General-relativistic wave$-$particle duality with torsion

We propose that the four-velocity of a Dirac particle is related to its relativistic wave function by $u^i=\bar{\psi}\gamma^i\psi/\bar{\psi}\psi$. This relativistic wave$-$particle duality relation is demonstrated for a free particle related to a plane wave in a flat spacetime. For a curved spacetime with torsion, the momentum four-vector of a spinor is related to a generator of translation, given by a covariant derivative. The spin angular momentum four-tensor of a spinor is related to a generator of rotation in the Lorentz group. We use the covariant conservation laws for the spin and energy$-$momentum tensors for a spinor field in the presence of the Einstein$-$Cartan torsion to show that if the wave satisfies the curved Dirac equation, then the four-velocity, four-momentum, and spin satisfy the classical Mathisson$-$Papapetrou equations of motion. We show that these equations reduce to the geodesic equation. Consequently, the motion of a particle guided by the four-velocity in the pilot-wave quantum mechanics coincides with the geodesic motion determined by spacetime. We also show how the duality and the operator form of the Mathisson$-$Papapetrou equations arise from the covariant Heisenberg equation of motion in the presence of torsion.