫ ×

∑

f1 (ω, r' )sin(nπx' /a)sin(mπy/b)sin(pπz ' /d)dx' dy ' dz ' d3 v +(8jμqω/abd)

∫ ×

(nπ/a)cos(nπx/a)sin(mπy/b)sin(pπz/d)

nmp

∑

cos(nπx/a)sin(mπy/b)sin(pπz/d)

nmp

vx f1 (ω, r' , v)cos(nπx' /a)sin(mπy ' /b)sin(pπz ' /d)dx' dy ' dz ' d3 v

We can express this as Ex (ω, r) = jωLx,1 f1 (ω, r) + Lx,2 f (ω, r) where Lx,1 and Lx,2 are linear integral operators defined by ∑ [cos(nπx/a)sin(mπy/b)sin(pπz/d) Lx,1 (r|r' , v ' ) = (8μq/abd)vx' nmp

General Relativity and Cosmology with Engineering Applications

619

×cos(nπx' /a)sin(mπy ' /b)sin(pπz ' /d)]/(k 2 − h[n, m, p]2 ) where

h[n, m, p]2 = π 2 (n2 /a2 + m2 /b2 + p2 /d2 )

and Lx,2 (r|r' ) = Lx,2 (r|r' , v) = −(8q/εabd)

∑

[(nπ/a)cos(nπx/a)sin(mπy/b)sin(pπz/d)

nmp

×sin(nπx' /a)sin(mπy ' /b)sin(pπz ' /d)]/(k 2 − h[n, m, p]2 ) Thus, we can write ∫ ∫ ' ' ' ' 3 ' 3 ' Ex (ω, r) = jω Lx,1 (r|r , v )f1 (ω, r , v )d r d v + Lx,2 (r|r' )f1 (ω, r' , v ' )d3 r' d3 v ' Likewise for the other components: ∑ Ly,1 (r|r' , v ' ) = (8μq/abd)vy' [sin(nπx/a)cos(mπy/b)sin(pπz/d) nmp

×sin(nπx' /a)cos(mπy ' /b)sin(pπz ' /d)]/(k 2 − h[n, m, p]2 ) where

h[n, m, p]2 = π 2 (n2 /a2 + m2 /b2 + p2 /d2 )

and Ly,2 (r|r' ) = Ly,2 (r|r' , v) = −(8q/εabd)

∑

[(mπ/b)sin(nπ/a)cos(mπy/b)sin(pπz/d)

nmp

×sin(nπx/a)sin(mπy/b)sin(pπz/d)]/(k 2 − h[n, m, p]2 ) and finally, Lz,1 (r|r' , v ' ) = (8μq/abd)vz'

∑

[sin(nπx/a)sin(mπy/b)cos(pπz/d)

nmp

×sin(nπx' /a)sin(mπy ' /b)cos(pπz ' /d)]/(k 2 − h[n, m, p]2 ) where

h[n, m, p]2 = π 2 (n2 /a2 + m2 /b2 + p2 /d2 )

and Lz,2 (r|r' ) = Lz,2 (r|r' , v) = −(8q/εabd)

∑

[(pπ/d)sin(nπ/a)sin(mπy/b)cos(pπz/d)

nmp

sin(nπx/a)sin(mπy/b)sin(pπz/d)]/(k 2 − h[n, m, p]2 ) We further define the kernels L1 (r, v|r' , v ' ) = Lx,1 (r|r' , v ' )vx + Ly,1 (r|r' , v ' )vy + Lz,1 (r|r' , v ' )vz

620

General Relativity and Cosmology with Engineering Applications

and

L2 (r, v|r' , v ' ) = Lx,2 (r|r' )vx + Ly,2 (r|r' )vy + Lz,2 (r|r' )vz

Then, the linearized Boltzmann equation becomes (jω + (v, ∇r ) + (q/m)(jωL1 + L2 ) + 1/τ )f1 (ω, r, v) = 0 and hence, formally, we can define the possible plasma oscillation frequencies by the solution of det(jω(1 + qL1 /m) + (v, ∇r ) + qL2 /m + 1/τ ) = 0

Reference:Pragya Shilpi, H.Parthasarathy and D.K.Upadhyay, ”Some theo­ retical studies in plasmonic waveguides”, Technical report, NSIT, 2017. [42] Some extra problems in the gtr. Cosmological metrics: dτ 2 = dt2 − S 2 (t)f (r)dt2 − S 2 (t)r2 (dθ2 + sin2 (θ)dφ2 ) This metric is to be determined when the energy-momentum tensor of matter is T μν = (ρ(t) + p(t))v μ v ν − p(t)g μν where v r = 0, r = 1, 2, 3. To check that this condition also called the co-moving solution is a solution to the geodesic equation, we have to check that d2 v r /dτ 2 + Γrμν v μ v ν = 0 In other words, we must check that Γr00 = 0, r = 1, 2, 3 or equivalently, in view of the diagonal nature of the metric, Γr00 = 0 ie g00,r = 0, r = 1, 2, 3 This is true since g00 = 1. We now calculate Sμν = Tμν − T gμν /2 First, T = gμν T μν = ρ + p − 4p = ρ − 3p, so S00 = T00 − T g00 /2 =

General Relativity and Cosmology with Engineering Applications

621

ρ + p − p − (ρ − 3p)/2 = (3p + ρ)/2 S11 = T11 − T g11 /2 = −pg11 − (ρ − 3p)g11 /2 (p − ρ)g11 /2 = (ρ − p)S 2 f /2 S22 = T22 − T g22 /2 = −pg22 − (ρ − 3p)g22 /2 = (p − ρ)g22 /2 = (ρ − p)r2 S 2 /2 S33 = T33 − T g33 /2 = (ρ − p)r2 S 2 sin2 (θ)/2 α α β R00 = Γα 0α,0 − Γ00,α − Γ00 Γαβ β +Γα 0β Γ0α

=

3 ∑ k=1

Γk0k,0 +

3 ∑

(Γk0k )2

k=1

Γk0k = g kk Γk0k = (log(gkk ),0 )/2 = S ' /S So

R00 = 3(S ' /S)' + 3(S ' /S)2 = 3S '' /S

The other linearly independent equation can either be obtained from R11 or equivalently using T:νμν = 0 in view of the Bianchi identity for the Einstein tensor. The other field equations will imply that f (r) = (1 − kr2 )−1 . So we shall assume this. We get ((ρ + p)v μ v ν ):ν − p,μ = 0 or ((ρ + p)v ν ):ν v μ + (ρ + p)v ν v:μν − p,μ = 0 so

√ √ ((ρ + p)v ν −g),ν = p,ν v ν −g √ √ which gives on substituting v 0 = 1, v k = 0, k = 1, 2, 3, −g = S 3 r2 sin(θ) f (r), ((ρ + p)S 3 )' = p' S 3 All ρ, p, S are functions of time only. This equation can be expressed as (ρS 3 )' = −3pS 2 S ' or equivalently,

(4πρS 3 /3)' = −4πS 2 pS ' − − − (1)

This equation has a simple physical interpretation: 4πρS 3 /3 is the total mass or equivalently energy inside a sphere of radius S(t) and its time derivative gives the rate of energy increase of matter within this sphere. On the other hand −4πS 2 p is the total force due to external pressure on the spherical surface

