Symplectic structure for general relativity and Einstein-Brillouin-Keller quantization

TL;DR

This paper develops a symplectic geometric framework for classical and quantum general relativity, applying geometrical quantization and Einstein-Brillouin-Keller conditions to explore quantum solutions and black hole spectra.

Contribution

It introduces a symplectic geometric approach to quantize pure gravity without matter, incorporating topological corrections and discussing black hole mass spectra.

Findings

Constructed a symplectic geometry for classical general relativity.

Applied geometrical quantization to pure gravity.

Discussed a possible mass spectrum for Schwarzschild black holes.

Abstract

The Hamiltonian system of general relativity and its quantization without any matter or gauge fields are discussed on the basis of the symplectic geometrical theory. A symplectic geometry of classical general relativity is constructed using a generalized phase space for pure gravity. Prequantization of the symplectic manifold is performed according to the standard procedure of geometrical quantization. Quantum vacuum solutions are chosen from among the classical solutions under the Einstein-Brillouin-Keller quantization condition. A topological correction of quantum solutions, namely the Maslov index, is realized using a prequantization bundle. In addition, a possible mass spectrum of Schwarzschild black holes is discussed.