Dirac constraint quantization of a parametrized field theory by anomaly-free operator representations of spacetime diffeomorphisms
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(..sigma..,M) and can thus be written as a functional curl of an anomaly potential on Emb(..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 Schroedinger embedding momenta. By smearing the Heisenberg and the Schroedinger 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 Schroedinger to the Heisenberg picture.
- Research Organization:
- Department of Physics, University of Utah, Salt Lake City, Utah 84112
- OSTI ID:
- 6447670
- Journal Information:
- Phys. Rev. D; (United States), Journal Name: Phys. Rev. D; (United States) Vol. 39:8; ISSN PRVDA
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
657002 -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
BANACH SPACE
DIFFERENTIAL EQUATIONS
DIRAC EQUATION
ENERGY-MOMENTUM TENSOR
EQUATIONS
HAMILTONIANS
HEISENBERG PICTURE
HILBERT SPACE
LIE GROUPS
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MINKOWSKI SPACE
PARTIAL DIFFERENTIAL EQUATIONS
PHASE SPACE
QUANTIZATION
QUANTUM OPERATORS
SCALAR FIELDS
SCHROEDINGER PICTURE
SPACE
SPACE-TIME
SYMMETRY GROUPS
TENSORS
WAVE EQUATIONS