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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 spacetime 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 energymomentum tensor of matter in a background curved metric
2.7 Maxwell’s equations in a background curved metric
2.8 The energymomentum 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 postNewtonian 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 KleinGordon 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 Energymomentum tensor of matter with viscous and thermal corrections
3.7 Energymomentum tensor of the electromagnetic field in a background curved spacetime
3.8 Relativistic Fermi fluid in a gravitational field
3.9 The postNewtonian approximation
3.10 EnergyMomentum tensor of matter with viscous and thermal corrections
3.11 Energymomentum 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 postNewtonian 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 spacetime
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 Energymomentum tensor of matter with viscous and thermal corrections
4.7 Energymomentum tensor of the electromagnetic field in a background curved spacetime
4.8 Relativistic Fermi fluid in a gravitational field
4.9 The postNewtonian 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 NavierStokes Fluid equations with gravitational self interaction
5.9 Wave equations in mechanics
5.10 Surface of revolution
5.11 1D Schrodinger equation
5.12 Lagrange’s triangle in mechanics
5.13 Number theory
5.14 Blurring of 3D 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 Reection 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 spacetime
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 KnillLaflamme 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 DeBroglie Duality of particle and wave properties of matter
2 Bohr’s correspondence principle
3 BohrSommerfeld’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 selfadjoint 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 GoriniKossakowskiSudarshanLindblad (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 HudsonParthasarathy 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 (SchrodingerHeisenberg) 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 HartreeFock apporoximate method for computing the wave functions of a many electron atom
40 The BornOppenheimer 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 LippmanSchwinger equation for the scattering matrix, Born scattering
48 Design of quantum gates using time dependent scattering theory
49 EvansHudson flows and its application to the quantization of the fluid dynamical equations in noise
50 Classical nonlinear filtering
51 Derivation of the extended Kalman filter (EKF) as an approximation to the Kushner filter
52 Belavkin’s theory of nondemolition measurements and quantum filtering in coherent states based on the Hudson Parthasarathy Boson Fock space theory of quantum noise, The quantum KallianpurStriebel 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 SudarshanLindblad equation for observables in an open quantum system
62 The YangMills 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 StroockVaradhan martingales
Appendix
References
Index
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, Delhi110078
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 334872742 © 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, 9787508400. 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 CataloguinginPublication Data A catalogue record for this book is available from the British Library Library of Congress CataloginginPublication Data A catalog record has been requested ISBN: 9781032001623(hbk) ISBN: 9781003173021(ebk)
Table of Contents
Part I:
The General Theory of Relativity and Some of Its Applications
Chapter 1: The Special Theory of Relativity 118
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 spacetime 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 energymomentum tensor of matter in a background
curved metric 2.7 Maxwell’s equations in a background curved metric
1944
19
19
20
21
22
37
38
VI
General Relativity and Cosmology with Engineering Applications
2.8 The energymomentum 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 postNewtonian equations of celestial mechanics,
gravitation and hydrodynamics
39
40
41
42
42
Chapter 3: Engineering Applications of General Relativity 45106
3.1 Applications of general relativity to global positioning systems 45
3.2 General relativistic corrections to the KleinGordon
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 Energymomentum tensor of matter with viscous and
thermal corrections 66
3.7 Energymomentum tensor of the electromagnetic field
in a background curved spacetime 69
3.8 Relativistic Fermi fluid in a gravitational field 70
3.9 The postNewtonian approximation 71
3.10 EnergyMomentum tensor of matter with viscous and
thermal corrections 75
3.11 Energymomentum 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 postNewtonian 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 spacetime 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) 107164
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 Energymomentum tensor of matter with viscous and
thermal corrections 126
4.7 Energymomentum tensor of the electromagnetic field
in a background curved spacetime 129
4.8 Relativistic Fermi fluid in a gravitational field 130
4.9 The postNewtonian 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 165252
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 NavierStokes Fluid equations
with gravitational self interaction 171
5.9 Wave equations in mechanics 172
5.10 Surface of revolution 172
5.11 1D 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 3D objects in random motion Commutators of products of matrices Path of a light ray in an medium having inhomogeneous
refractive index Reection 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 spacetime 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 KnillLaflamme 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 DeBroglie Duality of particle and wave properties of matter Bohr’s correspondence principle BohrSommerfeld’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 selfadjoint 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 GoriniKossakowskiSudarshanLindblad (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 HudsonParthasarathy 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
(SchrodingerHeisenberg) 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 HartreeFock apporoximate method for computing
the wave functions of a many electron atom The BornOppenheimer 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 LippmanSchwinger equation for the scattering matrix, Born scattering Design of quantum gates using time dependent
scattering theory EvansHudson flows and its application to the quantization
of the fluid dynamical equations in noise Classical nonlinear 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 nondemolition measurements
and quantum filtering in coherent states based on
the Hudson Parthasarathy Boson Fock space theory of
quantum noise, The quantum KallianpurStriebel 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 SudarshanLindblad equation for observables in an open
quantum system 378
The YangMills 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 StroockVaradhan martingales 395
Appendix
397653
References
654655
Index
656668
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 MichelsonMorley 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 nonrotating galaxies in the universe and nonaccelerating 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 spacetime 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 spacetime. This leads us to understand that a gravitational ﬁeld is simply a set of spacetime 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 spacetime metric and these geodesics are therefore curved paths on the spacetime 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 spacetime 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 postNewtonian 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 spacetime. 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 energymomentum 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 spacetime. 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, NavierStokes ﬂuid dynamical equations, the KleinGordon equation with Higgs potentials, the YangMills noncommutative gauge ﬁeld and matter ﬁeld equations involve partial derivatives with respect
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General Relativity and Cosmology with Engineering Applications to spacetime 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 electronpositron ﬁeld and the YangMills ﬁeld describing electroweak 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 ReissnerNordstorm solution involving the presence of a charge at the centre of the spherical distribution of matter and the KerrNewman 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 energymomentum 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 spacetime 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 spacetime 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 spacetime man ifolds that cannot be transformed away to ﬂat spacetime manifolds. Genuine ﬂat spacetime 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 spacetime 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 positionvelocity space is given when the background spacetime 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 energymomentum 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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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 HudsonParthasarathy 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 RobertsonWalker 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 energymomentum 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 Smatrix for interaction between photons, electrons, positrons, gravitons and perhaps other particles like quarks, leptons, mesons, pions and gauge bosons appearing in the YangMills 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 nonmoving 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, spacetime forms a uniform manifold, which means that the trans formation law (t, r) → (t' , r' ) of the spacetime 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 MichelsonMorley 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 spacetime 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 noninertial 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.
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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 spacetime 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 spacetime fourvector 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
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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
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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
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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 spacetime 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 spacetime 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.
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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 spacetime 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
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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 xaxis, then the spacetime 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
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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 xposition of the particle relative to K at time t and x ¯(t¯) the ¯ that xposition 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
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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 specialrelativistic version of the Euler equation (ie the NavierStokes 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 energymomentum 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 energymomentum 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
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and hence the total energymomentum 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 EulerLagrange 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 spacetime 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 spacetime 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
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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 spacetime 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 − gndt2 /c2 = (1 − g 2 /c2 )dt2 + 2g(n, dr)dt/c2 − dr2 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 spacetime, then we can still cancel it over an inﬁnitesimal region of spacetime by moving to a locally freely falling frame. This raises the important question as to given a metric in K which is nonMinkowskian, 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 spacetime 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 spacetime coordinates and if the gravitational ﬁeld is nonconstant, 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 spacetime 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
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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 spacetime.
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 spacetime. Let S be a pdimensional 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 EulerLagrange 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 EulerLagrange equations.
