Loops, Knots, Gauge Theories and Quantum Gravity

by Rodolfo Gambini , Jorge Pullin , Foreword by Abhay Ashtekar

• ISBN13: 9780521654753
• ISBN10: 0521654750
• Format: Paperback
• Copyright: 2000-07-03
• Publisher: Cambridge University Press

Summary

Now in paperback, this text provides a self-contained introduction to applications of loop representations and knot theory in particle physics and quantum gravity. Loop representations (and the related topic of knot theory) are of considerable current 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. This text begins with a detailed review of loop representation theory. It then goes on to describe loop representations in Maxwell theory, Yang-Mills theories as well as lattice techniques. Applications in quantum gravity are then discussed in detail. Following chapters move on to consider 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.

Table of Contents

Foreword	xiii
Preface	xv
Holonomies and the group of loops	1	(28)
Introduction	1	(2)
The group of loops	3	(4)
Infinitesimal generators of the group of loops	7	(18)
The loop derivative	8	(2)
Properties of the loop derivative	10	(7)
Connection derivative	17	(4)
Contact and functional derivatives	21	(4)
Representations of the group of loops	25	(2)
Conclusions	27	(2)
Loop coordinates and the extended group of loops	29	(23)
Introduction	29	(1)
Multitangent fields as description of loops	30	(2)
The extended group of loops	32	(6)
The special extended group of loops	33	(3)
Generators of the SeL group	36	(2)
Loop coordinates	38	(6)
Transverse tensor calculus	38	(4)
Freely specifiable loop coordinates	42	(2)
Action of the differential operators	44	(3)
Diffeomorphism invariants and knots	47	(3)
Conclusions	50	(2)
The loop representation	52	(36)
Introduction	52	(2)
Hamiltonian formulation of systems with constraints	54	(4)
Classical theory	54	(2)
Quantum theory	56	(2)
Yang-Mills theories	58	(5)
Canonical formulation	59	(2)
Quantization	61	(2)
Wilson loops	63	(9)
The Mandelstam identities	64	(3)
Reconstruction property	67	(5)
Loop representation	72	(15)
The loop transform	76	(4)
The non-canonical algebra	80	(5)
Wavefunctions in the loop representation	85	(2)
Conclusions	87	(1)
Maxwell theory	88	(25)
The Abelian group of loops	89	(2)
Classical theory	91	(1)
Fock quantization	92	(4)
Loop representation	96	(7)
Bargmann representation	103	(6)
The harmonic oscillator	103	(2)
Maxwell-Bargmann quantization in terms of loops	105	(4)
Extended loop representation	109	(3)
Conclusions	112	(1)
Yang-Mills theories	113	(18)
Introduction	113	(2)
Equations for the loop average in QCD	115	(3)
The loop representation	118	(6)
SU (2) Yang-Mills theories	118	(3)
SU (N) Yang-Mills theories	121	(3)
Wilson loops and some ideas about confinement	124	(6)
Conclusions	130	(1)
Lattice techniques	131	(30)
Introduction	131	(2)
Lattice gauge theories: the Z(2) example	133	(11)
Covariant lattice theory	133	(3)
The transfer matrix method	136	(2)
Hamiltonian lattice theory	138	(3)
Loop representation	141	(3)
The SU (2) theory	144	(12)
Hamiltonian lattice formulation	145	(2)
Loop representation in the lattice	147	(3)
Approximate loop techniques	150	(6)
Inclusion of fermions	156	(3)
Conclusions	159	(2)
Quantum gravity	161	(27)
Introduction	161	(3)
The traditional Hamiltonian formulation	164	(7)
Lagrangian formalism	164	(1)
The split into space and time	164	(4)
Constraints	168	(1)
Quantization	169	(2)
The new Hamiltonian formulation	171	(8)
Tetradic general relativity	172	(1)
The Palatini action	173	(1)
The self-dual action	174	(1)
The new canonical variables	175	(4)
Quantum gravity in terms of connections	179	(8)
Formulation	179	(1)
Triads to the right and the Wilson loop	180	(5)
Triads to the left and the Chern-Simons form	185	(2)
Conclusions	187	(1)
The loop representation of quantum gravity	188	(21)
Introduction	188	(1)
Constraints in terms of the T algebra	189	(3)
Constraints via the loop transform	192	(4)
Physical states and regularization	196	(12)
Diffeomorphism constraint	196	(1)
Hamiltonian constraint: formal calculations	197	(4)
Hamiltonian constraint: regularized calculations	201	(7)
Conclusions	208	(1)
Loop representation: further developments	209	(29)
Introduction	209	(1)
Inclusion of matter: Weyl fermions	209	(6)
Inclusion of matter: Einstein-Maxwell and unification	215	(3)
Kalb-Ramond fields and surfaces	218	(3)
The Abelian group of surfaces	218	(2)
Kalb-Ramond fields and surface representation	220	(1)
Physical operators and weaves	221	(9)
Measuring the geometry of space in terms of loops	222	(6)
Semi-classical states: the weave	228	(2)
2+1 gravity	230	(7)
Conclusions	237	(1)
Knot theory and physical states of quantum gravity	238	(37)
Introduction	238	(1)
Knot theory	239	(3)
Knot polynomials	242	(11)
The Artin braid group	243	(2)
Skein relations, ambient and regular isotopies	245	(4)
Knot polynomials from representations of the braid group	249	(2)
Intersecting knots	251	(2)
Topological field theories and knots	253	(11)
Chern-Simons theory and the skein relations of the Jones polynomial	254	(6)
Perturbative calculation and explicit expressions for the coefficients	260	(4)
States of quantum gravity in terms of knot polynomials	264	(10)
The Kauffman bracket as a solution of the constraints with cosmological constant	264	(1)
The Jones polynomial and a state with A = 0	265	(7)
The Gauss linking number as the key to the new solution	272	(2)
Conclusions	274	(1)
The extended loop representation of quantum gravity	275	(27)
Introduction	275	(2)
Wavefunctions	277	(3)
The constraints	280	(5)
The diffeomorphism constraint	281	(2)
The Hamiltonian constraint	283	(2)
Loops as a particular case	285	(4)
Solutions of the constraints	289	(2)
Regularization	291	(9)
The smoothness of the extended wavefunctions	292	(2)
The regularization of the constraints	294	(6)
Conclusions	300	(2)
Conclusions, present status and outlook	302	(7)
Gauge theories	302	(2)
Quantum gravity	304	(5)
References	309	(10)
Index	319