Some equilibrium properties in statistical quantum field theory (QFT), that is, relativistic QFT at finite temperature are reviewed. Study of QFT at finite temperature is motivated by cosmological problems, high energy heavy ion collisions, and speculations about possible phase transitions, also searched for in numerical simulations. In particular, the situation of finite temperature phase transitions, or the limit of high temperature (an ultra-relativistic limit where the temperature is much larger than the physical masses of particles) are discussed. The concept of dimensional reduction emerges, in many cases, statistical properties of finite-temperature QFT in (1, d − 1) dimensions can be described by an effective classical statistical field theory in (d − 1) dimensions. Dimensional reduction generalizes a property already observed in the non-relativistic example of the Bose gas, and indicates that quantum effects are less important at high temperature. The corresponding technical tools are a mode-expansion of fields in the Euclidean time variable, singling out the zero modes of boson fields, followed by a local expansion of the resulting (d − 1)-dimensional effective field theory (EFT). Additional physical intuition about QFT at finite temperature in (1, d−1) dimensions can be gained by considering it as a classical statistical field theory in d dimensions, with finite size in one dimension. This identification makes an analysis of finite temperature QFT in terms of the renormalization group (RG), and the theory of finite-size effects of the classical theory, possible. These ideas are illustrated with several simple examples, the φ4 field theory, the non-linear σ-model, the Gross–Neveu model and some gauge theories.
A vanishing theorem for supersymmetric quantum field theory and finite size effects in multiphase cluster expansions