A Hamiltonian Formulation of Nonsymmetric Gravitational Theories

The dynamics of a class of nonsymmetric gravitational theories is presented in Hamiltonian form. The derivation begins with the first-order action, treating the generalized connection coefficients as the canonical coordinates and the densitised components of the inverse of the fundamental tensor as conjugate momenta. The phase space of the symmetric sector is enlarged compared to the conventional treatments of General Relativity (GR) by a canonical pair that represents the metric density and its conjugate, removable by imposing strongly an associated pair of second class constraints and introducing Dirac brackets. The lapse and shift functions remain undetermined Lagrange multipliers that enforce the diffeomorphism constraints in the standard form of the NGT Hamiltonian. Thus the dimension of the physical constraint surface in the symmetric sector is not enlarged over that of GR. In the antisymmetric sector, all six components of the fundamental tensor contribute conjugate pairs for the massive theory, and the absence of additional constraints gives six configuration space degrees of freedom per spacetime point in the antisymmetric sector. For the original NGT action (or, equivalently, Einstein’s Unified Field Theory), the U(1) invariance of the action is shown to remove one of these antisymmetric sector conjugate pairs through an additional first class constraint, leaving five degrees of freedom. The restriction of the dynamics to GR configurations is considered, as well as the form of the surface terms that arise from the variation of the Hamiltonian. In the resulting Hamiltonian system for the massive theory, singular behavior is found in the relations that determine some of the Lagrange multipliers near GR and certain NGT spacetimes. What this implies about the dynamics of the theory is not clearly understood at this time.