The low energy limit of the non-commutative Wess-Zumino model

Abstract

The non-commutative Wess-Zumino model is used as a prototype for studying the low energy behavior of a renormalizable non-commutative field theory. We start by deriving the potentials mediating the fermion-fermion and boson-boson interactions in the non-relativistic regime. The quantum counterparts of these potentials are afflicted by ordering ambiguities but we show that there exists an ordering prescription which makes them hermitean. For space/space noncommutativity it turns out that Majorana fermions may be pictured as rods oriented perpendicularly to the direction of motion showing a lack of locality, while bosons remain insensitive to the efects of the non commutativity. For time/space noncommutativity bosons and fermions can be regarded as rods oriented along the direction of motion. For both cases of noncommutativity the scattering state describes scattered waves, with at least one wave having negative time delay signalizing the underlying nonlocality. The super field formulation of the model is used to compute the corresponding effective action in the one- and two-loop approximations. In the case of time/space noncommutativity, unitarity is violated in the relativistic regime. However, this does not preclude the existence of a unitary low energy limit.