Loops, Knots, Gauge Theories and Quantum Gravity

✍ Scribed by Rodolfo Gambini, Jorge Pullin

• Publisher: Cambridge University Press
• Year: 2023
• Tongue: English
• Leaves: 340
• Series: Cambridge Monographs on Mathematical Physics
• Edition: 1
• Category: Library

✦ Synopsis

This file comes from zlibrary

This volume provides a self-contained introduction to applications of loop representations, and the related topic of knot theory, in particle physics and quantum gravity. These topics are of considerable interest because they provide a unified arena for the study of the gauge invariant quantization of Yang-Mills theories and gravity, and suggest a promising approach to the eventual unification of the four fundamental forces. The book begins with a detailed review of loop representation theory and then describes loop representations in Maxwell theory, Yang-Mills theories as well as lattice techniques. Applications in quantum gravity are then discussed, with the following chapters considering knot theories, braid theories and extended loop representations in quantum gravity. A final chapter assesses the current status of the theory and points out possible directions for future research. First published in 1996, this title has been reissued as an Open Access publication on Cambridge Core.

✦ Table of Contents

Cover
series page
Half-title page
Title page
Copyright page
Dedication page
Contents
Foreword
Preface
1 Holonomies and the group of loops
1.1 Introduction
1.2 The group of loops
1.3 Infinitesimal generators of the group of loops
1.3.1 The loop derivative
1.3.2 Properties of the loop derivative
1.3.3 Connection derivative
1.3.4 Contact and functional derivatives
1.4 Representations of the group of loops
1.5 Conclusions
2 Loop coordinates and the extended group of loops
2.1 Introduction
2.2 Multitangent fields as description of loops
2.3 The extended group of loops
2.3.1 The special extended group of loops
2.3.2 Generators of the SeL group
2.4 Loop coordinates
2.4.1 Transverse tensor calculus
2.4.2 Freely specifiable loop coordinates
2.5 Action of the differential operators
2.6 Diffeomorphism invariants and knots
2.7 Conclusions
3 The loop representation
3.1 Introduction
3.2 Hamiltonian formulation of systems with constraints
3.2.1 Classical theory
3.2.2 Quantum theory
3.3 Yang-Mills theories
3.3.1 Canonical formulation
3.3.2 Quantization
3.4 Wilson loops
3.4.1 The Mandelstam identities
3.4.2 Reconstruction property
3.5 Loop representation
3.5.1 The loop transform
3.5.2 The non-canonical algebra
3.5.3 Wavefunctions in the loop representation
3.6 Conclusions
4 Maxwell theory
4.1 The Abelian group of loops
4.2 Classical theory
4.3 Fock quantization
4.4 Loop representation
4.5 Bargmann representation
4.5.1 The harmonic oscillator
4.5.2 Maxwell-Bargmann quantization in terms of loops
4.6 Extended loop representation
4.7 Conclusions
5 Yang-Mills theories
5.1 Introduction
5.2 Equations for the loop average in QCD
5.3 The loop representation
5.3.1 SU(2) Yang-Mills theories
5.3.2 SU(N) Yang-Mills theories
5.4 Wilson loops and some ideas about confinement
5.5 Conclusions
6 Lattice techniques
6.1 Introduction
6.2 Lattice gauge theories: the Z(2) example
6.2.1 Covariant lattice theory
6.2.2 The transfer matrix method
6.2.3 Hamiltonian lattice theory
6.2.4 Loop representation
6.3 The SU(2) theory
6.3.1 Hamiltonian lattice formulation
6.3.2 Loop representation in the lattice
6.3.3 Approximate loop techniques
6.4 Inclusion of fermions
6.5 Conclusions
7 Quantum gravity
7.1 Introduction
7.2 The traditional Hamiltonian formulation
7.2.1 Lagrangian formalism
7.2.2 The split into space and time
7.2.3 Constraints
7.2.4 Quantization
7.3 The new Hamiltonian formulation
7.3.1 Tetradic general relativity
7.3.2 The Palatini action
7.3.3 The self-dual action
7.3.4 The new canonical variables
7.4 Quantum gravity in terms of connections
7.4.1 Formulation
7.4.2 Triads to the right and the Wilson loop
7.4.3 Triads to the left and the Chern-Simons form
7.5 Conclusions
8 The loop representation of quantum gravity
8.1 Introduction
8.2 Constraints in terms of the T algebra
8.3 Constraints via the loop transform
8.4 Physical states and regularization
8.4.1 Diffeomorphism constraint
8.4.2 Hamiltonian constraint: formal calculations
8.4.3 Hamiltonian constraint: regularized calculations
8.5 Conclusions
9 Loop representation: further developments
9.1 Introduction
9.2 Inclusion of matter: Weyl fermions
9.3 Inclusion of matter: Einstein-Maxwell and unification
9.4 Kalb-Ramond fields and surfaces
9.4.1 The Abelian group of surfaces
9.4.2 Kalb-Ramond fields and surface representation
9.5 Physical operators and weaves
9.5.1 Measuring the geometry of space in terms of loops
9.5.2 Semi-classical states: the weave
9.6 2+1 gravity
9.7 Conclusions
10 Knot theory and physical states of quantum gravity
10.1 Introduction
10.2 Knot theory
10.3 Knot polynomials
10.3.1 The Artin braid group
10.3.2 Skein relations, ambient and regular isotopies
10.3.3 Knot polynomials from representations of the braid group
10.3.4 Intersecting knots
10.4 Topological field theories and knots
10.4.1 Chern-Simons theory and the skein relations of the Jones polynomial
10.4.2 Perturbative calculation and explicit expressions forthe coefficients
10.5 States of quantum gravity in terms of knot polynomials
10.5.1 The Kauffman bracket as a solution of the constraints with cosmological constant
10.5.2 The Jones polynomial and a state with A = 0
10.5.3 The Gauss linking number as the key to the new solution
10.6 Conclusions
11 The extended loop representation of quantum gravity
11.1 Introduction
11.2 Wavefunctions
11.3 The constraints
11.3.1 The diffeomorphism constraint
11.3.2 The Hamiltonian constraint
11.4 Loops as a particular case
11.5 Solutions of the constraints
11.6 Regularization
11.6.1 The smoothness of the extended wavefunctions
11.6.2 The regularization of the constraints
11.7 Conclusions
12 Conclusions, present status and outlook
12.1 Gauge theories
12.2 Quantum gravity
References
Index