Loop quantum gravity without the Hamiltonian constraint

Details Loop quantum gravity without the Hamiltonian constraint Loop quantum gravity without the Hamiltonian constraint

2012-03-29

arxiv.org/abs/1203.6525v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

5 pages

Scientific paper

We show that under certain technical assumptions, including a generalisation of CMC foliability and strict positivity of the scalar field, general relativity conformally coupled to a scalar field can be quantised on a partially reduced phase space, meaning reduced only with respect to the Hamiltonian constraint and a proper gauge fixing. More precisely, we introduce, in close analogy to shape dynamics, the generator of a local conformal transformation acting on both, the metric and the scalar field. A new metric, which is invariant under this transformation, is constructed and used to define connection variables which can be quantised by standard loop quantum gravity methods. Since this connection is invariant under the local conformal transformation, the generator of which is shown to be a good gauge fixing for the Hamiltonian constraint, the Dirac bracket associated with implementing these constraints coincides with the Poisson bracket for the connection. Thus, the well developed kinematical quantisation techniques for loop quantum gravity are available, while the Hamiltonian constraint has been solved (more precisely, gauge fixed) classically. The physical interpretation of this system is that of general relativity on a fixed spatial slice, the associated "time" of which is given by the value of the generator of local conformal transformations. While it is hard to address dynamical problems in this framework (due to the complicated "time" function), it seems, due to good accessibility properties of the gauge in certain situations, to be well suited for problems such as the computation of black hole entropy, where actual physical states can be counted and the dynamics is only of indirect importance. Also, the interpretation of the geometric operators gets an interesting twist, which exemplifies the deep relationship between observables and the choice of a time function.

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