小波及重正化WAVELETS AND RENORMALIZATION
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作者： G. Battle著
出 版 社： Pengiun Group (USA)
出版时间： 1999-12-1字数：版次： 1页数： 561印刷时间： 1999/05/01开本：印次： 1纸张： 胶版纸I S B N ： 9789810226244包装： 精装内容简介
WAVELETS AND RENORMALIZATION describes the role played by wavelets in Euclidean field theory and classical statistical mechanics. The author begins with a stream-lined introduction to quantum field theory from a rather basic point of view. Functional integrals for imaginary-time-ordered expectations are introduced early and naturally, while the connection with the statistical mechanics of classical spin systems is introduced in a later chapter.
A vastly simplified (wavelet) version of the celebrated Glimm-Jaffe construction of the f43 quantum field theory is presented. It is due to Battle and Federbush, and it bases an inductively defined cluster expansion on a wavelet decomposition of the Euclidean quantum field. The presentation is reserved for the last chapter, while the more basic aspects of cluster expansions are reviewed in the chapter on classical spin systems.
Wavelets themselves are studied from two different points of view arising from two disciplines. The mathematical point of view covers the basic properties of wavelets and methods for constructing well-known wavelets such as Meyer wavelets, Daubechies wavelets, etc. The physical point of view covers the renormalization group formalism, where there is a close connection between wavelets and Gaussian fixed points.
The book is heavily mathematical, but avoids the theorem-proof-theorem-proof format in the interests of preserving the flow of the discussion - i.e., it is written in the style of an old-fashioned theoretical physics book, but the major claims are rigorously proven. The minor themes of the book are reflection positivity, the combinatorics of cluster expansions, and the issue of phase transitions - themes which have nothing to do with wavelets, but which provide necessary cultural background for the physical context.
目录
Preface
1 Mathematical Sketches of Quantum Physics
1.1 Measurement and Dynamics
1.2 Imaginary Time Correlations
1.3 Correspondence Principle
1.4 The Harmonic Oscillator
1.5 The Feynman Kac Formula
1.6 Quantum Mechanics of One Relativistic Particle
1.7 Crisis in the Theory and Formalism
1.8 Canonical Formalism of Quantum Field Theory
1.9 Functional Integration for Quantum Field Theory
1.10 Axiomatic Quantum Field Theory
1.11 The Constructive Problem
1.12 A Tale of Two Dimensions
1.13 The Need for Phase Cell Analysis
2 Wavelets - Basic Theory and Construction
2.1 The Balian Low Theorem
2.2 Lemarie and Meyer Wavelets
2.3 Daubcchies Wavelets
2.4 Vanishing Moments
2.5 Uncertainty Relations for (P) = 0 Wavelet States
2.6 Further Constraints of Heisenberg Type
2.7 Variational Construction
2.8 Without Intrascale Orthogonality
2.9 Chui Wang Wavelets
2.10 Multi-Dimensional Wavelets
3 Equilibrium States of Classical Crystals
3.1 Classical Spin Systems
3.2 Phase Transitions and Ergodie Decomposition
3.3 Phase Transitions and Reflection Positivity
3.4 The Ising Model
3.5 The Ginzburg Landau Model
3.6 Polymer Expansions
3.7 Expansion for Nearest-Neighbor Interactions
3.8 Inductive Interpolation
3.9 A Generalization of the Polymer Estimation
3.10 Unit-Vector Spins with Long-Range Coupling
3.11 Combinatorial Properties of Interpolation Weights
3.12 Scalar Spins with Long Range Coupling
3.13 Extra 1/N! Factors in the Inductive Expansion
4 A Wavelet Introduction to the Renormalization Group
4.1 Averaging Transformations
4.2 The Renormalization Group Transformation
4.3 Minimizers and Orthogonality
4.4 Connection to the Continuum Wavelet Cutoff
4.5 Canonical Wavelet Manifold and ZF Transformation
4.6 Relevant, Marginal, and Irrelevant Parameters
4.7 Canonical Inverse-Limit Manifold
4.8 Wick Ordering and Linearized RG Analysis
4.9 The Second-Order RG Transformation
4.10 Wavelet Diagrams
4.11 The One-Loop-per-Scale Transformation
4.12 The Wilson Wavelet Recursion Formula
4.13 The Baker Dyson Wilson ttierarchical Model
4.14 The Hierarchical Model to Second Order
4.15 Hierarchical One-Loop-per-Scale Transformation
4.16 Wavelet Correction of the Hierarchical Model
5 Wavelet Analysis of 4/3
5.1 The Cutoff - A Finite Set of Modes
5.2 Wavelet Estimates
5.3 Inductive Operations
5.4 Estimates on Numerical Factors
5.5 Expansion Rules and Wavelet Diagrams
5.6 Organizing the Completed Expansion
5.7 The Phase Cell Polymer
5.8 Estimating the Activity h Stability and Quartic Positivity
5.9 Estimating the Activity Ih Internal Combinatories
5.10 Combinatorics for Summing Over Polymers
5.11 Strategy for Number Divergence Cancellation
5.12 From Infinitely Many Cases to Finitely Many
5.13 Wavelet Diagrams for Cases
5.14 How to Assign Numerical Factors
Bibliography
