Constraint algebra in loop quantum gravity reloaded. II. Toy model of an Abelian gauge theory: Spatial diffeomorphisms

In A. Henderson, A. Laddha, and C. Tomlin, preceding article, [Phys. Rev. D 88, 044028 (2013)], we initiated an approach towards quantizing the Hamiltonian constraint in loop quantum gravity by requiring that it generates an anomaly-free representation of constraint algebra of -shell. We investigated this issue in the case of a toy model of a ($2+1$)-dimensional $\mathrm{U}(1{)}^3$ gauge theory, which can be thought of as a weak coupling limit of Euclidean three-dimensional gravity. However, in paper I, we only focused on the most nontrivial part of the constraint algebra that involves the commutator of two Hamiltonian constraints. In this paper we continue with our analysis and obtain a representation of full constraint algebra in a loop quantized framework. We show that there is a representation of the diffeomorphism group with respect to which the Hamiltonian constraint quantized in paper I is diffeomorphism covariant. Our work can be thought of as a potential first step towards resolving some longstanding issues with the Hamiltonian constraint in canonical loop quantum gravity.

Figure 1

The $\delta$-ciliated graph ${\gamma }_{\sigma (\delta ,\tilde{c})}(\tilde{c})$ determined by $\tilde{c}$ in the neighborhood of the vertex $v_I$ with respect to the coordinate system $\{x_{v_I}\}=\{x^1,x^2\}$.