1208.0462 (T. Christodoulakis et al.)

Conditional Symmetries and the Canonical Quantization of Constrained
Minisuperspace Actions: the Schwarzschild case    [PDF]

T. Christodoulakis, N. Dimakis, Petros A. Terzis, G. Doulis, Th. Grammenos, E. Melas, A. Spanou

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 $\Psi$. 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"

View original: http://arxiv.org/abs/1208.0462