General Relativity and Cosmology with Engineering Applications
2.5.5
The general theory of connections and deﬁnition of the RiemannChristoﬀel 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 noncoordinate 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 Γsymbols 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
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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 RiemannChristoﬀel 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 spacetime 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 RiemannChristoﬀel curvature tensor measures the degree of noncommutativity between the co variant derivatives of a vector along two diﬀerent coordinate directions. This noncommutativity 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 antisymmetrization 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
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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 spacetime 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)
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3.6
Energymomentum tensor of matter with vis cous and thermal corrections
Assume ﬁrst the special relativistic case. The energymomentum 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 spacetime 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 spacetime 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
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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
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We now compute the ﬁrst few approximations for the NavierStokes equation. These approximations are obtained by ﬁrst setting up the exact general rela tivistic NavierStokes 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)
Energymomentum 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 energymomentum 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 energymomentum 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 spacetime 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 spacetime 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 ∫ +
3.9
V (x, y)ψa (x)∗ ψa (x)ψb (y)∗ ψb (y)d3 xd3 y
The postNewtonian approximation
Quantities are expanded in powers of the three velocity magnitude or equiva lently velocity relationship in Newtonian mechanics, √in view of the massorbital √ 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
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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 nondiagonal 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 spacetime 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 )
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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 )
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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 postNewtonian equations of celestial mechanics. We now derive the postNewtonian 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
Energymomentum tensor of matter with viscous and thermal corrections
The main idea here is to start with the energymomentum 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 energymomentum 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 energymomentum tensor correction due to viscous and thermal eﬀects. This ﬁnal corrected energymomentum tensor is used in calculating the
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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 energymomentum 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
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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 spacetime 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 spacetime 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
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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
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We now compute the ﬁrst few approximations for the NavierStokes equation. These approximations are obtained by ﬁrst setting up the exact general rela tivistic NavierStokes 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)
Energymomentum 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 energymomentum 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 energymomentum of the EM ﬁeld (assuming absence
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√ 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 spacetime 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 spacetime 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 postNewtonian approximation
Quantities are expanded in powers of the three velocity magnitude or equiva lently velocity relationship in Newtonian mechanics, √in view of the massorbital √ 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 nondiagonal 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 ,
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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 spacetime 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 )
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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 )
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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 postNewtonian equations of celestial mechanics. We now derive the postNewtonian 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
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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, xt' , x' ) =< T (ψ1 (t, x)ψ1 (t' , x' )∗ ) >, F (t, xt' , 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 =
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= −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 spacetime coordinates x is given by (Steven Weinberg, Gravitation and Cosmology) [γ a Vaμ (i∂μ + eAμ + iΓμ ) − m]ψ = 0
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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 )
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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 spacetime. 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 spacetime 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)
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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 2D 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 = dr2 = g11 (u, v)du2 + g22 (u, v)dv 2 + 2g12 (u, v)dudv
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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 RiemannChristoﬀel 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 = dr2 . 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 2D 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,
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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
KleinGordon Equation in a RobertsonWalker Uni verse
For the RobertsonWalker metric speciﬁed in the previous problem, write down the KleinGordon 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 KleinGordon 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
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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 energymomentum 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 spacetime 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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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
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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 energymomentum tensors of matter and radiation and f μν is the energymomentum 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 EinsteinMaxwellDirac 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 spacetime 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 energymomentum 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 spacetime using a multilayered neural network. Assume that the metric of spacetime 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)+
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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
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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 spacetime 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 energymomentum 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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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
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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 (DNNRNN). 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 spacetime 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 velocitydensitypressure of a ﬂuid moving in this
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curved spacetime or on electromagnetic ﬁelds satisfying the Maxwell equations in this curved spacetime, we wish to estimate θ. Alternatively, we can learn about the parameter θ even when it varies slowly with time using a DNNRNN 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 DNNRNN 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 EulerLagrange 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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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 GKSLHeisenberg 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
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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(−z2 /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(−z2 /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(−z2 /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(−z2 /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(rr' ), r, r' ∈ D satisfying (∇2r + k 2 )G(rr' ) = δ 3 (r − r' ), r, r' ∈ D and
G(rr' ) = 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(rr' )∇2 ψ(r)−ψ(r)∇2 G(rr' )] = D
[G(rr' )∂ψ(r)/∂n ˆ −ψ(r)∂G(rr' )/∂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 LaplaceBeltramiHelmholtz 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 nonnegative 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
Cqcoding theorem
A is the source alphabet and corresponding to each x ∈ A we have a density ρ(x). If u ∈ An , we write N (xu) for the number of times x appears in u and Pu (x) = N (xu)/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 (xu) − 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 (xu) ≤ 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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√
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 (xu) and correspondingly E(n, u, δ) = ⊗x∈A E(ρ(x)⊗N (xu) , δ) Then, from the above, it easily follows that √
2−N (xu)H(ρ(x))−O(
n)
E(ρ(x)⊗N (xu) , δ) ≤ ρ(x)⊗N (xu) E(ρ(x)⊗N (xu) , δ) ≤
2−N (xu)H(ρ(x))+O(
√
n)
E(ρ(x)⊗N (xu) , 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.
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113
Quantum theory of ﬁelds
The KleinGordon 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 KleinGordon 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
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∂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
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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,
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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) =< 0T {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) =< 0T {U (∞, −∞)φ(x)φ(y)}0 >= < 0U (∞, 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
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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(Kt−K.r))+¯ er (K, σ)a(K, sigma)∗ exp(i(Kt−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 ∫ =
2K2 e(K, σ)2 a(K, σ)∗ a(K, σ)d3 K
once we make use of the fact that K × e(K, σ) = Ke(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) = (2K)−1/2 in order to ensure that ∫ HF =
Ka(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)] = −iKa(t, K, σ), a∗ (t, K, σ),t = i[HF , a(t, K, σ)∗ ] = iKa∗ (t, K, σ)
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These equations imply a(t, K, σ),t = −K2 a(t, K, σ) a∗ (t, K, σ),tt = −K2 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 3vector (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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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 skewsymmetric 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
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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 nonrelativistic 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 nonrelativistic 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 Energymomentum tensor of matter with vis cous and thermal corrections Assume ﬁrst the special relativistic case. The energymomentum 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)
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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 spacetime 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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and the four tensor S μν = Qμ v ν + Qν v μ We then have at the spacetime 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 RobertsonWalker metric of homogeneous and isotropic spacetime. 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 spacetime) 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 energymomentum 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 RobertsonWalker universe.
General Relativity and Cosmology with Engineering Applications
4.7
129
Energymomentum 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 energymomentum 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 energymomentum 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 spacetime 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 spacetime 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 postNewtonian approximation
Quantities are expanded in powers of the three velocity magnitude or equiva lently in view of the massorbital 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 + ...,
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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 nondiagonal 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 spacetime 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
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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 postNewtonian equations of celestial mechanics. We now derive the postNewtonian 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, xt' , x' ) =< T (ψ1 (t, x)ψ1 (t' , x' )∗ ) >, F (t, xt' , x' ) =< T (ψ1 (t, x)ψ2 (t' , x' )) > where T is the time ordering operator.
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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 spacetime 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))
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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 spacetime. 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 spacetime 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 = dr2 = 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 RiemannChristoﬀel 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 = dr2 . 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 EinsteinMaxwell 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 energymomentum tensor of the matter ﬁeld and Sμν is the energymomentum 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 spacetime, 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 spacetime, 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.
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4.13.7
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Some aspects of group representation theory
The group under which the equations of general relativity are invariant is the group of all spacetime 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) >
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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.
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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 CauchyRiemann equations, (q1 , q2 ) form an orthogonal curvilinear co ordinate system in the xy plane. We have ∇2⊥ = ∂x2 + ∂y2 = 4 = 4g ' (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 spacetime 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 tz 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 PostNewtonian magnetohydrodynamics 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) + ...
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(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 EulerLagrange equations are satisﬁed by the superﬁeld for the Lagrangian density L, ∂μ N μ = ∂L δSa,μ ∂Sa,μ
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∂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 electronpositron ﬁeld of matter via the gravitational connection for the Dirac ﬁeld in curved spacetime. 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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(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 − uˆ 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 Gtransformations, 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 Ginvariant, 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 Gtransformations. 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 spacetime 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 skewsymmetric matrix ε and hence γ5 ε is a skewsymmetric matrix whose square is −I. Also, γ5 εγ μ γ5 ε = −γ μT (verify this). Thus, {La , Lb } = −2γ μ γ5 ε∂/∂xμ
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4.13.17
157
Quantum Belavkin ﬁltering and control for gen eral quantum Levy output measurements
In this work, we consider the HudsonParthasarathy noisy Schrodinger equation with single annihilation, creation and conservation processes and nondemolition 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 nonzero 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 VonNeumann algebra and further satisfy the nondemolition 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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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 3D 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
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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
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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 × rcos(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 oneone 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.
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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 rotationtranslationpermutation 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)]
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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
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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 )+
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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
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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.
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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 semisimple 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 oneone 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.
