Quantum fields in toroidal topology

TL;DR

This paper develops a framework using the standard c*-algebra representation to describe quantum fields in toroidal topologies, generalizing finite-temperature field theory and analyzing particle decay in compactified space-times.

Contribution

It introduces a novel modular representation for fields in toroidal topologies, generalizing the Fourier representation and eliminating the need for the 2x2 real-time formalism.

Findings

Describes a condensate in the ground state for compactified fields

Analyzes decay processes of particles in compactified space-time

Extends the formalism to non-abelian gauge theories

Abstract

The standard representation of c*-algebra is used to describe fields in compactified space-time dimensions characterized by topologies of the type $ \Gamma_{D}^{d}=(\mathbb{S}^{1})^{d}\times \mathbb{M}^{D-d}$. The modular operator is generalized to introduce representations of isometry groups. The Poincar\'{e} symmetry is analyzed and then we construct the modular representation by using linear transformations in the field modes, similar to the Bogoliubov transformation. This provides a mechanism for compactification of the Minkowski space-time, that follows as a generalization of the Fourier-integral representation of the propagator at finite temperature. An important result is that the $2\times2$ representation of the real time formalism is not needed. The end result on calculating observables is described as a condensate in the ground state. We analyze initially the free Klein-Gordon and Dirac fields, and then formulate non-abelian gauge theories in $\Gamma_{D}^{d}$. Using the S-matrix, the decay of particles is calculated in order to show the effect of the compactification.