In the U(1)^3 toy model of loop quantum gravity, the authors rearrange the quantum Hamiltonian constraint into a discrete evolution equation and, via a questionable continuum limit, a Schrödinger-like equation with a geometrical time operator.
Asymptotically flat boundary conditions for the $U(1)^3$ model for Euclidean Quantum Gravity
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abstract
A generally covariant $U(1)^3$ gauge theory describing the $G_N \to 0$ limit of Euclidean general relativity is an interesting test laboratory for general relativity, specially because the algebra of the Hamiltonian and diffeomorphism constraints of this limit is isomorphic to the algebra of the corresponding constraints in general relativity. In the present work, we study boundary conditions and asymptotic symmetries of the $U(1)^3$ model and show that while asymptotic spacetime translations admit well-defined generators, boosts and rotations do not. Comparing with Euclidean general relativity, one finds that exactly the non-Abelian part of the $SU(2)$ Gauss constraint which is absent in the $U(1)^3$ model plays a crucial role in obtaining boost and rotation generators.
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Geometrical Quantum Time in the $U(1)^3$ Model of Euclidean Quantum Gravity
In the U(1)^3 toy model of loop quantum gravity, the authors rearrange the quantum Hamiltonian constraint into a discrete evolution equation and, via a questionable continuum limit, a Schrödinger-like equation with a geometrical time operator.