Dynamical symmetries of supersymmetric oscillators

Abstract

In this paper, we describe the dynamical symmetries of classical supersymmetric oscillators in one and two spatial (bosonic) dimensions. Our main ingredient is a generalized Poisson bracket which is defined as a suitable classical counterpart to commutators and anticommutators. In one dimension, i.e., in the presence of one bosonic and one fermionic coordinate, the Hamiltonian admits a $U(1,1)$ symmetry for which we explicitly compute the first integrals. It is found that suitable forms of the supercharges emerge in a natural way as fermionic conserved quantities. Following this, we describe classical supercharge operators based on the generalized Poisson bracket and subsequently define supersymmetry transformations. We perform a straightforward generalization to two spatial dimensions where the Hamiltonian has an overall $U(2,2)$ symmetry. We comment on plausible supersymmetric generalizations of the Pais-Uhlenbeck and isotonic oscillators, and also present the possibility of defining a generalized Nambu bracket within the classical formalism.