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

TL;DR

This paper extends Koopman-von Neumann classical mechanics to poly-symplectic fields, introduces a Lorentz covariant Hamiltonian formulation, and generalizes classical and quantum commutation relations for field quantization.

Contribution

It develops a new covariant Hamiltonian framework for fields, connecting classical and quantum formalisms through generalized commutation relations.

Findings

Introduces a Lorentz covariant Hamiltonian density.

Provides generalized commutation relations for classical and quantum fields.

Clarifies the process of 1st and 2nd quantization via operator mapping.

Abstract

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. We clarify how 1st and 2nd quantization can be found by simply mapping between operators in classical and quantum commutator algebras.