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arXiv:2310.17581 · Physical review. D/Physical review. D.

Gravitational spin Hall effect of Dirac particle and the weak equivalence principle

Zhen-Lai Wang

Published Feb 23, 2024• 6 citations
arXivHTMLPDF

Abstract

We present a spin-induced non-geodesic effect of Dirac wave packets in a static uniform gravitational field. Our approach is based on the Foldy-Wouthuysen transformation of Dirac equation in a curved spacetime, which predicts a gravitational spin-orbit coupling. Due to this coupling, we find that the dynamics of the free-fall Dirac wave packets with opposite spin polarization will yield the transverse splitting in the direction perpendicular to spin orientation and gravity, which is known as the gravitational spin Hall effect. Even in a static uniform gravitational field, such effect suggests that the weak equivalence principle is violated for quantum particles.

DOI
10.1103/physrevd.109.044060

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We present a spin-induced non-geodesic effect of Dirac wave packets in a static uniform gravitational field. Our approach is based on the Foldy-Wouthuysen transformation of Dirac equation in a curved spacetime, which predicts a gravitational spin-orbit coupling. Due to this coupling, we find that the dynamics of the free-fall Dirac wave packets with opposite spin polarization will yield the transverse splitting in the direction perpendicular to spin orientation and gravity, which is known as the gravitational spin Hall effect. Even in a static uniform gravitational field, such effect suggests that the weak equivalence principle is violated for quantum particles.
We present a novel fundamental effect that for the matter waves the space-averaging free-fall point of quantum particles undergoes a spin-dependent transverse shift in the gravitational field of Earth. This effect is similar to the geometric spin Hall effect (GSHE) [Aiello et al., Phys. Rev. Lett. 103, 100401 (2009)] and can be called gravity-induced GSHE. This effect suggests that there might be violations of the universality of free fall (UFF) or weak equivalence principle (WEP) in the quantum domain.
We present a novel fundamental effect that for the matter waves the space-averaging free-fall point of quantum particles undergoes a spin-dependent transverse shift in the gravitational field of Earth. This effect is similar to the geometric spin Hall effect (GSHE) [Aiello et al., Phys. Rev. Lett. 103, 100401 (2009)] and can be called gravity-induced GSHE. This effect suggests that there might be violations of the universality of free fall (UFF) or weak equivalence principle (WEP) in the quantum domain.
The weak equivalence principle of gravity is examined at the quantum level in two ways.First, the position detection probabilities of particles described by a non-Gaussian wave packet projected upwards against gravity around the classical turning point and also around the point of initial projection are calculated.These probabilities exhibit mass dependence at both these points, thereby reflecting the quantum violation of the weak equivalence principle.Second, the mean arrival time of freely falling particles is calculated using the quantum probability current, which also turns out to be mass dependent.Such a mass dependence is shown to be enhanced by increasing the non-Gaussianity parameter of the wave packet, thus signifying a stronger violation of the weak equivalence principle through a greater departure from Gaussianity of the initial wave packet.The mass dependence of both the position detection probabilities and the mean arrival time vanishes in the limit of large mass.Thus, compatibility between the weak equivalence principle and quantum mechanics is recovered in the macroscopic limit of the latter.A selection of Bohm trajectories is exhibited to illustrate these features in the free fall case.
Investigation into the applicability of the equivalence principle in quantum mechanics has taken many forms, with varying conclusions. Here, a dynamical semiclassical description of a wave packet in terms of its center of mass and higher quantum fluctuations is applied to the case of a quantum particle in gravitational free fall. The analysis provides an intuitive account of the origin of mass dependence in quantum-gravitational dynamics through an effective potential that enforces the uncertainty principle. This potential has two implications: (i) The lowest order quantum fluctuations encoding the width and spreading of the wave packet obey an uncertainty relation whose observance is mass dependent. (ii) In an inhomogeneous gravitational field tidal effects couple the center of mass motion to the quantum fluctuations. The combined effect results in a clear demonstration of how some conceptions of the weak equivalence principle, based on mass dependence, are violated. The size of this violation is within sensitivities of current E\"otv\"os and clock-based return time experiments.
Abstract We propose an approach that allows to systematically take into account gravity in quantum particle physics. It is based on quantum field theory and the general principle of relativity. These are used to build a model for quantum particles in curved spacetime. We compute by its means a deviation from a classical geodesic in the Earth’s gravitational field. This shows that free fall depends on quantum- matter properties. Specifically, we find that the free-fall universality and the wave-packet spreading are mutually exclusive phenomena. We then estimate the Eötvös parameter for a pair of atoms freely falling near the Earth’s surface, provided that the wave-packet spreading is more fundamental than the weak equivalence principle.