Non-commutativity in polar coordinates

Abstract

We reconsider the fundamental commutation relations for non-commutative R 2 described in polar coordinates with non-commutativity parameter θ . Previous analysis found that the natural transition from Cartesian coordinates to the traditional polar system led to a representation of [r,φ] as an everywhere diverging series. In this article we compute the Borel resummation of this series, showing that it can subsequently be extended throughout parameter space and hence provide an interpretation of this commutator. Our analysis provides a complete solution for arbitrary r and θ that reproduces the earlier calculations at lowest order and benefits from being generally applicable to problems in a two-dimensional non-commutative space. We compare our results to previous literature in the (pseudo-)commuting limit, finding a surprising spatial dependence for the coordinate commutator when θ r 2 . Finally, we raise some questions for future study in light of this progress.

U(N) Yang-Mills in non-commutative space time

“…This product is just a deformation of the familiar algebra of functions [32] on R n (for discussion of non-commutative space described in polar coordinates see [33][34][35]) and makes use of a common notation where derivatives act in the direction of their overhead arrows. In Fourier space the Moyal product of a number of functions of the operatorsx i reads 1…”

Noncommutativity of four-dimensional axisymmetric spacetime in polar coordinate

“…However, at present, such applications of noncommutative gravity have not progressed so much, due in part to the difficulty of establishing consistent noncommutative spacetime description by an arbitrary coordinate selection. In the first place, even the polar coordinate system usually employed in gravity research has only recently begun to be studied in the field of noncommutativity [16][17][18][19]. a e-mail: r201770192ve@jindai.jp Noncommutative spacetime is characterized by the nontrivial commutation relation [ xµ , xν ] = iθ µν , which is usually defined in the Cartesian coordinate system.…”

“…In [17], the similar relation is derived from defining x = r cos θ , ŷ = r sin θ . Furthermore, in [18,19] using the Moyal deformation quantization [20], higher order term of commutation relation between real space coordinates r = x 2 + y 2 and φ = arctan(y/x) by the expansion of Θ was calculated. In particular, in [19], the behavior of the commutation relation in the limit when r → ∞ and r ≪ Θ was investigated by deformation quantization and Borel resummation.…”

“…Furthermore, in [18,19] using the Moyal deformation quantization [20], higher order term of commutation relation between real space coordinates r = x 2 + y 2 and φ = arctan(y/x) by the expansion of Θ was calculated. In particular, in [19], the behavior of the commutation relation in the limit when r → ∞ and r ≪ Θ was investigated by deformation quantization and Borel resummation. In any coordinate system, the original noncommutativity is hard to be seen in most cases as long as the methods described in the above paragraph are used.…”

“…In this paper, we showed that the generalized Moyal product gives consistent commutation relations in Cartesian and polar coordinate. In the foregoing studies of polar noncommutativity [16][17][18][19], only a commutation relation of two-dimensional space was derived, but we have specifically dealt with a commutation relation in four-dimensional spacetime. Furthermore, since the generalized Moyal product is available even in arbitrary spacetime, we investigated the effect of noncommutativity in the noncommutative Kerr spacetime.…”