Semiclassical analysis of Dirac fields on curved spacetime

Abstract

We present a semiclassical analysis for Dirac fields on an arbitrary spacetime background and in the presence of a fixed electromagnetic field. Our approach is based on a Wentzel-Kramers-Brillouin approximation, and the results are analyzed at leading and next-to-leading order in the small expansion parameter ℏ. Taking into account the spin-orbit coupling between the internal and external degrees of freedom of wave packets, we derive effective ray equations with spin-dependent terms. These equations describe the gravitational spin Hall effect of localized Dirac wave packets. We treat both massive and massless Dirac fields and show how a covariantly defined Berry connection and the associated Berry curvature govern the semiclassical dynamics. The gravitational spin Hall equations are shown to be particular cases of the Mathisson-Papapetrou equations for spinning objects.

Spin Hall effects in the sky

“…A similar spin-orbit coupling mechanism is expected to influence the propagation of wave packets in gravitational fields [24,25]. This leads to gravitational spin Hall effects, which have been predicted for electromagnetic [26][27][28][29], linearized gravitational [30,31] and Dirac [32][33][34][35] fields propagating on curved spacetime backgrounds. In this general relativistic context, wave packets carrying intrinsic angular momentum follow frequency-and polarization-dependent trajectories when propagating through inhomogeneous gravitational fields.…”

“…A similar transport equation defined in terms of a Berry connection was also obtained in the case of Dirac fields [32]. At this stage, spin-orbit interactions are not taken into account.…”

“…Spin-dependent effects on the propagation of wave packets and particles have been widely studied in general relativity using different methods, such as WKBtype approximations [25-28, 30, 32, 34, 35, 49-55], the Mathisson-Papapetrou equations for spinning objects [27,[56][57][58][59][60], approximation methods inspired from quantum mechanics [29,31,33] and others [61,62] (see also [63][64][65][66][67][68][69]). Furthermore, gravitational spin Hall effects are predicted for different fields propagating in curved spacetime, such as electromagnetic [26][27][28][29], linearized gravitational [30,31] and massive and massless Dirac fields [32,34,35] (spin Hall effects have also been predicted for more exotic particles such as massless particles with anyonic spin [70][71][72][73]).…”

“…In this section, we present the main steps of the derivation of the gravitational spin Hall effect. A general approach for the description of this effect has been given for electromagnetic waves in [26], and later the same method has been adapted for linearized gravitational waves in [30] and for Dirac fields in [32]. Here, we mainly focus on the derivation of the gravitational spin Hall effect of light, as given in [26,27].…”

“…Here, we mainly focus on the derivation of the gravitational spin Hall effect of light, as given in [26,27]. The derivation of the effect in the case of linearized gravitational waves [30] or Dirac fields [32] can be performed following similar steps.…”