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 ON 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 ON symmetric φ22 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 φ22 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 1N expansion
14.9 The exponent η at order 1N
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 φ22 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 1N expansion: Calculations
A5 Functional renormalization group: Complements
Bibliography
Index
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