Supersymmetry and the Odd Poisson Bracket

TL;DR

This paper explores the applications of the odd Poisson bracket in classical and quantum systems, including reformulating dynamics, describing hydrodynamics, and constructing quantum representations, with implications for supersymmetry and space-time structure.

Contribution

It introduces new applications of the odd Poisson bracket in classical, quantum, and geometric contexts, including a novel linear odd bracket related to Lie groups.

Findings

Reformulation of classical Hamiltonian dynamics using the odd bracket

Development of quantum representations for the odd bracket

Introduction of a linear odd bracket for semi-simple Lie groups

Abstract

Some applications of the odd Poisson bracket developed by Kharkov's theorists are represented, including the reformulation of classical Hamiltonian dynamics, the description of hydrodynamics as a Hamilton system by means of the odd bracket and the dynamics formulation with the Grassmann-odd Lagrangian. Quantum representations of the odd bracket are also constructed and applied for the quantization of classical systems based on the odd bracket and for the realization of the idea of a composite spinor structure of space-time. At last, the linear odd bracket, corresponding to a semi-simple Lie group, is introduced on the Grassmann algebra.