# Dimensional reduction of a finite-size scalar field model at finite temperature

## TL;DR

This paper explores how a finite-size spatial dimension in a thermal scalar field model can be related to a lower-dimensional finite temperature model, providing a practical prescription for dimensional reduction.

## Contribution

It establishes a relationship between finite-size and finite-temperature models in different dimensions, with a specific focus on one-loop calculations for any dimension D.

## Key findings

- Finite spatial size in D dimensions relates to finite temperature in D-1 dimensions.
- Strict dimensional reduction is not possible, but a valid prescription can be defined.
- Results are applicable for any dimension D, exemplified in D=4.

## Abstract

We investigate the process of dimensional reduction of one spatial dimension in a thermal scalar field model defined in $D$ dimensions (inverse temperature and $D-1$ spatial dimensions). We obtain that a thermal model in $D$ dimensions with one of the spatial dimensions having a finite size $L$ is related to the finite temperature model with just $D-1$ spatial dimensions and no finite size. Our results are obtained for one-loop calculations and for any dimension $D$. For example, in $D=4$ we have a relationship between a thin film with thickness $L$ at finite temperature and a surface at finite temperature. We show that, although a strict dimensional reduction is not allowed, it is possible to define a valid prescription for this procedure.