On Sat, Jul 28, 2018 at 4:24 PM, W. Edwin Clark <wclark@mail.usf.edu> wrote:
On Sat, Jul 28, 2018 at 3:31 PM, Mike Stay <metaweta@gmail.com> wrote:
On Sat, Jul 28, 2018 at 12:41 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Sedenions aren't a division algebra.
Octonions aren't associative. And your point is ...?
Furey isn't claiming that this is a Theory of Everything. She's just saying that if you take the Georgi–Glashow GUT, which uses SU(5), and then factor it as ℝ⊗ℂ⊗ℍ⊗𝕆 , which are the four normed division algebras,
SU(5) is a group (hence associative) of dimension 24 as a real manifold ( https://en.wikipedia.org/wiki/Special_unitary_group#Properties )
ℝ⊗ℂ⊗ℍ⊗𝕆 is a non-associative algebra over the reals of dimension 1*2*4*8= 64.
So how can the latter be a factorization of the former? Clearly there is something else going on.
Sorry, I phrased that badly. Here's the abstract from the relevant paper: We demonstrate a model which captures certain attractive features of SU(5) theory, while providing a possible escape from proton decay. In this paper we show how ladder operators arise from the division algebras ℝ, ℂ, ℍ, and 𝕆. From the SU(n) symmetry of these ladder operators, we then demonstrate a model which has much structural similarity to Georgi and Glashow's SU(5) grand unified theory. However, in this case, the transitions leading to proton decay are expected to be blocked, given that they coincide with presumably forbidden transformations which would incorrectly mix distinct algebraic actions. As a result, we find that we are left with Gsm=SU(3)C×SU(2)L×U(1)Y/ℤ6. Finally, we point out that if U(n) ladder symmetries are used in place of SU(n), it may then be possible to find this same Gsm=SU(3)C×SU(2)L×U(1)Y/ℤ6, together with an extra U(1)X symmetry, related to B−L. -- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com