From N = 2 supersymmetric classical to quantum mechanics and back: the SUSY WKB approximation

In this essay, written to commemorate the sixtieth birthday of J. Carinena, we discuss several elementary issues in one-dimensional supersymmetric quantum mechanics. The role of the Riccati equation in this framework has been thoroughly analyzed by Carinena and collaborators at the highest level of mathematical rigor by approaching topics such as the factorization method or shape invariance from a group-theoretical point of view, see (1), (2) and (3). Our purpose here is to approach these matters from a rather physical point of view. To construct a supersymmetric quantum mechanical system starting from a physical potential energy we shall be led to deal with the Hamilton-Jacobi or the Poisson equations, although in both cases there is an associated Riccati equation. We shall focus on studying the relationship between supersymmetric classical and quantum mechanical systems, following the standard References (4) and (5) and the more recent Lectures of A. Wipf (6). In particular, models where supersymmetry is unbroken and instantons exist will be analyzed at length. Aother issue to be treated with care is the semiclassical behavior of supersymmetric quantum systems, this done with the help of the enlightening paper of A. Comtet et al. (10).