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Physics Bulletin

Field Theory, the Renormalisation Group, and Critical Phenomena

M A Moore

Published Sep 1, 1979• 44 citations
Publisher

Abstract

Daniel J Amit 1978 Maidenhead, Berks: McGraw-Hill xiv + 366 pp price £13.35 This monograph starts with a formulation of the problems of phase transitions and quantum field theory in terms of functional integrals and then introduces in a clear and compact way Feynman diagrams, vertex functions, symmetry breaking and Goldstone's theorem, etc. A discussion of divergences leads into the renormalisation programme, counterterms and then finally the renormalisation group.

DOI
10.1088/0031-9112/30/9/025

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Daniel J Amit 1978 Maidenhead, Berks: McGraw-Hill xiv + 366 pp price £13.35 This monograph starts with a formulation of the problems of phase transitions and quantum field theory in terms of functional integrals and then introduces in a clear and compact way Feynman diagrams, vertex functions, symmetry breaking and Goldstone's theorem, etc. A discussion of divergences leads into the renormalisation programme, counterterms and then finally the renormalisation group.
Pertinent Concepts and Ideas in the Theory of Critical Phenomena Formulation of the Problem of Phase Transitions in Terms of Functional Integrals Functional Integrals in Quantum Field Theory Perturbation Theory and Feynman Graphs Vertex Functions and Symmetry Breaking Expansions in the Number of Loops and Components Renormalization The Renormalization Group and Scaling the Critical Region The Computation of the Critical Exponents Beyond Leading Scaling Universality Revisited Critical Behavior with Several Couplings Crossover Phenomena Critical Phenomena near Two Dimensions Real Space Methods Finite Size Scaling Monte Carlo Methods. Numerical Field Theory.
These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.
Abstract Introduced as a quantum extension of Maxwell's classical theory, quantum electrodynamic (QED) has been the first example of a quantum field theory (QFT). Eventually, QFT has become the framework for the discussion of all fundamental interactions at the microscopic scale except, possibly, gravity. More surprisingly, it has also provided a framework for the understanding of second order phase transitions in statistical mechanics. In fact, as hopefully this work illustrates, QFT is the natural framework for the discussion of most systems involving an infinite number of degrees of freedom with local couplings. These systems range from cold Bose gases at the condensation temperature (about ten nanokelvin) to conventional phase transitions (from a few degrees to several hundred) and high energy particle physics up to a TeV, altogether more than twenty orders of magnitude in the energy scale. Therefore, although excellent textbooks about QFT had already been published, I thought, many years ago, that it might not be completely worthless to present a work in which the strong formal relations between particle physics and the theory of critical phenomena are systematically emphasized. This option explains some of the choices made in the presentation. A formulation in terms of field integrals has been adopted to study the properties of QFT. The language of partition and correlation functions has been used throughout, even in applications of QFT to particle physics. Renormalization and renormalization group (RG) properties are systematically discussed. The notion of effective field theory (EFT) and the emergence of renormalizable theories are described. The consequences for fine-tuning and triviality issue are emphasized. This fifth edition has been updated and fully revised.
This chapter discusses a very popular method of studying critical phenomena analytically: the so-called ϵ-expansion. It describes the renormalisation group calculation of the critical exponents in the complete Landau theory, near d = 4. The chapter outlines the strategy of the calculation of 0, to prepare the reader for the algebraic horrors to follow. R. P. Feynman developed useful shorthand for the terms, which is to write down diagrams indicating how different fields are paired up. There is one-to-one correspondence between each Feynman diagram or Feynman graph and a term in the perturbation series. This correspondence is not unique: there are many ways to write down diagrams for the terms, and people have their own preferences. Once one has figured out this correspondence, then one only has to be able to draw all the appropriate diagrams to see what one should compute.