Are any of the non-division algebras similar to the grassmann algebra?
For example, Tony Smith has a webpage about zerodivisor algebra, where the Cl(7) Clifford algebra has numbers that square to zero, a2=0, just like grassmann numbers do.
And John Baez has a webpage about the Cayley-Dickson construction that gives an iterative process to get these Clifford algebras. One goes from real numbers to complex numbers to quaternions to octonions to sedenions to complexified sedenions to M(8,R) numbers to M(8,R)+M(8,R) numbers that square to zero, like grassmann numbers. Did I get this all right?
I find this curious because I'm instructed that grassmann numbers are used in supersymmetic field theories (SUSY), but it is not explained other than for convenience why one should use grassmann numbers. See Lenord Susskind video series on supersymmetry.
But, perhaps there is a natural reason why one should use grassmann or Cl(7) numbers. It seems that there is a Cayley-Dickson type construction in quantum theory. There are the reals used for probability, and complex numbers used in wavefunction, and quaternions used in SU(2) QFT, and octonions used in SU(3) QFT. These are all the division algebras. But can this patern be iterated into the non-division algebras, and do they have application in physics as well? One would then keep iterating and applying these higher dimensional algebras until you get to the Cl(7) numbers where you'd be describing SUSY, right?
But perhaps this begs a deeper question as to why only the division algebras find expression in physics. Could it be because the non-division algebras cannot be established over the reals? On wikipedia.org I found the Hurwitz theorem that states in effect that the only algebras that can be established over the reals are the real, complex, quaternion, octonions. (Did I get this right?) So if non-division algebras cannot be established over the reals, does that mean any theory based on non-division algebra could not propagate since spacetime consists of a real field? I have to confess that I've only just started learning what division and non-divsion algebras are, and I'm not sure what it means to say division algebra over the reals. I'm only guessing here and would appreciate any insight you might have.
The closest paper I could find that even has a chance of being relevant is the paper by John Baez and John Huerta, Division Algebras and Supersymmetry I.
Here's a link that explicitly states that the Grassmann algebra is a Clifford algebra:
Extended Grassmann and Clifford algebras. See last sentence of paragraph 1 of Section 1, Preliminaries.
I can't say I understand everything about this sentence. So I wonder, does this mean that the Clifford algebras that give us the reals Cl(0), the complex numbers Cl(1), the quaternions Cl(2), the octonions Cl(3), all division algebras used in physics,... can the Clifford algebras be extended to the non-division algebras, even unto Cl(7) of the Grassmann algebras?
They use grassmann numbers in supersymmetry, but the particles are not observed. I wonder if it might be because the supersymmetry algebra is not a division algebra and cannot be established over the reals so that the fields cannot propage in spacetime (which is real valued). Any thoughts on this would be appreciated.
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