Quantization of the Schwarzschild geometry

6th International Workshop DICE2012 Spacetime – Matter – Quantum Mechanics: From the Planck Scale to Emergent Phenomena

物理学;力学

Melas, Evangelos^1

University of Athens, Department of Economics, Unit of Mathematics and Informatics, Pesmazoglou 8, 10559 Athens, Greece^1

关键词: Casimir invariants;    Dynamical evolution;    Einstein-Hilbert action;    Integrability;    Parametrizations;    Reparametrization;    Schwarzschild geometry;    Symmetric geometry;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/442/1/012037/pdf
DOI  :  10.1088/1742-6596/442/1/012037

学科分类：力学，机械学

来源: IOP

【 摘 要 】

The conditional symmetries of the reduced Einstein-Hilbert action emerging from a static, spherically symmetric geometry are used as supplementary conditions on the wave function. Based on their integrability conditions, only one of the three existing symmetries can be consistently imposed, while the unique Casimir invariant, being the product of the remaining two symmetries, is calculated as the only possible second condition on the wave function. This quadratic integral of motion is identified with the reparametrization generator, as an implication of the uniqueness of the dynamical evolution, by fixing a suitable parametrization of the r-lapse function. In this parametrization, the determinant of the supermetric plays the role of the mesure. The combined Wheeler-DeWitt and linear conditional symmetry equations are analytically solved. The solutions obtained depend on the product of the two "scale factors".