Quantum Field Theory at Finite Temperature: an Introduction

In these notes we review some properties of Statistical Quantum Field Theory at equilibrium, i.e Quantum Field Theory at finite temperature. We explain the relation between finite temperature quantum field theory in (d, 1) dimensions and statistical classical field theory in d + 1 dimensions. This identification allows to analyze the finite temperature QFT in terms of the renormalization group and the theory of finite size effects of the classical theory. We discuss in particular the limit of high temperature (HT) or the situation of finite temperature phase transitions. There the concept of dimensional reduction plays an essential role. Dimensional reduction in some sense reflects the known property that quantum effects are not important at high temperature. We illustrate these ideas with several standard examples, φ 4 field theory, the non-linear σ model and the Gross–Neveu model, gauge theories. We construct the corresponding effective reduced theories at one-loop order, using the technique of mode expansion of fields in the imaginary time variable. In models where the field is a vector with N components, the large N expansion provides another specially convenient tool to study dimensional reduction.