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

TL;DR

This paper derives the Dirac equation from classical-quantum correspondence, extends it to gravitational fields using 4-vector transformations, and compares it with the standard gravitational Dirac equation, revealing key differences.

Contribution

It presents a novel derivation of the Dirac equation applicable to gravitational fields, replacing spinor transformations with 4-vector transformations for general coordinates.

Findings

Derivation applies to free, electromagnetic, and gravitational cases.

Reformulation uses 4-vector transformation, maintaining Lorentz covariance.

Resulting gravitational Dirac equation differs from the standard Fock-Weyl form.

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.