Moyal Quantization, Holography, and the Quantum Geometry of Surfaces

TL;DR

This paper introduces a generalized Moyal quantization method for 2D surfaces, revealing connections with quantum holography and offering new insights into mathematical theorems and models of membranes.

Contribution

It extends phase space quantization to surfaces, linking it with holography and providing novel perspectives on mathematical and physical theories.

Findings

Connection between Moyal quantization and holography on Riemann surfaces

Insights into the Torelli theorem via quantum holography

Potential model for a quantum theory of membranes

Abstract

An elementary introduction is provided to the phase space quantization method of Moyal and Wigner. We generalize the method so that it applies to 2-dimensional surfaces, where it has an interesting connection with quantum holography. In the case of Riemann surfaces the connection between Moyal quantization and holography provides new insights into the Torelli theorem and the quantization of non-linear integrable models. Quantum holography may also serve as a model for a quantum theory of membranes.