Poisson Bracket for Fermion Fields: Correspondence Principle, Second Class Constraints and Hamilton-Jacobi equation

TL;DR

This paper develops a symmetric Poisson bracket framework for classical fermion fields, enabling a direct classical-quantum correspondence without Grassmann variables, and connects classical Hamilton-Jacobi theory with quantum fermionic dynamics.

Contribution

It introduces a classical Poisson bracket for fermions that parallels bosonic cases, eliminating the need for Grassmann variables and clarifying the classical-quantum relationship for fermionic fields.

Findings

Classical fermion fields can be described without Grassmann variables.

The Dirac bracket effectively handles second class constraints.

Semiclassical limit recovers the Hamilton-Jacobi equation for fermions.

Abstract

We introduce a symmetric Poisson bracket that allows us to describe anticommuting fields on a classical level in the same way as commuting fields, without the use of Grassmann variables. By means of a simple example, we show how the Dirac bracket for the elimination of the second class constraints can be introduced, how the classical Hamiltonian equations can be derived and how quantization can be achieved through a direct correspondence principle. Finally, we show that the semiclassical limit of the corresponding Schroedinger equation leads back to the Hamilton-Jacobi equation of the classical theory. Summarizing, it is shown that the relations between classical and quantum theory are valid for fermionic fields in exactly the same way as in the bosonic case, and that there is no need to introduce anticommuting variables on a classical level.