Abstract
An introduction of embedding variables as physical fields locked to the metric by coordinate conditions helps to turn the quantum constraints into a many-fingered time Schrödinger equation. An attempt is made to generalize this process from noncanonical (Gaussian and harmonic) coordinate conditions to a canonical coordinate condition (the constant mean extrinsic curvature slicing). The dynamics of a scalar field is described by a Lagrangian whose field equations imply that the value of at is the mean extrinsic curvature of a hypersurface passing through . By adding to the Hilbert Lagrangian , the "extrinsic time field" is coupled to gravity. Its energy-momentum tensor has the structure of a perfect fluid (the reference fluid) which satisfies weak energy conditions. The canonical analysis of the total action is complicated by the changing rank of the Hessian. When a hypersurface is transverse to the foliation, this rank is higher, and one obtains only the standard super-Hamiltonian and supermomentum constraints. At stationary points of , the hypersurface becomes tangent to a leaf, the rank of the Hessian gets lower, and more constraints arise. At transverse points, the super-Hamiltonian constraint can be solved with respect to the momentum which is canonically conjugate to the time function . In this form, the constraint leads to a functional Schrödinger equation. As one approaches a stationary point of , additional constraints arise. They ultimately invalidate the functional Schrödinger equation. If the stationary points fill a region, some constraints become second class and must be eliminated before quantization. On the foliation itself, such elimination leads to a reduced system of first-class constraints: the supermomentum constraints, and a single Hamiltonian constraint describing evolution along the foliation. The constraint quantization yields an ordinary Schrödinger equation for the conformal three-geometry. For a critical value of the coupling, the second-class constraints further proliferate, and the system becomes either inconsistent or dynamically frozen.
- Received 16 January 1992
DOI:https://doi.org/10.1103/PhysRevD.45.4443
©1992 American Physical Society