相变与重正化群

1 Quantum field theory and the renormalization group.

1.1 Quantum electrodynamics: A quantum field theory.

1.2 Quantum electrodynamics: The problem of infinities

1.3 Renormalization.

1.4 Quantum field theory and the renormalization group

1.5 A triumph of QFT: The Standard Model

1.6 Critical phenomena: Other infinities

1.7 Kadanoff and Wilson’s renormalizationgroup

1.8 Effective quantum field theories

2 Gaussian expectation values. Steepest descent method

2.1 Generating functions

2.2 Gaussian expectation values.Wick’s theorem

2.3 Perturbed Gaussian measure. Connected contributions

2.4 Feynman diagrams. Connected contributions.

2.5 Expectation values. Generating function. Cumulants

1 Quantum field theory and the renormalization group.

1.1 Quantum electrodynamics: A quantum field theory.

1.2 Quantum electrodynamics: The problem of infinities

1.3 Renormalization.

1.4 Quantum field theory and the renormalization group

1.5 A triumph of QFT: The Standard Model

1.6 Critical phenomena: Other infinities

1.7 Kadanoff and Wilson’s renormalizationgroup

1.8 Effective quantum field theories

2 Gaussian expectation values. Steepest descent method

2.1 Generating functions

2.2 Gaussian expectation values.Wick’s theorem

2.3 Perturbed Gaussian measure. Connected contributions

2.4 Feynman diagrams. Connected contributions.

2.5 Expectation values. Generating function. Cumulants

2.6 Steepest descent method

2.7 Steepest descent method: Several variables， generating functions

Exercises

3 Universality and the continuum limit

3.1 Central limit theorem of probabilities

3.2 Universality and fixed points of transformations

3.3 Random walk and Brownian motion

3.4 Random walk: Additional remarks

3.5 Brownian motion and path integrals

Exercises

4 Classical statistical physics: One dimension

4.1 Nearest-neighbour interactions. Transfer matrix

4.2 Correlation functions

4.3 Thermodynamic limit

4.4 Connected functions and cluster properties

4.5 Statistical models: Simple examples

4.6 The Gaussian model924.7 Gaussian model: The continuumlimit

4.8 More general models: The continuumlimit

Exercises

5 Continuum limit and path integrals

5.1 Gaussian path integrals

5.2 Gaussian correlations.Wick’s theorem

5.3 Perturbed Gaussian measure

5.4 Perturbative calculations: Examples

Exercises

6 Ferromagic systems. Correlation functions

6.1 Ferromagic systems: Definition

6.2 Correlation functions. Fourier representation

6.3 Legendre transformation and vertex functions

6.4 Legendre transformation and steepest descent method

6.5 Two- and four-point vertex functions

Exercises145

7 Phase transitions: Generalities and examples

7.1 Infinite temperature or independent spins

7.2 Phase transitions in infinite dimension

7.3 Universality in infinite space dimension

7.4 Transformations， fixed points and universality

7.5 Finite-range interactions in finite dimension

7.6 Ising model: Transfer matrix

7.7 Continuous symmetries and transfer matrix

7.8 Continuous symmetries and Goldstone modes

Exercises

8 Quasi-Gaussian approximation: Universality， critical dimension.

8.1 Short-range two-spin interactions

8.2 The Gaussian model: Two-point function.

8.3 Gaussian model and random walk

8.4 Gaussian model and field integral

8.5 Quasi-Gaussian approximation

8.6 The two-point function: Universality

8.7 Quasi-Gaussian approximation and Landau’s theory

8.8 Continuous symmetries and Goldstone modes

8.9 Corrections to the quasi-Gaussian approximation

8.10 Mean-field approximation and corrections

8.11 Tricritical points

Exercises

9 Renormalization group: General formulation

9.1 Statistical field theory. Landau’s Hamiltonian

9.2 Connected correlation functions. Vertex functions

9.3 Renormalization group: General idea

9.4 Hamiltonian flow: Fixed points， stability

9.5 The Gaussian fixed point.2319.6 Eigen-perturbations: General analysis

9.7 A non-Gaussian fixed point: The ε-expansion

9.8 Eigenvalues and dimensions of local polynomials

10 Perturbative renormalization group: Explicit calculations.

10.1 Critical Hamiltonian and perturbative expansion

10.2 Feynman diagrams at one-loop order

10.3 Fixed point and critical behaviour

10.4 Critical domain

10.5 Models with O(N) orthogonal symmetry

10.6 Renormalization group near dimension 4

10.7 Universal quantities: Numerical results

11 Renormalization group: N-ponent fields

11.1 Renormalization group: General remarks

11.2 Gradient flow

11.3 Model with cubic anisotropy

11.4 Explicit general expressions: RG analysis

11.5 Exercise: General model with two parameters

Exercises

12 Statistical field theory: Perturbative expansion

12.1 Generating functionals

12.2 Gaussian field theory.Wick’s theorem

12.3 Perturbative expansion

12.4 Loop expansion

12.5 Dimensional continuation and regularization

Exercises

13 The σ4 field theory near dimension 4

13.1 Effective Hamiltonian. Renormalization

13.2 Renormalization group equations

13.3 Solution of RGE: The ε-expansion

13.4 Effective and renormalized interactions

13.5 The critical domain above Tc

14 The O(N) symmetric (φ2)2 field theory in the large N limit

14.1 Algebraic preliminaries

14.2 Integration over the field φ: The determinant

14.3 The limit N →∞: The critical domain

14.4 The (φ2)2 field theory for N →∞

14.5 Singular part of the free energy and equation of state

14.6 The λλ and φ2φ2 two-point functions

14.7 Renormalization group and corrections to scaling

14.8 The 1/N expansion

14.9 The exponent η at order 1/N

14.10 The non-linear σ-model

15 The non-linear σ-model

15.1 The non-linear σ-model on the lattice

15.2 Low-temperature expansion

15.3 Formal continuum limit

15.4 Regularization

15.5 Zero-momentum or IR divergences

15.6 Renormalization group

15.7 Solution of the RGE. Fixed points

15.8 Correlation functions: Scaling form

15.9 The critical domain: Critical exponents

15.10 Dimension 2

15.11 The (φ2)2 field theory at low temperature

16 Functional renormalization group

16.1 Partial field integration and effective Hamiltonian

16.2 High-momentum mode integration andRGE

16.3 Perturbative solution: φ4 theory

16.4 RGE: Standard form

16.5 Dimension 4

16.6 Fixed point: ε-expansion

16.7 Local stability of the fixed point

Appendix

A1 Technical results

A2 Fourier transformation: Decay and regularity

A3 Phase transitions: General remarks

A4 1/N expansion: Calculations

A5 Functional renormalization group: Complements

Bibliography

Index

《相变与重正化群(英文影印版)》详细讨论了相变与重正化群的关系。特别是相变中的连续极限、相干长度及标度律等等。本书适合所有物理学领域的科研工作者和研究生阅读。

相变无疑是物理学中的最重要的现象之一。对于相变的研究贯穿整个物理学，甚至是人类文明史。而现代物理学中，与相变息息相关的一个方法就是重正化群方法，其概念和思想已经渗透于物理学的各个领域。《相变与重正化群(英文影印版)》的引进，能够供所有物理学领域的工作者作为参考。