08 Dec 15
Originally posted by sonhouseIt's a slight surprise that this hasn't been looked at before. Non-commutativity is standard in quantum mechanics, [x, p] = ih (where h is the reduced Planck constant) is basically the starting point for the theory. We're used to the various symmetries being described by things like U(1) which is the group of unit complex numbers and also the circle group, and SU(2), the group of unit quaternions and the universal covering group of the group of rotations in ordinary 3 dimensional space. There's a mathematical construction called the Cayley-Dickson construction [1] where complex numbers are constructed from real numbers by forming a pair and defining multiplication. Then one can construct quaternions from complex numbers using the same procedure. The next step produces Octonions [2], which are non-associative. In fact one loses an algebraic property at each step - real numbers are self-conjugate, complex numbers (in general) are not, quaternions don't commute, quaternions, complex and real numbers are all associative, but octonions are not.
Quite an earful!
http://phys.org/news/2015-12-physicists-unusual-quantum-mechanics.html
These things have been known about for over a century and it would be strange if they did not have a physics application. On the other hand they don't form a group so it's not obvious to me how they fit in. I should read the paper [3], if I get through it I'll try and come back.
The unit objects are all important in physics, the unit real number is 1 and is the trivial group. The octonions are not associative so don't form a group, but is related to spin 7.
[1] https://en.wikipedia.org/wiki/Cayley-Dickson_construction
[2] https://en.wikipedia.org/wiki/Octonion
[3] http://arxiv.org/abs/1510.07559