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Symplectic Dirac-Kähler fields. (English) Zbl 0968.81037
Summary: 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 \(\text{Sp}(2N)\), 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.

MSC:
81S10 Geometry and quantization, symplectic methods
81T13 Yang-Mills and other gauge theories in quantum field theory
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
53C27 Spin and Spin\({}^c\) geometry
53D05 Symplectic manifolds (general theory)
58J90 Applications of PDEs on manifolds
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
81T20 Quantum field theory on curved space or space-time backgrounds
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