On the quantization of systems with anticommuting variables

Abstract

In this paper we consider the pseudomechanics, that is the mechanics of a system described by ordinary canonical variables and by Grassmann variables. We study the canonical formalism and in particular we define the Poisson brackets. We show that the algebra of the Poisson brackets is a graded Lie algebra. Using this fact as a hint for quantization we show that the corresponding quantized theory is the ordinary quantum theory with Fermi operators. It follows that the classical limit (ħ → 0) of the quantum theory is, in general, the pseudomechanics.