The Effective Action at Finite Temperature and Density With Application to Bose-Einstein Condensation

TL;DR

This paper introduces the effective action method in quantum field theory, applying it to finite temperature and density scenarios, including Bose-Einstein condensation, with detailed results for various gases and external conditions.

Contribution

It provides a comprehensive pedagogical framework for applying the effective action formalism to Bose-Einstein condensation and related phenomena at finite temperature and in curved space.

Findings

High temperature expansions for Bose gases derived

Meissner effect demonstrated for nonrelativistic gases

Effective potential formalism applied to curved space and external fields

Abstract

A simple pedagogical introduction to the effective action method of quantum field theory is given at a level suitable for beginning postgraduate students. It is shown how to obtain the effective potential at zero temperature from a regularized zero-point energy. The results are applicable to curved as well as to flat space. The generalization to finite temperatures is also given. It is shown how to obtain high temperature expansions of the thermodynamic potential for the neutral free Bose gas and the charged Bose gas in both the relativistic and nonrelativistic limits. The results are obtained for an arbitrary spatial dimension and in curved space. Results are also obtained for the self-interacting relativistic gas in three spatial dimensions. A detailed discussion of how the formalism may be applied to study Bose-Einstein condensation is given. The interpretation of Bose-Einstein condensation as symmetry breaking is discussed. Application is given to the study of charged bosons, both relativistic and nonrelativistic, in a constant magnetic field. The Meissner effect is obtained for the nonrelativistic gas in three spatial dimensions. The final application is to the study of nonrelativistic bosons in a harmonic oscillator confing potential trap. A number of analytical approaches to this are discussed.