Symplectic Dirac-K\"ahler Fields

TL;DR

This paper introduces symplectic Dirac-K"ahler fields defined on phase-space, linking them to metaplectic spinors and exploring their implications in quantum mechanics and lattice fermion problems.

Contribution

It develops a novel framework for phase-space fermions using symplectic geometry, extending Dirac-K"ahler fields beyond space-time.

Findings

Symplectic Dirac-K"ahler fields are equivalent to infinite families of metaplectic spinors.

The framework connects gauge theory of quantum mechanics with phase-space fermions.

An analogy between species doubling in lattice fermions and quantization issues is identified.

Abstract

For the description of space-time fermions, Dirac-K\"ahler fields (inhomogeneous differential forms) provide an interesting alternative to the Dirac spinor fields. In this paper we develop a similar concept within the symplectic geometry of phase-spaces. Rather than on space-time, symplectic Dirac-K\"ahler fields can be defined on the classical phase-space of any Hamiltonian system. They are equivalent to an infinite family of metaplectic spinor fields, i.e. spinors of Sp(2N), in the same way an ordinary Dirac-K\"ahler field is equivalent to a (finite) mulitplet of Dirac spinors. The results are interpreted in the framework of the gauge theory formulation of quantum mechanics which was proposed recently. An intriguing analogy is found between the lattice fermion problem (species doubling) and the problem of quantization in general.