Pati-Salam

One way of looking at all of this is as a geometric setting for Grand Unified theories.

The Georgi-Glashow model of SU(5) and its associated Spin(10) enlargement is more popular.

But the Pati-Salam model is also attractive when presented differently.

In the usual presentation of the Pati-Salam grand unified theory, the groups are given as SU(4) × SU(2) × SU(2) which suffers from ambiguities.

To begin with, there are no fewer simple factors in Pati-Salam theory than there are reductive factors in the Standard Model.

Further, there is the naming ambiguity as we have many names for the same objects:

However, if we accept that non-compactness is the price generally to be paid in any unified theory that incorporates both space and time, we should expect reduction to non-compact groups5 whose maximal compact subgroups 5We, years ago, remember following such reductions along the lines of Bar-Natan and Witten which involve incorporating an endomorphism of the non-compact complements into to the Hodge Star operators but have yet to successfully resurrect the technique, nor have we found our notes for this period.

Such problems of reconstruction over nearly 40 years are, lamentably, found throughout this document but they are likely to get worse rather than better by waiting to fix them. For those sensitive to errors of this type we recommend waiting for a future draft.

will always be Semi-simple or at least reductive. Thus, we view the semi-simple nature of Pati-Salam paradoxically as more of a blessing than curse given that we are trying to generate quantum numbers from the anthropic choice of Spin(1, 3) which ensures the existence of hyperbolic PDE dynamics unspoiled by ellipticity. To this end we have:

Pati-Salam Grand Unified Group

where we have made use of the low dimensional isomorphism SU(4) = Spin(6) which can be best understood through the study of sphere transitive Weyl spin representations via Clifford Algebras.

As for the SU(2)×SU(2) factors, we see that it is most advantageous to view this via low dimensional isomorphism with Spin(4) so as to obtain the following diagram:

indicating that the appearance of a phantom 10-dimensional representation is common to both the Georgi-Glashow and Pati-Salam theories and strongly suggests looking for a fundamental explanation. One disadvantage on finding oneself on the Pati-Salam branch of the above tree, is that it suggests non-compact groups bigger than itself which are difficult to accommodate in unitary Bosonic theories with bounded energy.

An advantage however is that it does not lead immediately to proton decay like the original SU(5) model of Georgi-Glashow. Further, by privileging both a compact structure group within Spin(6, 4) and a complex structure on the phantom 10-dimensional representation, the Standard Model group appears to be very close to being at the intersection of those requirements.