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

We generalize Koopman-von Neumann classical mechanics to poly symplectic fields and recover De Donder-Weyl's theory. Compared with Dirac's Hamiltonian density, it inspires a new Hamiltonian formulation with a canonical momentum field that is Lorentz-covariant with symplectic geometry. We provide commutation relations for the classical and quantum fields that generalize the Koopman-von Neumann and Heisenberg algebras. The classical algebra requires four fields that generalize spacetime, energy-momentum, frequency-wavenumber, and the Fourier conjugate of energy-momentum. We clarify how first and second quantization can be found by simply mapping between operators in classical and quantum commutator algebras.