Quaternionic Formulation of Bekenstein-Sanders Guage Fields for TeVeS

The Bekenstein-Sanders tensor-vector-scalar theory of gravitation (TeVeS) has been shown to account for the galactic rotation curves, lensing, and other cosmological phenomena without the significant presence of dark matter.

Treating the Bekenstein-Sanders field Bμ for which Bμ Bμ
= -1 as a gauge field requires that the field be non-Abelian. This structure was worked out in a previous publication by Horwitz, Gershon and Schiffer, where an equivalent Kaluza-Klein metric was found for an extended (5D) spacetime. In this paper, we study a quaternionic formulation of this theory with quaternionic gauge fields and quaternionic wave functions (as discussed in two seminal books by S.L. Adler), thereby establishing a connection between quaternionic quantum mechanics and general relativity. It was shown that TeVeS gravitational theory theory can be derived by a conformal transformation from a Hamiltonian form on a curved space for which the Bekenstein-Sanders vector field Bμ is a non-Abelian gauge field.