Spatio-temporal Chaos & Vacuum Fluctuations Of Quantized Fields

by Christian Beck

Overview

This book describes new applications for spatio-temporal chaotic dynamical systems in elementary particle physics and quantum field theories. The stochastic quantization approach of Parisi and Wu is extended to more general deterministic chaotic processes as generated by coupled map lattices. In particular, so-called chaotic strings are introduced as a suitable small-scale dynamics of vacuum fluctuations. This more general approach to second quantization reduces to the ordinary stochastic quantization scheme on large scales, but it also opens up interesting new perspectives: chaotic strings appear to minimize their vacuum energy for the observed numerical values of the free standard model parameters.

Product Details

ISBN-13:	9789810247980
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	04/30/2002
Series:	Advanced Series In Nonlinear Dynamics , #21
Pages:	292
Product dimensions:	6.10(w) x 8.60(h) x 0.70(d)

Table of Contents

	Preface	vii
	Introduction	xiii
Chapter 1	Chaotic quantization of field theories	1
1.1	Stochastic quantization	1
1.2	Dynamical generation of the noise	3
1.3	The free Klein-Gordon field with chaotic noise	6
1.4	Chaotic quantization in momentum space	9
1.5	Gauge fields with chaotic noise	11
1.6	Distinguished properties of Tchebyscheff maps	13
1.7	Graph theoretical method	17
1.8	Perturbative approach	23
Chapter 2	Chaotic strings	27
2.1	Motivation for chaotic strings	27
2.2	Anti-integrable limit of a continuum [phi superscript N+1]-theory	30
2.3	Possible generalizations	33
2.4	Yet another way to derive the chaotic string	35
2.5	Symmetry properties	38
2.6	Stability properties	41
2.7	Fixed points	44
2.8	Spatio-temporal patterns	47
Chapter 3	Vacuum energy of chaotic strings	57
3.1	Self energy of the N = 3 string	57
3.2	Self energy of the N = 2 string	60
3.3	Self energy for general N	62
3.4	Interaction energy of chaotic strings	65
3.5	Double strings	67
3.6	Rotating strings	69
Chapter 4	Phase transitions and spontaneous symmetry breaking	75
4.1	Some general remarks on phase transitions	75
4.2	Vacuum expectation on 1-dimensional lattices	79
4.3	Real scalar field on d-dimensional lattices	82
4.4	Complex scalar field with U (1) symmetry	90
4.5	Chaotic Higgs field with SU (2) symmetry	92
Chapter 5	Stochastic interpretation of the uncertainty relation	95
5.1	Fluctuations of momenta and positions	95
5.2	Newton's law and self interaction	97
5.3	Coulomb forces and Laplacian coupling	99
5.4	Duality of interpretations	103
5.5	Feynman webs	104
5.6	Physical interpretation of discrete string symmetries	106
5.7	Fluctuations of the metric and a 1+1 dimensional model of quantum gravity	108
Chapter 6	Generalized statistical mechanics approach	113
6.1	Heat bath of the vacuum	113
6.2	States of maximum information	116
6.3	States of minimum correlation	118
6.4	Nonextensive statistical mechanics	119
6.5	Energy dependence of the entropic index q	124
6.6	Fluctuations of temperature	126
6.7	Klein-Gordon field with fluctuating momenta	129
Chapter 7	Interaction energy of chaotic strings	131
7.1	Analogue of the Einstein field equations	131
7.2	The 3A string--electric interaction strengths of electrons and d-quarks	133
7.3	The 3B string--weak interaction strengths of neutrinos and u-quarks	136
7.4	High-precision prediction of the electroweak parameters	139
7.5	The 2A string--strong interaction strength at the W-mass scale	141
7.6	The 2B string--the lightest scalar glueball	144
7.7	The 2A[superscript -] and 2B[superscript -] strings--towards a Higgs mass prediction	145
7.8	Gravitational interaction	148
Chapter 8	Self energy of chaotic strings	151
8.1	Self interacting scalar field equations	151
8.2	The 3A string--weak and strong interactions of heavy fermion flavors	152
8.3	The 3B string--further mixed states of heavy fermion flavors	156
8.4	The 2A string--further bosons	158
8.5	The 2B string--Yukawa interaction of the top quark	160
8.6	Yukawa and gravitational interactions of all quarks and leptons	162
8.7	Neutrino mass prediction	168
8.8	The 2A[superscript -] and 2B[superscript -] strings--bosonic mass ratios	172
Chapter 9	Total vacuum energy of chaotic strings	175
9.1	Hadronization of free quarks	175
9.2	Mesonic states	179
9.3	Baryonic states	182
9.4	CP violation	186
9.5	Planck scale interpretation	186
9.6	Dark matter	187
Chapter 10	Grand unification	191
10.1	Supersymmetric versus non-supersymmetric theories	191
10.2	A supersymmetric scenario	194
10.3	A non-supersymmetric scenario	196
10.4	Final unification at the Planck scale	198
10.5	Simplification for sin[superscript 2] [theta subscript W] = 1/2	200
10.6	Bosons at the Planck scale	202
10.7	Some thoughts on supersymmetry	203
Chapter 11	11-dimensional space-time and quantum gravity	207
11.1	Chaotic dynamics in compactified dimensions	207
11.2	Quantized Einstein field equations	210
11.3	N = 1 strings and Minkowski space	213
11.4	Potentials for the N = 1 strings and inflation	215
11.5	Black holes, Hawking radiation, and duality	217
11.6	The limit E [right arrow] [infinity]	220
11.7	Brief history of the universe--as seen from chaotic strings	222
Chapter 12	Summary	229
12.1	Motivation and main results	229
12.2	The chaotic string dynamics	232
12.3	Vacuum energy of chaotic strings	234
12.4	Fixing standard model parameters	237
12.5	Numerical findings	240
12.6	Physical embedding	247
12.7	Conclusion	249
	Bibliography	253
	Index	267

Editorial Reviews

This book describes new applications for spatio-temporal chaotic dynamical systems in elementary particle physics and quantum field theories. The stochastic quantization approach of Parisi and Wu is extended to more general deterministic chaotic processes as generated by coupled map lattices. Chaotic strings are introduced as spatio- temporal chaotic models of vacuum fluctuations. The book is written for an interdisciplinary readership of graduate students and researchers interested in nonlinear dynamics, stochastic processes, statistical mechanics, quantum field theory, and string theory. The author is affiliated with the School of Mathematical Sciences at Queen Mary, University of London. Annotation c. Book News, Inc., Portland, OR