Thomas Precession, Berry potential and the Meron

TL;DR

This paper explores the geometric origin of Thomas precession in the Dirac equation, linking it to a nonabelian Berry potential generated by a meron, a topological object with instanton number 1/2, in parameter space.

Contribution

It identifies the meron as the source of the nonabelian Berry potential responsible for Thomas precession in the nonrelativistic Dirac equation.

Findings

Thomas precession linked to nonabelian Berry potential.

Meron characterized as the topological object producing the potential.

Degeneracy points correspond to the meron center.

Abstract

We begin with a prior observation by one of us that Thomas precession in the nonrelativistic limit of the Dirac equation may be attributed to a nonabelian Berry vector potential. We ask what object produces the nonabelian potential in parameter space, in the same sense that the abelian vector potential arising in the adiabatic transport of a nondegenerate level is produced by a monopole, (centered at the point where the level becomes degenerate with another), as shown by Berry. We find that it is a {\em meron}, an object in four euclidean dimensions with instanton number ${1 \over 2}$, centered at the point where the doubly degenerate positive and negative energy levels of the Dirac equation become fourfold degenerate.