Quantum Fields at Non-zero Temperature

Franco Strocchi

The general structure discussed above provides a neat and unique prescription for the quantization of relativistic fields at non-zero temperature ( thermofield theory ).

Wim Schoenmaker, Roger Horsley

Theoretical Physics

Tohru Koma

We study a quantum antiferromagnetic Heisenberg model on a hypercubic lattice in three or higher dimensions $d\ge 3$. When a phase transition occurs with the continuous symmetry breaking, the nonvanishing spontaneous magnetization which is obtained by applying the infinitesimally weak symmetry breaking field is equal to the maximum spontaneous magnetization at zero or non-zero low temperatures. In addition, the transverse correlation in the infinite-volume limit exhibits a Nambu-Goldstone-type slow decay. In this paper, we assume that the transverse correlation decays by power law with distance. Under this assumption, we prove that the power is equal to $2-d$ at non-zero low temperatures, while it is equal to $1-d$ at zero temperature. The method is applied also to a quantum XY model and a classical Heisenberg model at non-zero low temperatures in three or higher dimensions. The resulting power is given by the same $2-d$ at non-zero low temperatures.

Michel Le Bellac

We now start what will be the subject of this book: quantum field theory at non-zero temperature and/or non-zero chemical potential. Two competing formalisms have been used at zero temperature in order to study field theory: the operator formalism, which is the older one, and the path integral formalism, which represents the more modern approach. Actually, it may often be illuminating to look at a given problem from both points of view, although in some cases one of the formalisms may prove to be definitely superior to the other: for example, path integrals are much simpler when quantizing gauge theories. At finite temperature, both formalisms are useful, and it is instructive to be able to switch from one approach to the other. Since the simplest case of field theory is field theory with zero space dimension, or, in other words, quantum mechanics, we shall begin with a short description of quantum mechanics at finite temperature, or, equivalently, quantum statistical mechanics. We first deal with the path integral approach, and later on revert to the more conventional operator formalism. We shall be particularly interested in time-ordered products of position operators, which will generalize to time-ordered products of field operators in field theory.

Henk T. C. Stoof, D. B. M. Dickerscheid, K. B. Gubbels