Particle spin dynamics as the grassmann variant of classical mechanics

Abstract

A generalization of classical mechanics is presented. The dynamical variables (functions on the phase space) are assumed to be elements of an algebra with anticommuting generators (the Grassmann algebra). The action functional and the Poisson brackets are defined. The equations of motion are deduced from the variational principle. The dynamics are also described by means of the Liouville equation for the phase-space distribution. The canonical quantization leads to the Fermi (anticommutator) commutation relations. The phase-space path integral approach to the quantum theory is also formulated. The theory is applied to describe the particle spin. In the nonrelativistic case, the elements of the phase-space are anticommuting three-vectors ξ, transformed to the Pauli matrices after the quantization: ξ = ( & ̵ h ̷ 2 ) 1 2 σ . Classical description of the spin precession and of the spin-orbital forces is given. To introduce the relativistic spin in an invariant manner one needs a five-dimensional phase space (a four-vector plus a scalar). The Lagrangian is singular and there is a constraint, resulting from a “supersymmetry.” The quantized phase-space elements are proportional to the Dirac matrices γ 5 γ μ and γ 5, while the constraint is transformed to the Dirac equation. The phase-space distribution and the interaction with an external field are also considered.