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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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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 nonGaussian. In this case, the Edgeworth series expansion for nonGaussian 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 nonstationary 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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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 ”Mstep”, 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, Chising Leung, Gilbert H.Young and WingKay 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+1k] = E[x[k+1]Yk ] = E(fk (x[k])Yk ) ≈ fk (ˆ x[kk])+(1/2)fk'' (ˆ x[kk])V ec(Px [kk])
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where ˆ[kk])(x[k] − x ˆ[kk])T Yk ] Px [kk] = Cov(x[k]Yk ) = E[(x[k] − x Let x[k + 1k])) x ˆ[k + 1k + 1] = x ˆ[k + 1k] + 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 + 1k + 1] 2 Yk ] is a minimum. This means that we must minimize ψ(K[k]) = x[k + 1k))) T r(Cov(x[k + 1] − x ˆ[k + 1k] − K[k](hk (x[k]) + εy [k] − hk (ˆ ≈ T r(Cov(ex [k + 1k] − K[k](h'k (ˆ x[k + 1k])ex [k + 1k] + εy [k + 1]))) = T r[(I − K[k]Hk )Px [k + 1k](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 + 1k](I − K[k]Hk )T + Pεy [k + 1]K[k]T = 0 where
x[k + 1k]) ∈ Rd×n Hk = h'k (ˆ
Equivalently, K[k] = Px [k + 1k]HkT (Hk Px [k + 1k]HkT + Pε,y [k + 1])−1 Application of the matrix inversion lemma gives K[k] = Px [k + 1k]HkT (Pε,y [k + 1]−1 − Pε,y [k + 1]−1 Hk (Px [k + 1k]−1 + HkT Pε,y [k + 1]−1 Hk )−1 HkT Pε,y [k + 1]−1 ) = Px [k + 1k]HkT Pε,y [k + 1]−1 − Px [k + 1k]HkT Pε,y [k + 1]−1 Hk (Px [k + 1k]−1 + HkT Pε,y [k + 1]−1 Hk )−1 HkT Pε,y [k + 1]−1 = Px [k + 1k]HkT Pε,y [k + 1]−1 − Px [k + 1k](In − Px [k + 1k]−1 (Px [k + 1k]−1 + HkT Pε,y [k + 1]−1 Hk )−1 HkT Pε,y [k + 1]−1 = (Px [k + 1k]−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 + 1k + 1] in terms of Px [k + 1k] and Px [k + 1k] in terms of Px [kk]. To do this, we observe that ˆ[k + 1k] = ex [k + 1k] = x[k + 1] − x x[kk]) − (1/2)fk'' (ˆ x[kk])V ec(Px [kk]) = fk (x[k]) + εx [k + 1] − fk (ˆ
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and hence forming the conditional covariance on both sides given Yk gives us Px [k + 1k] = Fk Px [kk]FkT + Pε,x [k + 1] where
Fk = f ' (ˆ x[kk]) ∈ Rn×n
Finally, we observe that ex [k + 1k + 1] = x[k + 1] − x ˆ[k + 1k + 1] = x[k + 1k])) x[k + 1] − x ˆ[k + 1k] + K[k](y[k + 1] − hk (ˆ x[k + 1k])) = ex [k + 1k] − K[k](hk (x[k + 1]) + εy [k + 1] − hk (ˆ x[k + 1k])ex [k + 1k] + εy [k + 1]) ≈ ex [k + 1k] − K[k](h'k (ˆ and hence (this calculation followed by tracing has already been computed above) Px [k +1k +1] = (In −K[k]Hk )Px [k +1k](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 + 1k] = fk (ˆ x[kk], w ˆ[k]) x[k + 1k], w ˆ[k]) x ˆ[k + 1k + 1] = x ˆ[k + 1k] + K[k](y[k + 1] − hk (ˆ where K[k] = Px [k + 1k]−1 + HkT Pε,y [k + 1]−1 Hk )−1 HkT Pε,y [k + 1]−1 where Px [k + 1k] = Fk Px [kk]FkT + Pε,x [k + 1] and Hk = hk,x (ˆ x[k + 1k], w ˆ[k]), Fk = fk,x (ˆ x[kk], w ˆ[k]) Px [k +1k +1] = (In −K[k]Hk )Px [k +1k](In −K[k]Hk )T +K[k]Pε,y [k +1]K[k]T
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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[mm − 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[mm − 1], w ˆ[m − 1]) − Hm,w δw 2
m=0
Here, x[mm − 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 [mm − 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 [mm − 1]]
m=0
where x[mm − 1], w ˆ[m − 1]) ey [mm − 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 [mm − 1]
m=0
Thus,
w ˆ[k + 1] = w ˆ[k] + P [k + 1]−1 b[k + 1]
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Now, P [k + 1] = λP [k] + HkT+1,w Hk+1,w , T b[k + 1] = λb[k] + Hk+1,w ey [k + 1k]
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 + 1k] 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+1k] 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+1k]−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 [kk]) x ˆ[k+1k] = f (Aˆ[k]ˆ x[kk]+B x[kk]+B
ˆ [k]ˆ ˆ [k]u[k] + Fˆ [k])) x ˆ[k + 1k + 1] = x ˆ[k + 1k] + K[k](y[k + 1] − h(D x[k + 1k] + E ˆ [k]h' (D ˆ [k]ˆ ˆ [k]u[k] + Fˆ [k]) x[k + 1k] + E Hk = D ˆ [k]u[k] + Cˆ [k]) x[kk] + B Fk = Aˆ[k]f ' (Aˆ[k]ˆ K[k] = Px [k + 1k]−1 + HkT Pε,y [k + 1]−1 Hk )−1 HkT Pε,y [k + 1]−1 where Px [k + 1k] = Fk Px [kk]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 + 1k + 1] = (In − K[k]Hk )Px [k + 1k](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+1k]−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 + 1k] = y[k + 1] − hk+1 (D x[k + 1k] + E ∂ hk+1 (Dˆ x[k + 1k] + 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 + 1k] + E(w)u[k] + F (w))( +
∂D(w) x ˆ[k + 1k] ∂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 + 1k] = a ˆ[k]ˆ x[kk] + bu[k] + c + α(ˆ a[k]ˆ x[kk] + bu[k] + c)2 , Hk = a ˆ[k] + 2αa ˆ[k](ˆ a[k]ˆ x[k + 1k] + bu[k + 1] + c) K[k] = (1/Px [k + 1k] + Hk2 /Pε,y )−1 Hk /Pε,y = Hk Px [k + 1k]/(Px [k + 1k]Hk2 + Pε,y ) x ˆ[k+1k+1] = x ˆ[k+1k]+K[k](y[k+1]−dˆ[k]ˆ x[k+1k]−eu[k]−f −β(dˆ[k]ˆ x[k+1k]+eu[k]+f )2 )
Px [k + 1k] = Fk2 Px [kk] + Pε,x
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where ˆ[k] + 2βa ˆ[k](ˆ a[k]ˆ x[kk] + bu[k] + c) Fk = a a ˆ[k](1 + 2β(ˆ a[k]ˆ x[kk] + bu[k] + c)) Px [k + 1k + 1] = (1 − K[k]Hk )2 Px [k + 1k] + K[k]2 Pε,y Note that K[k], Hk , Px [k + 1k], Px [kk] are all scalars, ie, 1 × 1 matrices. Also 1 − K[k]Hk = Pε,y /(Px [k + 1k]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+1k]−Hk+1,w δw ˆ[k])
Here, x[k+1k]+eu[k+1]+f )−β(dˆ[k]ˆ x[k+1k]+eu[k+1]+f )2 , ey [k+1k] = 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 + 1k] + 2βx ˆ[k + 1k](dˆ[k]ˆ x[k + 1k] + 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 + 1k] − 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+1k]+2βx ˆ[k+1k](dˆ[k]ˆ x[k+1k]+eu ˆ[k+1]+f ))2
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General Relativity and Cosmology with Engineering Applications = qdd [k]ˆ x[k + 1k]2 (1 + 2β(dˆ[k]ˆ x[k + 1k] + eu[k] + f )2 ) T Q[k]Hk+1,w = (ˆ x[k+1k]+2βx ˆ[k+1k](dˆ[k]ˆ x[k+1k]+eu[k+1]+f ))[qad [k], qdd [k]]T
=x ˆ[k + 1k](1 + 2β(dˆ[k]ˆ x[k + 1k] + 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 + 1k]. Doing so, we get Hk+1,w = [(dˆ[k]+2βdˆ[k](dˆ[k]ˆ x[k+1k]+eu[k+1]+f ))ˆ x[k](1+2α(ˆ a[k]ˆ x[kk]+bu[k+1]+c),
x ˆ[k + 1k] + 2βx ˆ[k + 1k](dˆ[k]ˆ x[k + 1k] + 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 + 1k] = a ˆ[kk]ˆ x[kk] + bu[k] + c + α(ˆ a[kk]ˆ x[kk] + bu[k] + c)2 , a ˆ[k + 1k] = a ˆ[kk], dˆ[k + 1k] = dˆ[kk] Pξ [k + 1k] = Fk Pξ [kk]FkT + Pε,ξ [k + 1] where
Fk = fk' (ξˆ[kk]) 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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so that (
Fk = (
a ˆ[kk] + 2αa ˆ[kk](ˆ a[kk]ˆ x[kk] + bu[k] + c) 0 0
Note that
(
Pxx [k + 1k] Pξ [kk] = ( Pxa [k + 1k] Pxd [k + 1k]
) c + 2αx ˆ[kk](ˆ a[kk]ˆ x[kk] + bu[k] + c) 0 1 0 ) 0 1
Pxa [k + 1k] Paa [k + 1k] Pad [k + 1k]
) Pxd [k + 1k] Pad [k + 1k] ) Pdd [k + 1k]
Pε,ξ [k] = diag[Pε,x [k], 0, 0] ξ[k + 1k + 1] = [ˆ x[k + 1k + 1], a ˆ[k + 1k + 1], dˆ[k + 1k + 1]]T = [ˆ x[kk], a ˆ[kk], dˆ[kk]]T +K[k](y[k+1]−(dˆ[k+1k]ˆ x[k+1k]+eu[k+1]+f +β(dˆ[k+1k]ˆ x[k+1k]+eu[k+1]+f )2 )) Hk = h'k (ξˆ[k + 1k]) = [dˆ[k + 1k] + 2βdˆ[k + 1k](dˆ[k + 1k]ˆ x[k + 1k] + eu[k] + f ), 0, x ˆ[k + 1k] + 2βx ˆ[k + 1k](dˆ[k + 1k]ˆ x[k + 1k] + eu[k] + f )] ∈ R1×3 K[k] = Pξ [k + 1k]HkT (Hk Pξ [k + 1k]HkT + Pε,y [k + 1])−1 ∈ R3×1 Pξ [k+1k+1] = Pξ [k+1k+1] = (In −K[k]Hk )Pξ [k+1k](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 + 1k] = a ˆ[kk]ˆ x[kk] + bu[k] + c + α(ˆ a[kk]ˆ x[kk] + bu[k] + c)2 2 Px [k + 1k] = Fxk Px [kk] + Pε,x [k + 1]
where Fxk = a ˆ[kk] + 2αa ˆ[kk](ˆ a[kk]ˆ x[kk] + bu[k] + c) 2 Kx [k] = Px [k + 1k]Hxk (Hxk Px [k + 1k] + Pε,y [k + 1])−1
Hxk = dˆ[kk] + 2βdˆ[kk](dˆ[kk]ˆ x[kk] + eu[k] + f ) x ˆ[k+1k+1] = x ˆ[k+1k]+Kx [k](y[k+1]−(dˆ[kk]ˆ x[k+1k]+eu[k+1]+f +β(dˆ[kk]ˆ x[k+1k]+eu[k+1]+f )2 )
Px [k + 1k + 1] = (1 − Kx [k]Hxk )2 Px [k + 1k] + Kx [k]2 Pε,y [k + 1] a ˆ[k + 1k] = a ˆ[kk], dˆ[k + 1k] = dˆ[kk], or equivalently, w ˆ[k + 1k] = w[kk] ˆ
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since w[k] = [a[k], d[k]]T w ˆ[k + 1k + 1] = w ˆ[k + 1k] + Kw [k](y[k + 1] − (dˆ[k + 1k]ˆ x[k + 1k] + eu[k + 1] + f ) −β(dˆ[k + 1k]ˆ x[k + 1k] + eu[k + 1]) where T T (Hwk Pw [k + 1k]Hwk + Pε,y [k + 1])−1 Kw [k] = Pw [k + 1k]Hwk
Hwk = [0, x ˆ[k + 1k] + 2βx ˆ[k + 1k](dˆ[k + 1k]ˆ x[k + 1k] + eu[k + 1] + f )]T and ﬁnally, Pw [k+1k+1] = (I2 −Kw [k]Hwk )Pw [k+1k](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 + 1k] = E[ξ[k + 1]Yk ] = E(fk (ξ[k])Yk ) ≈ fk (ξˆ[kk]) Pξ [kk] = Cov(ξ[k]Yk ) = E[(ξ[k] − ξˆ[kk])(x[k] − ξˆ[kk])T Yk ] We have ξˆ[k + 1k + 1] = ξˆ[k + 1k] + K[k](y[k + 1] − hk+1 (ξˆ[k + 1k])) approximately. Let
Hk = h'k+1 (ξˆ[k + 1k]) ∈ Rd×n
Then, from previous analysis K[k] = Pξ [k + 1k]HkT (Hk Pξ [k + 1k]HkT + Pε,y [k + 1])−1 = (Pξ [k + 1k]−1 + HkT Pε,y [k + 1]−1 Hk )−1 HkT Pε,y [k + 1]−1 Pξ [k + 1k] = Fk Pξ [kk]FkT + Pε,ξ [k + 1] where
Fk = f ' (ξˆ[kk]) ∈ Rn×n
