From: Terry Pilling <terry@offshell.phys.ndsu.nodak.edu>
Subject: Re: Applications of Non-Associative Algebras?
Date: Thu, 31 Aug 2000 09:36:43 -0400
Newsgroups: sci.physics.research
Summary: Physical manifestations of Bott periodicity 

There are several interesting statements in Okubo's book which 
indicate applications of non-associative algebras.
In particular the last several chapters.
He says in a couple of remarks:

"It appears that we can generalize the Einstein theory into an 
octonionic theory."

and 

"The standard Yang-Mills gauge theory can also be recast in 
Chern-Simon form, with some non-associative algebra."

There are lots of other cool things about the Octonions in particular
though.
I have been interested in non-associative algebras for some time
after having read and been inspired by JB's interesting Week 104 of 
"this weeks finds".
I have especially been intrigued by the connections with Bott periodicity
and its relationship to the division algebras. 
The Hurwitz theorem of the division algebras is connected to Bott
periodicity and Bott periodicity is intimately related to Clifford 
algebras and Clifford algebras are connected to gauge theories (Dirac
operators). To see the connection with Clifford algebra you could
refer to Atiyah, Bott and Shapiro's "Clifford Modules", Topology, Vol 3.
1964. (I have had a tough time trying to go through that paper though).

A recent paper by Nieto and Alejo-Armenta hep-th/000184 proves Hurwitz
theorem directly from tensor analysis on the structure constants of a
division algebra and then proceeds to prove that the spheres S^1, S^3 
and S^7 are parallelizable. So the parallelizablity of these spheres is
a direct result of the division algebras. This is all well known stuff
but it is interesting that these authors proved it using simple tensor
analysis and didn't have to bring in the high powered math.

Another thing that is neat is that for any topological space, the qth
homotopy group \pi_q(X) is defined as the set of base-point preserving 
continuous maps f:S^q -> X. When X is a compact Lie group you see the
periodicity theorem (when N+1 > (q+2)/d below). The following is a table
of the qth homotopy groups for 3 of the more important Lie groups over
a range of q. Note that Z denotes the integers and Z_2 is integers modulo
2.

       BOTT PERIODICITY TABLE
         for N+1 > (q+2)/d

         d=2      d=1      d=4
 ---------------------------------
|  q  |  U(N)  |  O(N)  |  Sp(N)  | PHYSICAL MANIFESTATION
|=================================| ----------------------
|  0  |   0    |  Z_2   |   0     | (parity)
|---------------------------------|
|  1  |   Z    |  Z_2   |   0     | (Electromagnetism and Spin)
|---------------------------------|
|  2  |   0    |  0     |   0     |
|---------------------------------|
|  3  |   Z    |  Z     |   Z     | (Yang-Mills Instantons)
|---------------------------------|
|  4  |   0    |  0     |   Z_2   |
|---------------------------------|
|  5  |   Z    |  Z     |   Z     |
|---------------------------------|
|  6  |   0    |  0     |   0     |
|---------------------------------|
|  7  |   Z    |  Z     |   Z     | (Octonionic Instantons)?
|---------------------------------|
|  8  |   0    |  Z_2   |   Z     |
|---------------------------------|
|  9  |   Z    |  Z_2   |   0     |
|---------------------------------|
| 10  |   0    |  0     |   0     |
|---------------------------------|
| etc |        |        |         |


It seems like most of the non-trivial ones have a physical
manifestation and I have indicated several of the interesting
ones above. In particular notice the spheres which are 
parallelizable via our division algebra argument. 
The fact that the sphere is parallelizable means that you can 
construct a global line bundle out of it over the group manifold 
and due to this you can have twisting in the map to the group and the
number of twists is the Z in the homotopy groups for S^1, S^3 and S^7 above.

I have mentioned that the maps from S^7 -> X corresponding to \pi_7(X)
above are non-trivial and the physical manifestation is octonionic
instantons. I am going out on a limb with that statement but I think it
has to be true. For more info on Octonionic instantons there are quite a
number of papers, some of which are:

Fubini and Nicolai, Physics Letters, June 1983
Jeff Harvey, Phys. Rev. Lett. 66,5, 1991 (549)
Dereli, et. al., Physics Letters, 126B, 1983 (33)
hep-th/0002155
hep-th/9910003
and many more...

The really interesting thing right now (for me) is the connection
to supersymmetry. Quite a number of people have worked on this as
a search on the archives will show. I have also found some very
astonishing facts but don't have time or space here for them. I will 
state them in a more precise way elsewhere.



-Ter