Supersymmetric probability distributions

Abstract

We use anticommuting variables to study probability distributions of random variables, that are solutions of Langevin's equation. We show that the probability density always enjoys "worldpoint supersymmetry". The partition function, however, may not. We find that the domain of integration can acquire a boundary, that implies that the auxiliary field has a non-zero expectation value, signalling spontaneous supersymmetry breaking. This is due to the presence of "fermionic" zeromodes, whose contribution cannot be cancelled by a surface term. This we prove by an explicit calculation of the regularized partition function, as well as by computing the moments of the auxiliary field and checking whether they satisfy the identities implied by Wick's theorem. Nevertheless, supersymmetry manifests itself in the identities that are satisfied by the moments of the scalar, whose expressions we can calculate,for all values of the coupling constant. We also provide some quantitative estimates concerning the visibility of supersymmetry breaking effects in the identities for the moments and remark that the shape of the distribution of the auxiliary field can influence quite strongly how easy it would be to mask them, since the expectation value of the auxiliary field doesn't coincide with its typical value.