622

General Relativity and Cosmology with Engineering Applications and hence −4πS 2 pS ' is the rate at which pressure forces do work on the matter within the sphere. Thus (1) is a simple energy equation in Newtonian mechanics. On the other hand the Einstein field equation R00 = −8πGS00 gives

3S '' /S = −8πG(3p + ρ)/2

or

S '' = −4πG(p + ρ/3)S − − − (2)

We now assume Newtonian mechanics for a particle of mass m located on the spherical surface having radius S(t). The energy conservation equation for this mass is ' mS 2 /2 − G(4πρS 3 /3S)m = E gives

'

S 2 /2 − 4πGρS 2 /3 = E/m − − − (3)

If we neglected pressure in (2), then we would get S '' = −4πGρS/3 or equivalently,

S '' = −(G.4πρS 3 /3S 2 ) − − − (4)

which is the Newtonian equation of motion of a particle placed on the surface of the sphere. We shall now see how the Einstein field equation for R11 determines the constant E/m in (3). We calculate the R11 component of the Ricci tensor: α α β α β R11 = Γα 1α,1 − Γ11,α − Γ11 Γαβ + Γ1β Γ1α 1 2 3 Γα 1α = Γ11 + Γ12 + Γ13

= (1/2)

3 ∑

(loggkk ),1 = (1/2)(f ' /f + 4/r) = f ' /2f + 2/r

k=1

and so

' ' 2 Γα 1α,1 = (f /2f ) − 2/r 0 1 Γα 11,α = Γ11,0 + Γ11,1 =

(−1/2)g11,00 + (f ' /2f )' = (SS ' )' f + f ' /2f So α Γα 1α,1 − Γ11,α =

−2/r2 − (SS ' )' f β Γα 11 Γαβ =

Γ011 Γk0k + Γ111 Γk1k =

General Relativity and Cosmology with Engineering Applications

623

(−1/4)g11,0 (loggkk ),0 + (1/4)(log(g11 ),1 )(loggkk ),1 ) '

'

= (3SS ' f )(S ' /S) + (f ' /4f )(f ' /f + 4/r) = 3S 2 f + f 2 /4f 2 + f ' /rf β Γα 1β Γ1α = 3 2 ) = (Γ111 )2 + (Γ212 )2 + (Γ13 0 +2Γ110 Γ11

= (1/4)(((logg11 ),1 )2 + ((logg22 ),1 )2 + ((logg33 ),1 )2 ) − (1/2)(log(g11 )),0 g11,0 '

= (1/4)(f 2 /f 2 + 8/r2 ) + (1/2)(2S ' /S)(2SS ' f ) Combining all this, we get finally, '

R11 = −(SS ' )' f − S 2 f − f ' /rf Also,

S11 = (ρ − p)S 2 f /2

so the second Einstein field equation becomes on using f ' /rf = −2k/(1 − kr2 ) = −2kf, '

−(SS ' )' − S 2 + 2k = −4πG(ρ − p)S 2 or equivalently,

'

2S 2 + SS '' − 2k − 4πG(ρ − p)S 2 = 0

Combining this with the previous Einstein field equation S '' = −4πG(p + ρ/3)S gives us

'

2S 2 − 4πG(p + ρ/3)S 2 − 2k − 4πG(ρ − p)S 2 = 0

which simplifies to

'

S 2 − 8πGρS 2 /3 = 2k

which can also be expressed as '

S 2 /2 − 4πGρS 2 /3 = k which is the same as the Newtonian result (3) provided we identify k with E/m. Thus, Newtonian cosmology yields the full results of general relativity. [43] Quantum communication theory. Lower bound on M in the greedy algorithm: ∑ p(n) (u)T r(ρ(u)D) T r(¯ ρ⊗n D) = u∈An

624

General Relativity and Cosmology with Engineering Applications ∑

≥

p(n) (u)T r(ρ(u)D)

u∈T (n,p,δ)

≥ γp(n) (T (n, p, δ)) ≥ γ(1 − 1/δ 2 ) on using T r(ρ(u)D) > gamma, ∀u ∈ T (n, p, δ) and the Chebyshev inequality in the form p(n) (T (n, p, δ)) > 1 − 1/δ 2 . Further, noting that ∑ E(¯ ρ⊗n , δ) = |u >< u| u∈T (n,Pρ¯,δ)

and using the fact that u ∈ T (n, Pρ¯, δ) implies √ |N (x|u) − nPρ¯(x)| < nPρ¯(x)(1 − Pρ¯(x)), x ∈ I where ρ¯ =

∑

|x > Pρ¯(x) < x|

x∈I

is the spectral representation of ρ¯, it follows that √ √ nPρ¯(x)−δ nPρ¯(x)(1 − Pρ¯(x)) < N (x|u) < nPρ¯(x)+δ nPρ¯(x)(1 − Pρ¯(x)), x ∈ I and hence ¯ 1δ 2−nH(ρ)−K

√

where K1 = −

n

¯ 1δ < Πx∈I Pρ¯(x)N (x|u) < 2−nH(ρ)+K

√

n

∑√ Pρ¯(x)(1 − Pρ¯(x))log(Pρ¯(x)) x∈I

This is the same as ¯ 1δ 2−nH(ρ)−K

√

n

(n)

¯ 1δ < Pρ¯ (u) < 2−nH(ρ)+K

√

n

where u ∈ T (n, Pρ¯, δ), we get ¯ 1δ 2−H(ρ)−K

√

n

¯ 1δ E(¯ ρ⊗n , δ) < ρ¯⊗n E(¯ ρ⊗n , δ) < 2−nH(ρ)+K

Taking traces gives us ¯ 1δ T r(E(¯ ρ⊗n , δ)) ≤ 2nH(ρ)+K

√

n

and Chebyshev’s inequality gives (n)

T r(¯ ρ⊗n E(¯ ρ⊗n , δ)) = Pρ¯ (T (n, Pρ¯, δ)) > 1 − 1/δ 2

√

n

E(¯ ρ⊗n , δ)

General Relativity and Cosmology with Engineering Applications

625

This is in fact a part of the Schumacher quantum compression theorem. Now, applying the lemma to the inequality ¯ 1δ ρ¯⊗n E(¯ ρ⊗n , δ) ≤ 2−nH(ρ)+K

√

n

E(¯ ρ⊗n , δ)

gives us ¯ 1δ T r(D) ≥ 2nH(ρ)−K

√

n

(T r(¯ ρ⊗n D) − T r(¯ ρ⊗n (1 − E(¯ ρ⊗n , δ))

¯ 1δ ≥ (γ(1 − 1/δ 2 ) − 1/δ 2 )2nH(ρ)−K

√

n

and hence using M.2n

∑ x∈A

√ p(x)H(ρ(x))+K2 δ n

≥

M ∑

T r(E(n, uk , δ)) ≥

k=1

we get

¯ M ≥ A(δ)2n(H(ρ)−

M ∑

T r(Dk ) = T r(D)

k=1 ∑ x∈A

√ p(x)H(ρ(x))−K3 δ n

where A(δ)√> 0 provided that n is sufficiently large and δ grows with n, but not as fast as n, say as n1/2−σ where σ > 0. This result then gives us by writing Mn in place of n that liminfn→∞ n−1 log(Mn ) ≥ I(p, ρ) where I(p, ρ) = H(¯ ρ) −

