General-relativistic wave$-$particle duality with torsion

Abstract

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.

arXiv comment14 pages, 1 figure; published version

References in corpus (14)

• Cosmology with torsion: An alternative to cosmic inflation
• Nonsingular, big-bounce cosmology from spinor-torsion coupling
• Can Bohmian mechanics be made relativistic?
• Semiclassical limit for Dirac particles interacting with a gravitational field
• Radial motion into an Einstein-Rosen bridge
• Universe in a black hole in Einstein$-$Cartan gravity
• Big bounce from spin and torsion
• Nonsingular Dirac particles in spacetime with torsion
• Big bounce and closed universe from spin and torsion
• Classical Physics: Spacetime and Fields
• Analysis of big bounce in Einstein--Cartan cosmology
• Gravitational collapse of a fluid with torsion into a universe in a black hole
• Relativistic Bohmian trajectories and Klein-Gordon currents for spin-0 particles
• Noncommutative momentum and torsional regularization

Cited by in corpus (3)

• Black holes in the expanding Universe
• Classical characters of spinor fields in torsion gravity
• Madelung Structure of the Dirac Equation