ADM-like Hamiltonian formulation of gravity in the teleparallel geometry

Abstract

We present a new Hamiltonian formulation of the teleparallel equivalent of general relativity (TEGR) meant to serve as the departure point for canonical quantization of the theory. TEGR is considered here as a theory of a cotetrad field on a spacetime. The Hamiltonian formulation is derived by means of an ADM-like 3 + 1 decomposition of the field and without any gauge fixing. A complete set of constraints on the phase space and their algebra are presented. The formulation is described in terms of differential forms.

Hamiltonian and primary constraints of new general relativity

“…where we introduced the components ξ A of the normal vector n to the x 0 = const hypersurfaces in the dual tetrad basis [10]…”

“…This can be done best in terms of a full-fledged Hamiltonian analysis in terms of the Dirac-Bergmann algorithm for constrained Hamiltonian systems. It is known that the teleparallel equivalent of general relativity (TEGR), which yields the same dynamics and solutions for the metric defined by the tetrads as general relativity and contains no additional degrees of freedom, is self-consistent and ghost-free [10][11][12][13][14][15][16]. The hope is that this is not the only contender of the class of healthy teleparallel theories of gravity in this sense.…”

“…The main difference between the previous studies and the approach we present in this article lies in the method which is employed in order to implement the vanishing curvature of the teleparallel connection. Previous studies can mainly be divided into two groups, either assuming a vanishing spin connection (which is known as the Weitzenböck gauge) [10,[15][16][17]25], or an arbitrary spin connection, whose curvature is then enforced to vanish by using Lagrange multipliers in the action functional [12,13]. Here we use a different ansatz, by allowing for a non-vanishing spin connection, as mandated by the covariant formulation of teleparallel gravity [1,2], which is obtained explicitly by applying a local Lorentz transformation to the vanishing Weitzenböck gauge spin connection.…”

Variables suitable for constructing quantum states for the teleparallel equivalent of general relativity II

“…Throughout the paper this manifold will represent a space-like slice of a spacetime. The phase space of TEGR is a Cartesian product P × Θ, where [3] 1. Θ consists of all quadruplets of one-forms (θ A ) on Σ such that the metric…”

“…To explain what we mean by "useful" let us recall that TEGR is a constrained system (see e.g. [4,5,6,3]) and since the constraints on the phase space of TEGR are too complicated to be solved classically we are going to apply the Dirac's approach to quantize TEGR. According to the Dirac's approach one first constructs a space of quantum states corresponding to the unconstrained phase space, that is, a space of kinematic quantum states.…”