Thomas Precession, Berry Potential, and the Meron

The Thomas precession in the nonrelativistic limit of the Dirac equation may be attributed to a non-Abelian Berry vector potential. We ask what object produces the non-Abelian 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 meron, an object in four Euclidean dimensions with instanton number {}, centered at the point where the doubly degenerate positive and negative energy levels of the Dirac equation become fourfold degenerate.