Pξ [k +1k +1] = (In −K[k]Hk )Pξ [k +1k](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]
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The dual EKF for this system can be implemented as follows. ˆ 1 [t], ..., W ˆ k [t], x[t + 1], U zˆ1 [tt])), k = 1, 2, ..., N + 1 zˆk [t + 1t] = ψk (W zˆk [t + 1t + 1] = zˆk [t + 1t] + K[t](y[t + 1] − zˆN +1 [t + 1t]) Pz [t + 1t] = Ft Px [tt]FtT + Pε,z [t + 1] ˆ [t])[g1 , 0, 0, .., 0] ∈ RN +1×N +1 z [t + 1t], W Ft = g,z (ˆ ˆ [t]), ..., gN +1,z (ˆ ˆ [t])]T g1 = [g1,z1 (ˆ z1 [t + 1t], W z1 [tt], W 1 K[t] = (Pz [t + 1t]−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 + 1t]−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[mm − 1], w ˆ[m − 1] + δw) 2
m=0
We approximate: hm (ˆ x[mm − 1], w ˆ[m − 1] + δw) ≈
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General Relativity and Cosmology with Engineering Applications hm (ˆ x[mm − 1], w ˆ[m − 1]) +
q ∑ ∂ r hm (ˆ x[mm − 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 [mm − 1] −
m=0
q ∑
Hm,w,r (δw)⊗r 2
r=1
Now,  ey [mm − 1] −
q ∑
Hm,w,r (δw)⊗r 2
r=1
= ey [mm−1] 2 +
∑
q T (δw)⊗sT Hm,w,s Hm,w,r (δw)⊗r −2
r,s=1
∑
q
ey [mm−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 [mm − 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 [mm − 1] −
m=0
= ey [mm−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 [mm−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 [mm − 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 [mm − 1] r=1 m=0
m=0
which is easily solved by matrix inversion.
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Estimating the weights in the presence of white nonGaussian 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[mm − 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) = A1/2 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 meanvariance 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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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(kk)) which is In the EKF, the state predictor is constructed as x ˆ(k+1k) = 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 nonGaussian. Hence, to eﬀectively approximate the above conditional expectation, the unscented Kalman ﬁlter uses x ˆ(k + 1k) = N −1
N ∑
Fk (ˆ x(kk) + ξr )
r=1
where ξr are vectors chosen so that they approximately represent independent samples of x(k) − x ˆ(kk). 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+1k+1) ˆ(k+1k)+Pxy (k+1k)Pyy (k+1k)−1 (y(k+1)−yˆ(k+1k)) = E(x(k+1)Yk+1 ) ≈ x but to compute Pxy and Pyy , we use N ∑ Pxy (k+1k) = N −1 (Fk (ˆ x(kk)+ξr )−x ˆ(k+1k))(hk+1 (ˆ x(k+1k)+ηr )−yˆ(k+1k)) r=1
where yˆ(k + 1k) = N
−1
N ∑
hk+1 (ˆ x(k + 1k) + ηr )
r=1
and further, Pyy (k+1k) = N −1
N ∑
[(hk+1 (ˆ x(k+1k)+ηr )−yˆ(k+1k))(hk+1 (ˆ x(k+1k)+ηr )−yˆ(k+1k))T ]
r=1
We note that to calculate the samples ξr of e(kk) = x(k) − x ˆ(kk), the UKF makes use of the corresponding error covariance P (kk) while to compute sam ˆ(k + 1k), the UKF makes use of the error ples ηr of e(k + 1k) = x(k + 1) − x covariance P (k + 1k). 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 +1k) = N −1
N ∑
(Fk (ˆ x(kk)+ξr )−x ˆ(k +1k))(Fk (ˆ x(kk)+ξr )−x ˆ(k +1k))T
r=1
P (k + 1k + 1) = P (k + 1k) − Pey (k + 1k)Pyy (k + 1k)−1 Pey (k + 1k)T Pey (k + 1k) = Pxy (k + 1k) 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 2D 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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General Relativity and Cosmology with Engineering Applications
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 feedforward 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 feedforward 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 backpropagation 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 feedforward 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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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 backpropagation 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 spacetime 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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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 HahnBanach 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 selfadjoint operator.
5.29 Estimating the metric parameters from geodesic measurements Suppose that the metric of spacetime 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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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 3D 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
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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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is the Hamiltonian density. The noiseless classical Hamilton equations for the ﬁeld that are equivalent to the above EulerLagrange 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 spacetime 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] KleinGordon ﬁeld in a background metricHamiltonian form [d2] Einstein ﬁeld equations for gravitation in Hamiltonian form. We express the metric of spacetime 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 EinsteinHilbert 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 KleinGordon 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
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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 selfadjoint 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 oneone 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 spacetime
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 spacetime 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
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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 HudsonParthasarathy 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 HudsonParthasarathy 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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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 HudsonParthasarathy 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 HudsonParthasarathy 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
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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 spacetime in such a way that a desired unitary gate is formed after time T .
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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
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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 dAφ(u) >= u(t)dtφ(u) >, dΛφ(u) >= (dA∗ dA/dt)φ(u) >= u(t)dA∗ φ(u) > and dA∗ φ(u) >= dBφ(u) > −dAφ(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 dψ(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) >
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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 spacetime 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 ndimensional spacetime, 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 NavierStokes 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 = kn−1 dkdΩ(kˆ) with obvious meanings. Let
λ = k2 = k 2
λ represents the kinetic energy H0 of the projectile. We write with kˆ = ω ∈
S n−1 ,
√ f˜(k) = f˜(kω) = f˜( λω)
Then, 2
 f  = ˜ f 2 =
∫
√ √ λ(n−1)/2 d λdωf˜( 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 HilbertSchmidt 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 GramSchmidt 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 Xxp+1 ], [Xxp+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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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 spacetime 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
KnillLaﬂamme 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 KnillLaﬂamme 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 Eψ0 >→ Eψk > 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 KnillLaﬂamme 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 Eψk > is expressible as a linear combination of Ea ψk >, a = 1, 2, ..., q and hence we get from the above equation that l ∑
Rj Eψk >< 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 YangMills 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 RiemannChristoﬀel curvature tensor obtained by using the gravitational spin connection. Unlike the YangMills 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 spacetime. −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 spacetime with coordinates X μ . Let g˜μν denote the metric w.r.t. the spacetime 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
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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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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
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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 KnillLaﬂamme 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.
