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

TL;DR

This paper explores how conditional symmetries in constrained minisuperspace models, specifically the Schwarzschild case, can serve as quantum conditions to select physically meaningful wave functions during canonical quantization.

Contribution

It introduces the concept of conditional symmetries as variational symmetries and applies this framework to the canonical quantization of the Schwarzschild minisuperspace model, identifying unique quantum conditions.

Findings

Only one symmetry can be imposed consistently.

The Casimir invariant serves as a second quantum condition.

Explicit solutions depend on the product of scale factors.

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 $\Psi$. 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"