Generalized Gauge Theories with Nonunitary Parallel Transport: General Relativity with Cosmological Constant as an Example

Abstract

In gauge theories parallel transporters (PTs) 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 $\frac{3}{16πGΛ}\int tr(F \wedge F i γ_5)$ 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.