Dimensional reduction of the Dirac equation in arbitrary spatial dimensions

TL;DR

This paper explores how the Dirac equation simplifies when reducing the number of spatial dimensions, revealing different forms depending on whether the higher-dimensional space has even or odd dimensions, using Hadamard's descent method.

Contribution

It introduces a systematic approach to dimensional reduction of the Dirac equation applicable in arbitrary spatial dimensions, including explicit representation hierarchies.

Findings

Dirac equation reduces to a single or two decoupled equations depending on dimension parity.

Hadamard's method of descent effectively relates high- and low-dimensional theories.

Explicit hierarchies of representations facilitate the reduction process.

Abstract

We investigate the general properties of the dimensional reduction of the Dirac theory, formulated in a Minkowski spacetime with an arbitrary number of spatial dimensions. This is done by applying Hadamard's method of descent, which consists in conceiving low-dimensional theories as a specialization of high-dimensional ones that are uniform along the additional space coordinate. We show that the Dirac equation reduces to either a single Dirac equation or two decoupled Dirac equations, depending on whether the higher-dimensional manifold has even or odd spatial dimensions, respectively. Furthermore, we construct and discuss an explicit hierarchy of representations in which this procedure becomes manifest and can easily be iterated.