This research dissertation has three objectives. The first objective is to develop a Lagrangian formulation of the canonical proper-time many-particle theory for classical electrodynamics. The original approach was based on the Hamiltonian formulation, assuming that the retarded (scalar) potentials were symmetric and the vector potentials were anti-symmetric (upon quantization the Hamiltonian formulation leads to Schrodinger and Heisenberg representations, while a Lagrangian approach leads to the Feynman path integral formulation). In general, the symmetry conditions on the potentials are incorrect, so that we plan to formulate the Lagrangian with the correct symmetry conditions for the global approach to interacting particles and later find the corresponding Euler-Lagrange equations. This will lead to a different canonical proper-time force for these two cases as compared to the Hamiltonian theory.

The second objective is to investigate and compare the canonical proper-time gauge conditions as applied to the Maxwell’s equations, which are expressed in terms of scalar and vector potentials. The third objective is to apply the canonical proper-time formulation of classical electrodynamics to the study of highly focused electromagnetic beams and pulses. The current approach is to modify the complex source method to include both advanced and retarded potentials (see Fedotov et al. [1]). This approach eliminates the singularities brought about by point sources and implements counter-propagating beams or pulses. We then extract a single beam out of the system by means of a Fourier transform. Since with the proper-time theory the corresponding differential equation takes the form of the telegraph equation, we plan to study and investigate solutions to this equation.

Results are in the form of derived equations and are determined in a qualitative way. Therefore we do not seek to determine or apply results quantitatively through data or other means. Such results may be applied in future work or research as an extension of what has been derived thus far.