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

Abstract

We present the second (and final) part of an analysis aimed at introducing variables which are suitable for constructing a space of quantum states for the Teleparallel Equivalent of General Relativity. In the first part of the analysis we introduced a family of variables on the "position" sector of the phase space. In this paper we distinguish differentiable variables in the family. Then we define momenta conjugate to the differentiable variables and express constraints of the theory in terms of the variables and the momenta. Finally, we exclude variables which generate an obstacle for further steps of the Dirac's procedure of canonical quantization of constrained systems we are going to apply to the theory. As a result we obtain two collections of variables on the phase space which will be used (in a subsequent paper) to construct the desired space of quantum states.Keywords Teleparallel equivalent of general relativity · Canonical quantization · Space of kinematic quantum states

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

“…Fortunately, as it will be proven in [13], there exist exactly two closely related elements of the family {(ξ I , θ J )} for which the problem just described does not appear-the elements are closely related in this sense that functions {ι} distinguishing them differ from each other by a factor −1. Using one of these two elements we will construct in [12] a space D of kinematic quantum states for TEGR.…”

“…To carry out the desired deformation of the coframe (θ ′I ) we proceed as follows: by virtue of the compactness of g 0 we can cover Int g 0 by a finite number of open subsets {W β } such that each W β admits existence of an almost characteristic function φ β on it 13 13 To satisfy this requirement W β may be defined as an open coordinate ball of non-zero radius. and γ ∩ β W β = Int g 0 .…”

“…In Section 4 we present the family of new variables. Section 5 contains a short summary and an outline of the analysis to be presented in the accompanying paper [13]. In Appendix we prove two very important lemmas which guarantee that both kinds of variables considered in this paper provide d.o.f.…”

On Teleparallel Quantum Gravity in Schwarzschild Space-Time

“…Teleparallel gravity has been developed and tested over the years in what concerns its classical features [17][18][19][20][21][22][23] and in our opinion it seems to be a plausible theory of gravitation. However there are few attempts to quantize this theory, for instance we refer the set of papers [24][25][26][27] which were developed as an application of Dirac's method to TEGR. Hence we intent to give our contribution in this process by analyzing the quantum version of Schwarzschild's solution of field equations that arises from the identification H = e t (0)0 in the Weyl's prescription.…”

ADM-like Hamiltonian formulation of gravity in the teleparallel geometry: derivation of constraint algebra

“…In our previous paper [1] we presented a Hamiltonian formulation of the Teleparallel Equivalent of General Relativity (TEGR) regarded as a theory of cotetrad fields on a spacetime-the formulation is meant to serve as a point of departure for canonical quantization à la Dirac of the theory (preliminary stages of the quantization are described in [2,3,4]). In [1] we found a phase space, a set of (primary and secondary) constraints on the phase space and a Hamiltonian.…”