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Quantum scalar field in quantum gravity: the vacuum in the spherically symmetric case

We study gravity coupled to a scalar field in spherical symmetry using loop quantum gravity techniques. Since this model has local degrees of freedom, one has to face ‘the problem of dynamics’, that is, diffeomorphism and Hamiltonian constraints that do not form a Lie algebra. We tackle the problem using the ‘uniform discretization’ technique. We study the expectation value of the master constraint and argue that among the states that minimize the master constraint is one that incorporates the usual Fock vacuum for the matter content of the theory.