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 renormalization group

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 Thermody namics limit

4.4 Connected functions and cluster properties

4.5 Statistical models：Simple examples

4.6 The Gaussian model

4.7 Gaussian model：The continuum limit

4.8 More general models：The continuum limit

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 Ferromagnetic systems.Correlation functions

6.1 Ferromagnetic 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

Exercises

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(RG)：General idea

9.4 Hamiltonian flow：Fixed points,stability

9.5 The Gaussian fixed point

9.6 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 RG near dimension 4

10.7 Universal quantities：Numerical results

11 Renormalization group：N-component fields

11.1 RG：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 dimensional regularization

Exercises

13 Theσ4 field theory near dimension 4

13.1 Effective Hamiltonian.Renormalization

13.2 RG equations

13.3 Solution ofRG equations：Theε-expansion

13.4 The critical domain above Tc

13.5 RG equations for renormalized vertex functions

13.6 Effective and renormalized interactions

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

14.1 Algebraic preliminaries

14.2 Integration over the fieldφ：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 RG and corrections to scaling

14.8 The 1/N expansion

14.9 The exponent ηat order 1/N

14.10 The non-linearσ-model

15 The non-linearσ-model

15.1 The non-linearσ-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 and RG equations

16.3 Perturbative solution：φ4 theory

16.4 RG equations：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 flow equations：Additional considerations

Bibliography

Index