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Quantum Field Theory and Critical Phenomena$
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Jean Zinn-Justin

Print publication date: 2002

Print ISBN-13: 9780198509233

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198509233.001.0001

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Quantum Field Theory At Finite Temperature: Equilibrium Properties

Quantum Field Theory At Finite Temperature: Equilibrium Properties

Quantum Field Theory and Critical Phenomena


Oxford University Press

This chapter reviews some equilibrium properties in Statistical Quantum Field Theory, that is, relativistic Quantum Field Theory (QFT) at finite temperature, a relativistic extension of the statistical quantum theories discussed in Sections 5.5, 5.6. It discusess, in particular, the limit of high temperature or the situation of finite temperature phase transitions. It emphasizes that additional physical intuition about QFT at finite temperature in (1; d - 1) dimensions can be gained by realizing that it can also be considered as a classical statistical field theory in d dimensions with finite size in one dimension. This identification allows, in particular, an analysis of finite temperature QFT in terms of the renormalization group and the theory of finite size effects of the classical theory. These ideas are illustrated with several standard examples, the φ4 field theory, the non-linear σ model, the Gross–Neveu model, some gauge theories. The corresponding effective reduced theories are constructed at one-loop order. In models where the field is a N-component vector, the large N expansion provides a specially convenient tool to study the complete crossover between low and high temperature, and, therefore, dimensional reduction.

Keywords:   finite temperature phase transitions, equilibrium, quantum field theory, Gross–Neveu model, gauge theories

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