Non-commutative oscillators and the commutative limit

It is shown in first order perturbation theory that anharmonic oscillators in non-commutative space behave smoothly in the commutative limit just as harmonic oscillators do. The non-commutativity provides a method for converting a problem in degenerate perturbation theory to a non-degenerate problem. In the last few years theories in non-commutative space [1]-[5] have been studied extensively. While the motivation for this kind of space with noncommuting coordinates is mainly theoretical, it is possible to look experimentally for departures from the usually assumed commutativity among the space coordinates [6]-[8]. So far no clear departure has been found, but it is clear that any experiment can only provide a bound on the amount of non-commutativity, and that more precise experiments in future can reveal a small amount. Meanwhile, there are some theoretical issues which have arisen in the course of these investigations. If one calls the non-commutativity parameter θ, so that two spatial coordinates x,y satisfy the relation [x,y] = iθ, one would expect ordinary commutative space to emerge in the limit θ → 0. In many field theoretical and quantum mechanical problems, however, the passage from the non-commutative space to its commutative limit has not appeared to be smooth [9]-[14]. The literature is replete with expressions where θ appears in the denominator. The simplest example is the two-dimensional harmonic oscillator. As in commutative space, this quantum mechanical problem is exactly ∗e-mail [email protected] †e-mail [email protected]