SCALE TRANSFORMATIONS ON THE NONCOMMUTATIVE PLANE AND THE SEIBERG–WITTEN MAP

Abstract

We write down three kinds of scale transformations (i)–(iii) on the noncommutative plane. Transformation (i) is the analogue of standard dilations on the plane, transformation (ii) is a rescaling of the noncommutative parameter θ, and transformation (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 θ. 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 (i) and (ii) depend on the choice of star product, and show the relation of (ii) to Seiberg–Witten transformations. Because transformation (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 θ is not fixed.