The electromagnetic Lorentz condition problem and symplectic properties of Maxwell- and Yang–Mills-type dynamical systems

Abstract

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.

The Analysis of Lagrangian and Hamiltonian Properties of the Classical Relativistic Electrodynamics Models and Their Quantization

“…Electrodynamics is essentially characterized by its Lorentz invariance from a theoretical perspective, and this very important property has had a revolutionary influence [7,16,21,37,45] on the whole development of physics. In spite of the breadth and depth of theoretical understanding of electromagnetics, there remain several fundamental open problems and gaps in comprehension related to the true physical nature of Maxwell's theory when it comes to describing electromagnetic waves as quantum photons in a vacuum: These start with the difficulties in constructing a successful Lagrangian approach to classical electrodynamics that is free of the Dirac-Fock-Podolsky inconsistency [9,16,17], and end with the problem of devising its true quantization theory without such artificial constructions as a Fock space with "indefinite" metrics, the Lorentz condition on "average", and regularized "infinities" [7] of S-matrices. Moreover, there are the related problems of obtaining a complete description of the structure of a vacuum medium carrying the electromagnetic waves, and deriving a theoretically and physically valid Lorentz force expression for a moving charged point particle interacting with and external electromagnetic field.…”

On the Classical Maxwell–Lorentz Electrodynamics, the Electron Inertia Problem, and the Feynman Proper Time Paradigm

“…In this section, we will develop further the vacuum field theory approach within the Feynman proper time paradigm, devised before in [12,10], to the electromagnetic J.C. Maxwell and H. Lorentz electron theories and show that they should be suitably modified: namely, the basic Lorentz force equations should be generalized following the Landau-Lifschitz least action recipe [37], taking also into account the pure electromagnetic field impact. When applied the devised vacuum field theory approach to the classical electron shell model, the resulting Lorentz force expression appears to satisfactorily explaine the electron inertial mass term exactly coinciding with the electron relativistic mass, thus confirming the well known assumption [31,53] by M. Abraham and H. Lorentz. As was reported by F. Dyson [15,16], the original Feynman approach derivation of the electromagnetic Maxwell equations was based on an a priori general form of the classical Newton type force, acting on a charged point particle moving in three-dimensional space R 3 endowed with the canonical Poisson brackets on the phase variables, defined on the associated tangent space T (R 3 ).…”

“…This fact was stressed [17,18,43,47,49] by Einstein, Minkowski and Poincaré, and later exhaustively analyzed by R. Feynman, who argued [20] that the dynamical equation of a moving point charged particle is physically sensible only with respect to its proper time reference frame. This is Feynman's proper time reference frame paradigm, which was recently further elaborated and applied both to the electromagnetic Maxwell equations in [22,23,24] and to the Lorentz type equation for a moving charged point particle under external electromagnetic field in [10,12,11,7]. As it was there argued from a physical point of view, the least action principle should be applied only to the expression (2.11) written with respect to the proper time reference frame K τ , whose temporal parameter τ ∈ R is independent of an observer and is a closed differential one-form.…”

The Quantum Fermionic Charged Particle Self-Interaction Problem within the Fock Multitime and Feynman Proper Time Paradigms

“…In the present work, we mostly concentrate on the detailed quantum and classical analyses of the self-interacting shell model charged particle within the Fock multitime approach [21,31] time paradigm [22,23,28,29] subject to deriving the electromagnetic Maxwell equations and the related expression for a Lorentz-like force within the vacuum field theory approach devised in works [8,10,[15][16][17][18][32][33][34]. Furthermore, we will explain and apply the obtained results to treating the classical Lorentz-Abraham [1,44,45,[48][49][50][51][52]54,57,58,60,65,67,73,74,76,79] electromagnetic mass origin problem.…”