General-relativistic wave$-$particle duality with torsion
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arXiv
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| Format: | Preprint |
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2022
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| author | Guedes, Francisco Ribeiro Benard Popławski, Nikodem Janusz |
| author_facet | Guedes, Francisco Ribeiro Benard Popławski, Nikodem Janusz |
| contents | We propose that the four-velocity of a Dirac particle is related to its relativistic wave function by $u^i=\barψγ^iψ/\barψψ$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_03234 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | General-relativistic wave$-$particle duality with torsion Guedes, Francisco Ribeiro Benard Popławski, Nikodem Janusz General Relativity and Quantum Cosmology High Energy Physics - Theory Quantum Physics We propose that the four-velocity of a Dirac particle is related to its relativistic wave function by $u^i=\barψγ^iψ/\barψψ$. 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. |
| title | General-relativistic wave$-$particle duality with torsion |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory Quantum Physics |
| url | https://arxiv.org/abs/2211.03234 |