Four-vector vs. four-scalar representation of the Dirac wave function

Mayeul Arminjon (1) , Frank Reifler (2)

1 3SR - Laboratoire sols, solides, structures - risques [Grenoble]
2 MS2 137-205 - Division Maritime Systems and Sensors

In a Minkowski spacetime, one may transform the Dirac wave function under the spin group, as one transforms coordinates under the Poincaré group. This is not an option in a curved spacetime. Therefore, in the equation proposed independently by Fock and Weyl, the four complex components of the Dirac wave function transform as scalars under a general coordinate transformation. Recent work has shown that a covariant complex four-vector representation is also possible. Using notions of vector bundle theory, we describe these two representations in a unified framework. We prove theorems that relate together the different representations and the different choices of connections within each representation. As a result, either of the two representations can account for a variety of inequivalent, linear, covariant Dirac equations in a curved spacetime that reduce to the original Dirac equation in a Minkowski spacetime. In particular, we show that the standard Dirac equation in a curved spacetime, with any choice of the tetrad field, is equivalent to a particular realization of the covariant Dirac equation for a complex four-vector wave function.

Mots clés: Dirac equation; curved spacetime; vector bundle; spinor bundle; hyperbolic partial differential equations

Domaines: Relativité Générale et Cosmologie Quantique [gr-qc]

Mayeul Arminjon, Frank Reifler. Four-vector vs. four-scalar representation of the Dirac wave function. International Journal of Geometric Methods in Modern Physics, 2012, 9 (4), pp.1250026. ⟨10.1142/S0219887812500260⟩. ⟨hal-00683400⟩