Quantization of Pseudoclassical Systems in the Schr\"odinger Realization

TL;DR

This paper explores the quantization of pseudoclassical systems with anticommuting variables in the Schr"odinger picture, extending standard quantum mechanics methods to include spin models and superselection sectors.

Contribution

It introduces a generalized Schr"odinger quantization approach for pseudoclassical systems, including spin models, with a consistent norm and adjointness properties, and relates physical states to Dirac-K"ahler fermions.

Findings

Quantization methods extend to pseudoclassical systems with anticommuting variables.

Physical state space corresponds to spinor wave functions with superselection sectors.

Physical states are isomorphic to Dirac-K"ahler fermions, with a different inner product.

Abstract

We examine the quantization of pseudoclassical dynamical systems, models that have classically anticommuting variables, in the Schr\"odinger picture. We quantize these systems, which can be viewed as classical models of particle spin, using the generalized Gupta-Bleuler method as well as the reduced phase space method in even dimensions. With minimal modifications, the standard constructions of Schr\"odinger quantum mechanics of constrained systems work for pseudoclassical systems. We generalize the standard Schr\"odinger norm and implement the correct adjointness properties of observables and constraints. We construct the state space corresponding to spinors as physical wave functions of anticommuting variables, finding that there are superselection sectors in both the physical and ghost subspaces. The physical states are isomorphic to those of the Dirac-K\"ahler formulation of fermions though the inner product in Dirac-K\"ahler theory is not equivalent to ours.