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We generalize Koopman-von Neumann classical mechanics to poly-symplecticfields and recover De Donder-Weyl theory. Comparing with Dirac's Hamiltoniandensity inspires a new Hamiltonian formulation with a canonical momentum fieldthat is Lorentz covariant with symplectic geometry. We provide commutationrelations for the classical and quantum fields that generalize the Koopman-vonNeumann and Heisenberg algebras. The classical algebra requires four fieldsthat generalize space-time, energy-momentum, frequency-wavenumber, and theFourier conjugate of energy-momentum. We clarify how 1st and 2nd quantizationcan be found by simply mapping between operators in classical and quantumcommutator algebras.

Quantization of a New Canonical, Covariant, and Symplectic Hamiltonian Density