Lecture Notes In Topics In Path Integrals And String Representations

by Luiz C L Botelho

Overview

'All are every interesting topics treated with a high level of mathematical sophistication. One of the very useful tricks the author repeatedly resorts to is the introduction of one-parameter families of operators interpolating between two operators which appear naturally in the formalism. From this one-parameter family a differential equation for the determinant (or ratio of determinants) or for correlation functions is derived, which can then be solved. This is a very simple, elegant and powerful technique.'
Mathematical Reviews ClippingsFunctional Integrals is a well-established method in mathematical physics, especially those mathematical methods used in modern non-perturbative quantum field theory and string theory. This book presents a unique, original and modern treatment of strings representations on Bosonic Quantum Chromodynamics and Bosonization theory on 2d Gauge Field Models, besides of rigorous mathematical studies on the analytical regularization scheme on Euclidean quantum field path integrals and stochastic quantum field theory. It follows an analytic approach based on Loop space techniques, functional determinant exact evaluations and exactly solubility of four dimensional QCD loop wave equations through Elfin Botelho fermionic extrinsic self avoiding string path integrals.

Product Details

ISBN-13:	9789813143463
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	03/20/2017
Pages:	244
Product dimensions:	6.10(w) x 9.10(h) x 0.90(d)

Table of Contents

Preface vii

1 Bosonic Free Strings and Non-supersymetric QCD(SU(∞)): A constant gauge field path integral study 1

1.1 Introduction 1

1.2 The Static Confining Potential for the Eguchi Kawai Model on the Continuum 3

1.3 The Luscher correction to inter quark potential on the reduced model 7

1.4 Some path integral dynamical aspects of the reduced QCD as a path integral dynamics of euclidean strings 12

1.5 Appendix A 20

1.6 References 22

2 Basics Polyakov's quantum surface theory on the formalism of functional integrals and applications 25

2.1 Introduction 25

2.2 Elementary results on the Classical (Bosonic) Surface theory 26

2.3 Path integral quantization on Polyakov's theory of surface (or how to quantize 2D massless scalar fields in the presence of 2D quantmn gravity) 32

2.4 Path-Integral quantization of the Nambu-Goto theory of random surfaces 39

2.5 References 42

2.6 Appendix A: 2D Abelian Dirac Determinant On the Formal Evaluation of the Euclidean Dirac Functional Determinant on Two-dimensions 42

2.7 Appendix B: On Atyah-Singer Index Theorem in the Framework of Feynman Pseudo-Classical Path Integrals -Author's Original Remarks 44

2.8 References 47

2.9 Appendix C: Path integral bosonisation for the Thirring model in the presence of vortices 48

2.10 References 52

2.11 Appendix D: 53

2.12 References 57

2.13 Appendix E: Path-Integral Bosonization for the Abelian Thirring Model on a Riemann Surface - The QCD(SU(N)) String 58

2.14 References 64

3 Critical String Wave Equations and the QCD(U(N_C)) String 65

3.1 Introduction 65

3.2 The Critical Area-Diffusion String Wave Equations 65

3.3 A Bilinear Fermion Coupling on a Self-Interacting Bosonic Random Surface as Solution of QCD(U(N_C)) Migdal-Makeenko Loop Equantion 69

3.4 Appendix A: A Reduced Covariant String Model for the Extrinsic String 74

3.5 Appendix B: The Loop Space Program i the Bosouic λφ⁴ - O(N)-Field Theory and the QCD Triviality for R^D. D > 4 77

3.6 References 82

4 The Formalism of String Functional Integrals for the Evaluation of the Interquark Potential and Non Critical Strings Scattering Amplitudes 83

4.1 Introduction 83

4.2 Basics Results on the Classical Bosonic Surface Theory and the Nambu-Goto String Path Integral 84

4.3 The Nambu-Goto Extrinsic Path String 91

4.4 Studies on the perturbative evaluation of closed Scattering Amplitude in a Higher order Polyakov's Bosonic String Model 96

4.5 Appendix A: The distributional limit of the Epstein function 103

4.6 Appendix B: Integral Evaluation 104

4.7 Appendix C: On the perturbative evaluation of the bosonic: string closed scattering amplitude on Polyakov's framework 104

4.8 References 107

5 The D → -∞ saddle-point spectrum analysis of the open bosonic Polyakov string in R^D x SO(N) - The QCD(SC(∞)) string 109

5.1 Introduction 109

5.2 The non-tachyouic spectrum and scalar amplitudes at D → -∞ 110

5.3 References 116

6 The Electric Charge Confining in Abelian Rank two Tensor Field Model 117

6.1 Introduction 117

6.2 The interquark potential evaluation 117

6.3 Appendix A: The dynamics of the QCD(SU(∞)) tensor fields from strings 121

6.4 References 124

6.5 Appendix B: Path-integral bosonization for a nonrenormalizable axial four-dimensional fermion model 124

6.6 Introduction 124

6.7 The model 125

6.8 Appendix C 129

6.9 References 130

7 Infinities on Quantum Field Theory: A Functional Integral Approach 133

7.1 Introduction 133

7.2 Infinities on Quantum Field Theory on the Functional Integral Formalism 134

7.3 On the cut-off remotion on a two-dimensional Euclidean QFT model 139

7.4 On the construction of the Wiener Measure 145

7.5 On the Geometrodynamical Path Integral 147

7.6 Appendix A 150

7.7 Appendix B 151

7.8 Appendix C 153

7.9 References 155

8 Some comments on rigorous finite-volume euclidean quantum field path integrals in the analytical regularization scheme 157

8.1 Introduction 157

8.2 Some rigorous finite-volume quantum field path integral in the Analytical regularization scheme 158

8.3 References 167

8.4 Appendix A: Some Comments on the Support of Functional Measures in Hilbert Space 168

8.5 Appendix B 171

9 On the Rigorous Ergodic Theorem for a Class of Non-Linear Klein Gordon Wave Propagations 173

9.1 Introduction 173

9.2 On the detailed mathematical proof of the R.A.G.E. theorem 174

9.3 On the Boltzman Ergodic Theorem in Classical Mechanics as a result of the R.A.G.E theorem 177

9.4 On the invariant ergodic functional measure for non-linear Klein-Condon wave equations with kinetic trace class operators 180

9.5 An Ergodic theorem in Banach Spaces and Applications to Stochastic-Langevin Dynamical Systems 184

9.6 Appendix A: The existence and uniqueness results for some polinomial wave motions in 2D 187

9.7 Appendix B: The Ergodic theorem for Quantized wave propagations 191

9.8 Appendix C: A Rigorous Mathematical proof of the Ergodic theorem for Wide-Sense Stationary Stochastic Process 193

9.9 References 195

10 A Note on Feynman-Kac Path Integral Representations for Scalar Wave Motions 197

10.1 Introduction 197

10.2 On the path integral representation 198

10.3 Appendix A: The Acoustic Case 209

10.4 Appendix B: A Toy model for stable numerics on wave propagation 211

10.5 Appendix C 213

10.6 Appendix D: The Causal Propagator - The Retarded Potential 215

10.7 Appendix E: The Causal Propagator - The Damped Case 216

10.8 References 217

11 A Note on the extrinsic phase Space path Integral Method for quantization on Riemannian Manifold Particle Motions - An application of Nash Embedding Theorem 219

11.1 Introduction 219

11.2 The Phase Space Path Integral Representation 220

11.3 References 226

Index 227