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| Quantum field theory and the renormalization group | |
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| Quantum electrodynamics: A quantum field theory | |
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| Quantum electrodynamics: The problem of infinities | |
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| Renormalization | |
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| Quantum field theory and the renormalization group | |
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| A triumph of QFT: The Standard Model | |
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| Critical phenomena: Other infinities | |
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| Kadanoff and Wilson's renormalization group | |
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| Effective quantum field theories | |
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| Gaussian expectation values. Steepest descent method | |
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| Generating functions | |
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| Gaussian expectation values. Wick's theorem | |
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| Perturbed Gaussian measure. Connected contributions | |
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| Feynman diagrams. Connected contributions | |
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| Expectation values. Generating function. Cumulants | |
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| Steepest descent method | |
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| Steepest descent method: Several variables, generating functions | |
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| Exercises | |
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| Universality and the continuum limit | |
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| Central limit theorem of probabilities | |
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| Universality and fixed points of transformations | |
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| Random walk and Brownian motion | |
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| Random walk: Additional remarks | |
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| Brownian motion and path integrals | |
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| Exercises | |
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| Classical statistical physics: One dimension | |
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| Nearest-neighbour interactions. Transfer matrix | |
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| Correlation functions | |
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| Thermodynamic limit | |
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| Connected functions and cluster properties | |
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| Statistical models: Simple examples | |
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| The Gaussian model | |
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| Gaussian model: The continuum limit | |
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| More general models: The continuum limit | |
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| Exercises | |
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| Continuum limit and path integrals | |
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| Gaussian path integrals | |
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| Gaussian correlations. Wick's theorem | |
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| Perturbed Gaussian measure | |
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| Perturbative calculations: Examples | |
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| Exercises | |
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| Ferromagnetic systems. Correlation functions | |
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| Ferromagnetic systems: Definition | |
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| Correlation functions. Fourier representation | |
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| Legendre transformation and vertex functions | |
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| Legendre transformation and steepest descent method | |
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| Two- and four-point vertex functions | |
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| Exercises | |
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| Phase transitions: Generalities and examples | |
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| Infinite temperature or independent spins | |
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| Phase transitions in infinite dimension | |
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| Universality in infinite space dimension | |
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| Transformations, fixed points and universality | |
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| Finite-range interactions in finite dimension | |
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| Ising model: Transfer matrix | |
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| Continuous symmetries and transfer matrix | |
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| Continuous symmetries and Goldstone modes | |
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| Exercises | |
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| Quasi-Gaussian approximation: Universality, critical dimension | |
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| Short-range two-spin interactions | |
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| The Gaussian model: Two-point function | |
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| Gaussian model and random walk | |
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| Gaussian model and field integral | |
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| Quasi-Gaussian approximation | |
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| The two-point function: Universality | |
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| Quasi-Gaussian approximation and Landau's theory | |
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| Continuous symmetries and Goldstone modes | |
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| Corrections to the quasi-Gaussian approximation | |
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| Mean-field approximation and corrections | |
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| Tricritical points | |
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| Exercises | |
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| Renormalization group: General formulation | |
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| Statistical field theory. Landau's Hamiltonian | |
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| Connected correlation functions. Vertex functions | |
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| Renormalization group: General idea | |
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| Hamiltonian flow: Fixed points, stability | |
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| The Gaussian fixed point | |
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| Eigen-perturbations: General analysis | |
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| A non-Gaussian fixed point: The [epsilon]-expansion | |
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| Eigenvalues and dimensions of local polynomials | |
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| Perturbative renormalization group: Explicit calculations | |
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| Critical Hamiltonian and perturbative expansion | |
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| Feynman diagrams at one-loop order | |
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| Fixed point and critical behaviour | |
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| Critical domain | |
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| Models with O(N) orthogonal symmetry | |
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| Renormalization group near dimension 4 | |
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| Universal quantities: Numerical results | |
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| Renormalization group: N-component fields | |
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| Renormalization group: General remarks | |
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| Gradient flow | |
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| Model with cubic anisotropy | |
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| Explicit general expressions: RG analysis | |
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| Exercise: General model with two parameters | |
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| Exercises | |
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| Statistical field theory: Perturbative expansion | |
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| Generating functionals | |
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| Gaussian field theory. Wick's theorem | |
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| Perturbative expansion | |
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| Loop expansion | |
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| Dimensional continuation and regularization | |
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| Exercises | |
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| The [sigma superscript 4] field theory near dimension 4 | |
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| Effective Hamiltonian. Renormalization | |
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| Renormalization group equations | |
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| Solution of RGE: The [epsilon]-expansion | |
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| Effective and renormalized interactions | |
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| The critical domain above T[subscript c] | |
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| The O(N) symmetric ([phi superscript 2])[superscript 2] field theory in the large N limit | |
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| Algebraic preliminaries | |
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| Integration over the field [phi]: The determinant | |
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| The limit N to [infinity]: The critical domain | |
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| The ([phi superscript 2])[superscript 2] field theory for N to [infinity] | |
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| Singular part of the free energy and equation of state | |
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| The [left angle bracket lambda lambda right angle bracket] and [left angle bracket phi superscript 2 phi superscript 2 right angle bracket] two-point functions | |
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| Renormalization group and corrections to scaling | |
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| The 1/N expansion | |
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| The exponent [eta] at order 1/N | |
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| The non-linear [sigma]-model | |
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| The non-linear [sigma]-model | |
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| The non-linear [sigma]-model on the lattice | |
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| Low-temperature expansion | |
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| Formal continuum limit | |
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| Regularization | |
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| Zero-momentum or IR divergences | |
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| Renormalization group | |
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| Solution of the RGE. Fixed points | |
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| Correlation functions: Scaling form | |
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| The critical domain: Critical exponents | |
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| Dimension 2 | |
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| The ([phi superscript 2])[superscript 2] field theory at low temperature | |
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| Functional renormalization group | |
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| Partial field integration and effective Hamiltonian | |
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| High-momentum mode integration and RGE | |
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| Perturbative solution: [phi superscript 4] theory | |
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| RGE: Standard form | |
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| Dimension 4 | |
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| Fixed point: [epsilon]-expansion | |
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| Local stability of the fixed point | |
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| Appendix | |
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| Technical results | |
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| Fourier transformation: Decay and regularity | |
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| Phase transitions: General remarks | |
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| 1/N expansion: Calculations | |
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| Functional renormalization group: Complements | |
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| Bibliography | |
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| Index | |