Dirac Equation on a Curved 2+1 Dimensional Hypersurface

TL;DR

This paper derives an effective Dirac equation for a relativistic spin 1/2 particle constrained to a curved 2+1 dimensional surface, relevant for understanding electron behavior in systems like graphene.

Contribution

It introduces a method to describe relativistic particles on curved surfaces without referencing external dimensions, extending previous work with a new derivation using the thin layer approach.

Findings

Derived an effective Dirac equation on curved 2+1 surfaces

Compared results with previous models by Burgess and Jensen

Provided insights into relativistic particles in curved geometries

Abstract

Interest on 2 + 1 dimensional electron systems has increased considerably after the realization of novel properties of graphene sheets, in which the behaviour of electrons is effectively described by relativistic equations. Having this fact in mind, the following problem is studied in this work: when a spin 1/2 particle is constrained to move on a curved surface, is it possible to describe this particle without giving reference to the dimensions external to the surface? As a special case of this, a relativistic spin 1/2 particle which is constrained to move on a 2 + 1 dimensional hypersurface of the 3 + 1 dimensional Minkowskian spacetime is considered, and an effective Dirac equation for this particle is derived using the so-called thin layer method. Some of the results are compared with those obtained in a previous work by M. Burgess and B. Jensen.