The DHE on a RCST and the Meaning of Active Local Lorentz Invariance

Abstract

In this chapter we give a formulation of the Dirac-Hestenes equation on a Riemann-Cartan manifold $(M,\boldsymbol{g},\nabla,\tau _{\boldsymbol{g}},\uparrow )$ using the Clifford and spin-Clifford bundles formalism. We show that the obtained equation which follows for a properly chosen Lagrangian density (heuristically based on the principle of minimum coupling) agrees with the one proposed by some authors using the standard concept of covariant spinor fields. However, we do more: we show that postulating invariance under active rotational gauge transformations of the Dirac-Hestenes Lagrangian implies in the equivalence (in a precise sense) of torsion free and non torsion free connections. Such a result suggests that the choice of a particular connection in order to formulate spacetime field theories (which includes the gravitational field) is somewhat arbitrary. This issue is deeply investigated in Chap. 11