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## paper Niels Gresnigt’s, “Relativistic Physics in the Clifford Algebra Cl(1, 3)”

Posted by peeterjoot on July 11, 2009

I recently stumbled on the PhD thesis of Niels Gresnigt.

Chapter II of this thesis has an excellent introduction to Clifford algebras. He starts in the logical way, with a perpendicularity argument, much preferable than the other common approach (ie: the ad-hoc definition of the vector product $a b = a \cdot b + a \wedge b$ in terms of the dot and wedge). Niels’s treatment is like the axiomatic approach found in many other places, but done nicely without the formality.

I’m not sure I buy his argument for why the spatially negative metric is natural. Correspondence with a right handed rule or the quaternions both seem like arbitrary choices to me. His main problem is one of motivating either the choice of signature $(+,-,-,-)$ or $(-,+,+,+)$, and the light like pathlength argument he makes does that up to the sign. After that it doesn’t seem to me that there’s any good physical reason for either pick. Having proper time expressible directly in in terms of the vector square does work nicely for relativistic dynamics, so it probably doesn’t matter too much how he gets to that choice.

I have to admit skipping most of chapter III and IV. I’ve examined translation to and from the Pauli matrix algebra and clifford algebra formalisms in as much detail as I care to, and then looked at the Dirac matrix algebra in the same way. The Null vector corresponance between light cone paths and non-invertable algebraic elements elaborated on in chapter IV is something that doesn’t seem unnatural enough to me to warrent to spending the effort to read this chapter (especially since it builds on his matrix work of chapter III). What is interesting there are the sneak peek references to a generalized Maxwell equation where the potential includes trivector and vector elements.

In chapter V (the Lorentz force from energy considerations) the examples of interaction energies in this chapter are quite nice. It is interesting to see how only the coupled energy terms lead to interaction forces. Reading this chapter (it was interesting, but incomplete seeming, and also not quite relavent seeming to the thesis) left me with a desire to read the reference Lagrangian interaction.

I can make it part way into chapter 6 before it gets too complex for me. Once proca equations, and spin, and the generalized maxwell equation gets into the picture I can’t make sense of things with what I currently know.