Clifford Algebra and Dirac equation for TE, TM in waveguide

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 if a (first order) Dirac equation is involved. Making use of Clifford Algebra, by first it is shown that a spinor ψ satisfying Dirac equation describes, trough the relativistic energy impulse four vector, the energy propagation of the electromagnetic field into a waveguide and in free space. At the same time ψ 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 all the time. The equivalence is embedded in the usual V and I mode description [1]. The Dirac equation for TE, TM modes opens new interesting interpretations. For example the effect on ψ of a gauge transformation with the electromagnetic gauge group generator ( 3 σ i in the Hestenes notation [2]) 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 suggest an analogy with electron, positron and possibly neutrino for the TEM. [1] S. Ramo, J. R. Whinnery, T. van Duzer, “Fields and Waves in Communication Electronics”, John Wiley (1994) [2] D. Hestenes, “Space-time structure of weak and electromagnetic interactions”, Found. Phys. 12, 153-168 (1982) Clifford Algebra and Dirac equation for TE, TM in waveguide. Introduction 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, taking into account only the dependence from propagation coordinate, obey to a second order equation which mathematically speaking is the Klein Gordon equation, as for a relativistic particle. Since this is a relativistic equation of 2nd order one wonders if there are, and what are the corresponding relativistic equations of 1st order. In the electromagnetic theory or in the theory of waveguides such kind of equations for TE, TM modes do not exist. We have Maxwell equations of course, but they give the second order wave equation and not the Klein Gordon equation. In analogy with the electron we suppose that such equations are the Dirac equations. This is, in fact, true. To show this, Clifford Algebra is employed. (Note: useful references (electromagnetism, Clifford Algebra etc.) are in [1]...... [7], and in [8]......[14] for electron models, Dirac equation and so on). It is also shown that this description with Dirac equation has been implicit in the waveguide theory all the time. The equivalence is embedded in the usual (see for example [7]) V and I mode description. A Dirac spinor ψ describes TE, TM modes in such a way that only global characteristics are accounted, I mean energy, impulse and polarization. Practically the action of ψ is to give the relativistic energy impulse four vector of the mode (the total energy-momentum vector), and also polarization. The ψ solutions for TE, TM (and TEM) modes corresponds to the electron, positron (and neutrino) plane wave solutions of the Dirac equations. But obviously the Dirac equation for TE, TM modes opens new interesting interpretations. For example the effect on ψ of a gauge transformation with the electromagnetic gauge group generator ( 3 σ i in the Hestenes notation [12]) 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. Maxwell equations with Clifford Algebra Maxwell equations are obtained introducing the Clifford number (see A4) or “even number”: (1) ( ) ( ) l t l t jH H Tji jE E F + + + = (in MKSA units E ε e H μ ). The analyticity condition: (2) 0 * = ∂ F (3) τ ∂ ∂ Τ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ z j y i x *