Comments on interactions in the SUSY models

We consider special supersymmetry (SUSY) transformations with m generators ← s_α, for some class of models and study the physical consequences when making the Grassmann-odd transformations to form an Abelian supergroup with finite parameters and a set of group-like elements with finite parameters being functionals of the field variables. The SUSY-invariant path integral measure within conventional quantization scheme leads to the appearance of the Jacobian under a change of variables generated by such SUSY transformations, which is explicitly calculated. The Jacobian implies, first of all, the appearance of trivial interactions in the transformed action, and, second, the presence of a modified Ward identity which reduces to the standard Ward identities in the case of constant parameters. We examine the case of the N = 1 and N = 2 supersymmetric harmonic oscillators to illustrate the general concept by a simple free model with (1, 1) physical degrees of freedom. It is shown that the interaction terms U_tr have a corresponding SUSY-exact form: U_tr = (V₍₁₎ ← s; V₍₂₎ ← anti s ← s) generated naturally under such generalized formulation. We argue that the case of a non-trivial interaction cannot be obtained in such a way. (orig.)