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

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.

Text of a talk given at the DICE2014 Workshop (Castiglioncello (Livorno), Italy). Submitted to the Proceedings (H.T. Elze et al., eds.). in Seventh International Workshop DICE2014: Spacetime - Matter - Quantum Mechanics, Sep 2014, Castiglioncello (Provincia di Livorno), Italy

References in corpus (5)

• Uniqueness and Self-Conjugacy of Dirac Hamiltonians in arbitrary Gravitational Fields
• Energy expectation values of a particle in nonstationary fields
• General reference frames and their associated space manifolds
• Some remarks on quantum mechanics in a curved spacetime, especially for a Dirac particle
• Absence of the non-uniqueness problem of the Dirac theory in a curved spacetime. Spin-rotation coupling is not physically relevant

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