Phase transitions and renormalization group / / Jean Zinn-Justin

• Autore: Zinn-Justin Jean
• Pubblicazione: Oxford, : Oxford University Press, 2007
• Edizione: 1st ed.
• Descrizione fisica: xii, 452 p
• Disciplina: 530.414
• Soggetto topico: Phase transformations (Statistical physics); Renormalization (Physics)
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- Contents -- 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 Thermodynamic 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: 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 & -- #949 -- -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-component 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 & -- #963 -- [sup(4)] field theory near dimension 4 -- 13.1 Effective Hamiltonian. Renormalization -- 13.2 Renormalization group equations -- 13.3 Solution of RGE: The & -- #949 -- -expansion -- 13.4 Effective and renormalized interactions -- 13.5 The critical domain above T[sub(c)] -- 14 The O(N) symmetric (& -- #934 -- [sup(2)])[sup(2)] field theory in the large N limit -- 14.1 Algebraic preliminaries -- 14.2 Integration over the field & -- #934 -- : The determinant -- 14.3 The limit N & -- #8594 & -- #8734 -- : The critical domain -- 14.4 The (& -- #934 -- [sup(2)][sup(2)] field theory for N & -- #8594 & -- #8734 -- 14.5 Singular part of the free energy and equation of state -- 14.6 The (& -- #955 -- & -- #955 -- ) and (& -- #934 -- [sup(2)] & -- #934 -- [sup(2)]) 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 & -- #963 -- -model -- 15 The non-linear & -- #963 -- -model -- 15.1 The non-linear & -- #963 -- -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 (& -- #934 -- [sup(2)])[sup(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 RGE -- 16.3 Perturbative solution: & -- #934 -- [sup(4)] theory -- 16.4 RGE: Standard form -- 16.5 Dimension 4 -- 16.6 Fixed point: & -- #949 -- -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 -- A -- B -- C -- D -- E -- F -- G -- H -- I -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W.
• Sommario/riassunto: The renormalization group is one of most important theoretical concepts that has emerged in physics during the twentieth century. It explains important properties of fundamental interactions at the microscopic scale, as well as universal properties of continuous macroscopic phase transitions.
• Titolo autorizzato: Phase transitions and renormalization group
• ISBN: 0191527742; 9780191527746
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910814182203321
• Lo trovi qui: Univ. Federico II
• Serie: Oxford graduate texts.