Zeta function regularization for a scalar field in a compact domain

TL;DR

This paper derives formulas relating the zeta function of a Laplacian on a product space to that on a compact manifold, enabling precise calculations of thermodynamic functions for scalar fields in compact domains.

Contribution

It provides a new expression for the zeta function on product spaces and exact formulas for specific geometries, facilitating rigorous analysis of physical models at finite temperature.

Findings

Derived formulas for zeta functions on product spaces.

Exact zeta function formulas for boxes and tori.

Enabled precise computation of thermodynamic functions.

Abstract

We express the zeta function associated to the Laplacian operator on $S^1_r\times M$ in terms of the zeta function associated to the Laplacian on $M$, where $M$ is a compact connected Riemannian manifold. This gives formulas for the partition function of the associated physical model at low and high temperature for any compact domain $M$. Furthermore, we provide an exact formula for the zeta function at any value of $r$ when $M$ is a $D$-dimensional box or a $D$-dimensional torus; this allows a rigorous calculation of the zeta invariants and the analysis of the main thermodynamic functions associated to the physical models at finite temperature.