Dirac electron in space-times with torsion: Spinor propagation, spin precession, and nongeodesic orbits

The WKB limit of the noniterated Dirac equation in a Riemann-Cartan space-time is discussed. It is shown that within this framework the behavior of a Dirac particle is dominated by the new connection ${{\Gamma }_{\mu \nu }^{*}}^{\lambda }=\left\{\left\{\lambda \right\}\left\{\mu \nu \right\}\right\}-3\mathrm{}{K}_{\left[\mu \nu \epsilon \right]}{g}^{\epsilon \lambda }$ formed from the Christoffel connection and the contortion. The relevant effects are the following: (i) The normalized Dirac spinor is parallel propagated by ${{\Gamma }_{\mu \nu }^{*}}^{\lambda }$ along the particle's orbit. (ii) The same is true for the spin vector. By this the gyrogravitational ratio is specified as well. (iii) The particle orbit is nongeodesic. The respective "force" is of the usual form with the spin coupled to the curvature tensor $R^{*\alpha \beta \gamma \delta }\left({\Gamma }^{*}\right)$ of the connection ${{\Gamma }_{\mu \nu }^{*}}^{\lambda }$. The orbit is thereby defined by the streamlines of the conserved convection four-current obtained from the Dirac current by a Gordon decomposition. The cumulative effects (ii) and (iii) can in principle be used to detect torsion by measuring the spin precession of a massive spin-½ particle or by measuring its orbit in a Stern-Gerlach type of experiment.