The Electromagnetic Lorentz Condition Problem and Symplectic Properties of Maxwell and Yang-Mills Type Dynamical Systems

TL;DR

This paper develops a symplectic reduction framework for Maxwell and Yang-Mills systems that incorporates the Lorentz condition, addressing longstanding issues in their Hamiltonian formulations.

Contribution

It introduces a novel symplectic reduction approach that naturally includes the Lorentz condition in Maxwell's equations and extends to Yang-Mills systems.

Findings

Formulated a symplectic reduction theory for Maxwell equations including the Lorentz condition.

Derived symplectically reduced Poisson structures for Yang-Mills equations.

Provided a geometric solution to the Dirac-Fock-Podolsky problem.

Abstract

Symplectic structures associated to connection forms on certain types of principal fiber bundles are constructed via analysis of reduced geometric structures on fibered manifolds invariant under naturally related symmetry groups. This approach is then applied to nonstandard Hamiltonian analysis of of dynamical systems of Maxwell and Yang-Mills type. A symplectic reduction theory of the classical Maxwell equations is formulated so as to naturally include the Lorentz condition (ensuring the existence of electromagnetic waves), thereby solving the well known Dirac -Fock - Podolsky problem. Symplectically reduced Poissonian structures and the related classical minimal interaction principle for the Yang-Mills equations are also considered. 1.