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

Understand

We generalize Koopman-von Neumann classical mechanics to poly-symplectic fields and recover De Donder-Weyl theory.

• Comparing with Dirac's Hamiltonian density 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 space-time, energy-momentum, frequency-wavenumber, and the Fourier conjugate of energy-momentum.