Scale Transformations on the Noncommutative Plane and the Seiberg-Witten Map

TL;DR

This paper explores three types of scale transformations on the noncommutative plane, analyzing their properties, relations to the Seiberg-Witten map, and potential as symmetries in noncommutative field theories.

Contribution

It introduces and characterizes three scale transformations on the noncommutative plane, linking them to gauge invariance and the Seiberg-Witten map, and discusses their role as symmetries.

Findings

Transformations depend on star product choice.

Transformation ii) relates to Seiberg-Witten transformations.

Transformation iii) preserves algebraic relations and can be a symmetry.

Abstract

We write down three kinds of scale transformations {\tt i-iii)} on the noncommutative plane. {\tt i)} is the analogue of standard dilations on the plane, {\tt ii)} is a re-scaling of the noncommutative parameter $\theta$, and {\tt iii)} is a combination of the previous two, whereby the defining relations for the noncommutative plane are preserved. The action of the three transformations is defined on gauge fields evaluated at fixed coordinates and $\theta$. The transformations are obtained only up to terms which transform covariantly under gauge transformations. We give possible constraints on these terms. We show how the transformations {\tt i)} and {\tt ii)} depend on the choice of star product, and show the relation of {\tt ii)} to Seiberg-Witten transformations. Because {\tt iii)} preserves the fundamental commutation relations it is a symmetry of the algebra. One has the possibility of implementing it as a symmetry of the dynamics, as well, in noncommutative field theories where $\theta$ is not fixed.