Dirac-type equations in a gravitational field, with vector wave function

TL;DR

This paper explores how different choices of coordinate systems and connections in a gravitational field lead to distinct Dirac-type equations, affecting their covariance and adherence to the equivalence principle.

Contribution

It introduces new Dirac-type equations derived from classical Hamiltonian mechanics, highlighting their differences from the standard Fock-Weyl equation in curved spacetime.

Findings

Two distinct Dirac equations depend on the chosen connection.

One equation obeys the equivalence principle in an extended sense.

The equations transform the wave function as a four-vector.

Abstract

An analysis of the classical-quantum correspondence shows that it needs to identify a preferred class of coordinate systems, which defines a torsionless connection. One such class is that of the locally-geodesic systems, corresponding to the Levi-Civita connection. Another class, thus another connection, emerges if a preferred reference frame is available. From the classical Hamiltonian that rules geodesic motion, the correspondence yields two distinct Klein-Gordon equations and two distinct Dirac-type equations in a general metric, depending on the connection used. Each of these two equations is generally-covariant, transforms the wave function as a four-vector, and differs from the Fock-Weyl gravitational Dirac equation (DFW equation). One obeys the equivalence principle in an often-accepted sense, whereas the DFW equation obeys that principle only in an extended sense.