Pauli-type coupling between spinors and curved spacetime

Abstract

In this study we prove that the Pauli interaction -- which is associated with a length parameter -- emerges when the minimal coupling recipe is applied to the non-degenerate version of the Dirac Lagrangian. The conventional Dirac Lagrangian is rendered non-degenerate if supplemented by a particular term quadratic in the derivatives of the spinors. For dimensional reasons, this non-degenerate Dirac Lagrangian is associated with a length parameter $\ell$. It yields the standard free Dirac equation in Minkowski space. However, if the Dirac spinor is minimally coupled to gauge fields, then the length parameter~$\ell$ becomes a physical coupling constant yielding novel interactions. For the U($1$) symmetry the Pauli coupling of fermions to electromagnetic fields arises, modifying the fermion's magnetic moment. In a second step we then investigate how analogous "Pauli-type" couplings of gravity and matter arise if fermions are embedded in curved spacetime. Minimal coupling of the Dirac field to the gauge field of gravity, the spin connection, leads to an anomalous spin-torsion interaction and a curvature-dependent mass correction. The relation of the latter to Mach's Principle is discussed. Moreover, it is found for a totally anti-symmetric torsion that an upper limit for the "strength" of the torsion exists in order for a solution to remain causal, while causality for a vector torsion requires a lower limit for its amplitude. We calculate the mass correction in the De~Sitter geometry of vacuum with the cosmological constant $\Lambda$. Possible implications for the existence of effective non-zero rest masses of neutrinos are addressed. Finally, an outlook on the impact of mass correction on the physics of "Big Bang" cosmology, black holes, and of neutron stars is provided.