Generalized gauge theories with nonunitary parallel transport: * General relativity with cosmological constant as an example

Abstract.

In gauge theories parallel transporters (PTs) $\mathcal{U}(C)$ along paths C play an important role. Traditionally they are unitary or pseudoorthogonal maps between vector spaces. We propose to abandon unitarity of parallel transporters and with it the a priori assumption of metricity in general relativity. A *-operation on parallel transporters serves as a substitute for it, and this *-operation is proven to be unique on group theoretical grounds. The vierbein and the spin connection appear as distinguishable parts of a single de Sitter gauge field with field strength F. The action takes the form $\frac{3}{16\pi G\Lambda}\int\text{tr}(\boldsymbol{F}\wedge\boldsymbol{F}i\boldsymbol{\gamma}_{5})$ and both the Einstein field equations with arbitrarily small but nonvanishing cosmological constant Λ and the condition of vanishing torsion are obtained from it. The equation of motion for classical massive bodies turns out to be de Sitter covariant.