Quantum field theory on toroidal topology: algebraic structure and applications

TL;DR

This paper reviews the foundations of quantum field theory on a torus, highlighting its algebraic structure and applications in particle and condensed matter physics, with a focus on finite size effects.

Contribution

It provides a unified presentation of the algebraic foundations of compactified quantum field theory on a torus and discusses its diverse applications.

Findings

Enhanced understanding of quantum fields on toroidal topology.

Applications to finite size effects in physics.

Framework for future research in particle and condensed matter physics.

Abstract

The development of quantum theory on a torus has a long history, and can be traced back to the 1920s, with the attempts by Nordstr\"om, Kaluza and Klein to define a fourth spatial dimension with a finite size, being curved in the form of a torus, such that Einstein and Maxwell equations would be unified. Many developments were carried out considering cosmological problems in association with particles physics, leading to methods that are useful for areas of physics, in which size effects play an important role. This interest in finite size effect systems has been increasing rapidly over the last decades, due principally to experimental improvements. In this review, the foundations of compactified quantum field theory on a torus are presented in a unified way, in order to consider applications in particle and condensed matted physics.