I here append Michael's comment..
Author Michael James Goodband replied on Dec. 7, 2012 @ 16:44 GMT
I just want to point out that a (S0, S1, S3, S7) universe is not just a proposal. I claim that it is the conclusion when we follow the spirit of Rick's standard of letting the algebras do the talking. However, Rick's choice of listening to the octonions violates the meta-principle of "make no preference". Instead, we should listen to all the normed division algebras R, C, H, O and let the octonions tell us their role. Now as a physicist, I want some experimental facts, but I will only need the Standard Model table for left-handed particles and the mismatch with their right-handed counterparts. table
I claim that you can read-off the structure of the reality in which these particles exist from the properties of the normed division algebras given in John Baez's paper - attached for easy reference. I really do mean read-off, as in number of calculations = 0. The first things to note are the left and right-handed spinor irreducible representations in 4 (quaternions) and 8 dimensions (octonions) in Table 4 (p161 or pg17 of PDF), and the normed trialities of irreducible representations that give the quaternions and octonions on p162 or pg18 of PDF. Specifically note the triality of the irreducible representations V_8, S_8, S-_8 and their relationship to the Dynkin diagram D4 (on p163 or pg19 of PDF) of Spin(8) described on p162 (pg18 of PDF).
Now count up the number of particles in the SM table: in each column there are 2 quarks with 3 colours plus a lepton and lepton neutrino, giving 8. This matches the triality of the 8-dim irreducible representations V_8, S_8, S-_8 and so the octonions tells us that they are about the different particle charges: octonions=particle space, just as quatenions=real space. Returning to Table 4, we see that left and right handed spinors only occur for the octonions and quaternions. Now to get the 8-dim octonion spinors to have the spatial handedness of quaternion spinors - and so match the chirality of the SM particle table - the octonion space would need to be mapped to the quaternion space so as to acquire their spatial handedness. This requires first picking out a 3-vector from the octonion space - which is something that has to be done in order to define the cross-product in 7D space residing in the octonions - and then map it to spatial 3-space. This mapping of a 3-vector in the octonion space to 3-space gives the 3 8-dim irreducible representations a spatial chirality, and breaks the symmetry of the octonion space. Thus the normed division algebras have just told us what the Higgs field is really all about.
Returning to the D4 Dynkin diagram on p163 (pg19 of PDF) to consider what the symmetry breaking for the SM particles must be, using the Dynkin diagrams: SU(2) is 1 node; SU(3) is 2 linked nodes; and SU(4) is 3 linked nodes. If we imagine breaking all the links of D4, we would have a central SU(2) and triality involving an outer SU(2) ~Spin(3), plus a U(1) symmetry between the central and outer SU(2) groups. Now symmetry breaking in a Dynkin diagram involves removing a node, which here would be the central SU(2), leaving intact the triality over the outer SU(2)~Spin(3) and the U(1) symmetry. A 3-dim colour representation - needed to get particles in the 8-dim of V_8, S_8, S-_8 - selects Spin(3) over SU(2), giving the symmetry breaking encoded in the SM particle table as:
Spin(3)*SU(2)*U(1) -> Spin(3)*U(1)
The normed division algebras are telling us that the colour group cannot be SU(3), but is actually Spin(3). The condition for getting particle-like objects in any symmetry breaking pattern demands closed spaces, which here means S7 -> S3*S3*S1 and the monopole homotopy group PI6(S2) = Z4*Z3 (S6 being left after the unbroken S1 is put to one side) confirms the triality of 3 families of 4 particles in the SM table. The homotopy group PI7(S3) =Z2 also confirms the chirality of the mapping from the closed octonion space (particle space S7) to the closed quaternion space (closed spatial universe S3). These homotopy groups obviously come from the structure of the corresponding algebras.
Given the SM particle table, the normed division algebras are screaming out the structure of reality to anyone who is listening. Arguing against this conclusion is not arguing against me, but arguing against the fundamental fermions and the structure of the normed division algebras - that's an argument that's lost before it's even begun. Structure of reality: DONE.
That's the easy way of how we arrive at what the structure of reality is, now how did the universe arrive at the answer?
MichaelAttachment #1: baezocto.pdf