Propagation of spinors on a noncommutative spacetime: equivalence of the formal and the effective approach

Abstract

AbstractSome noncommutative (NC) theories posses a certain type of dualities that are implicitly built within their structure. In this paper we establish still another example of this kind, and we do this perturbatively in the first order of the Seiberg–Witten expansion. More precisely, we show that a particular model of noncommutative $$U(1)_{\star } $$ U ( 1 ) ⋆ gauge field coupled to a NC scalar field and to a classical geometry of the Reissner–Nordström (RN) type is to a first order in deformation completely equivalent at the level of equations of motion to the commutative U(1) gauge theory coupled to a commutative scalar field and to a classical geometry background, different from the starting RN background. The new (effective) metric is obtained from the RN metric by switching on an additional nonvanishing $$r-\phi $$ r - ϕ component. Using this first order duality between two theories and physical systems they describe, we formulate an effective approach to studying a dynamics of spin $$\frac{1}{2}$$ 1 2 fields on the curved background of RN type with an abiding noncommutative structure. As opposed to that, we also investigate in a more formal way a dynamics of spin $$\frac{1}{2}$$ 1 2 fields, and we do this perturbatively, within a first order in deformation parameter, by studying a semiclassical theory which describes the NC $$U(1)_{\star } $$ U ( 1 ) ⋆ gauge field coupled to NC spin $$\frac{1}{2}$$ 1 2 field and also coupled to gravity, which is however treated classically. Upon utilising the Seiberg–Witten (SW) map in order to write the NC spinor and NC gauge fields in terms of their corresponding commutative degrees of freedom, we find that the equation of motion for the fermion field obtained within the formal approach exactly coincides with the equation of motion obtained within the effective approach that utilises first order noncommutative duality. Therefore, linearized equations of motion for a spinor field in SW expansion turn out to be the same as equations of motion in a perturbed metric. We then use these results to analyze the problem of stability of solutions of the equations of motion and the associated issue of superradiance, as related to fermions in RN spacetime with an all-pervasive noncommutative structure.