Abstract
A general classical theorem is presented according to which all invariant relations among the space time metric scalars, when turned into functions on the Phase Space of full Pure Gravity (using the Canonical Equations of motion), be-come weakly vanishing functions of the Quadratic and Linear Constraints. The implication of this result is that (formal) Dirac consistency of the Quantum Operator Constraints (annihilating the wave Function) suffices to guarantee space time covariance of the ensuing quantum theory: An ordering for each invariant relation will always exist such that the emanating operator has an eigenvalue identical to the classical value. The example of 2+1 Quantum Cosmology is explicitly considered: The four possible Cosmological Solutions -two for pure Einstein’s equations plus two more when a Λ term is present- are exhibited and the corresponding models are quantized. The invariant relations describing the geometries are explicitly calculated and promoted to operators whose eigenvalues are their corresponding classical values.
Highlights
A general classical theorem is presented according to which all invariant relations among the space time metric scalars, when turned into functions on the Phase Space of full Pure Gravity, be-come weakly vanishing functions of the Quadratic and Linear Constraints
The implication of this result is that Dirac consistency of the Quantum Operator Constraints suffices to guarantee space time covariance of the ensuing quantum theory: An ordering for each invariant relation will always exist such that the emanating operator has an eigenvalue identical to the classical value
The problem of space time covariance of a Quantum Theory of Gravity (Isham, 1995) within the context of Canonical Quantization can essentially be described as follows: Classically, Einstein's Field equations are known to be equivalent to the Hamiltonian and Momentum constraints plus the Canonical Equations of motion. This is understandable since, the canonical analysis uses objects defined on the hypersurface, the momenta involve the extrinsic curvature and carry the information of the geometrical meaning of the classical constraints is maintained at the quantum level, one is justified to expect space time covariance of the ensuing theory
Summary
The fate of space time covariance is somewhat obscure. if one takes special care so that the. The problem of space time covariance of a Quantum Theory of Gravity (Isham, 1995) within the context of Canonical Quantization can essentially be described as follows: Classically, Einstein's Field equations are known to be (not manifestly but explicitly) equivalent to the Hamiltonian and Momentum constraints plus the Canonical Equations of motion. This is understandable since, the canonical analysis uses objects defined on the hypersurface, the momenta involve the extrinsic curvature and carry the information of the geometrical meaning of the classical constraints is maintained at the quantum level, one is justified to expect space time covariance of the ensuing theory. In this sense it is proven that the two pairs of state vectors are equivalent
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