DeDonder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory

TL;DR

This paper introduces a covariant hypercomplex quantum framework for field theory based on the DeDonder-Weyl formulation, replacing complex numbers with Clifford algebra, and demonstrates its classical limit and Ehrenfest correspondence.

Contribution

It presents a novel hypercomplex extension of quantum mechanics for covariant field theory using Clifford algebra, unifying classical and quantum descriptions.

Findings

Derivation of a covariant hypercomplex Schrödinger equation

Classical limit recovers DeDonder-Weyl Hamilton-Jacobi equation

Expectation values satisfy DeDonder-Weyl canonical equations

Abstract

A quantization of field theory based on the DeDonder-Weyl covariant Hamiltonian formulation is discussed. A hypercomplex extension of quantum mechanics, in which the space-time Clifford algebra replaces that of the complex numbers, appears as a result of quantization of Poisson brackets of differential forms put forward for the DeDonder-Weyl formulation earlier. The proposed covariant hypercomplex Schr\"odinger equation is shown to lead in the classical limit to the DeDonder-Weyl Hamilton-Jacobi equation and to obey the Ehrenfest principle in the sense that the DeDonder-Weyl canonical field equations are satisfied for the expectation values of properly chosen operators.