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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.
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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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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), BD(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)
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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 selfadjoint. 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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Combining these two equations gives us Gr(A¯) = Gr(A∗ ) which implies that
A¯ = A∗
It follows from the selfadjointness 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 selfadjoint. Thus, we have proved that a symmetric operator A is selfadjoint 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 VonNeumann 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 electronpositron 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 SchwingerTomonaga and Dyson while Feynman developed the path integral approach to such calculations with ﬁnally Dyson proving the equivalence of the theories of Feynman and SchwignerTomonaga. 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
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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 YangMills group theo retic generalization of the Dirac or KleinGordon 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 spacetime 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 Gtransformations of the matter ﬁeld and the gauge ﬁeld provided that the Lagrangian is a Ginvariant func tion of its agruments. Owing to the noncommutativity 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 nonAbelian 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 spacetime vari ables. The corresponding gauge transformation of the gauge ﬁeld potentials,
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namely, the electromagnetic four potential simpliﬁes to the standard Lorentz gauge transformation Aμ → Aμ + ∂μ φ. Based on the YangMills nonAbelian generalization of electromagnetism, Salam,Weinberg and Glashow were able to unify the electromagnetic forces and the weak forces, calling it the electroweak theory. Their idea was to start with a nonmassive 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 nonderivative 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 nonderivative quadratic components of the broken part while it contains nonderivative 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 Ginvariant. 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 nonderivative 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
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family of noncommuting 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 HudsonParthasrathy (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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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 GlauberSudarshan nonorthogonal 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 GlauberSudarshan 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 atomicbath 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
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viewed as an linearalgebrafunctional analytic approach to probability theory with generalizations to the noncommutative 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 VonNeumann algebra of observables that satisﬁes the nondemolition 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 nondemolition 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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equation for πt (X) is the fundamental quantum ﬁltering equation and is the noncommutative generalization of Kushner’s equation for classical nonlinear 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
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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 nonHamiltonian 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 nondemolition 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 nondemolition 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 Sφin >=< 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 ' Rλ, 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 ' Rλ, 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 electronpositron ﬁeld of matter and if possible other YangMills 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 spacetime 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 YangMills ﬁeld in curved spacetime. This Lagrangian density is added to the EinsteinHilbert Lagrangian density of the gravitational ﬁeld and to the Lagrangian density of the electromagentic and YangMills 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 YangMills 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 nonAbelian SU (N ) gauge ﬁelds for the YangMills 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/absorptionemission processes involving gravitons, photons, electrons, positrons, and particles associated with the YangMills 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 KleinGordon and Higgs, the YangMills 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 spacetime 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 superspace 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 spacetime 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 spacetime 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 KleinGordon Lagrangian density is a quadratic function of the spacetime 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 Dcomponent 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 KleinGordon 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 nonAbelian 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 mattergauge 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 YangMills 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 DeBroglie 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 DeBroglie, 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 DeBroglie 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 DeBroglie’s matterwave 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] BohrSommerfeld’s quantization rules. If q is a canonical position coordinate vector and p the canonical momentum vector, then for cyclic motion, the BohrSommerfeld’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 actionangle 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 BohrSommerfeld 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 nonrelativistic 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 + 2A1 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 DeBroglie matterwave 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 = hk1 −k2 /2π according to DeBroglie 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
dψ(t) > = H(t)ψ(t) > dt
where H(t) is a possibly time varying selfadjoint 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 selfadjoint 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)Xψ(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)Xψ(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 noncommutative 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 SchrodingerLiouville 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
iψk' (t) >= H(t)ψk (t) > and hence
−i < ψk' (t) =< ψk (t)H(t)
so that iρ' (t) =
∑ (iψk' (t) > pk < ψk (t) + iψk (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)Xψk (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 =< ψW (t)∗ (U0 (t)∗ XU0 (t))W (t)ψ > where
∫ W (t) = T {exp(−i
t
V˜ (τ )dτ )} 0
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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) >=< ψU (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 nondegenerate 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
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[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 ≤< Ea∗ aE >=< EH − ω/2E >= 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 aE >= 0. In other words, we have aω/2 >= 0 and hence (d/dq + mωq) < qω/2 >= 0 or equivalently,
< qω/2 >= C.exp(−mω 2 q 2 /2)
with C being the normalizing constant chosen so that ∫ ∫ 1 =< ω/2ω/2 >=  < qω/2 > 2 = C2 exp(−mω 2 q 2 )dq R
R
= C2 = (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 nondegenerate 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 Hψ >= Eψ > 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, kV1 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, kV1 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
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∑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, rψ1 > + < l, rV1 ψ0 >= E1 < l, rψ0 >= 0 so that
< l, rV1 ψ0 > E0 (m) − E0 (l) ∑ which implies that the unperturbed state ψ0 >= r cs (m, r)m, r > gets perturbed to ψ0 > +εψ1 > +O(ε2 ) < l, rψ1 >=
where ψ1 >=
∑
l, r >< l, rV1 ψ0 > /(E0 (m) − E0 (l))
l/=m,r=1,2,...,d(m)
and ∑
d(m)
< l, rV1 ψ0 >=
cs (m, k) < l, rV1 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),
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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 < mU (T )n >= −i
r ∑
εk < mUk (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 + 1nk + 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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∑
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 pdimensional 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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[27] The quantum Ito formula of Hudson and Parthasarathy. [28] The general theory of quantum stochastic diﬀerential equations. [29] The HudsonParthasarathy 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 FeynmanKac 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 (SchrodingerHeisenberg) and La grangian (path integral) approaches to quantum mechanics. The SchrodingerHeisenberg approaches to quantum mechanics are Hamilto nian approaches. Feynman proposed an alternative approach to nonrelativistic 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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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) =< 0T {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) =< 0T {U (∞, −∞)φ(x)φ(y)}0 >= < 0U (∞, 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 matterﬁeld 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 3dimensional 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
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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 3D 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 )
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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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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/4πr − 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/4πr 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(Kt−K.r))+¯ er (K, σ)a(K, sigma)∗ exp(i(Kt−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
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General Relativity and Cosmology with Engineering Applications ∫ =
2K2 e(K, σ)2 a(K, σ)∗ a(K, σ)d3 K
once we make use of the fact that K × e(K, σ) = Ke(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) = (2K)−1/2 in order to ensure that ∫ HF =
Ka(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)] = −iKa(t, K, σ), a∗ (t, K, σ),t = i[HF , a(t, K, σ)∗ ] = iKa∗ (t, K, σ) These equations imply a(t, K, σ),t = −K2 a(t, K, σ) a∗ (t, K, σ),tt = −K2 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.