∑

p(x)H(ρ(x))

x∈A

This is a Cq channel with source probability p, source alphabet A and quantum channel x → ρ(x) that maps A into the set of states in the Hilbert space H. Remark: If u ∈ T (n, p, δ), then √ √ np(x) − δ np(x)(1 − p(x)) < N (x|u) < np(x) + δ np(x)(1 − p(x)), x ∈ A This implies that T r(E(n, u, δ)) = Πx∈A T r(E(ρ(x)⊗N (x|u) , δ)) ≤ Πx∈A 2N (x|u)H(ρ(x))+K3 δ ≤ 2n

∑

√

n

√

x∈A

p(x)H(ρ(x))+K4 δ n

[44] Simulation of quantum stochastic differential equations. Abstract: In this problem, we consider the problem of simulating a quantum stochastic differential equation (qsde) in the sense of Hudson and Parthasarathy as a model for the noisy Schrodinger equation by constructing an truncated or­ thonormal basis for the Boson Fock space of the noisy bath using coherent vectors. We then simulate the Belavkin quantum filtering equation when the

626

General Relativity and Cosmology with Engineering Applications

measurements consists of a superposition of quantum Brownian motions and quantum Poisson processes. The filter dynamics is simulated as a matrix dif­ ference equation and finally, we simulate a control algorithm for removing a part of the GKSL noise using a technique developed by Luc Bouten. The paper concludes with some formulas for the rate of entropy increase for the GKSL equation, for the Belavkin filter and for the controlled Belavkin filter. I.The problem statement. The HP qsde is given by dU (t) = (−(iH + P )dt + L1 dA(t) + L2 dA(t)∗ + SdΛ(t))U (t) − − − (1) where H, P, L1 , L2 , S are system operators with H, P Hermitian and a relation­ ship exists between these operators so as to guarantee unitarity of the evolution, ie d(U ∗ U ) = 0. This condition is derived using the quantum Ito’s formula [KRP book] dAdA∗ = dt, (dΛ)2 = dΛ, dA.dΛ = dA, dΛ.dA∗ = dA∗ − − − (2) and all the other products of differentials of these three processes are zero. The system Hilbert space is chosen as h = Cp and the noise Hilbert space is the Boson Fock space Γs (L2 (R+ )). We choose the standard orthonormal basis |ηk >, k = 1, 2, ..., p for h and choose distinct vectors u1 , ..., uN ∈ L2 (R+ ), for example, uk (t) = (2/T )1/2 sin(2πkt/T )χ[0,T ] (t). We define the exponential vectors |e(u) >∈ Γs (L2 (R+ )) in the usual way and by Gram-Schmidtting the vectors |e(uk ) >, k = 1, 2, ...., N obtain an orthonormal set |ξr >=

r ∑

c(r, s)|e(us ) >, 1 ≤ r ≤ N

s=1

span{|e(ur ) >: 1 ≤ r ≤ N } = span{|ξr >: 1 ≤ r ≤ N } = M is a subspace of the Boson Fock space and if N is large enough, we can regard it as an approximate onb for the Boson Fock space. We now rewrite the HP equation (1) in this approximate basis after truncation as follows: U (t) is replaced by the pN × pN matrix with (N (a − 1) + r, N (b − 1) + s)th entry given by U (t, a, r; b, s) =< ηa ⊗ ξr |U (t)|ηb ⊗ ξs >, 1 ≤ a, b ≤ p, 1 ≤ r, s ≤ N We find that < ηa ⊗ ξr |dU (t)|ηb ⊗ ξs >= dU (t, a, r; b, s), ∑

< ηa ⊗ ξr |(−iH + P )U (t)|ηb ⊗ ξs >= < ηa ⊗ ξr |(iH + P )|ηc ⊗ ξm >< ηc ⊗ ξm |U (t)|etab ⊗ ξs >

c,m

=

p ∑ (iH + P )(a, c)U (t, c, r : b, s) c=1

General Relativity and Cosmology with Engineering Applications ∑

< ηa ⊗ ξr |L1 dA(t)U (t)|ηb ⊗ ξs >= L1 (a, c) < ηc ⊗ ξr |U (t)dA(t)|ηb ⊗ ξs >

c

∑

=

627

L1 (a, c) < ηc ⊗ ξr |U (t)|ηb ⊗ ξs' > c(s, k)d(k, s' )uk (t)dt

c,k,s'

=

∑

L1 (a, c)U (t, c, r; b, s' )c(s, k)d(k, s' )uk (t)dt

c,k,s'

where we have used ∑ c(s, k)|e(uk ) >, dA(t)|e(uk ) >= uk (t)dt|e(uk ) >, |ξs >= k

|e(uk ) >=

∑

d(k, s' )|ξs' >

s'

where ((d(k, s)) is the inverse of the matrix ((c(k, s)). Further, ∑

< ηa ⊗ ξr |L2 dA(t)∗ U (t)|ηb ⊗ ξs >= L2 (a, c)¯ c(r, k)d¯(k, r' )¯ ur' (t)U (t, c, r' ; b, s)dt

c,k,r '

by the same logic as used above. Finally, < ηa ⊗ ξr |SdΛ(t)U (t)|ηb ⊗ ξs >= ∑

S(a, c)¯ c(r, k)d(k, r' )¯ c(s, m)d¯(m, s' )¯ uk (t)um (t)U (t, c, r' ; b, s' )

c,k,m,r ' ,s'

where we use < e(u)|dΛ(t)|e(v) >= u ¯(t)v(t)dt < e(u)|e(v) >, and

¯(t)dt. < e(u)|e(v) > < e(u)|dA(t)∗ |e(v) >= u

In this way after discretizing time, we get from the HP equation. a difference equation for the evolution matrix entries U (t, a, r; b, s). This has been simulated in our work []. The Belavkin filter and stochastic Schrodinger equations simulations: When non-demolition measurements of the form Yo (t) = U (t)∗ Yi (t)U (t), Yi (t) = cA(t) + c¯A(t)∗ + Λ(t) are made, then the Belavkin filter has the form dπt (X) = Ft (X)dt +

inf ∑ty k=1

Gkt (X)(dYo (t))k

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General Relativity and Cosmology with Engineering Applications

