  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.
