Nonpertubative Effects of Extreme Localization in Noncommutative Geometry

“Extremely” localized wavefunctions in noncommutative geometry have disturbances that are localized to distances smaller than √ θ, where θ is the “area” parameter that measures noncommutativity. In particular, distributions such as the sign function or the Dirac delta function are limiting cases of extremely localized wavefunctions. It is shown that Moyal star products of extremely localized wavefunctions cannot be correctly computed perturbatively in powers of θ. Nonperturbative effects as a function of θ are explicity displayed through exact computations in several examples. In particular, for distributions, star products end up being functions of θ and have no expansion in positive powers of θ. This result provides a warning for computations in noncommutative space that often are performed with perturbative methods. Furthermore, the result may have interesting applications that could help elucidate the role of noncommutative geometry in several areas of physics. This research was partially supported by the US Department of Energy under grant number DE-FG03-84ER40168. 1 1 Star-commutators with distributions For simplicity we will limit our discussion in this note to a two dimensional noncommutative plane (generalizations are immediate). The two noncommutative coordinates are denoted as x1 = x and x2 = p as a reminder of the close relation between noncommutative geometry and quantum mechanics, but we have in mind various applications of noncommutative geometry in physics, including the quantum Hall effect, strings in large background fields, and string field theory. The noncommutativity parameter θ has dimensions of “area”, i.e. units of x times units of p, and its meaning depends on the specific physical application. As we will make explicit, localization to distances shorter than √ θ produce nonperturbative effects as a function of the parameter θ in computations involving the noncommutative geometry. Consider functions in the noncommutative plane Λ (p, x) . In particular, consider the sign step-function, ε (p) = p |p| , which takes the values ±1 for p ≷ 0 respectively. Its derivative is the Dirac delta function ∂ ∂p ε (p) = 2δ (p) . (1) As is well known, it can be represented as an integral ε (p) = ∫ ∞ −∞ dq πi e (