Loops, Knots, Gauge Theories and Quantum Gravity

340 Pages·2009·66.633 MB·English

by  Rodolfo Gambini, Jorge Pullin

This text provides a self-contained introduction to applications ofloop repre
sentations and knot theory in particle physics and quantum gravity.
Loop representations (and the related topic of knot theory) are of con
siderable current interest because they provide a unified arena for the study of
the gauge invariant quantization ofY ang-Mills theories and gravity, and sug
gest a promising approach to the eventual unification of the four fundamental
forces. This text begins with a review of calculus in loop space and the funda
mentals ofloop representations. It then goes on to describe loop representa
tions in Maxwell theory and Yang-Mills theories as well as lattice techniques.
Applications in quantum gravity are then discussed in detail. Following chap
ters move on to consider knot theories, the braid algebra and extended loop rep
resentations in quantum gravity. A final chapter assesses the current status of
the theory and points out possible directions for future research.
This self-contained introduction will be of interest to graduate stu
dents and researchers in theoretical physics and applied mathematics.
CAMBRIDGE MONOGRAPHS ON MATHEMATICAL PHYSICS
General editors: P. V. Landshoff, D. R. Nelson, D. W. Sciama, S. Weinberg
LOOPS, KNOTS, GAUGE THEORIES AND QUANTUM GRAVITY
Cambridge Monographs on Mathematical Physics
J. AmbjI'Jrn, B. Durhuus and T. Jonsson Quantum Geometry: A Statistical Field Theory Approach
A. M. Anile Relative Fluids and Magneto-Fluids
J. A. de Azcarraga and J. M. Izquierdo Lie Groups, Lie Algebras, Cohomology and Some Applications
in Physicst
J. Bernstein Kinetic Theory in the Early Universe
G. F. Bertsch and R. A. Broglia Oscillations in Finite Quantum Systems
N. D. Birrell and P. C. W. Davies Quantum Fields in Curved Spacet
D. M. Brink Semiclassical Methods in Nucleus-Nucleus Scattering
+
S. Carlip Quantum Gravity in 2 1 Dimensions
J. C. Collins Renormalizationt
P. D. B. Collins An Introduction to Regge Theory and High Energy Physicst
M. Creutz Quarks, Gluons and Lattices t
P. D. D'eath Supersymmetric Quantum Cosmology
F. de Felice and C. J. S. Clarke Relativity on Curved Manifoldst
B. De Witt Supermanifolds, second edition t
P. G. O. Freund Introduction to Supersymmetryt
F. G. Friedlander The Wave Equation on a Curved Space-Time
J. Fuchs Affine Lie Algebras and Quantum Groupst
J. A. H. Futterman, F. A. Handler and R. A. Matzner Scattering from Black Holes
R. Gambini and J. Pullin Loops, Knots, Gauge Theories and Quantum Gravityt
M. GiX:keler and T. Schucker Differential Geometry, Gauge Theories and Gravityt
C. Gomez, M. Ruiz Altaba and G. Sierra Quantum Groups in Two-dimensional Physics
M. B. Green, J. H. Schwarz and E. Witten Superstring Theory, volume 1: INTRODUCTIONt
M. B. Green, J. H. Schwarz and E. Witten Superstring Theory, volume 2:
LOOP AMPLITUDES, ANOMALIES AND PHENOMENOLOGyt
S. W. Hawking and G. F. R. Ellis The Large-Scale Structure of srace-Timet
F. Iachello and A. Arima The Interacting Boson Model
F. Iachello and P. van !sacker The Interacting Boson-Fermion Model
C. Itzykson and J.-M. DroutTe Statistical Field Theory, volume 1:
FROM BROWNIAN MOTION TO RENORMALIZATION AND LATTICE GAUGE THEORyt
C. Itzykson and J.-M. DroutTe Statistical Field Theory, volume 2:
STRONG COUPLING, MONTE CARLO METHODS, CONFORMAL FIELD THEORY, AND
RANDOM SYSTEMSt
J. I. Kapusta Finite-Temperature Field Theoryt
V. E. Korepin, A. G. Izergin and N. M. Boguliubov The Quantum Inverse Scattering Method and
Co"elation Functionst
D. Kramer, H. Stephani, M. A. H. MacCallum and E. Herlt Exact Solutions of Einstein's Field
