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General Relativity and Quantum Cosmology, gr-qc,High Energy Physics - Theory, hep-th
Abstract:
In [1] we initiated an approach towards quantizing the Hamiltonian constraint
in Loop Quantum Gravity (LQG) by requiring that it generates an anomaly-free
representation of constraint algebra off-shell. We investigated this issue in
the case of a toy model of a 2+1-dimensional $U(1)^{3}$ gauge theory, which can
be thought of as a weak coupling limit of Euclidean three dimensional gravity.
However in [1] we only focused on the most non-trivial part of the constraint
algebra that involves commutator of two Hamiltonian constraints. In this paper
we continue with our analysis and obtain a representation of full constraint
algebra in loop quantized framework. We show that there is a representation of
the Diffeomorphism group with respect to which the Hamiltonian constraint
quantized in [1] is diffeomorphism covariant. Our work can be thought of as a
potential first step towards resolving some long standing issues with the
Hamiltonian constraint in canonical LQG.
