Lagrangian mechanics for fields

An introduction to Lagrangian methods for classical fields in flat spacetime and then in curved spacetime. The Euler-Lagrange equations for Lagrangian densities are obtained, and applied to the wave, Klein-Gordan, Weyl, Dirac, Maxwell and Proca equations. The canonical energy tensor is obtained. Conservation laws and Noether’s theorem are described. An example of the treatment of Interactions is given by presenting the the QED Lagrangian. Finally, covariant Lagrangian methods are described, and the Einstein field eqution is derived from the Einstein-Hilbert action.

London-Proca-Hirsch Equations for Electrodynamics of Superconductors on Cantor Sets

In a recent paper published at Advances in High Energy Physics (AHEP) journal, Yang Zhao et al. derived Maxwell equations on Cantor sets from the local fractional vector calculus. It can be shown that Maxwell equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. However, so far there is no derivation of equations for electrodynamics of superconductor on Cantor sets. Therefore, in this paper I present for the first time a derivation of London-Proca-Hirsch equations on Cantor sets. The name of London-Proca-Hirsch is proposed because the equations were based on modifying Proca and London-Hirsch’s theory of electrodynamics of superconductor. Considering that Proca equations may be used to explain electromagnetic effects in superconductor, I suggest that the proposed London-Proca-Hirsch equations on Cantor sets can describe electromagnetic of fractal superconductors. It is hoped that this paper may stimulate further investigations and experiments in particular for fractal superconductor. It may be expected to have some impact to fractal cosmology modeling too.

A Derivation of Proca Equations on Cantor Sets: A Local Fractional Approach

In a recent paper published at Advances in High Energy Physics (AHEP) journal, Yang Zhao et al. derived Maxwell equations on Cantor sets from the local fractional vector calculus. It can be shown that Maxwell equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. Using the same approach, elsewhere Yang, Baleanu & Tenreiro Machado derived systems of Navier-Stokes equations on Cantor sets. However, so far there is no derivation of Proca equations on Cantor sets. Therefore, in this paper we present for the first time a derivation of Proca equations and GravitoElectroMagnetic (GEM) Proca-type equations on Cantor sets. Considering that Proca equations may be used to explain electromagnetic effects in superconductor, We suggest that Proca equations on Cantor sets can describe electromagnetic of fractal superconductors; besides GEM Proca-type equations on Cantor sets may be used to explain some gravitoelectromagnetic effects of superconductor for fractal media. It is hoped that this paper may stimulate further investigations and experiments in particular for fractal superconductor. It may be expected to have some impact to fractal cosmology modeling too.

Field equations for the massive vector boson from Dirac and Weinberg formalisms

This paper is a follow-up to an earlier paper that discussed the single-particle quantum mechanics of massless bose particles. In the present paper we extend the analysis to the massive vector (j = 1) bosons that occur in electroweak interactions. As in the previous paper we make a connection between a generalization of the Dirac equation and the equations obtained by Weinberg from S-matrix field theory. The starting point is the Bargmann–Wigner generalization of the Dirac equation. This leads to the Proca equations for a vector potential field, then to Maxwell’s equations, which we finally relate to Weinberg’s equations. We spend some time analyzing the quantity Tr(Ψ(x)*Ψ(x)), where Ψ(x) is the Bargmann–Wigner wave function (a symmetric four by four matrix). Using Lagrangian and Hamiltonian density equations, we show that the trace has the interpretation of being the Hamiltonian density for the vector potential field. We also use the Lagrangian analysis to construct a conserved current via Noether’s theorem.