where if X is a system observable and jt is the star unital homomorphism from the Banach space of bounded system observables B(h) into the space of observables defined on the space B(h ⊗ Γs (L2 ([0, t]) defined by the HP noisy Heisenberg evolution jt (X) = U (t)∗ XU (t) = U (t)∗ (X ⊗ I)U (t) then we define the conditional expectation πt (X) = E(jt (X)|ηt] ) where ηt] is the algebra generated by the Abelian family of operators {Yo (s) : s ≤ t}. All expectations and conditional expectations are taken in the system plus bath state |f ⊗ φ(u) >, where |f >∈ h is the system state and |φ(u) >= exp(− || u ||2 /2)|e(u) > is a normalized coherent state of the bath. Remark: The fact that ηt] is an Abelian family follows from the identity Yo (t) = U (T )∗ Yi (t)U (T ), T ≥ t and the fact that Yi (t) commutes with all the system operators. The above identity can be verified by calculating the differential of U (T )∗ Yi (t)U (T ) with respect to T and using the unitarity condition on U (T ) that is expressed entirely in terms of system observables which commutes with the purely bath observable Yi (t). Remark: We can give a general technique for constructing a family nondemolition processes in the sense of Belavkin, ie, process which jointly form an Abelian family and whose values at time t commute with HP evolved system observables js (X) for times s ≥ t. The procedure is to choose our bath Hilbert space as Γs (L2 (R) ⊗ Cp ) and on this space define the quantum noise processes Λab (t), 0 ≤ a, b ≤ p satisfying the quantum Ito rule dΛab (t).dΛcd (t) = εad dΛcb (t) where εad is one if a = d = / 0 and zero otherwise. It is immediately recognized that Λa0 (t) = Aa (t), Λ0a (t) = Aa (t)∗ , a ≥ 1 are annihilation and creation operators and Λab (t) for a, b ≥ 1 are the quantum Poisson processes defined in the notation of [KRP, book] by Λab (t) = Λ|b>< φ(u)|dM udM u ¯ ∫

We have [Hs , ρ(t)] =

[Hs , C(t, u, u ¯)] ⊗ |φ(u) >< φ(u)|dudu ¯

Hb |φ(u) >< φ(u) >= =

∑

∑

ωk uk a∗k |φ(u) < φ(u)|

k

uk + ∂/∂uk )(|φ(u) >< |phi(u)|) ωk uk (¯

k

∑

|φ(u) < φ(u)|Hb = ωk u ¯k (uk + ∂/∂ u ¯k )(|φ(u) >< φ(u)|)

k

and hence ∑

[Hb , |φ(u) >< φ(u)|] = ωk (uk ∂/∂uk − u ¯k ∂/∂ u ¯k )(|φ(u) >< φ(u)|)

k

∑

¯) ⊗ |φ(u) >< φ(u)|) = HI (t)(C(t, u, u fk (t)uk Xk C ⊗ (|φ(u) >< φ(u)|) + f¯k (t)Xk∗ C ⊗ (¯ uk + ∂/∂uk )(|φ(u) >< φ(u)|)

k

Thus, integration by parts gives HI (t)ρ(t) =

∑ k

∫ fk (t)Xk

uk C(t, u, u ¯) ⊗ (|φ(u) >< φ(u)|)dudu ¯

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General Relativity and Cosmology with Engineering Applications +

∑

f¯k (t)Xk∗

∫ (¯ uk − ∂/∂uk )C(t, u, u ¯) ⊗ (|φ(u) >< φ(u)|)dudu ¯

k

Likewise, ρ(t)HI (t) =

∑

∫ C(t, uu ¯)Xk ⊗ (uk + ∂/∂ u ¯k )(|φ(u) >< φ(u)|)dudu ¯

fk (t)

k

+

∑ k

=

∑

∫ f¯k (t) ∫

(uk − ∂/∂ u ¯k )C(t, u, u ¯)Xk ⊗ (|φ(u) >< φ(u)|)dudu ¯

fk (t)

k

+

C(t, u, u ¯)Xk∗ uk ⊗ (|φ(u) >< φ(u)|)dudu ¯

∑

∫ f¯k (t)

u ¯k C(t, u, u ¯)Xk∗ ⊗ (|φ(u) >< φ(u)|)dudu ¯

k

Thus, Schrodinger’s equation reduces to the complex variable pde i∂C(t, u, u ¯)/∂t = ∑

ωk (¯ uk ∂C(t, u, u ¯)/∂u ¯k − uk ∂C(t, u, u ¯)/∂uk )

k

¯)] + +[Hs , C(t, u, u +

∑

∑

(uk fk (t)[Xk , C(t, u, u ¯)] − u ¯k f¯k (t)[C(t, u, u ¯), Xk∗ ])

k

[fk (t)(∂C(t, u, u ¯)/∂u ¯k )Xk − f¯k (t)Xk∗ ∂C(t, u, u ¯)/∂u ¯k ]

k

Reference: Preeti and H.Parthasarathy, Technical report, NSIT, 2 [47] Observer based control in discrete time. The state and output equations are x[n + 1] = fn (x[n]) + gn (x[n])w[n + 1] y[n] = hn (x[n]) + v[n] The desired trajectory xd [n] satisfies xd [n + 1] = fn (xd [n]) The observer/state estimator is x[n]) + L[n](y[n] − hn (ˆ x[n])) x ˆ[n + 1] = fn (ˆ The feedback controller to the state equation is given by ˆ[n]) x[n + 1] = fn (x[n]) + gn (x[n])w[n + 1] + K[n](xd [n] − x

General Relativity and Cosmology with Engineering Applications

637

The aim is to design the optimal feedback controller matrix K[n] as well as the optimal observer feedback matrix L[n] so that optimal tracking as well as optimal observer is achieved. We define the tracking error by e[n] = xd [n] − x[n] and the observer error by k[n] = x[n] − x ˆ[n] Then, we have approximately, x[n + 1] − x ˆ[n + 1] = k[n + 1] = fn (x[n]) + gn (x[n])w[n + 1] + K[n](e[n] + k[n])] −fn (ˆ x[n]) − L[n](hn (x[n]) + v[n] − hn (ˆ x[n])) =

fn' (ˆ x[n])k[n]

+ gn (ˆ x[n])w[n + 1] + K[n](e[n] + k[n])

x[n])k[n] + v[n]) −L[n](h'n (ˆ and e[n + 1] = xd [n + 1] − x[n + 1] = x[n])) fn (xd [n]) − fn (x[n]) − gn (x[n])w[n] − K[n](hn (x[n]) + v[n] − hn (ˆ = fn' (ˆ x[n])e[n] − gn (ˆ x[n])w[n + 1] − K[n](h'n (ˆ x[n])k[n] + v[n]) Equivalently, x[n])+K[n]−L[n]h'n (ˆ x[n]))k[n]+gn (ˆ x[n])w[n+1]+K[n]e[n]−L[n]v[n], k[n+1] = (fn' (ˆ e[n + 1] = fn' (ˆ x[n])e[n] − gn (ˆ x[n])w[n + 1] − K[n]h'n (ˆ x[n])k[n] − K[n]v[n] This can be expressed in block matrix form as ( ) k[n + 1] = e[n + 1] (

)( ) fn' (ˆ x[n]) + K[n] − L[n]h'n (ˆ x[n]) K[n] k[n] x[n]) fn' (ˆ x[n]) e[n] −K[n]h'n (ˆ ( )( ) gn (ˆ x[n]) −L[n] w[n + 1] + x[n]) −K[n] v[n] −gn (ˆ

The matrices L[n], K[n] are to be chosen so that if Q is a positive definite matrix of appropriate dimensions, then E[[k[n + 1]T , e[n + 1]T ]Q[k[n + 1]T , e[n + 1]T ]T |Yn ] is a minimum subject to certain quadratic constraints on L[n], K[n]. We write this difference equation as ξ[n + 1] = A[n]ξ[n] + B[n]ε[n + 1]

