3 de agosto de 2012

1208.0462 Conditional Symmetries and the Canonical Quantization of Constrained Minisuperspace Actions: the Schwarzschild case

T. Christodoulakis∗, N. Dimakis†, Petros A. Terzis‡
Nuclear and Particle Physics Section, Physics Department,
University of Athens, GR 157–71 Athens
G. Doulis§
Department of Mathematics and Statistics, University of Otago
P.O. Box 56, Dunedin 9010, New Zealand
Th. Grammenos¶
Department of Civil Engineering, University of Thessaly,
GR 383–34 Volos
E. Melask
Logistics Department, GR 32-200, Thiva
Technological Educational Institution of Chalkida
A. Spanou∗∗
School of Applied Mathematics and Physical Sciences,
National Technical University of Athens, GR 157–80, Athens

Abstract: A conditional symmetry is defined, in the phase-space of a quadratic in velocities constrained action, as a simultaneous conformal symmetry of the supermetric and the superpotential. It is proven that such a symmetry corresponds to a variational (Noether) symmetry.The use of these symmetries as quantum conditions on the wave-function entails a kind of selection rule. As an example, the minisuperspace model ensuing from a reduction of the Einstein - Hilbert action by considering static, spherically symmetric configurations and r as the independent dynamical variable, is canonically quantized. The conditional symmetries of this reduced action are used as supplementary conditions on the wave function. Their integrability conditions dictate, at a first stage, that only one of the three existing symmetries can be consistently imposed. At a second stage one is led to the unique Casimir invariant, which is the product of the remaining two, as the only possible second condition on. The uniqueness of the dynamical evolution implies the need to identify this quadratic integral of motion to the reparametrisation generator. This can be achieved by fixing a suitable parametrization of the r-lapse function, exploiting the freedom to arbitrarily rescale it. In this particular parametrization the measure is chosen to be the determinant of the supermetric. The solutions to the combined Wheeler - DeWitt and linear conditional symmetry equations are found and seen to depend on the product of the two “scale factors”.