Dimensional reduction of the Dirac theory

TL;DR

This paper explores the dimensional reduction of the Dirac theory from three to two spatial dimensions, analyzing free and coupled cases, and reveals how invariance and specific solutions influence the resulting theories.

Contribution

It extends Hadamard's method of descent to include electromagnetic coupling, deriving simplified 2+1 dimensional theories and identifying conditions for decoupling.

Findings

Invariance along one spatial direction leads to decoupled free Dirac equations.

Electromagnetic coupling results in more complex 2+1 dimensional theories.

Decoupling of physical sectors depends on specific solution classes.

Abstract

We perform a reduction from three to two spatial dimensions of the physics of a spin-1/2 fermion coupled to the electromagnetic field, by applying Hadamard's method of descent. We consider first the free case, in which motion is determined by the Dirac equation, and then the coupling with a dynamical electromagnetic field, governed by the Dirac-Maxwell equations. We find that invariance along one spatial direction splits the free Dirac equation in two decoupled theories. On the other hand, a dimensional reduction in the presence of an electromagnetic field provides a more complicated theory in 2+1 dimensions, in which the method of descent is extended by using the covariant derivative. Equations simplify, but decoupling between different physical sectors occurs only if specific classes of solutions are considered.