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General Relativity and Cosmology with Engineering Applications

with obvious meanings, ie A[n], B[n] are functions of K[n], L[n], x ˆ[n]. Then E[ξ[n + 1]T Qξ[n + 1]|Yn ] = T r(QA[n]Rξ [n + 1|n]A[n]T ) + T r(QB[n]Rε B[n]T ) and we have to minimize this w.r.t K[n], L[n] subject to quadratic constraints of the form T r(S1 (K[n] ⊗ K[n])) + T r(S2 (L[n] ⊗ L[n])) + T r(S3 (K[n] ⊗ L[n])) = E This is an elementary quadratic optimization problem and the result is the value of K[n], L[n] as functions of x ˆ[n]. We leave it as an exercise to the reader. Note that we are defining Rξ [n + 1|n] as the covariance of ξ[n + 1] = [k[n + 1]T , e[n + 1]T ]T given Yn = {y[k] : k ≤ n}. [48] DEKF-RLS applied to multivariate linear systems. The system is de­ fined by the stochastic difference equation x[n] = A1 x[n − 1] + A2 x[n − 2] + w[n] and the measured output by y[n] = Cx[n] + v[n] Here x[n] is an N ] × 1 vector and A1 , A2 are N × N matrices. y[n] is a p × 1 vector so that C becomes a p×n matrix. We assume that A1 , A2 are expressible as q q ∑ ∑ wk [n]Pk , A2 = wk [n]Qk A1 = k=1

k=1

where wk [n] are the weights to be estimated along with the state. These weights are constants so they satisfy the trivial difference equations wk [n + 1] = wk [n] Pk , Qk are known N × N matrices. We now design an EKF for x[n] assuming that the weight estimates w ˆk [n] are known and then after obtaining the estimates x ˆ[n + 1|n] and x ˆ[n + 1|n + 1] from the newly arrived measurement data y[n + 1], we apply the RLS to update the weight estimates to w ˆk [n + 1]. Before doing this, we must transform the state equations to first order difference equations so that the EKF can be directly applied. Thus, we define ξ[n] = [x[n]T , x[n − 1]T ]T ∈ R2N Thus, the state dynamics becomes ξ[n + 1] =

General Relativity and Cosmology with Engineering Applications (

A1 IN

)

A2 0 (

We define M=

(

where Rk =

(

A1 IN

A2 0

Pk IN

Qk 0

) =

∑

w[n + 1]

wk [n]Rk

k

)

(

and

)

B[n] 0

ξ[n] +

G[n] =

, k = 1, 2, ..., q B[n] 0

)

Then, the state equations can be expressed as ξ[n + 1] = M [n]ξ[n] + G[n]w[n + 1] and the output equation as y[n] = Dξ[n] + v[n] where D = [C, 0] Applying the EKF to ξ[n] gives ˆ [n]ˆ ξˆ[n + 1|n] = M x[n|n] ˆ [n]Pξ [n|n]M ˆ [n]T + G[n]Pw G[n]T Pξ [n + 1|n] = M ˆ [n]ξˆ[n + 1|n]) ξˆ[n + 1|n + 1] = ξˆ[n + 1|n] + K[n](y[n + 1] − M where

K[n] = (Pξ [n + 1|n]−1 + DT Pv −1D)−1 DT Pv−1

Pξ [n + 1|n + 1] = (I − K[n]D)Pξ [n + 1|n](I − K[n]D)T + K[n]Pv K[n]T In these expressions, ˆ [n] = M

p ∑

w ˆk [n]Rk

k=1

Now we discuss the RLS for obtaining w ˆk [n + 1]. For that, we note that y[n] = Dξ[n] = D(M [n − 1]ξ[n − 1] + G[n − 1]w[n]) This suggests that we calculate w ˆk [n + 1] by minimizing En+1 ({ws }) =

n+1 ∑ m=1

λn+1−m (y[m]

639

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General Relativity and Cosmology with Engineering Applications

−D

q ∑

ws Rs ξˆ[m − 1|m − 1])T W [m − 1](y[m] − D

s=1

q ∑

ws Rs ξˆ[m − 1|m − 1])

s=1

where W [n − 1] is a positive definite matrix. For the ml estimate, we choose W [m − 1] = (DG[n − 1]G[n − 1]T DT )−1 and if this inverse does not exist, we replace it by the pseudo-inverse, ie, the Moore-Penrose inverse. Setting the derivative of En+1 w.r.t. ws to zero at w ˆs [n + 1] gives us the optimal normal equations n+1 ∑

T T T ˆ λn+1−m ξ[m−1|m−1] Rs D W [m−1](y[m]−D

m=1

∑

qw ˆl [n+1]Rl ξˆ[m−1|m−1]) = 0

l=1

ˆl [n + 1], we find that Writing wl in place of w q n+1 ∑ ∑ ( λn+1−m x ˆ[m − 1|m − 1]T RsT DT W [m − 1]DRl ξˆ[m − 1|m − 1])wl l=1 m=1 n+1 ∑

=

λn+1−m ]ξˆ[m − 1|m − 1]T RsT DT W [m − 1]y[m], s = 1, 2, ..., q

m=1

Now define the q × q matrix Xn+1 by Xn+1 [s, l] =

n+1 ∑

λn+1−m ξˆ[m−1|m−1]T RsT DT W [m−1]DRl ξˆ[m−1|m−1], 1 ≤ s, l ≤ q

m=1

and the q × 1 vector ηn+1 by ηn+1 [s] =

n+1 ∑

λn+1−m ]ξˆ[m − 1|m − 1]T RsT DT W [m − 1]y[m], s = 1, 2, ..., q

m=1

Then, we get

−1 w ˆ[n + 1] = (w ˆs [n + 1]) = Xn+1 ηn+1

We can now develop an RLS for computing this. First note that Xn+1 [s, l] = λXn [s, l] + ξˆ[n|n]T RsT DT W [n]DRl ξˆ[n|n] ηn+1 [s] = ληn [s] + ξˆ[n|n]T RsT DT W [n]y[n + 1] We write S[n + 1] = ((ξˆ[n|n]T RsT DT W [n]T DRl ξˆ[n|n]))1≤s,l≤q ∈ Rq×q and s[n + 1] = ((ξˆ[n|n]T RsT DT W [n]y[n + 1]))1≤s≤q ∈ Rq

