General-relativistic wave–particle duality with torsion

Abstract

Abstract We propose that the four-velocity of a Dirac particle is related to its relativistic wave function by u i = ψ ˉ γ i ψ / ψ ˉ ψ . 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.