The Pati-Salam model is one of the first Grand Unified Theories (GUTs) to be built, following the establishment of the Standard Model in the early 1970s. The basic idea was to extend the strong gauge group SU(3) to SU(4), and at the same time extend the weak gauge group Spin(3) to Spin(4). Hence they extended the weak-strong gauge group Spin(3) x SU(3) to Spin(4) x SU(4).
The first thing to say about this is that Spin(3) x SU(3) contains only 11 of the 12 degrees of freedom in the Standard Model gauge group. To get all 12 we need Spin(3) x U(3), which suggest extending to Spin(4) x U(4). This gives us 22 degrees of freedom instead of the Pati-Salam 21.
The second thing to say is that weak Spin(3) is actually treated as a complex Lie group in the Standard Model, not a real Lie group. In other words, an extension from Spin(3) to Spin(3,1) is already used in the Standard Model. It therefore seems superfluous to extend from Spin(3) to Spin(4) = Spin(3) x Spin(3) as well, particular as the extra copy of Spin(3) has never had a satisfactory physical interpretation.
I therefore propose that, instead of extending Spin(3) to Spin(4), we extend to Spin(3,1). Similarly, I propose that, instead of extending U(3) to U(4), we extend to U(3,1). The reason for this is so that we can clearly separate the three colours of quarks, in U(3,0), from the colour of the leptons, in U(0,1).
Putting the Lorentz group in as well, we get a group Spin(3,1) x Spin(3,1) x U(3,1) with 6+6+16=28 degrees of freedom altogether. This contains the Standard Model Spin(3,1) x Spin(3) x U(3), with 18 degrees of freedom, plus 10 parameters, which consist of 9 boosts and one rotation. The rotation parameter is probably the Weinberg angle, and the 9 boosts relate to the 9 fundamental masses (three electrons and six quarks, or whatever you think the fundamental masses are).
Now all we are missing is 4+4=8 parameters (mixing angles) in the CKM and PMNS matrices, and 3 coupling constants for the three forces. Adding in the mixing angles gives us 36 degrees of freedom, with signature (12,24), and therefore the group Sp(3,1). The coupling constants arise from adding in the quaternion scalars.
Our first task then, is to mix the two copies of Spin(3,1) into a copy of U(3,1). It is certainly possible to choose two disjoint copies of Spin(3,1) in U(3,1), so that they cover 12 of the 16 dimensions of the Lie algebra. The remaining 4 are rotations, one in each of the four quaternionic coordinates, so they split 3+1 like everything else. The mixing here does not involve the strong force, so does not involve quarks, so the mixing angles are not in the CKM matrix, so they must be in the PMNS matrix. In the usual formalism, three of them mix the three generations of neutrinos, while the last is an overall “phase” that is very difficult to measure experimentally.
Our second task is to mix the resulting copy of U(3,1) with the other (colour/strong) copy of U(3,1). Again, there is no problem in choosing two such groups, covering 32 of the 36 dimensions of the Lie algebra of Sp(3,1). The remaining four are again rotations, and again there is one for each coordinate, so again they split into 3 generation mixing angles and one overall phase. This time the strong force and the quarks are involved, and these parameters define the CKM matrix.
Our third task is identify the coupling constants. These come from quaternionic scalar multiplications, which act by (say) left-multiplications, while the gauge group Sp(3,1) acts by right multiplication by quaternionic matrices. We have to match up the left-multiplications with the right-multiplications in some way. This is an exercise in calculating quaternionic eigenvalues of a quaternionic matrix, but the eigenvalues are not well-defined, because they depend on the basis chosen for the quaternions. In the (complex) Standard Model, we only have complex scalar multiplications, and the (complex) eigenvalues are well-defined. The coupling constants come from breaking the quaternionic scalar symmetry down to a complex scalar symmetry. But this symmetry-breaking of scalars has no physical effect, because the quaternionic scalars do not interact with matter, or even with spacetime. Ultimately, the coupling constants disappear from the model.
You might think that we still need neutrino masses – but the three boosts that might be used for this purpose are being used for Lorentz transformations. We still have the PMNS matrix, which describes neutrino oscillations, and therefore contains the same physical information as the neutrino masses, so we do not need both.
That’s it, I suppose. That’s how to modify the Pati-Salam model to produce a GUT based on Sp(3,1), that contains the Standard Model plus 9 masses and 9 mixing angles. Next time I’ll tell you how to interpret (part of) the exact same model as curvature of spacetime and General Relativity. That re-defines the model as a Theory of Everything (TOE). Moreover, it explains what is missing from General Relativity, and how the quaternionic (non-commutative, or symplectic) geometry completes GR to a theory of gravity that works on all scales, and is therefore “renormalizable”, and quantised. The usual real or complex geometry is not scale-invariant, and is therefore not renormalizable, and cannot be quantised.