Classical characters of spinor fields in torsion gravity

We consider the problem of having relativistic quantum mechanics re-formulated with hydrodynamic variables, and specifically the problem of deriving the Mathisson–Papapetrou–Dixon equations (describing the motion of a massive spinning body moving in a gravitational field) from the Dirac equation. The problem will be answered on a general manifold with torsion and gravity. We will demonstrate that when plane waves are considered the MPD equations describe the general relativistic wave-particle duality with torsion (Guedes and Popławski 2024 Class. Quantum Grav.41 065011), but we will also see that in such a form the MPD equations become trivial.