## Towards A Moyal Quantization Program of the Membrane

**Authors:** Carlos Castro

A Moyal deformation quantization approach to a spherical membrane
(moving in flat target backgrounds) in the light cone gauge is presented.
The physical picture behind this construction relies in viewing the two
spatial membrane coordinates σ_{1}, σ_{2} as the two phase space variables
q, p, and the temporal membrane coordinate τ as time. Solutions to
the Moyal-deformed equations of motion are explicitly constructed in
terms of elliptic functions. A knowledge of the Moyal-deformed light-cone
membrane's Hamiltonian density H(q, p, τ ) allows to construct a timedependent
Wigner function ρ(q, p, τ ) as solutions of the Moyal-Liouville
equation, and from which one can obtain the expectation values of the operator
< H > = Trace (ρH) that define the quantum average values of the
energy density configurations of the membrane at any instant of time. It
is shown how a time-dependent quartic oscillator with q^{4}, p^{4}, q^{2}p^{2} terms
plays a fundamental role in the quantum treatment of membranes and
displays an important p ↔ q duality symmetry.

**Comments:** 15 Pages. This article has been submitted to the J. Phys. A : Math.

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### Submission history

[v1] 14 Nov 2009

[v2] 17 Nov 2009

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