Physics
Scientific paper
Jun 1992
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1992phrvd..45.4443k&link_type=abstract
Physical Review D (Particles, Fields, Gravitation, and Cosmology), Volume 45, Issue 12, 15 June 1992, pp.4443-4457
Physics
9
Canonical Formalism, Lagrangians, And Variational Principles
Scientific paper
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 T(X) is described by a Lagrangian LT whose field equations imply that the value of T at X is the mean extrinsic curvature K of a K=const hypersurface passing through X. By adding LT to the Hilbert Lagrangian LG, the ``extrinsic time field'' T(X) 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 X(x) is transverse to the K=const foliation, this rank is higher, and one obtains only the standard super-Hamiltonian and supermomentum constraints. At stationary points of T(x), the hypersurface becomes tangent to a K=const 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 P(x) which is canonically conjugate to the time function T(x). In this form, the constraint leads to a functional Schrödinger equation. As one approaches a stationary point of T(x), 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 K=const foliation itself, such elimination leads to a reduced system of 3∞3+1 first-class constraints: the 3∞3 supermomentum constraints, and a single Hamiltonian constraint describing evolution along the K=const 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 pro- liferate, and the system becomes either inconsistent or dynamically frozen.
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