On the Hamiltonian and energy operators in a curved spacetime, especially for a Dirac particle

TL;DR

This paper examines the definition and gauge dependence of Hamiltonian and energy operators for wave equations, especially Dirac particles, in curved spacetime, highlighting ambiguities and the importance of reference frames.

Contribution

It clarifies the gauge and representation dependence of Hamiltonian and energy operators in curved spacetime, emphasizing their proper definition in specific reference frames.

Findings

Hamiltonian depends on reference frames and gauge choices.

Energy operator's Hermitian part is gauge-dependent.

Proper energy operator should be well-defined in a fixed reference frame.

Abstract

The definition of the Hamiltonian operator H for a general wave equa-tion in a general spacetime is discussed. We recall that H depends on the coordinate system merely through the corresponding reference frame. When the wave equation involves a gauge choice and the gauge change is time-dependent, H as an operator depends on the gauge choice. This dependence extends to the energy operator E, which is the Hermitian part of H. We distinguish between this ambiguity issue of E and the one that occurs due to a mere change of the "represen-tation" (e.g. transforming the Dirac wave function from the "Dirac representation" to a "Foldy-Wouthuysen representation"). We also assert that the energy operator ought to be well defined in a given ref-erence frame at a given time, e.g. by comparing the situation for this operator with the main features of the energy for a classical Hamilto-nian particle.