Conditional symmetries and the canonical quantization of constrained minisuperspace actions: The Schwarzschild case

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. Iris 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 the 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 Psi. The uniqueness of the dynamical evolution implies the need to identify this quadratic integral of motion to the reparametrization 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". (C) 2013 Elsevier B.V. All rights reserved.