Clifford Algebra and Dirac Equation for TE, TM in Waveguide

Abstract

Following Hestenes and others we explore the possibility that the electron is a (sort of) bound electromagnetic wave. To do this a waveguide analogy is considered. The E, H field components in waveguide satisfy the second-order Klein-Gordon equation. The question is whether a (first-order) Dirac equation is involved. Making use of Clifford Algebra, it is firstly shown that a spinor satisfying Dirac equation describes, through the relativistic energy impulse four-vector, the energy propagation of the electromagnetic field in a waveguide and in free space. At the same time automatically describes TE and TM modes (TEM in free space), each with Right or Left polarization. It is shown that this description with Dirac equation has been implicit in the waveguide theory at all times. The equivalence is embedded in the usual V and I mode description (See, e.g., S. Ramo, J. R. Whinnery, T. van Duzer, “Fields and Waves in Communication Electronics”, John Wiley [1994]). The Dirac equation for TE, TM modes opens new interesting interpretations. For example, the effect on of a gauge transformation with the electromagnetic gauge group generator (see, D. Hestenes, “Space-time structure of weak and electromagnetic interactions”, Found. Phys. 12, 153-168 [1982]) is readily interpreted as a modification of the TE, TM group velocity. This acts as the electromagnetic force on a charge, and requires two opposite sign of (fictitious) charges for TE or TM. Obviously, this suggests an analogy with electron, positron and possibly neutrino for the TEM.