Abstract
The definition of the Hamiltonian operator H for a general wave equation 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 asan 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 “representation” (e.g. transforming the Dirac wave function from the “Dirac representation” to a “Foldy-Wouthuy senre presentation”). We also assert that the energy operator ought to be well defined in a given reference frame at a given time, e.g. by comparing the situation for this operator with the main features of the energy for a classical Hamiltonian particle.
Highlights
The quantum effects in the classical gravitational field which have been observed on Earth for neutrons and for atoms (e.g. [1, 2, 3, 4, 5]) are quantummechanical effects
This extends the previous results [9, 10] from the covariant Dirac equation to any quantum wave equation admitting a gauge choice
The same is true for the quantum-mechanical energy as it arises from the energy operator E, as long as there is no gauge choice
Summary
The quantum effects in the classical gravitational field which have been observed on Earth for neutrons and for atoms (e.g. [1, 2, 3, 4, 5]) are quantummechanical effects. The two realizations of the covariant Dirac equation that are got with two different tetrad fields are equivalent, at least locally [8]. We show that, when the gauge transformation depends on the time coordinate, H as an operator depends on the gauge choice This extends the previous results [9, 10] from the covariant Dirac equation to any quantum wave equation admitting a gauge choice. We distinguish between this ambiguity issue and the one that occurs due to a mere change of the “representation” — e.g. from the “Dirac representation” to a “Foldy-Wouthuysen representation”. The same is true for the quantum-mechanical energy as it arises from the energy operator E, as long as there is no gauge choice
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