Wavelets and Renormalizations

by Battle, G.

• ISBN13: 9789810226244
• ISBN10: 9810226241
• Format: Hardcover
• Copyright: 1999-05-01
• Publisher: World Scientific Pub Co Inc

Summary

"the author has succeeded in giving a vivid and pedagogical presentation of a monumental work in recent mathematical physics, illustrating the mutual influence between wavelet analysis and renormalization group techniques of Euclidean field theory."Mathematical Reviews, 2001"This book is a great achievement. It is difficult to read, but very rewarding for those who read it in depth."Monatshefte f r Mathematik, 2001

Table of Contents

Preface	ix
Mathematical Sketches of Quantum Physics	1	(104)
Measurement and Dynamics	6	(5)
Imaginary Time Correlations	11	(4)
Correspondence Principle	15	(8)
The Harmonic Oscillator	23	(7)
The Feynman-Kac Formula	30	(6)
Quantum Mechanics of One Relativistic Particle	36	(4)
Crisis in the Theory and Formalism	40	(7)
Canonical Formalism of Quantum Field Theory	47	(9)
Functional Integration for Quantum Field Theory	56	(8)
Axiomatic Quantum Field Theory	64	(9)
The Constructive Problem	73	(9)
A Tale of Two Dimensions	82	(15)
The Need for Phase Cell Analysis	97	(8)
Wavelets--Basic Theory and Construction	105	(74)
The Balian-Low Theorem	110	(7)
Lemarie and Meyer Wavelets	117	(6)
Daubechies Wavelets	123	(8)
Vanishing Moments	131	(4)
Uncertainty Relations for (P) = 0 Wavelet States	135	(5)
Further Constraints of Heisenberg Type	140	(10)
Variational Construction	150	(6)
Without Intrascale Orthogonality	156	(7)
Chui-Wang Wavelets	163	(6)
Multi-Dimensional Wavelets	169	(10)
Equilibrium States of Classical Crystals	179	(102)
Classical Spin Systems	185	(9)
Phase Transitions and Ergodic Decomposition	194	(7)
Phase Transitions and Reflection Positivity	201	(13)
The Ising Model	214	(6)
The Ginzburg-Landau Model	220	(8)
Polymer Expansions	228	(9)
Expansion for Nearest-Neighbor Interactions	237	(7)
Inductive Interpolation	244	(10)
A Generalization of the Polymer Estimation	254	(3)
Unit-Vector Spins with Long-Range Coupling	257	(6)
Combinatorial Properties of Interpolation Weights	263	(8)
Scalar Spins with Long Range Coupling	271	(4)
Extra 1/N! Factors in the Inductive Expansion	275	(6)
A Wavelet Introduction to the Renormalization Group	281	(120)
Averaging Transformations	290	(7)
The Renormalization Group Transformation	297	(6)
Minimizers and Orthogonality	303	(7)
Connection to the Continuum Wavelet Cutoff	310	(8)
Canonical Wavelet Manifold and ZF Transformation	318	(7)
Relevant, Marginal, and Irrelevant Parameters	325	(7)
Canonical Inverse-Limit Manifold	332	(8)
Wick Ordering and Linearized RG Analysis	340	(8)
The Second-Order RG Transformation	348	(8)
Wavelet Diagrams	356	(4)
The One-Loop-per-Scale Transformation	360	(9)
The Wilson Wavelet Recursion Formula	369	(7)
The Baker-Dyson-Wilson Hierarchical Model	376	(6)
The Hierarchical Model to Second Order	382	(7)
Hierarchical One-Loop-per-Scale Transformation	389	(5)
Wavelet Correction of the Hierarchical Model	394	(7)
Wavelet Analysis of ϕ4/3	401	(112)
The Cutoff - A Finite Set of Modes	407	(5)
Wavelet Estimates	412	(5)
Inductive Operations	417	(6)
Estimates on Numerical Factors	423	(8)
Expansion Rules and Wavelet Diagrams	431	(12)
Organizing the Completed Expansion	443	(8)
The Phase Cell Polymer	451	(5)
Estimating the Activity I: Stability and Quartic Positivity	456	(8)
Estimating the Activity II: Internal Combinatorics	464	(7)
Combinatorics for Summing Over Polymers	471	(7)
Strategy for Number Divergence Cancellation	478	(5)
From Infinitely Many Cases to Finitely Many	483	(9)
Wavelet Diagrams for Cases	492	(12)
How to Assign Numerical Factors	504	(9)
Bibliography	513