Deformations of Spacetime Symmetries: Gravity, Group-Valued Momenta, and Non-Commutative Fields 9783662630952, 9783662630976

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Deformations of Spacetime Symmetries: Gravity, Group-Valued Momenta, and Non-Commutative Fields
 9783662630952, 9783662630976

Table of contents :
About the Authors
Part IFrom Gravity to Curved Momentum Space and Non-commutative Spacetime
1 Invitation: Gravity, Point Particles, and Group-Valued Momenta
1.1 Introduction
1.2 2+1 Gravity, Particles, and Curvature of Momentum Space
2 Gravity in 2+1 Dimensions as a Chern–Simons Theory
2.1 Pure Gravity in 2+1 Dimensions
2.2 Particle Coupling
2.3 Effective Deformed Single Particle Action
2.4 Properties of the Deformed Lagrangian
2.5 The Case of Many Particles
2.6 κ-Carrollian Deformation
2.7 Deformed Spacetime Symmetries of 2+1 Quantum Gravity
3 Gravity in 3+1 Dimensions, Particles, and Topological Limit
3.1 From 3+1 to 2+1 Dimensions via Dimensional Reduction
3.2 Gravity as a Constrained BF Theory
3.3 Particles Coupled to Gravity
3.4 Equations for the Particle–Gravity System
3.5 The Topological Vacuum
3.6 Effective Deformed Particle
Part IIDeformed Particles and Their Symmetries
4 Deformed Classical Particles: Phase Space and Kinematics
4.1 From Symplectic Manifolds to Poisson–Lie Groups
4.1.1 From Symplectic Manifolds to Poisson Manifolds
4.1.2 Lie Groups as Phase Spaces
4.1.3 Deforming Phase Spaces: The Classical Doubles
4.2 Relativistic Spinless Particle: Flat Momentum Space
4.3 Deforming Momentum Space to the AN(n) Group
4.4 Deforming Momentum Space to the Group SL(2,mathbbR)
4.5 Relativistic Hamiltonian Mechanics with Curved Momentum Space
4.6 κ-Poincaré Particle—The Free Case
4.7 The System of Two κ-Deformed Particles
4.7.1 Translations
4.7.2 Lorentz Symmetry
5 Hopf Algebra Relativistic Symmetries: The κ-Poincaré Algebra
5.1 Hopf Algebra Structures in Quantum Theory
5.2 The κ-Poincaré Hopf Algebra
Part IIIAn Introduction to κ-Deformed Fields
6 Classical Fields, Symmetries, and Conserved Charges
6.1 Preliminaries
6.1.1 More on the κ-Poincaré Algebra
6.1.2 Non-commutative Calculus
6.1.3 Weyl Maps and -Product
6.1.4 Summary of the Technical Tools
6.2 Action and Field Equations
6.3 Complex Scalar Field and Momentum Space Action
6.4 Poincaré Symmetry of the Action
6.5 Symplectic Structure
6.6 Conserved Charges
7 Free Quantum Fields and Discrete Symmetries
7.1 Free κ-Deformed Quantum Fields: The Feynman Propagator
7.1.1 The κ-Deformed Free Field Partition Function
7.1.2 The Feynman Propagator
7.2 Creation/Annihilation Operators Algebra
7.3 Discrete Symmetries
7.3.1 Parity calP
7.3.2 Time Reversal calT
7.3.3 Charge Conjugation calC
7.4 Properties of One-Particle States

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