I don't understand any of the physics, but this is an interesting math idea that I'd never encountered before. I think this refers to the tensor product of the four real division algebras: R (x) C (x) H (x) O (where R = reals, C = complexes, H = quaternions, O = octonions, and (x) denotes tensor product over the real numbers. Tensor products are one of the two basic ways to combine algebras over a ring. But if I recall how I learned about this in a course called 18.26, I think everything was commutative. (Which the quaternions and octonions aren't, but I'm not sure it matters.) I will attempt to unwind the definition of tensor product of algebras, since I'd like to know what this weird thing is. *****-----until further notice, R means any commutative ring-----***** Let's say A and B are algebras over a commutative ring R. Then their tensor product A (x) B will be the quotient of the free algebra over R generated by the elements {a_j (x) b_j | a_j in A, b_j in B}, i.e., A*B = {Sum r_j (a_j (x) b_j), a_j in A, b_j in B, r_j in R, Sum is finite}. To get the tensor product we need to factor out by the subring Z generated by elements of the form ((ra)(x)b) - (a(x)(rb)), r in R, a in A, b in B and the like that give the tensor product its bilinearity. ----- *****-----From now on R is just the reals again-----***** Dimensions of tensor products multiply, so the dimension of R (x) C (x) H (x) O where (x) is always understood as tensor with respect to R=reals, is 1 x 2 x 4 x 8 = 64. Hmm, weird. Anyone care to explain (anything about) the physics? —Dan Henry Baker wrote: ----- ... ... Combined as $latex \mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, the four number systems form a 64-dimensional abstract space. ... ... -----