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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 3vector (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
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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 skewsymmetric 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 nonzero 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 spinﬁeld 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 spacetime as Dμν (x, y) =< 0T {Aμ (x)Aν (y)}0 > and the electron propagator as Slm (x, y) =< 0T {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
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and this equation has solutions ∫
√ √ [a(K, σ)eμ (K, σ)exp(−ik.x)/ 2K+a(K, σ)∗ e¯μ (K, σ)exp(ik.x)/ 2K]d3 K
Aμ (x) =
where k.x = kμ xμ = Kt − 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, < 0a(K, σ)∗ = 0, we have ∫ eν (K ' , σ ' ) Dμν (x, x' ) = θ(t − t' ) < 0a(K, σ)a(K ' , σ ' )∗ 0 > eμ (K, σ)¯
∫ +
×(exp(−i(k.x − k ' .x' ))/2K)d3 Kd3 K ' θ(t' −t) < 0a(K ' , σ ' )a(K, σ)∗ 0 > eν (K ' , σ ' )¯ eμ (K, σ)(exp(i(k.x−k ' .x' ))/2K)d3 Kd3 K '
= θ(t−t' )
∫
+θ(t' −t)
eμ (K, σ)¯ eν (K ' , σ ' )δσ,σ' δ 3 (K−K ' )/2K)(exp(−i(k.x−k ' .x' ))/2K)d3 Kd3 K '
∫ ∫
=
eν (K ' , σ ' )¯ eμ (K, σ)δσ,σ' δ 3 (K−K ' )(exp(i(k.x−k ' .x' ))/2K)d3 Kd3 K ' [eμ (K, σ)¯ eν (K, σ)θ(t − t' )exp(−iK(t − t' ) + iK.(r − r' ))+
eμ (K, σ)θ(t' − t)exp(iK(t − t' ) − iK.(r − r' ))]d3 K/2K 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 (2K)−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 − K2 ) = 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 − K2 ) = iπexp(−iK(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 − K2 + iε and to preserve Lorentz invariance, we may assume ∫ ημν ημν ˆ Dμν (k) = Dμν (x).exp(−ik.x)d4 x = 02 = 2 k − K2 + iε k + iε where
k 2 = k 02 − K2 = 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 )
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= 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 pseudoinverse 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/pseudoinverse 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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the electron propagator is given by Slm (x, y) =< 0T (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 electronpositronphoton 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
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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 − ge >< 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, Hψ >= Eψ > gives H0 ψ > −ge >< eψ >= Eψ > or
ψ >= g(H0 − E)−1 e >< eψ > ∑ =g k > (Ek − E)−1 < ke >< eψ > k
The normalization condition < ψψ >= 1 then implies ∑ g2 (Ek − E)−2  < ke > 2  < eψ > 2 = 1 k
This implies that  < eψ > 2 = (g 2
∑
(Ek − E)2  < ke > 2 )−1
k
If the above sum is divergent, then we would get < eψ >= 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 < ke > 2 )−1 k:Ek ≤Λ
and get with this cutoﬀ imposed that  < eψ > 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
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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 spacetime and ηab the Minkowski metric of ﬂat spacetime. 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 spacetimes 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 spacetime: (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 ,
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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 spacetime 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 noninertial metric to the inertial metric. So we assume our curved spacetime 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
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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 GlauberSudarshan rep resentation. Representing any state of the laser interacting with the spin of an atom using the GlauberSudarshan representation having matrix coeﬃcients. Expressing the evolution equation for the density operator of the laseratom system using the GlauberSudarshan 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)∗ )∗
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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 GaluberSudarshan 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 ptuple 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(< uz >) 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) >=< uz > e(z) >, ak e(z) >= zk e(z) > Also, a∗k e(z) >=
∑
z n a∗k a∗n0 > /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∗ )
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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 ze(v) >= φ(z)exp(< uz > + < zv >)d2p z = 1 We can express the joint state of the atom and the ﬁeld in the interaction representation as a GlauberSudarshan 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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335
(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 3D 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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General Relativity and Cosmology with Engineering Applications
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 nondegeneracy), 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
General Relativity and Cosmology with Engineering Applications
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 HartreeFock 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 < ψHψ > 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)}] =< ψHψ > − λ(k, j)(< ψk ψj > −δkj ) 1≤k≤j≤N
We observe that < ψHψ >=
∑
< 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 < ψHψ > 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 BornOppenheimer 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 /2rkl − rmj  Vee = (k,l)/=(m,j)
is the total electronelectron interaction potential energy. ∑ VN N = Z 2 e2 /Rk − Rm  k/=m
is the total nucleonnucleon interaction potential energy. ∑ Ze2 /Rk − rml  VeN = − k,m,l
is the total electronnucleon 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 electronnucleon 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 (Xηti )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 (Xηti )U (t))U (t)∗ ZU (t)] = E[U (t)∗ XZU (t) − U (t)∗ Et (Xηti )ZU (t)] = E[U (t)∗ (XZ − Et (Xηti )Z)U (t)] = Et (XZ − Et (XZηti )) = 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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General Relativity and Cosmology with Engineering Applications
Then, we claim that EF (t) (Xηti ) = 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)Zηti )/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 HudsonParthasarathy 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
General Relativity and Cosmology with Engineering Applications
359
The EvansHudson ﬂow corresponding to this HP equation is obtained by taking a selfadjoint 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 VonNeumann 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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ie the output measurements are nondemolition processes. Let ηt = σ(Yk (s) : s ≤ t, k = 1, 2, ..., r). Then ηt is an Abelian VonNeumann 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 ] =
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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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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 HudsonParthasarathy 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 GoriniKossakowski 0 SudarshanLindblad (GKSL) generator on the system operator space. More generally, for general nondemolition 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)
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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 unitaryantiunitary 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 quasiinvariant 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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= (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)
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General Relativity and Cosmology with Engineering Applications since [S(t), S ' (t)] is a cnumber 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 cnumber 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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= 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 VonNeumann 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 (KnillLaﬂamme): 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 >< ψEs∗ 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, < φψ >= 0, we have ∑  < φRk Es ψ > 2 = λ < φψ > 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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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 KnillLaﬂamme 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 >
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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)) =
∑
< aW (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
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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 ) Postmultiplying both sides of this equation by their adjoints gives us P (F ∩ gE ∩ gh−1 F ) = λ2 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∗ = λ2 Px or equivalently,
Pax = λ2 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
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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(AB) where D(AB) = T r(A(logA − logB)) is the relative entropy between A and B. Thus, the ﬁrst condition gives R = D(AB). 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(AB) 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(AB) So, taking log(c) = D(AB) − δ gives log(1 − P2 (c)) ≥ −log(c) + s(log(c) + log(T r(A1−s B s ))/s) ≥ −D(AB) + δ + s(D(AB) − δ − D(AB)) = −D(AB) + (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(AB) − δ + D(AB) + 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(AB)/s + D(AB)/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(AB)/s2 +(D(AB)2 /s)+(D(AB)/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(AB)/s2 + (D(AB)2 /s) +(D(AB)/s2 ) − 2D(AB)/s2 The above equals T r(A(log(A))2 + B(log(B))2 − 2Alog(A)log(B))/s + (D(AB)2 /s which is positive. Hence, g(s) for s ≥ 0 attains its minimum value of D(AB) at s = 0. [61] The SudarshanLindblad 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 YangMills 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 skewHermitian 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
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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 nonnegative 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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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)+
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((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
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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
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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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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
)     )
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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 nondiagonable matrix A. If A is nondiagonable,
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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 nonsingular 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 selfadjoint, ie S¯ is selfadjoint. Further if R(S + i) and R(S − i) are both exactly H, then S is selfadjoint. 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 selfadjoint, 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
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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) > xX(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) > xX(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 (xy) = −
d P (X(t) > xX(0) = y) = (2πt)−1/2 (exp(−(x−y)2 /2t) dx −exp(−(x+y)2 /2t)), x, y > 0
Also P (X(t) = 0X(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) ≥ 0X(0) = y) = qt (xy)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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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 semiMartingales. Let Xt be continuous semimartingale. By the DoobMeyer 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 RiemannStieltjes 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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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 selfadjoint 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 nonGaussian 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 nondemolition 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 nonuniformities 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 spacetime S00 = Now
