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

Abstract

The Lagrangian and Hamiltonian properties of classical electrodynamics models and their associated Dirac quantizations are studied. Using the vacuum field theory approach developed in (Prykarpatsky et al. Theor. Math. Phys. 160(2): 1079-1095, 2009 and The field structure of a vacuum, Maxwell equations and relativity theory aspects. Preprint ICTP) consistent canonical Hamiltonian reformulations of some alternative classical electrodynamics models are devised, and these formulations include the Lorentz condition in a natural way. The Dirac quantization procedure corresponding to the Hamiltonian formulations is developed. The crucial importance of the rest reference systems, with respect to which the dynamics of charged point particles is framed, is explained and emphasized. A concise expression for the Lorentz force is derived by suitably taking into account the duality of electromagnetic field and charged particle interactions. Finally, a physical explanation of the vacuum field medium and its relativistic properties fitting the mathematical framework developed is formulated and discussed.

The Maxwell Electromagnetic Equations and the Lorentz Type Force Derivation—The Feynman Approach Legacy

“…As for the general case of (3.1) results analogous to the above results hold, as described in detail in [15][16][17][18]. We need only mention that the Hamiltonian structure of the general equation (3.1) results naturally from its least action representation (3.7) with respect to the rest reference frame K(τ, q).…”

“…His result was analyzed by many authors [3-9, 11, 24] from different points of view, including its relativistic generalization [10]. As this problem is completely classical, we reanalyze the Feynman's derivation from the classical Hamiltonian dynamics point of view on the coadjoint space T * (N ), N ⊂ R 3 , and construct its nontrivial generalization compatible with results [15,16] of Sect. 1, based on a recently devised vacuum field theory approach [15,18].…”

“…As this problem is completely classical, we reanalyze the Feynman's derivation from the classical Hamiltonian dynamics point of view on the coadjoint space T * (N ), N ⊂ R 3 , and construct its nontrivial generalization compatible with results [15,16] of Sect. 1, based on a recently devised vacuum field theory approach [15,18]. Having further obtained the classical Maxwell electromagnetic equations we supply the complete legacy of Feynman's approach to the Lorentz force and demonstrate its compatibility with the relativistic generalization presented in [15][16][17][18]20].…”

“…1, based on a recently devised vacuum field theory approach [15,18]. Having further obtained the classical Maxwell electromagnetic equations we supply the complete legacy of Feynman's approach to the Lorentz force and demonstrate its compatibility with the relativistic generalization presented in [15][16][17][18]20].…”

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

“…It is well known [5,3,9,12,13,11] that the Hamiltonian formulation of Maxwell's electromagnetic field equations involves a very important classical problem; namely, to intrinsically introduce the Lorentz condition, which guarantees the wave structure of propagating quanta and the positivity of energy. Unfortunately, in spite of extensive classical studies by Dirac, Fock and Podolsky [10], the problem remains open.…”