General Relativity and Cosmology with Engineering Applications

641

so that w ˆ[n + 1] = (λXn + S[n + 1])−1 (ληn + s[n + 1]) which using the matrix inversion lemma can be expressed as w ˆ[n + 1] = [λ−1 Xn−1 −λ−1 Xn−1 (S[n + 1]−1 + λ−1 Xn−1 )−1 λ−1 Xn−1 ] ×[ληn + s[n + 1]] = Xn−1 ηn + λ−1 Xn−1 s[n + 1] − Xn−1 (λS[n + 1]−1 + Xn−1 )−1 Xn−1 ηn −λ−1 Xn−1 (λS[n + 1]−1 + Xn−1 )−1 Xn−1 s[n + 1] = w[n] ˆ + λ−1 Xn−1 s[n + 1] − Xn−1 (λS[n + 1]−1 + Xn−1 )−1 w ˆ[n] −λ−1 Xn−1 (I − (λS[n + 1]−1 ) + Xn−1 )−1 λS[n + 1]−1 )s[n + 1] = w[n] ˆ + Xn−1 (λS[n + 1]−1 + Xn−1 )−1 S[n + 1]−1 (s[n + 1] − S[n + 1]w[n]) ˆ Reference: Vijyant Agarwal and H.Parthasarathy, ”Dual EKF-RLS applied to state and weight estimation in linear multivariable stochastic systems. [49] Viscous and thermal effects in special-relativistic hydrodynamics. The energy-momentum tensor of the fluid is given by T μν = T (0)μν + ΔT μν where T (0)μν = (ρ + p)v μ v ν − pg μν is the energy-momentum tensor of the fluid without taking viscous and thermal effects and ΔT μν is the contribution to the energy-momentum tensor of the fluid due to viscous and thermal effects. Let n(x) denote the particle number density and ρ(x) the density. The entropy per particle is denoted by σ(x). We have by applying the first law of thermodynamics to each fluid particle, T dσ = d(ρ/n) + pd(1/n) = d((ρ + p)/n) − dp/n Thus, T σ,μ v μ = ((ρ + p)/n),μ v μ − p,μ v μ /n = ρ,μ v μ /n − (ρ + p)n,μ v μ /n2 μ = ρ mu v μ /n + (ρ + p)v,μ /n μ = (ρv μ ),μ /n + pv,μ /n

where we have used the particle number conservation in the form (nv μ ),μ = 0

642

General Relativity and Cosmology with Engineering Applications

We now use the energy-momentum conservation (T (0)μν + ΔT μν ),ν = 0 to get ((ρ + p)v ν ),ν − p,ν v ν + ΔT,νμν vμ = 0 or equivalently, ν + ΔT,νμν vμ = 0 (ρv ν ),ν + pv,ν

so that T σ,μ v μ = −ΔT,νμν vμ /n which can be rearranged using (nv μ ),μ = 0 as (nσv μ + ΔT μν vμ /T ),ν = ΔT μν (vμ /T ),ν The LHS is a perfect 4-divergence and hence can be interpreted as the rate of generation of entropy per unit volume of the fluid. This must be positive in accordance with the second law of thermodynamics. Thus, we must have ΔT μν (vμ /T ),ν ≥ 0 We now move to a frame in which the fluid is instantaneously at rest, ie, v r = 0, r = 1, 2, 3 at a given point. Then noting that ΔT 00 = 0 since viscous and thermal conduction cannot contribute anything to the energy density, it follows from the above that ΔT rs (vr /T ),s + ΔT r0 ((vr /T ),0 + (v0 /T ),r ) ≥ 0 (Note that ΔT μν = ΔT νμ must always hold). Since vr = 0 at that point, it follows from the above that (ΔT rs /2T )(vr,s + vs,r ) + (ΔT r0 /T )(vr,0 + v0,r − T,r )/T ) ≥ 0 at that point. (Note that v r = 0 implies v 0 = v0 = 1). Note that the equation ∑ v02 − vr2 = 1 r

at all space-time points implies that v0 v0,k +

∑

vr vr,k = 0

r

so that at that point and in the specified frame, vr = 0 which gives v0,k = 0 so that the above requirement reduces to (ΔT rs /2T )(vr,s + vs,r ) + (ΔT r0 /T )(vr,0 − T,r )/T ) ≥ 0

General Relativity and Cosmology with Engineering Applications

643

Consider now the tensor H μν = g μν − v μ v ν where g μν is the Minkowski metric. In the frame at that point where the fluid is instantly at rest, we have v r = 0 and hence in this frame that that point we have H 00 = H 0r = 0, H rs = −δ rs Thus, if we define S1μν = χ1 (T )H μα H νβ (vα,β + vβ,α ) where χ1 (T ) is a positive temperature dependent scalar, we then get that S1μν is a tensor and at that point in this frame where that particle is instantly at rest, we get S1rs = χ1 (T )(vr,s + vs,r ), S1r0 = S10r = 0, S100 = 0 Now define α μν S2μν = χ2 (T )v,α g

where χ2 (T ) is another positive temperature dependent scalar, then we have at that point in the same frame, k rs S2rs = −χ2 (T )v,k δ , S2r0 = S20r = 0, S200 = 0

Also, r δ rs (vr,s + vs,r ) = −2v,r

Thus, these definitions imply that at that point in the specified frame, (S1μν + S2μν )(vμ,nu + vν,μ ) = (S1rs + S2rs )(vr,s + vs,r ) ≥ 0 Finally, we define S3μν = χ3 (T )(H μα v ν + H να v μ )T,α Then, we have in the specified frame at that point, S3rs = 0, and

S30r = S3r0 = −χ3 (T )T,r

Thus, defining ΔT μν = S1μν + S2μν + S3μν we get the desired positivity for the rate of entropy increase in the given frame simultaneously ensuring that ΔT μν is a tensor. Expanded in full, we have ΔT μν =

644

General Relativity and Cosmology with Engineering Applications α μν χ1 (T )H μα H νβ (vα,β + vβ,α ) + χ2 (T )v,α g + χ3 (T )(H μα v ν + H να v μ )T,α

While using this formula in the gtr, all partial derivatives occurring here should be replaced by covariant derivatives. While studying the evolution of galaxies, ie, inhomogeneities in our homogeneous isotropic expanding universe taking into account viscous and thermal effects, we must perturb the Einstein field equations. This gives us the following equations: (0) δRμν = −8πG(δTμν + δΔTμν − δ((T (0) + ΔT )gμν /2))

To expand this expression further, we observe that δ(T (0) ) = δρ − 3δp, ΔT = gμν ΔT μν = χ1 gμν H μα H νβ (vα:β + vβ:α ) +4χ2 v:αα + 2χ3 H μα v μ T,α Now, gμν H μα H νβ (vα:β + vβ:α ) = = 2gμν H μα H νβ vα:β = 2gμν (g μα − v μ v α )(g νβ − v ν v β )vα:β = 2v:αα − v α v β vα:β = 2v:αα Thus,

(0) δTμν + δΔTμν

= (ρ + p)(vν δvμ + vμ δvν ) + vμ vν (δp + δρ)+ χ1 (H μα H νβ (δvα:β + δvβ:α ) +χ1 (vα:β + vβ:α )(H μα δH νβ + H νβ δH μα ) +χ2 (g μν δv:αα + v:αα δg μν ) + χ3 (δT,α )(H μα v ν + H να v μ ) +χ3 T,α (H μα δv ν + v ν δH μα + H να δv μ + v μ δH να ) Note that the unperturbed T,r , r = 1, 2, 3 should be taken as zero in agreement with the fact that the unperturbed universe is homogeneous and isotropic. In other words, the unperturbed temperature T is a function of time t only. Also note that the unperturbed metric is the Robertson-Walker metric: g00 = 1, g0r = 0, g11 = −S 2 (t)f (r), g22 = −S 2 (t)r2 , g33 = −S 2 (t)r2 sin2 (θ) Further the unperturbed metric satisfies the comoving property, ie, four veloci­ ties of the form (1, 0, 0, 0) are geodesics in this metric. In other words, we may take our unperturbed velocity field as v 0 = 1, v r = 0, r = 1, 2, 3

