The relation between the canonical Hamilton-Jacobi equation and the covariant Hamilton-Jacobi equation for Maxwell's electrodynamics

TL;DR

The paper investigates how the canonical Hamilton-Jacobi equation for Maxwell fields, expressed in terms of a functional of field configurations, relates to the covariant De Donder–Weyl Hamilton-Jacobi equation defined on a finite-dimensional space of potentials and spacetime coordinates. By performing a spacetime split and restricting to initial data, the authors derive the canonical HJ equation and the Gauss law from the covariant formulation without resorting to the canonical Hamiltonian framework. This demonstrates a concrete link between canonical and precanonical (covariant) approaches to field quantization and clarifies how quasi-classical limits might connect different quantization schemes. The results point to extensions to non-Abelian gauge theories and TEGR, with potential implications for quantum gravity and gauge theory formalisms.

Abstract

The aim of this paper is to understand the relation between the canonical Hamilton-Jacobi equation for Maxwell's electrodynamics, which is an equation with variational derivatives for a functional of field configurations, and the covariant (De Donder-Weyl) Hamilton-Jacobi equation, which is a partial derivative equation on a finite dimensional space of vector potentials and spacetime coordinates. We show that the procedure of spacetime splitting applied to the latter allows us to reproduce both the canonical Hamilton-Jacobi equation and the Gauss law constraint in the Hamilton-Jacobi form without a recourse to the canonical Hamiltonian analysis. Our consideration may help to analyze the quasi-classical limit of the connection between the standard quantization in field theory based on the canonical Hamiltonian formalism with a preferred time dimension and the precanonical quantization that uses the De Donder-Weyl Hamiltonian formulation where space and time dimensions treated equally.