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

• DOI: 10.1016/j.geomphys.2013.04.009
• Publication Date: 2013-05-07T22:02:10Z
• Authors: T. Christodoulakis; N. Dimakis; Petros A. Terzis; G. Doulis; Th. Grammenos; E. Melas; A. Spanou
• Keywords: Quantum Physics; 0103 physical sciences; FOS: Physical sciences; General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph); Quantum Physics (quant-ph); 01 natural sciences; General Relativity and Quantum Cosmology; Mathematical Physics

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"<br/>20 pages, LaTeX2e source file, no figures<br/>