General Relativity and Cosmology with Engineering Applications

645

Then, we evaluate the energy-momentum tensor perturbations as follows: δvα:β = δvα,β − Γραβ δvρ − δΓ0αβ In particular,

0 k 0 δv0 − Γrs δvk − δΓrs δvr:s = δvr,s − Γrs

Note that g μν vμ vν = 1 gives along with vr = 0, r = 1, 2, 3, v0 = 1, δv0 + δg 00 = 0 or

δv0 = −δg 00 = 0

where we have assumed that δg0μ = 0, μ = 0, 1, 2, 3 This is possible because we are free to make a small change in our coordinate system to ensure this condition. Thus, k δvk − δΓ0rs δvr:s = δvr,s − Γrs

Now,

δΓ0rs = δ(g 00 Γ0rs + g 0k Γkrs ) = δΓ0rs = −δgrs /2

Further, Γkrs = g kk (gkr,s + gks,r − grs,k )/2 = (loggkk ),s δkr /2 + (loggkk ),r δks /2 − g kk grr,k δrs /2 Thus, δvr:s = δvr,s −(loggrr ),s δvr /2 − (loggss ),r /2 − g kk grr,k δvk δrs /2 where in the last term, summation over the spatial index k is assumed. Further, δv0:r = −Γk0r δvk − δΓ00r since δv0 = 0 implies δv0,r = 0. Now, δΓ00r = δ(g 00 Γ00r + g 0k Γk0r ) = δ(g 0k Γk0r ) = Γk0r δ(g 0k ) = 0 since

δg 0k = −g 00 g ks δg0s = 0

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General Relativity and Cosmology with Engineering Applications

since by our choice of coordinates, δg0s = 0. This gives δv0:r = −Γk0r δvk Now,

δvr:0 = δvr,0 − Γkr0 δvk − δΓ0r0

Now, Γkr0 = g kk Γkr0 = (g kk /2)gkr,0 = δkr g kk gkk,0 /2 = δkr (loggkk ),0 /2 = (S '' (t)/S(t))δkr δΓ0r0 = 0 Thus, and

δvr:0 = δvr,0 − S '' (t)δvr /S(t) δv0:r = −Γk0r δvk = −(S '' (t)/S(t))δvr 0 δv0:0 = δv0,0 − δ(Γr00 vr + Γ00 v0 ) = 0

since δv0 = 0, δg0μ = 0 and g00 = 1. We next calculate the perturbation to the Ricci tensor components: α δRμν = δΓα μα:ν − δΓμν:α

where covariant derivatives are w.r.t. the unperturbed metric. Note that al­ though the Christoffel connection symbols are not tensors, their perturbations are tensors since the difference of two Christoffel symbols forms a tensor. Exercise: Calculate δΓα μν:β as a linear function of δhrs , hrs,μ , hrs,μν with coef­ ficients being expressed as functions of S(t),S’(t),S”(t), fk , fk,m , fk,ml , ψkml , ψkml,r where gkk = S 2 (t)fk , f1 = −1/(1 − kr2 ), f2 = −r2 , f3 = −r2 sin2 (θ), x0 = t, x1 = r, x2 = θ, x3 = φ Γkmr = S 2 (t)ψkmr so that ψkmr = (1/2)(fk,r δkm + fk,m δkr − fr,k δrm ) Note that fk , ψkmr are functions of only the spatial variables x1 , x2 . Hence, calculate δRμν for (μ, ν) = (0, 0), (μ, ν) = (r, 0), (μ, ν) = (r, s). Note the con­ vention used: Greek indices like μ, ν, ρ, σ, α, β take values 0, 1, 2, 3, ie, all spacetime indices while Roman indices like k, m, r, l, s, p, q only take values 1, 2, 3, ie, all spatial indices. Further, we have assumed a small change in the coordinate system so as to guarantee the ”gauge conditions” h0μ = 0 where hμν = δgμν . Explicitly, we can express δRμν = C1 (μνrs, x)hrs +C2 (μνrs, x)h'rs +C3 (μνrsk, x)hrs,k +C4 (μνrskm, x)hrs,km

General Relativity and Cosmology with Engineering Applications

647

+C5 (μν, rsk, x)hrs,k0 + C6 (μν, rs, x)hrs,00 where the summation is over the repeated spatial indices r, s and the coefficients Cj (μνrs, x) are functions of x = (t, r, θ) expressed as functions of fk , fk,r , fk,rm , S(t), S ' (t), S '' (t).

Derive explicit formulas for these coefficients. Now setup the ten perturbed Ein­ stein field equations δRμν = −8πG(δTμν − (δT gμν + hμν )/2) for the ten space-time functions hrs , δvr , δρ, 1 ≤ r ≤ s ≤ 3. Note that these perturbed field equations are linear second order pde’s for the stated functions. Develop a MATLAB programme to solve these perturbed field equations by discretizing the spatial indices only. Specifically, discretize the r variable as r = nΔ, n = 0, 1, 2, ..., N − 1, θ = πm/N, m = 0, 1, ..., N − 1, φ = 2πs/N, s = 0, 1, ..., N − 1. Then the resulting differential equations for the 6N 3 × 1 vector h(t) obtained by spatially discretizing hrs , 1 ≤ r ≤ s ≤ 3 and the 4N 3 × 1 vector ξ(t) obtained by discretizing the density and velocity perturbations ξ(t) = (δρ, δvr , r = 1, 2, 3) have the form A0 (t)h'' (t) + A1 (t)h' (t) + A2 (t)h(t) = A3 (t)ξ(t) where Ak (t), k = 0, 1, 2 are 10N 3 × 6N 3 matrices and A3 (t) is a 10N 3 × 4N 3 matrix. We partition the matrices A0 , A1 , A2 , A3 as follows: ( ) Ak1 (t) Ak (t) = , k = 0, 1, 2, 3 Ak2 (t) where Ak1 is of size 6N 3 × 6N 3 and Ak2 is of size 4N 3 × 4N 3 for k = 0, 1, 2 while A31 is of size 6N 3 × 4N 3 and A32 is of size 4N 3 × 4N 3 . Then, we get A01 (t)h'' (t) + A11 (t)h' (t) + A21 (t)h(t) = A31 (t)ξ(t) A02 (t)h'' (t) + A12 (t)h' (t) + A22 (t)h(t) = A32 (t)ξ(t) The second equation can be inverted to give ξ(t) = A32 (t)−1 (A02 (t)h'' (t) + A12 (t)h' (t) + A22 (t)h(t)) and substituting this into the first equation gives us (A01 )(t) − A31 (t)A32 (t)−1 A02 (t))h'' (t) + (A11 (t) − A31 (t)A32 (t)−1 A12 (t))h' (t) +(A21 (t) − A31 (t)A32 (t)−1 A22 (t))h(t) = 0 This is a system of coupled 6N 3 second order homogeneous ordinary differential equations of second degree for the 6N 3 ×1 vector h(t) and oscillations/oscillations with damping can be derived from it. [50] The Einstein field equations in the presence of radially moving matter distribution with radial symmetry. The four velocity field has the form (v 0 (t, r), v 1 (t, r), 0, 0)