1 Fαβ F αβ − F0α F0α 4
Fαβ F αβ = −2F0r F0r + Frs Frs = −2E2 + 2B2 F0α F0α = −F0r F0r = −E2
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General Relativity and Cosmology with Engineering Applications
so
1 1 (−E2 + B2 ) + E2 = (E2 + B2 ) 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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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 spacetime 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 spacetime is comoving, ie, v μ(0) = (1, 0, 0, 0) deﬁne geodesics in the unperturbed spacetime. 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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General Relativity and Cosmology with Engineering Applications (0)
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
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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
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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 = 4z ' (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, KleinGordon 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 KnillLaﬂamme 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] PostNewtonian 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(δX2 ) [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 KnillLaﬂamme 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 nonoverlapping 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
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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
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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 = [(2K)−1/2 a(K, σ)er (K, σ)exp(−i(Kt − K.r))+ (2K)−1/2 a∗ (K, σ)¯ er (K, σ)exp(i(Kt − 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(Kt − K.r))− (K/2)1/2 a∗ (K, σ)¯ er (K, σ)exp(i(Kt − K.r))]d3 K or equivalently in three vector notation, ∫ E = i [K/2)1/2 a(K, σ)e(K, σ).exp(−i(Kt − K.r))
422
General Relativity and Cosmology with Engineering Applications −(K/2)1/2 a∗ (K, σ)¯ e(K, σ)exp(i(Kt − K.r))]d3 K The magnetic ﬁeld is given by ∫ B = curlA = i [(2K)−1/2 a(K, σ)K × e(K, σ)exp(−i(Kt − K.r))− (2K)−1/2 a∗ (K, σ)K × e(K, σ)exp(i(Kt − K.r))]d3 K The em ﬁeld of the image can be regarded as the output of a spatiotemporal 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 (tθ) =
p ∑
(Gk (tθ) ⊗ ak + Gk (tθ)∗ ⊗ a∗k )
k=1 N ×N
, θ is the image parameter vector. The joint density of where Gk (tθ) ∈ C the atom and the image em ﬁeld at time t can be expressed using the GlauberSudarshan representation: ∫ ρ(t) = C exp(−z2 )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 (tθ), ρ(t)] and this translates to ∫ i A,t (t, z) ⊗ e(z) >< e(z)exp(−z2 )d2p z = ∑∫ ( (Gk (tθ)A(t, z) ⊗ ak e(z) >< e(z) + Gk (tθ)∗ A(t, z) ⊗ a∗k e(z) >< e(z) k
−A(t, z)Gk (tθ)⊗e(z) >< e(z)ak −A(t, z)Gk (tθ)∗ ⊗e(z) >< e(z)a∗k )exp(−z2 )d2p z) ∑∫ ∂ ( (Gk (tθ)A(t, z)zk ⊗e(z) >< e(z)+Gk (tθ)∗ A(t, z)⊗( e(z) >< e(z)) = ∂zk k
∂ e(z) >< e(z))−A(t, z)Gk (tθ)∗ z¯k ⊗e(z) >< e(z))exp(−z2 )d2p z) ∂z¯k ∑∫ ∂ = ( (zk Gk (tθ)A(t, z) − (Gk (tθ)∗ ( − z¯k )A(t, z)
−A(t, z)Gk (tθ)⊗(
∂zk
k
∂ − zk )A(t, z)Gk (tθ) − z¯k A(t, z)Gk (tθ)) ⊗ e(z) >< e(z)d2p z) ∂z¯k Thus, we get iA,t (t, z) = (T (tθ)A)(t, z) +(
where T (tθ) is a diﬀerential operator acting on the space of matrix valued functions of the complex variable z ∈ Cp deﬁned by T (tθ)X(z) =
∑
(zk Gk (tθ)X(z) − z¯k X(z)Gk (tθ) − Gk (tθ)∗ (
k
∂ − zk )X(z)Gk (tθ)) ∂z¯k The formal solution to this partial diﬀerential equation is ∫ t A(t, z) = τ {exp(−i T (sθ)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 submartingales. Let Mk , k = 0, 1, 2, ... be a nonnegative submartingale. 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 stoptime. 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) ≤ EMt /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 RadonNikodym 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 [np] = Pn,p ξn , eb [n − 1p] = Pn,p z ξn
We have the obvious formula based on orthogonal direct sum decompositions: ⊥ ⊥ Pn,p+1 = Pn,p − Psp{eb [n−1p]}
and hence ef [np + 1] = ef [np] − Kf [np]eb [n − 1p] − − − (1) and likewise, e˜b [np + 1] = eb [n − 1p] − Kb [np]ef [np − − − (2)] where
⊥ −p−1 e˜b [np + 1] = Psp{ξ ξn −1 ξ ,...,z −p ξ } z n ,z n n
It is easy to see that ( eb [np + 1] =
e˜b [np + 1] 0
)
428
General Relativity and Cosmology with Engineering Applications Here, the forward reﬂection coeﬃcient is Kf [np] =< eb [n − 1p], ef [np > /Eb [n − 1p] and the backward reﬂection coeﬃcient is Kb [np] =< eb [n − 1p], ef [np] > /Ef [np] where
Ef [np] = ef [np] 2 , Eb [n − 1p] = eb [n − 1p] 2
We thus have 0 ≤ Kf [np]Kb [np] =  < eb [n − 1p], ef [np] > 2 /Ef [np]Eb [n − 1p] ≤ 1 From (1) and (2), we get Ef [np + 1] = Ef [np] + Kf2 [np]Eb [n − 1p] − 2Kf [np]2 Eb [n − 1p] = Ef [np] − Kf2 [np]Eb [n − 1p] = (1 − Kf [np]Kb [np])Ef [np] and likewise, Eb [np + 1] = (1 − Kf [np]Kb [np])Eb [n − 1p] 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) > +δψn(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 eigenequation 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
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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
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∑
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 nonrelativistic 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 )
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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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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 HudsonParthasarathy 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 nonuniform transmission lines by expanding the distributed parameters and the line voltage and current as Fourier series in the spatial
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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 nonuniform 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 LandauLifshitz 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 nonlinearmemory 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 GoriniKossakowskiSudarshanLindblad (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 HudsonParthasarathy 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, HarishChandra 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 electronpositron 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 electronpositron 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 positionmomentum 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 SchwingerTomonaga 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 Ginvariant 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 Ginvariant Lagrangian density. This is what happens in the electroweak theory. The electroweakstrong uniﬁcation was achieved by GellMann and NeeMann 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 Gtransformation 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 Gactions 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 noncommutative YangMills 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 spacetime, then this tetrad can be understood as a lo cal transformation of curved spacetime 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 spacetime 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 spacetime, 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 spacetime and when matter moves in such a curved space time, follows geodesics which are shortest paths on the curved four dimensional manifold of spacetime. 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 nonconjugate Cartan algebras. The reprsentation theory for real semisimple Lie algebras was developed almost sin gle handedly by the great Indian mathematician HarishChandra 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 nonzero 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 nondegenerate 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 spacetime 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 spacetime in quantum general relativity. Gravitational redshift: Let U (r) be the gravitational potential. Then the approximate (Newtonian) metric of spacetime 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 spacetime 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 nonlinear 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 nonrandom 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, yx' , y ' )(δφ(x' , y ' )+gs (x' , y ' )δθs )dxdydx' dy ' )
where Q(x, yx' , y ' ) is the inverse Kernel of R(x, yx' , 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, yx' 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, yx' , y ' )g(x, y)g(x' , y ' )T dxdydx' dy ' ]−1 [
∫
Q(x, yx' , y ' )g(x, y)δφ(x' , y ' )dxdydx' dy ' ]
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A simple computation gives us the covariance of this parameter vector estimator: ∫ Cov(δθˆ) = [ Q(x, yx' , 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, nθ) =
∑
A(a, b, θ) < ψm , ∂xa ∂yb ψn >
a,b
The above equation can then be expressed as ∑ P (m, nθ)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, nθ0 )c(n, φ0 ) = c(m, s), n
∑ ∑ (( Pr (m, nθ0 )δθr )(¸n, φ0 ) + P (m, nθ0 )δc(n)) = c(m, w) n
r
where Pr (m, nθ0 ) =
∂P (m, nθ0 ) ∂θr
and δc(n) = c(n, φ0 + δφ) − c(n, φ0 )
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We write c0 (n) = c(n, φ0 ), Pr (m, n) = Pr (m, nθ0 ), P0 (m, n) = P (m, nθ0 ) 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 spacetime 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 nonAbelian 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
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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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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 StoneVonNeumann 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)
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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 v2 /2)U (−vτ, v) Thus, we get from the ﬁrst equation W1 (τ )V2 (v)W1 (−τ ) = exp(iτ v2 /2)V1 (−vτ )V2 (v) = exp(iτ v2 /2)exp(iτ v.P )V2 (v) For inﬁnitesimal τ , this gives −i[H, exp(−iv.Q)] = (iv2 /2 + iv.P )exp(−iv.Q) or equivalently, H − exp(−iv.Q).H.exp(iv.Q) = −v2 /2 − v.P − − − (1) The O(v) term of this equation gives i[v.Q, H] = −v.P or equivalently, i[H, Qj ] = Pj
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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. Nparticle 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)
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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)
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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 < vHt 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(< vexp(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
458
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 (km) = 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, mρI k, m > PI (km) = ∑ r < r, mρI 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 ˜ mρI k, ˜ m> ˆ m, k, m] < k, I[k, ∑ =∑ ˆ ˜ mρI ˜ 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 >< mm' >< 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 >< rTj∗' l' ⊗ j >< lm >< sl' > m >< j '  =
∑ jj '
Tjm k >< rTj∗' 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
E2 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
E2 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 ∫ ∇ × A2 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
∇ × A2 dxdydz =
∫ (ε/2)
E2 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 E2 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 nonGaussian 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 xk 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) = AΠM i=1 f ((Ax)i ) where A is an M × M matrix and x ∈ RM . We have ψ(x) = (2π)−M/2 Aexp(−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
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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(xy), ie, the MAP estimate. We have p(xy)py (y) = p(yx)px (x) = pw (y − Ax)px (x) Discrete time nonlinear 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 nonsingular 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, vYn )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, vYn )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, vYn )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, vYn )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, vYn )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, vYn )dv ∫ F (n, Yn+1 , vˆ0 [n + 1]) = ψ2 (n, y[n + 1], vˆ0 [n + 1], v)p(n, vYn ) 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 nonpositive 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) = α2 .