Equations
M. Le Bellac Thermal Field Theoryt
N. H. March Liquid Metals: Concepts and Theory
I. M. Montvay and G. Munster Quantum Fields on a Latticet
L. O'Raifeartaigh Group Structure of Gauge Theories t
A. Ozorio de Almeida Hamiltonian Systems: Chaos and Quantizationt
R. Penrose and W. Rindler Spinors and Space-time, volume 1:
TWo-SPINOR CALCULUS AND RELATIVISTIC FIELDSt
R. Penrose and W. Rindler Spinors and Space-time, volume 2:
SPINOR AND TWISTOR METHODS IN SPACE-TIME GEOMETRyt
S. Pokorski Gauge Field Theories t
J. Polchinski String Theory, volume 1
J. Polchinski String Theory, volume 2
V. N. Popov Functional Integrals and Collective Excitations t
R. Rivers Path Integral Methods in Quantum Field Theoryt
R. G. Roberts The Structure of the Protont
W. C. Saslaw Gravitational Physics of Stellar and Galactic Systemst
J. M. Stewart Advanced General Relativity
A. Vilenkin and E. P. S. Shellard Cosmic Strings and other Topological Defects t
R. S. Ward, R. O. Wells Jr Twister Geometry and Field Theoriest
t Issued as paperback
Loops, Knots,
Gauge Theories and
Quantum Gravity
RODOLFO GAMBINI
Universidad de la Republica, Montevideo
JORGE PULLIN
Pennsylvania State University
CAMBRIDGE
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DOI: 10.1017/9781009290203
© Rodolfo Gambini, Jorge Pullin and Abhay Ashtekar 2022
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Contents
Foreword xiii
Preface xv
1 Holonomies and the group of loops 1
1.1 Introduction 1
1.2 The group of loops 3
1.3 Infinitesimal generators of the group of loops 7
1.3.1 The loop derivative 8
1.3.2 Properties of the loop derivative 10
1.3.3 Connection derivative 17
1.3.4 Contact and functional derivatives 21
1.4 Representations of the group of loops 25
1.5 Conclusions 27
2 Loop coordinates and the extended group of loops 29
2.1 Introduction 29
2.2 Multitangent fields as description of loops 30
2.3 The extended group of loops 32
2.3.1 The special extended group of loops 33
2.3.2 Generators of the SeL group 36
2.4 Loop coordinates 38
2.4.1 Transverse tensor calculus 38
2.4.2 Freely specifiable loop coordinates 42
2.5 Action of the differential operators 44
2.6 Diffeomorphism invariants and knots 47
2.7 Conclusions 50
3 The loop representation 52
3.1 Introduction 52
3.2 Hamiltonian formulation of systems with constraints 54
3.2.1 Classical theory 54
3.2.2 Quantum theory 56
3.3 Yang-Mills theories 58
IX
x Contents
3.3.1 Canonical formulation 59
3.3.2 Quantization 61
3.4 Wilson loops 63
3.4.1 The Mandelstam identities 64
3.4.2 Reconstruction property 67
3.5 Loop representation 72
3.5.1 The loop transform 76
3.5.2 The non-canonical algebra 80
3.5.3 Wavefunctions in the loop representation 85
3.6 Conclusions 87
4 Maxwell theory 88
4.1 The Abelian group of loops 89
4.2 Classical theory 91
4.3 Fock quantization 92
4.4 Loop representation 96
4.5 Bargmann representation 103
4.5.1 The harmonic oscillator 103
4.5.2 Maxwell-Bargmann quantization in terms of loops 105
4.6 Extended loop representation 109
4.7 Conclusions 112
5 Yang-Mills theories 113
5.1 Introduction 113
5.2 Equations for the loop average in QCD 115
5.3 The loop representation 118
5.3.1 SU(2) Yang-Mills theories 118
5.3.2 SU(N) Yang-Mills theories 121
5.4 Wilson loops and some ideas about confinement 124
5.5 Conclusions 130
6 Lattice techniques 131
6.1 Introduction 131
6.2 Lattice gauge theories: the Z(2) example 133
6.2.1 Covariant lattice theory 133
6.2.2 The transfer matrix method 136
6.2.3 Hamiltonian lattice theory 138
6.2.4 Loop representation 141
6.3 The SU(2) theory 144
6.3.1 Hamiltonian lattice formulation 145
6.3.2 Loop representation in the lattice 147
6.3.3 Approximate loop techniques 150
6.4 Inclusion of fermions 156