The Hamiltonian formulation of N-bein, Einstein-Cartan, gravity in any dimension: the Progress Report (Extended version of a talk given on CAIMS-2009, June 11-14, London, Canada)

Abstract

The Hamiltonian formulation of N-bein, Einstein-Cartan, gravity, using its first order form in any dimension higher than two, is analyzed. This Hamiltonian formulation allows to explicitly show where peculiarities of three dimensional case (\textit{A.M.Frolov et al, 0902.0856 [gr-qc]}) occur and make a conjecture, based on presented in this report results, that there is one general for \textit{all} dimensions characteristic of N-bein formulation of gravity: after elimination of second class constraints the algebra of Poisson brackets among remaining first class secondary constraints is the Poincare algebra and in all dimensions N-bein, Cartan-Einstein, gravity \textit{is the Poincare gauge theory}. The gauge symmetry corresponding to the algebra of first class constraints has two parameters - rotational (Lorentz) and translational. Translational invariance is common to all dimensions but some terms in general expressions for gauge transformations of N-beins and connections are zero in a particular, three dimensional, case. The proof of our conjecture is outlined in detail. Some straightforward but tedious calculations remain to be completed to call our conjecture - a theorem and will be reported later.