On the Hamiltonian nature of semiclassical equations of motion in the presence of an electromagnetic field and Berry curvature

TL;DR

This paper demonstrates that semiclassical equations of motion with electromagnetic fields and Berry curvature are Hamiltonian, clarifying the relation between canonical and covariant variables and explaining phase space volume nonconservation.

Contribution

It establishes the Hamiltonian nature of these equations and derives the relations between different variable descriptions, addressing previous ambiguities.

Findings

Equations are shown to be Hamiltonian.

Relations between canonical and covariant variables are explicitly derived.

The Jacobian explains phase space volume nonconservation.

Abstract

We consider the semiclassical equations of motion of a particle when both an external electromagnetic field and the Berry gauge field in the momentum space are present. It is shown that these equations are Hamiltonian and relations between the canonical and covariant variables are determined through a consistent account of all components of the Berry connection. The Jacobian of the canonical-to-covariant-variables transformation describes the nonconservation of the 'naive' phase space volume in the covariant coordinates (D.Xiao, J.Shi, and Q.Niu, Phys. Rev. Lett. 95, 137204 (2005)).