Gauge Invariance and Classical Electrodynamics

Abstract

A Lagrangian for deriving the Maxwell-Lorentz field equations is obtained by starting from a general tensor of the second rank ${\mathcal{G}}_{\ensuremath{\mu}\ensuremath{\nu}}$, which is expressed as the sum of a symmetric and an antisymmetric part ${\mathcal{G}}_{\ensuremath{\mu}\ensuremath{\nu}}=\ensuremath{\lambda}{g}_{\ensuremath{\mu}\ensuremath{\nu}}+{F}_{\ensuremath{\mu}\ensuremath{\nu}}$. The symmetric part is identified with the metric tensor and the antisymmetric part with the electromagnetic field. A relationship between this second rank tensor and the gauge invariant Ricci-Einstein tensor is established by means of the gauge invariant theories of Weyl and Eddington. This relationship leads directly to the Klein-Gordon relativistic wave equation for a point charge moving in an electromagnetic field provided the function $\ensuremath{\lambda}$ is properly chosen.The Lagrangian density is defined as the quantity ${(\ensuremath{-}|{\mathcal{G}}_{\ensuremath{\mu}\ensuremath{\nu}}|)}^{\frac{1}{2}}$, where $|{\mathcal{G}}_{\ensuremath{\mu}\ensuremath{\nu}}|$ is the determinant associated with the tensor ${\mathcal{G}}_{\ensuremath{\mu}\ensuremath{\nu}}$. This choice is made for the Lagrangian density since ${(\ensuremath{-}|{\mathcal{G}}_{\ensuremath{\mu}\ensuremath{\nu}}|)}^{\frac{1}{2}}d\ensuremath{\tau}$ is the simplest generalized invariant volume element. Since the Lagrangian density is nonlinear and irrational as it stands, it is first rationalized by means of the Dirac matrices. If the Lagrangian that is obtained in this way is varied with respect to the vector and scalar potentials, one obtains a set of Maxwell-Lorentz equations for a charge-current distribution that is defined in terms of the field potentials. These equations are almost identical with those recently obtained by Dirac in his new classical electrodynamics. It is shown from the field equations that the velocity of the charge distribution is given by the relation $\mathrm{v}=(\frac{\mathrm{A}}{\ensuremath{\varphi}})c,$ where A is the vector potential, $\ensuremath{\varphi}$ is the scalar potential, and $c$ is the speed of light. This is identical with the result obtained from the retarded potentials of a point charge. For the static case it is shown that the field equations lead to solutions for the fields and potentials that are finite everywhere, and the self-energy of the point charge is finite. The classical radius of the electron, the Compton wavelength and the fine structure constant come into the theory quite naturally.