## 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.

**Download:** **PDF**

### Submission history

[v1] 14 Nov 2009

[v2] 17 Nov 2009

**Unique-IP document downloads:** 158 times

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*