## Clifford Algebra and Dirac Equation for TE, TM in Waveguide.

**Authors:** Giuliano Bettini

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 ( iσ_{3} 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.

**Comments:** 51 pages, V1 and v3 in Italian, V2 and v4 in English, (slightly amended, corrected formula (123)) .

**Download:** **PDF**

### Submission history

[v1] 30 Oct 2009

[v2] 5 Nov 2009

[v3] 20 Feb 2010

[v4] 21 Feb 2010

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