Critical Properties of Phi Theories

Hagen Kleinert, Verena Schulte-Frohlinde

490 pages, parution le 06/12/2001

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Résumé

This book explains in detail how to perform perturbation expansions in quantum field theory to high orders, and how to extract the critical properties of the theory from the resulting divergent power series. These properties are calculated for various second-order phase transitions of three-dimensional systems with high accuracy, in particular the critical exponents observable in experiments close to the phase transition.

Beginning with an introduction to critical phenomena, this book develops the functional-integral description of quantum field theories, their perturbation expansions, and a method for finding recursively all Feynman diagrams to any order in the coupling strength. Algebraic computer programs are supplied on accompanying World Wide Web pages. The diagrams correspond to integrals in momentum space. They are evaluated in 4-e dimensions, where they possess pole terms in 1/e. The pole terms are collected into renormalization constants.

The theory of the renormalization group is used to find the critical scaling laws. They contain critical exponents which are obtained from the renormalization constants in the form of power series. These are divergent, due to factorially growing expansion coefficients. The evaluation requires resummation procedures, which are performed in two ways: (1) using traditional methods based on Padé and Borel transformations, combined with analytic mappings; (2) using modern variational perturbation theory, where the results follow from a simple strong-coupling formula. As a crucial test of the accuracy of the methods, the critical exponent a governing the divergence of the specific heat of superfluid helium is shown to agree very well with the extremely precise experimental number found in the space shuttle orbiting the earth (whose data are displayed on the cover of the book).

The f⁴-theories investigated in this book contain any number N of fields in an O(N)-symmetric interaction, or in an interaction in which O(N)-symmetry is broken by a term of a cubic symmetry. The crossover behavior between the different symmetries is investigated. In addition, alternative ways of obtaining critical exponents of f⁴-theories are sketched, such as variational perturbation expansions in three rather than 4-e dimensions, and improved ratio tests in high-temperature expansions of lattice models.

Contents

Definition of f⁴-Theory
Feynman Diagrams
Diagrams in Momentum Space
Structural Properties of Perturbation Theory
Diagrams for Multicomponent Fields
Scale Transformations of Fields and Correlation Functions
Regularization of Feynman Integrals
Renormalization
Renormalization Group
Recursive Subtraction of UV-Divergences via R-Operation
Zero-Mass Approach to Counterterms
Calculation of Momentum Space Integrals
Generation of Diagrams
Results of the Five-Loop Calculation
Basic Resummation Theory
Critical Exponents of O(N)-Symmetric Theory
Cubic Anisotropy
Variational Perturbation Theory
Critical Exponents from Other Expansions
New Resummation Algorithm
Conclusion: Diagrammatic R-Operation Up to Five Loops
Contributions to Renormalization-Constants

Caractéristiques techniques

	PAPIER
Éditeur(s)	World Scientific
Auteur(s)	Hagen Kleinert, Verena Schulte-Frohlinde
Parution	06/12/2001
Nb. de pages	490
Format	19 x 25,8
Couverture	Broché
Poids	1050g
Intérieur	Noir et Blanc
EAN13	9789810246594