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

Abstract

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 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 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 "representation" (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 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. IntroductionThe 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 quantum-mechanical effects. That is, they were predicted (before their observation) by using "first-quantized" theory. They are, to this author's knowledge, the only observed effects of the gravity-quantum coupling. Thus, QM in a curved spacetime covers all currently available experiments about the interaction between gravity and the quantum. Neutrons are spin half particles, and spin half particles are normally described by the Dirac equation. Moreover, besides observing directly the quantum behaviour of particles in a gravitational field, one also would like to account for the influence of the gravitational field on, for example, the quantum behaviour of particles in an electromagnetic field, e.g. the behaviour of electrons in an atom. (This influence is very small in the very weak gravitational field that we have in the solar system, of course. Yet it is expected to be important in strong gravitational fields.) Therefore, QM of the (generally-)covariant Dirac equation is quite an important chapter of curved-spacetime QM. The covariant Dirac equation involves the choice, at any point X in spacetime (and depending smoothly on X), of an orthonormal basis of the tangent space at that point X. I.e., it involves the choice of a tetrad field [6,7]. This plays the role of a gauge field. Indeed, the two realizations of the covariant Dirac equation that are got with two different tetrad fields are equivalent, at least locally [8]. However, it has been proved in recent years that the Hamiltonian operator associated, in a given coordinate system, with the covariant Dirac equation, does depend on the choice of the tetrad field. In particular, the energy spectrum and the energy mean values depend on that choice [9,10], and so does the presence or absence of Mashhoon's spin-rotation coupling term [11,12].

Local Lorentz transformations and Thomas effect in general relativity

“…In accordance with Refs. [8,9], these fields do not coincide even for different tetrads belonging to the Schwinger gauge (see the example given in Sec. IV B).…”

“…A tetrad field in the Schwinger gauge is not unique and different tetrads may lead to different equations of motion. The corresponding Hamiltonians do not coincide either [9,48]. However, the forces and torques caused by a difference of the tetrads are fictitious and are not felt by an observer.…”

“…The corresponding Hamiltonians also differ [8,9]. However, appropriate coordinate transformations establish connections between them.…”

“…All tetrads are equivalent and the use of any tetrad is possible. Nevertheless, Hamiltonians and equations of motion of a test particle do not coincide even for different tetrads belonging to the Schwinger gauge [8,9] (see Subsec.…”

Chaotic dynamics of piezoelectric mems based on maximal Lyapunov exponent and Smaller Alignment Index computations

“…In many cases the dynamical behavior of physical systems can be modeled by Hamiltonian systems. Over the years the Hamiltonian formulation has been successfully applied in numerous areas of physics such as statistical mechanics [Brody et al, 2008], classical physics [Rohrlich, 1979], quantum mechanics [Arminjon, 2015] and many other fields . In general, Hamiltonian systems can be divided in two broad categories: conservative and non-conservative systems.…”