Dirac constraint quantization of a parametrized field theory by anomaly-free operator representations of spacetime diffeomorphisms.

Abstract

We construct a consistent Dirac constraint quantization of a parametrized massless scalar field propagating on a two-dimensional cylindrical Minkowskian background. The constraints are taken in the form of ``diffeomorphism Hamiltonians'' whose Poisson-brackets algebra is homomorphic to the Lie algebra of spacetime diffeomorphisms. The fundamental canonical variables are represented by operators acting on an embedding-dependent Fock space H which is based on the Heisenberg modes that are geometrically specified with respect to the Killing vector structure of the background. In the Heisenberg picture, the constraints become the Heisenberg embedding momenta and their Abelian Poisson algebra is homomorphically mapped into the operator commutator algebra without any anomaly. The algebra of normal-ordered Heisenberg evolution generators (which propagate the field operators) develops a covariantly defined anomaly. This anomaly is an exact two-form on the space of embeddings Emb(\ensuremath{\Sigma},M) and can thus be written as a functional curl of an anomaly potential on Emb(\ensuremath{\Sigma},M). By subtracting this potential from the normal-ordered Heisenberg generators (which amounts to their embedding-dependent reordering) we arrive at a commuting set of operators which we identify with the Schr\"odinger embedding momenta. By smearing the Heisenberg and the Schr\"odinger embedding momenta by spacetime vector fields we obtain a pair of anomaly-free operator representations of L DiffM. The diffeomorphism Hamiltonians annihilate the physical states and the smeared reordered Heisenberg evolution generators propagate the fields. We present the operator transformation from the Schr\"odinger to the Heisenberg picture. The two operator representations of L DiffM, by diffeomorphism Hamiltonians and by smeared Heisenberg evolution generators, guarantee that the Dirac constraint quantization is consistent, covariant, and leads to foliation-independent dynamics both in the Heisenberg and in the Schr\"odinger pictures.The appropriate factor ordering of the Hamiltonian flux operator and of the constraints is rewritten in terms of the fundamental Schr\"odinger variables with help of a normal-ordering kernel which is reconstructed from the intrinsic metric and the extrinsic curvature on a given embedding. All operators are defined and dynamics takes place on a single function space which is then restricted by the constraints to the space of physical states with a Hilbert-space structure.