Dirac equation from the Hamiltonian and the case with a gravitational field

Abstract

Starting from an interpretation of the classical-quantum correspondence, we derive the Dirac equation by factorizing the algebraic relation satisfied by the classical Hamiltonian, before applying the correspondence. This derivation applies in the same form to a free particle, to one in an electromagnetic field, and to one subjected to geodesic motion in a static metric, and leads to the same, usual form of the Dirac equation--in special coordinates. To use the equation in the static-gravitational case, we need to rewrite it in more general coordinates. This can be done only if the usual, spinor transformation of the wave function is replaced by the 4-vector transformation. We show that the latter also makes the flat-space-time Dirac equation Lorentz-covariant, although the Dirac matrices are not invariant. Because the equation itself is left unchanged in the flat case, the 4-vector transformation does not alter the main physical consequences of that equation in that case. However, the equation derived in the static-gravitational case is not equivalent to the standard (Fock-Weyl) gravitational extension of the Dirac equation.

27 pages, standard LaTeX. v2: minor style changes, accepted for publication in Found. Phys. Letters

References in corpus (3)

• Measurement of quantum states of neutrons in the Earth's gravitational field
• On the interaction of mesoscopic quantum systems with gravity
• A scale-relativistic derivation of the Dirac Equation

Cited by in corpus (6)

• Equivalent forms of Dirac equations in curved spacetimes and generalized de Broglie relations
• Dirac-type equations in a gravitational field, with vector wave function
• A non-uniqueness problem of the Dirac theory in a curved spacetime
• Post-Newtonian equation for the energy levels of a Dirac particle in a static metric
• Four-vector vs. four-scalar representation of the Dirac wave function
• Representations of the Dirac wave function in a curved spacetime