Supersymmetric Odd Mechanical Systems and Hilbert Q-Module Quantization

Abstract

Supersymmetry can be implemented within a particle model in two ways. The first one is commonly exploited and assumes that we use conventional graded Lie algebra approach in the sense that on the classical level we have a ℤ2-graded Lie-Poisson algebra of observables which after quantization is replaced by a ℤ2-graded Lie algebra of operators. Both, graded Poisson bracket and ℤ2-graded commutator are even mappings. The second way of realization of supersymmetry in a particle model is related to the anti-bracket algebras. In this case Lagrangian as well as Hamiltonian of the supersymmetric system is an odd Grassmann algebra valued function and the Grassmannian parity of canonical momenta is opposite to the parity of related coordinates. The anti-bracket is an odd mapping. Realizations of the mentioned type we shall call the even supersymmetric mechanics and the odd supersymmetric mechanics, respectively [1, 2]. The odd mechanics allows particular deformation of geometry of the configuration superspace. The realization of the supersymmetry algebra after the passage to phase superspace in terms of the Dirac anti-bracket remains conventional [3]. The canonical quantization of both types of models can be done in parallel but in the case of the odd systems one can introduce a new ℤ2-graded algebra generalizing complex numbers in such a sense that we introduce additional imaginary unit of the odd Grassmannian parity [4, 5]. Such a structure we shall call oddons (referring to the name of quaternions, octonions etc.). The formalism, in both cases, allows to mimic the approach known from the harmonic analysis on the Heisenberg group [6]. In the even sector it is done for a pair—Heisenberg group and Fermionic Heisenberg group. In the odd sector it is done for so called Odd Heisenberg group.