On the Hamiltonian formalism of the tetrad-connection gravity

Abstract

We present a detailed analysis of the Hamiltonian constraints of the d-dimensional tetrad-connection gravity where the non-dynamic part of the spatial connection is fixed to zero by an adequate gauge transformation. This new action leads to a coherent Hamiltonian formalism where the Lorentz, scalar and vectorial first-class constraints obeying a closed algebra in terms of Poisson brackets. This algebra closes with structure constants instead of structure functions resulting from the Hamiltonian formalisms based on the A.D.M. decomposition. The same algebra of the reduced first-class constraints, where the second-class constraints are eliminated as strong equalities, is obtained in terms of Dirac brackets. These first-class constraints lead to the same physical degrees of freedom of the general relativity.

On the Hamiltonian formalism of the tetrad-gravity with fermions

“…Note that as opposed in the pure gravity where the Poisson bracket of scalar constraint with itself vanishes strongly [1], in presence of the fermionic matter, this Poisson bracket vanishes only weakly. This shows that the set of constraints is complete and closed meaning that the total Hamiltonian (28) is consistent.…”

“…In this chapter we will extent the analysis of the Hamiltonian formalism of the d−dimensional tetrad-gravity to the fermionic field by fixing the non-dynamic part of the spatial connection to zero. The fixing of the non-dynamic part of the connection to zero is necessary to avoid the constraints resulting from the evolution equations of the non-dynamic part of the spatial connection which are difficult to analyze [1].…”

“…The consistency requires that these primary constraints must be preserved under the time evolution given by the total Hamiltonian (11). In order to satisfy the Jacobi identities, the calculation is given in terms of Poisson brackets projected by P P dQ 1KaL when projected elements of the phase space ω 1aKL and P aKL 1 are involved [1]:…”

“…In this paper we extent the Hamiltonian formalism of d−dimensional tetrad-gravity minimally coupled to the fermionic field, where the non-dynamic part of the spatial connection is fixed to zero [1]. This analysis is performed without the A.D.M.…”

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We extend the analysis of the Hamiltonian formalism of the ddimensional tetrad-connection gravity to the fermionic field by fixing the non-dynamic part of the spatial connection to zero [1]. Although the reduced phase space is equipped with complicated Dirac brackets, the first-class constraints which generate the diffeomorphisms and the Lorentz transformations satisfy a closed algebra with structural constants analogous to that of the pure gravity.

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