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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 VandderMonde matrix ((< u, uk >n ))1≤k,m≤p is nonsingular 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 ] )
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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) >≤ maxt−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 maxt−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) >
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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. HartreeFock 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 HartreeFock 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)
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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(−x2 /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(−mx2 /2it)exp((x, ∇Q ))ψ(Q)dx exp(−mx2 /2it)ψ(Q + x)d3 x exp(−mQ − x2 /2it)ψ(x)d3 x
Deﬁne the Kernel function Kt (Q) = (2πit/m)−3/2 exp(−mQ2 /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 ))
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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 RadonNikodym derivatives d < uE0 (λ)u > /dλ and d < f E0 (λ)f > /dλ exist and are ﬁnite. We have for V = u >< u with < uu >= 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 >< uexp(−itH0 )f > so  V.exp(−itH0 )f >=  < uexp(−itH0 )f >  ∫ =  (d < uE0 (λ)f > /dλ)exp(−iλt)dλ R
This is the magnitude of the Fourier transform of the function λ → d < uE(λ)f > /dλ. We note that  < udE(λ)f >  ≤< udE) (λ)u >1/2 < f dE0 (λ)f >1/2 Hence, d < uE0 (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 RadonNikodym 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 < uφ(H)E0 (λ)f >
exp(−isφ(λ))φ(λ)d < uE0 (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 < uφ(H)E0 (φ−1 (λ))f > /dλ)dλ ∫
−
exp(−isλ)λ(d < uE0 (φ−1 (λ))f > /dλ)dλ
provided we assume that the concerned RadonNikodym 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
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with the constraints < ψ1 ψ1 >=< ψ2 ψ2 >= 1, < ψ1 ψ2 >= 0 As in all eigenvalue problems we extremize S =< ψHψ > −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 eψ2 (r2 )2 . The second term in the above expression represents the eﬀect of the spin interaction between the two electrons,
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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 =< ψψ > 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 ⊗IVσ −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 integropartial 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 CauchyRiemann 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 = ∇u2 = ∇v2 = 1/z2 = 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 χZs 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)
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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 selfadjoint 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
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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 LippmanSchwinger 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 LippmanSchwinger 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 LippmanSchwinger 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
501
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 energymomentum 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 energymomentum 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' )
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< 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 energymomentum 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
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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
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< 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 ) =
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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),
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{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, xs, 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 (txsy) = 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 '
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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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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 χSn /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 nonrandom 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 supersymmetry 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 rightinvariant 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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(
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 HudsonParthasarathy 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 3D 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 HudsonParthasarathy 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 selfadjoint. 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)})
524
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 nondemolition 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)
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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 KleinGordon 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
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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 >) =
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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 >
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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 . SchrodingerVonNeumannLiouville 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 (xu)/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 (xui )/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
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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 (xv) − Πx∈A P (x)N (xv)  ∑
= e
x
=≤ exp(
N (xv)log(Pu (x))
∑
−e
∑ x
N (xv)log(P (x))

N (xv)(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 (xv)log(1 + f (x)/P (x) n)) − 1 =
x
√ N (xv)(f (x)/P (x) n − f (x)2 /2P (x)2 n + ...)) − 1
x
√ If v ∈ / T (P, n, δ), then N (xv) < nP (x) + O( n) in which case we see that ∑ √ √ N (xv)(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 (xv) = nP (x) + g(x) n where g(x) < 1. Hence, in this case, ∑ √ N (xv)(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 (xv)log(P (x)) = = N (xv)log(Pu (x)) − x

∑
x
√ N (xv)(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
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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(yx) = 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 ”FeinsteinKhinchin 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
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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] Quasiclassical quantum mechanics in a curved spacetime 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 feedforward 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 feedforward 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
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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 dAφ(u) >= u(t)dtφ(u) >, dA∗ φ(u) >= (dB(t) − u(t)dt)φ(u) > −, dΛφ(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 dψ(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
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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)) = (dψ(t) >) < ψ(t)+ψ(t) > d < ψ(t)+(dψ(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, dψt (B) >= (−(iH + P )dt + u(t)L1 dt + (dB(t) − u(t)dt)L2 + u(t)(dB(t) − u(t)dt)S)ψt (B) >
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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 backpropagation 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 feedforward 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 feedforward, ie, no backpropagation 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 feedforward, ∂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(Kct − K.r))]d3 K hμν (t, r) = [aμν (K)exp(i(Kct − K.r)) + a
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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 spacetime 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 spacetime 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 selfadjoint 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 ].
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General Relativity and Cosmology with Engineering Applications
[28] FeinsteinKhinchin 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(yx), 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
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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 (xu) denote the number of times x ∈ X appears in the sequence u. We deﬁne Pu (x) = N (xu)/n. Then, Pu is a probability distribution on X. T (N (xu), νx , δ) is the ⊗N (xu) set of all vx ∈ Y N (xu) for which log(νx (vx ))/N (xu) + H(Y x) < δ. In other words, vx ∈ T (N (xu), νx , δ) iﬀ 2−N (xu)(H(Y x)+δ) < νx⊗N (xu) (vx ) < 2−N (xu)(H(Y x)−δ) Writing v = ∪x∈X vx , it follows that if vx ∈ T (N (xu), 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 (xu), 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 (yvx )−N (xu)νx (y) < δ N (xu)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 (xu) − 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 (xu) ≤ 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 (xu)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 (xu)/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 (xu)
and so log(μn (u)) =
∑
N (xu)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 (yvx ) 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 (yv) − nq(y) =  N (yvx ) − np(x)νx (y) =
∑
x∈X
N (yvx ) −
x∈X
≤
∑
∑
x∈X
N (xu)νx (y)
x∈X
N (yvx ) − N (xu)νx (y) ≤
x∈X
≤
∑ √ δ N (xu)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 CauchySchwarz 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 (xu) (T˜(N (xu), 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
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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 energymomentum 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)
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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 =
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(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 nonrelativistic 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 interparticle 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 integrodiﬀerential 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)r2 /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)r2 /2c2 + 3R' (t)r4 /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 KleinGordon equation. [35] Estimating the metric of spacetime 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 spacetime. 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 ,
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(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
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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 < qη¯kn (r)p > D [uk < qηkn (r)p > Dn (θ(t), θ' (t)) + u kn
= F1 (θ(t), θ' (t)) say. Also let
+
∑
ξ3 (t) =
∫ [uk < qρkn (r)p >
kn
t 0
En (θ(s), θ' (s))ds+¯ uk < qρ¯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 < qρkn (r)p > En (θ(t), θ' (t))+¯ uk < qρ¯kn (r)p > E
kn
= F2 (θ(t), θ' (t)) say. We have ∫ PT (p → qu) = 
T
0
so deﬁning
exp(−iEt)ξ1 (t)dt2 , E = Eq − Ep ∫
ξ5 (t) =
t 0
exp(−iEs)ξ1 (s)ds
we have Pt = Pt (p → qu) = 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 spacetime 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 energymomentum 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 spacetime 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 3D 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 3D surface Σt at time t = x0 described by coordinates (xa , a = 1, 2, 3) inside the four dimensional spacetime R4 described by coordinates (X μ (x), μ = 0, 1, 2, 3). The unit normal to this 3D surface is denoted by nμ . Tangent vectors to Σt relative to the Xsystem are given by μ (X,a )μ , a = 1, 2, 3
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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 xsystem. 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
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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 )
611
612
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
615
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 integrodiﬀerential 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 (rr' , 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 (rr' ) = Lx,2 (rr' , 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 (rr , v )f1 (ω, r , v )d r d v + Lx,2 (rr' )f1 (ω, r' , v ' )d3 r' d3 v ' Likewise for the other components: ∑ Ly,1 (rr' , 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 (rr' ) = Ly,2 (rr' , 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 (rr' , 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 (rr' ) = Lz,2 (rr' , 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, vr' , v ' ) = Lx,1 (rr' , v ' )vx + Ly,1 (rr' , v ' )vy + Lz,1 (rr' , v ' )vz
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General Relativity and Cosmology with Engineering Applications
and
L2 (r, vr' , v ' ) = Lx,2 (rr' )vx + Ly,2 (rr' )vy + Lz,2 (rr' )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 energymomentum 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 comoving 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 =
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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
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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 (xu) − 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 (xu) < 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 (xu) < 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)
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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 (xu) < 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 (xu) , δ)) ≤ Πx∈A 2N (xu)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
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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 GramSchmidtting 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)dte(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 nondemolition 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) = Λb>< 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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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 + 1n]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 + 1n] as the covariance of ξ[n + 1] = [k[n + 1]T , e[n + 1]T ]T given Yn = {y[k] : k ≤ n}. [48] DEKFRLS 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 + 1n] and x ˆ[n + 1n + 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 + 1n] = M x[nn] ˆ [n]Pξ [nn]M ˆ [n]T + G[n]Pw G[n]T Pξ [n + 1n] = M ˆ [n]ξˆ[n + 1n]) ξˆ[n + 1n + 1] = ξˆ[n + 1n] + K[n](y[n + 1] − M where
K[n] = (Pξ [n + 1n]−1 + DT Pv −1D)−1 DT Pv−1
Pξ [n + 1n + 1] = (I − K[n]D)Pξ [n + 1n](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]
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General Relativity and Cosmology with Engineering Applications
−D
q ∑
ws Rs ξˆ[m − 1m − 1])T W [m − 1](y[m] − D
s=1
q ∑
ws Rs ξˆ[m − 1m − 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 pseudoinverse, ie, the MoorePenrose 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−1m−1] Rs D W [m−1](y[m]−D
m=1
∑
qw ˆl [n+1]Rl ξˆ[m−1m−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 − 1m − 1]T RsT DT W [m − 1]DRl ξˆ[m − 1m − 1])wl l=1 m=1 n+1 ∑
=
λn+1−m ]ξˆ[m − 1m − 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−1m−1]T RsT DT W [m−1]DRl ξˆ[m−1m−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 − 1m − 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[nn]T RsT DT W [n]DRl ξˆ[nn] ηn+1 [s] = ληn [s] + ξˆ[nn]T RsT DT W [n]y[n + 1] We write S[n + 1] = ((ξˆ[nn]T RsT DT W [n]T DRl ξˆ[nn]))1≤s,l≤q ∈ Rq×q and s[n + 1] = ((ξˆ[nn]T RsT DT W [n]y[n + 1]))1≤s≤q ∈ Rq
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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 EKFRLS applied to state and weight estimation in linear multivariable stochastic systems. [49] Viscous and thermal eﬀects in specialrelativistic hydrodynamics. The energymomentum 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 energymomentum tensor of the ﬂuid without taking viscous and thermal eﬀects and ΔT μν is the contribution to the energymomentum 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
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General Relativity and Cosmology with Engineering Applications
We now use the energymomentum 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 4divergence 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 spacetime 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
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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 RobertsonWalker 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
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Then, we evaluate the energymomentum 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 spacetime 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 nonzero components of the energymomentum 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 nonzero 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 nonzero. 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 )
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General Relativity and Cosmology with Engineering Applications
The modiﬁed energymomentum tensor of the matter distribution is Sμν = Tμν − T gμν /2 We compute its nonzero 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
651
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