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)

TL;DR

This paper analyzes the Hamiltonian formulation of N-bein, Einstein-Cartan gravity in any dimension, revealing a universal Poincare gauge structure and exploring the peculiarities of three-dimensional cases.

Contribution

It proposes a general conjecture that N-bein, Einstein-Cartan gravity always exhibits Poincare gauge symmetry after constraint elimination across all dimensions.

Findings

The algebra of constraints forms the Poincare algebra in all dimensions.

Translational invariance is present in all dimensions, with some terms vanishing in 3D.

The conjecture is supported by detailed calculations, pending formal proof.

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.