Finite size effects in the thermodynamics of a free neutral scalar field

TL;DR

This paper provides exact analytical lattice results for the thermodynamics of a free neutral scalar field, analyzing finite size effects and their convergence to continuum limits across different temperatures and volumes.

Contribution

It derives explicit lattice expressions for the partition function and thermodynamic quantities of a free scalar field in one and three dimensions, highlighting finite size effects.

Findings

Exact lattice results match continuum only at small temperature and volume.

Finite volume corrections significantly affect thermodynamic quantities.

Continuum limit is approached at high temperatures or large volumes.

Abstract

The exact analytical lattice results for the partition function of the free neutral scalar field in one spatial dimension in both the configuration and the momentum space were obtained in the framework of the path integral method. The symmetric square matrices of the bilinear forms on the vector space of fields in both configuration space and momentum space were found explicitly. The exact lattice results for the partition function were generalized to the three-dimensional spatial momentum space and the main thermodynamic quantities were derived both on the lattice and in the continuum limit. The thermodynamic properties and the finite volume corrections to the thermodynamic quantities of the free real scalar field were studied. We found that on the finite lattice the exact lattice results for the free massive neutral scalar field agree with the continuum limit only in the region of small values of temperature and volume. However, at these temperatures and volumes the continuum physical quantities for both massive and massless scalar field deviate essentially from their thermodynamic limit values and recover them only at high temperatures or/and large volumes in the thermodynamic limit.