Moyal Quantization, Holography and the QuantumGeometry of Surfaces

An elementary introduction is provided to the phase space quantization method ofMoyal and Wigner. We generalize the method so that it applies to 2-dimensional surfaces, whereit has an interesting connection with quantum holography. In the case of Riemann surfaces theconnection between Moyal quantization and holography provides new insights into the Torellitheorem and the quantization of non- linear integrable models. Quantum holography may alsoserve as a model for a quantum theory of membranes. 1998 Elsevier Science Ltd.