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General Relativity and Cosmology with Engineering Applications

so that if the metric has the form dτ 2 = A(t, r)dt2 − B(t, r)dr2 + 2D(t, r)dtdr − C(t, r)(dθ2 + sin2 (θ)dφ2 ) We can eliminate D by making a transformation t = f0 (t' , r' ), r = f1 (t' , r' ) so that

dt2 = f02,0 (dt' )2 + f02,1 (dr' )2 + 2f0,0 f0,1 dt' dr' , dr2 = f12,0 (dt' )2 + f12,1 (dr' )2 + 2f1,0 f1,1 dt' dr' , dtdr = f0,0 f1,0 (dt' )2 + f0,1 f1,1 (dr' )2 + (f0,0 f1,1 + f0,1 f1,0 )dt' dr'

These are substituted into the quadratic form Adt2 − Bdr2 + 2Ddtdr and the coefficient of dtdr is set equal to zero. This gives us a pde for the functions f0 , f1 . Thus, without loss of generality, we may assume the metric to be of the form dτ 2 = A(t, r)dt2 − B(t, r)dr2 − C(t, r)(dθ2 + sin2 (θ)dφ2 ) then

g00 = A, g11 = −B, g22 = −C, g33 = −C.sin2 (θ)

and implies where

Av 02 − Bv 12 = 1 v0 =

√

(1 + Bv 2 )/A, v0 = g00 v 0 =

√ A(1 + Bv 2 )

v 1 = v = v(t, r)

The non-zero components of the energy-momentum tensor are T00 = (ρ(t, r) + p(t, r))v02 − p(t, r)A, T11 = (ρ + p)v 2 + pB, T01 = T10 = (ρ + p)v0 v, T22 = pC, T33 = pC.sin2 (θ) The non-zero components of the Ricci tensor are R00 , R11 , R22 , R33 , R01 and thus, we get five Einstein field equations for the six functionsρ, p, v, A, B, C. The additional equation is the equation of state p(t, r) = p(ρ(t, r)). We shall first compute R01 and show that it is non-zero. α α β α β R01 = Γα 0α,1 − Γ01,α − Γ01 Γαβ + Γ0β Γ1α 0 1 2 3 Γα 0α = Γ00 + Γ01 + Γ02 + Γ03 ,

General Relativity and Cosmology with Engineering Applications = A,0 /2A + B,0 /2B + C,0 /C α Γ01,α = Γ001,0 + Γ101,1

= (A,1 /2A),0 + (B,0 /2B),1 β Γα 01 Γαβ =

Γ001 Γβ0β + Γ101 Γβ1β 0 1 2 3 = Γ01 (Γ000 + Γ01 + Γ02 + Γ03 ) 1 2 3 +Γ101 (Γ010 + Γ11 + Γ12 + Γ13 ) β Γα 0β Γ1α = 0 0 1 1 2 3 3 Γ10 + Γ01 Γ11 + Γ202 Γ12 + Γ03 Γ13 = Γ00

Thus, α Γα 0α,1 − Γ01,α =

(A,0 /2A + B,0 /2B + C,0 /C),1 − ((A,1 /2A),0 + (B,0 /2B),1 ) = (logC),01 β α β Γα 01 Γαβ − Γ0β Γ1α = 1 2 3 = Γ001 (Γ000 + Γ01 + Γ02 + Γ03 ) 1 2 3 +Γ101 (Γ010 + Γ11 + Γ12 + Γ13 ) 0 1 1 2 3 3 0 −(Γ00 + Γ01 Γ11 + Γ202 Γ12 + Γ03 Γ13 ) Γ10 0 1 0 2 0 = 2Γ01 Γ01 + Γ01 Γ02 + Γ01 Γ303 1 3 2 2 +Γ01 Γ212 + Γ101 Γ13 − Γ02 Γ12 3 −Γ303 Γ13

= g 00 g 11 g00,1 g11,0 /2 + g 00 g 22 g00,1 g22,0 /4 + g 00 g 33 g00,1 g33,0 /4 +g 11 g 22 g11,0 g22,1 /4 + g 11 g 33 g11,0 g33,1 /4 − (g 22 )2 g22,0 g22,1 /4 −(g 33 )2 g33,0 g33,1 /4 = (logA),1 (logB),0 /2 + (logA),1 (logC),0 /2+ +(logB),0 (logC),1 /2 − (logC),0 .(logC),1 /2 For simplicity of notation, we define 2a = logA, 2b = logB, 2c = logC These are functions of (t, r) only. Then, we get R01 = 2(c,01 − a,1 b,0 − a,1 c,0 − b,0 c,1 − c,0 c,1 )

649

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General Relativity and Cosmology with Engineering Applications

The modified energy-momentum tensor of the matter distribution is Sμν = Tμν − T gμν /2 We compute its non-zero components, namely S00 , S11 , S22 , S33 , S01 : T = g μν Tμν = ρ − 3p, S00 = T00 − T g00 /2 = (ρ + p)v02 − pA − (ρ − 3p)A/2 = (ρ + p)A(1 + Bv 2 ) + (p − ρ)A/2 = (3p + ρ)A/2 + (ρ + p)ABv 2 S01 = S10 = T01 = (ρ + p)A(1 + Bv 2 ) − pA = ρA(1 + Bv 2 ) + pABv 2 S11 = T11 − T g11 /2 = (ρ + p)v 2 + pB + (ρ − 3p)B/2 = (ρ − p)B/2 + (ρ + p)v 2 S22 = T22 − T g22 /2 = pC + (ρ − 3p)C/2 = (ρ − p)C/2 S33 = S22 sin2 (θ) We now compute R00 , R11 , R22 , R33 and set up the Einstein field equation for this problem. α α β α β R00 = Γα 0α,0 − Γ0,α − Γ00 Γαβ + Γ0β Γ0α

[51] The prime number theorem (Chebyshev’s proof) Theorem: Let x > 0 and denote by π(x) the number of primes ≤ x. Then, there exist finite positive real numbers A < B such that A.x/log(x) ≤ π(x) ≤ Bx/log(x), ∀x > 0 In other words, limsupx→∞ π(x)/(x/log(x)), liminfx→∞ π(x)/(x/log(x)) are finite positive real numbers (Actually, these are both equal to one, but the proof of that is harder). For x > 0 define ∑ ∑ θ(x) = log(p), ψ(x) = log(p) p≤x

(k,p):pk ≤x

where p runs over primes and k over positive integers. Lemma 1: θ(x) ≤ 2x.log(2), x > 0 Proof: We have 22m= (1 + 1)m =

2m ) (2m r ∑

r=0

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General Relativity and Cosmology with Engineering Applications (

and hence

Also since

)

2m m

(

)

2m m

=

≤ 22m

2m(2m − 1)...(m + 1) m!

it follows that if p is any prime in {m + 1, ..., 2m}, then p divides ( ) 2m ≤ 22m Πm