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arXiv:2305.08864

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

David Chester, Xerxes D. Arsiwalla, Louis Hirsch Kauffman, Michel R. P. Planat, Klee Irwin

Published May 5, 2023• 1 citation
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2305.08864

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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.
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.
We propose a manifestly covariant canonical method of field quantization based on the classical De Donder-Weyl covariant canonical formulation of field theory. Owing to covariance, the space and time arguments of fields are treated on an equal footing. To achieve both covariance and consistency with standard noncovariant canonical quantization of fields in Minkowski spacetime, it is necessary to adopt a covariant Bohmian formulation of quantum field theory. A preferred foliation of spacetime emerges dynamically owing to a purely quantum effect. The application to a simple time-reparametrization invariant system and quantum gravity is discussed and compared with the conventional noncovariant Wheeler-DeWitt approach.
Elements of the quantization in field theory based on the covariant polymomentum Hamiltonian formalism (the De Donder-Weyl theory), a possibility of which was originally discussed in 1934 by Born and Weyl, are developed. The approach is based on a recently proposed graded Poisson bracket on differential forms in field theory (see e.g. hep-th/9709229). A covariant analogue of the Schrödinger equation for a hypercomplex wave function on the space of field and space-time variables is put forward. It is shown to lead to the De Donder-Weyl Hamilton-Jacobi equations in quasiclassical limit. A possible relation to the functional Schrödinger picture in quantum field theory is outlined.
In a previous work by the author it was shown that the Koopman-von Neumann formulation of classical mechanics (KvN) exhibits quantum interference. As such it was there claimed that KvN ought to be considered as a proper quantum mechanical theory in a similar sense as ordinary non-relativistic quantum mechanics (OQM) is. In this article this claim is made manifest by showing that KvN and OQM can be taken as merely inequivalent representations resulting from one and the same quantization scheme. Of course, as the typical notion of quantization necessitates the canonical commutation relations (CCR) to hold, which they do not in KvN, the concept needs to be properly revised. To justify such a revision the typical reasons for necessitation the CCR, reductionism and quantization as 'turning brackets to commutators', are examined and found insufficient.