Critical Properties of [Greek Letter Phi]4-theoriesThis 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-î dimensions, where they possess pole terms in 1/î. 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 Pad 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 à 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 phi4-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 phi4-theories are sketched, such as variational perturbation expansions in three rather than 4-î dimensions, and improved ratio tests in high-temperature expansions of lattice models. |
Contents
Renormalization | 9 |
1 | 11 |
5 | 23 |
A Diagrammatic ROperation up to Five Loops | 30 |
Definition of 4Theory | 32 |
Notes and References | 40 |
Diagrams in Momentum Space | 53 |
1 | 59 |
431 | 254 |
Generation of Diagrams | 257 |
Notes and References | 285 |
A 2 | 294 |
Critical Exponents of ONSymmetric Theory | 304 |
Cubic Anisotropy | 314 |
Notes and References | 326 |
B 4 | 337 |
Notes and References | 76 |
1 | 91 |
Appendix | 140 |
Notes and References | 153 |
Notes and References | 185 |
ZeroMass Approach to Counterterms | 197 |
Appendix 12A Proof of Interchangeability of Differentiation and ROperation | 223 |
Reduction Algorithms | 238 |
Appendix 13A Fourier Transformation of Simple Powers in D Dimensions | 250 |
Other editions - View all
Critical Properties of [Greek Letter Phi]4-theories Hagen Kleinert,Verena Schulte-Frohlinde No preview available - 2001 |
Common terms and phrases
4-theory analytic asymptotic B-function Borel Borel sum calculated connected connected correlation functions convergence correlation functions counterterms coupling constant critical exponents derivative diagrammatic differentiation dimensional regularization dimensions divergent equation expansion coefficients external lines external momenta extrapolation ɛ-expansion Feynman diagrams Feynman integrals field theory five-loop fixed point fluctuations formula four-point diagrams Inserting IR-divergence IR-rearrangement Kleinert large-order behavior Lett logarithmic loops mass massless matrix momentum space O(N)-symmetric obtain Padé approximants parameter permutations perturbation expansions Phys plot pole terms power series propagator R-operation re-expanded renormalization constants renormalization group resummation right-hand side scaling Scub Section shown in Fig strong-coupling expansion strong-coupling limit subdiagram subtraction symmetry factors temperature tensor three-loop transformation two-loop two-point diagrams UV-divergence vacuum diagrams values variational perturbation theory vertex functions vertices weight factors yields zero Zm² αβγδ