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On the Supersymmetry Group of the Classical Bose-Fermi Oscillator

Supermechanics is the classical counterpart of Quantum Field Theories involving Bose and Fermi fields. The most prominent use of supermechanics from a mathematical perspective is the role of the classical free particle Lagrangian in the supersymmetric proofs of various index theorems. There has also been some interest in making the geometric description of supermechanics mathematically rigorous, both from a Lagrangian and Hamiltonian point of view11–13. In this note, we are concerned with the classical one-dimensional supersymmetric harmonic oscillator, or Bose-Fermi oscillator. By “classical”, we mean that we treat it as a supermechanical system, defined on a supersymplectic flat manifold. It is a simple but nontrivial example of a system with supersymmetries, that is, symmetries that mix the fermionic and the bosonic degrees of freedom. It first appeared as one example of a supersymmetric quantum mechanical system in Witten’s ground-breaking 1981 paper and was further investigated in the 1980’s and 1990’s. The infinitesimal supersymmetry transformations of the harmonic oscillator were initially introduced in an ad hoc way. It was later realized that the stabilizer algebra of the dynamics is the orthosymplectic superalgebra osp(2|2). In this note, we use the concept of a momentum map to investigate the supersymmetries of the harmonic oscillator. We show how the ad hoc supersymmetry transformations can be derived in a systematic way. Namely, we construct a Lie superalgebra bf(2|2) whose odd part consists of these transformations and show that bf(2|2) is the intersection of osp(2|2) and the Lie superalgebra of linear supersymplectic transformations. Alternatively, bf(2|2) is the Lie superalgebra of the largest connected subgroup of OSp(2|2) that admits a momentum map. This is essentially parallel to the ungraded case, since the group of canonical transformations admits a momentum map in any dimension. The paper is organized as follows: In the sections II and III, we review some facts about the theory of supermanifolds, in particular super Lie groups and the super version of Lie’s first theorem. In section IV, we give a short description of the Bose-Fermi oscillator and write down a class of supersymmetries, taken from DeWitt’s book. We construct the supersymmetry Lie algebra of these supersymmetries and name it bf(2|2). The corresponding group is BF (2|2). We then give a general definition of the concept of a momentum map in supermechanics, and show that it is preserved under the flow of Hamiltonians that are invariant under the group action. This is a generalization of the variant of Noether’s theorem in ungraded classical Hamiltonian mechanics. In the last section, we show that BF (2|2) has the properties stated above. We also verify that the momentum map is equivariant and read off the conserved quantities.