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.
The effective dynamics of weak coupling loop quantum gravity
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abstract
By taking the limit that Newton's Gravitational constant tends to zero, the weak coupling loop quantum gravity can be formulated as a $U(1)^3$ gauge theory instead of the original $SU(2)$ gauge theory. In this paper, a parametrization of the $SU(2)$ holonomy-flux variables by the $U(1)^3$ holonomy-flux variables is introduced, and the Hamiltonian operator based on this parametrization is obtained for the weak coupling loop quantum gravity. It is shown that the effective dynamics obtained from the coherent state path integrals in $U(1)^3$ and $SU(2)$ loop quantum gravity respectively are consistent to each other in the weak coupling limit, provided that the expectation values of the Hamiltonian operators on the coherent states in these two theories coincide with their classical expressions respectively.
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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.