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This question may fit better in the discussion of "Why Complex variables are required by QM?", but it also relates to the extent to which arguments by Hestenes are accepted in mainstream physics and may deserve its own discussion?

It seems to me that in the paper "The zitterbewegung interpretation of Quantum Mechanics " Foundations of Physics , 20 1213-1232 Prof Hestenes argues that spin is an actual precession of something physical and that complex variables exist in Quantum Mechanics because of this. I have also heard that Geometric Algebra, which I think was invented by Prof Hestenes can clearly explain why complex variables are required in QM. If these are both the case (spin is a physical phenomenon and GA explains why complex variables are used in QM) can anyone explain why QM is not taught beginning with the assumptions of Geometric Algebra?

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it's called the fundamental theorem of algebra –  Ryan Thorngren Oct 11 '12 at 15:42

2 Answers 2

The reason it is not taught is because geometric algebra (Clifford algebra to the rest of the world) is a specialized tool for producing certain representations of the rotation/Lorentz group, and it does not have a distinguished place as a defining algebra of space-time.

What Hestenes does is reformulate everything using Clifford algebras instead of the usual coordinates. This is like taking the "slash" of every vector. You can do this, but it is difficult to motivate. Hestenes' motivation is to make an algebra out of the space-time coordinates. But it is not at all clear that one should be able to multiply two vectors and get something sensible out. Why should it be so physically? The reason is clear when you have Dirac matrices, but to introduce this as an axiom is unmotivated from a physical point of view, and I do not think will help pedagogically.

Further, this formalism, while it naturally incorporates forms, has a hard time with symmetric tensors, which are just as natural as antisymmetric ones. You can represent symmetric tensors, of course, using spin indices, but there is no reason to prefer the Clifford algebra way, because you can use other formalisms with equivalent content.

To produce an algebra out of vectors and to claim that it is physical, requires an argument that vectors should multiply together to make antisymmetric tensors plus a scalar. This argument is lacking--- the scalar product and the wedge product are two separate ideas which do not need to be combined into a Clifford algebra unless you are motivated by spin-1/2.

I think that this idea is cute, but it is not pedagogically useful in itself. It might be useful as a way of motivating the Ramond construction in string theory (this is just a hunch, I don't know how to do this), or other otherwise mysterious Dirac matrix intuitions.

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Supersymmetry comes from considering these kind of handy approaches to constructing representations. Is there a sensible way to see/estimate what one might miss if one only takes these specific tools? How to specifiy the compliment of a method to construct objects (represenations) like that? –  NikolajK Oct 11 '12 at 14:59
@NickKidman: This is not a handy approach, it's a truncation of Dirac algebra. If you use general tensor methods, then you are fine, but that's not GA. GA truncates, and truncation is always bad. Generality is good. SUSY does not use GA, it uses superspace, and this is a trick that illuminates and obscures in nearly equal measures. One needs to learn the different superspaces along with a component approach to get a handle on it. –  Ron Maimon Oct 11 '12 at 17:47

Complex numbers certainly appear in QM and are very convenient, but, as I wrote here before, it is not obvious that they (or pairs of real numbers) are necessary. For example, one can make a scalar wavefunction real by a gauge transform (Schrödinger (Nature (London) 169, 538 (1952))). A similar result was obtained for the Dirac equation without any use of geometric algebra (http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf (my article published in the Journal of Mathematical Physics) or http://arxiv.org/abs/1008.4828 ).(EDITED)

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Dear akhmeteli, For your information, Physics.SE has a policy that it is OK to cite oneself, but it should be stated clearly and explicitly in the answer itself, not in attached links. –  Qmechanic Oct 11 '12 at 14:01
Thank you for pointing that out. I think my belated editing of this particular answer would not be advisable though, unless you tell me otherwise. –  akhmeteli Oct 11 '12 at 14:31
I don't see why not. Please make the change. –  Qmechanic Oct 11 '12 at 14:37
Thank you again. Done. –  akhmeteli Oct 11 '12 at 14:44

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