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