Quasi-plane waves for a particle with spin 1/2 on the background of Lobachevsky geometry: simulating of a special medium

TL;DR

This paper constructs exact solutions for spin-1/2 particles in Lobachevsky space, revealing how curved space solutions relate to flat space plane waves and simulating a quantum barrier scenario.

Contribution

It provides a complete set of solutions for Dirac and Weyl equations in Lobachevsky space using separation of variables, introducing an extended helicity operator.

Findings

Solutions reduce to plane waves in flat space limit

Lobachevsky space simulates a rising potential barrier

Method can be applied to quantum particles in curved backgrounds

Abstract

In the paper complete systems of exact solutions for Dirac and Weyl equations in the Lobachevsky space are constructed on the base of the method of separation of the variables in quasi-cartesian coordinates. An extended helicity operator is introduced. It is shown that solution constructed when translating to the limit of vanishing curvature coincide with common plane wave solutions on Minkowski space going in opposite z-directions. It is shown the problem posed in Lobachevsky space simulates a situation in the flat space for a quantum-mechanical particle of spin 1/2 in a 2-dimensional potential barrier smoothly rising to infinity on the right.