The Finite Temperature Effective Potential for Local Composite Operators

TL;DR

This paper develops a method to compute the finite temperature effective potential for scalar fields with self-interaction, using composite operators, and demonstrates rapid convergence to exact results in a one-dimensional quantum system.

Contribution

It applies the effective action method for composite operators to finite temperature scalar field theory, providing a systematic expansion and numerical validation in one dimension.

Findings

Quick convergence of approximations to the exact free energy.

Successful application of the method to a one-dimensional quantum anharmonic oscillator.

Effective mass and coupling determined by gap equations.

Abstract

The method of the effective action for the composite operators $\Phi^2(x)$ and $\Phi^4(x)$ is applied to the termodynamics of the scalar quantum field with $\lambda\Phi^4$ interaction. An expansion of the finite temperature effective potential in powers of $\hbar$ provides successive approximations to the free energy with an effective mass and an effective coupling determined by the gap equations. The numerical results are studied in the space-time of one dimension, when the theory is equivalent to the quantum mechanics of an anharmonic oscillator. The approximations to the free energy show quick convergence to the exact result.