Symplectic Dirac–Kähler fields

For the description of space–time fermions, Dirac–Kähler 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–Kähler 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 $\mathrm{Sp}\text{(2N),}$ in the same way an ordinary Dirac–Kähler field is equivalent to a (finite) multiplet 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.