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General Relativity and Quantum Cosmology, gr-qc,High Energy Physics - Theory, hep-th
Abstract:
The G -->0 limit of Euclidean gravity introduced by Smolin is described by a
generally covariant U(1)xU(1)xU(1) gauge theory. The Poisson bracket algebra of
its Hamiltonian and diffeomorphism constraints is isomorphic to that of
gravity. Motivated by recent results in Parameterized Field Theory and by the
search for an anomaly-free quantum dynamics for Loop Quantum Gravity (LQG), the
quantum Hamiltonian constraint of density weight 4/3 for this U(1)xU(1)xU(1)
theory is constructed so as to produce a non-trivial LQG-type representation of
its Poisson brackets through the following steps. First, the constraint at
finite triangulation, as well as the commutator between a pair of such
constraints, are constructed as operators on the `charge' network basis. Next,
the continuum limit of the commutator is evaluated with respect to an operator
topology defined by a certain space of `vertex smooth' distributions. Finally,
the operator corresponding to the Poisson bracket between a pair of Hamiltonian
constraints is constructed at finite triangulation in such a way as to generate
a `generalised' diffeomorphism and its continuum limit is shown to agree with
that of the commutator between a pair of finite triangulation Hamiltonian
constraints. Our results in conjunction with the recent work of Henderson,
Laddha and Tomlin in a 2+1-dimensional context, constitute the necessary first
steps toward a satisfactory treatment of the quantum dynamics of this model.
