3.5 The Question of Compatibility

We now have two routes to the theory. In Section 3.2 we saw how to reach it via the theory:

Our work in that section and in Section 3.1 showed that this diagram commutes, which is a way of saying that the theory extends the Standard Model.

In Section 3.4 we saw another route to the
theory, which goes through
:

Our work in that section and Section 3.3 showed that this diagram commutes as well. So, we have another way to extend the Standard Model and get the theory.

Drawing these two routes to
together gives us a cube:

Are these two routes to theory the same? That is, does the cube commute?

Proof.
We have already seen in Sections 3.1-3.4 that the
vertical faces commute. So, we are left with two questions involving the
horizontal faces. First: does the top face of the cube

commute? In other words: does a symmetry in go to the same place in no matter how we take it there? And second: does the bottom face of the cube commute? In other words: does this triangle:

commute?

In fact they both do, and we can use our affirmative answer to the
second question to settle the first.
As we remarked in Section 3.4, applying the map
to the Pati-Salam binary code given in Table 6, we get the
binary code given in Table 4. Thus, the linear maps
and agree on a basis, so this triangle commutes:

This in turn implies that the bottom face of the cube commutes, from which we see that the two maps from to going around the bottom face are equal:

The work of Section 3.1 through Section 3.4
showed that the vertical faces of the cube commute.
We can thus conclude from diagrammatic reasoning that the two maps from
to
going around the top face are equal:

Since the Dirac spinor representation is faithful, the map is injective. This means we can drop it from the above diagram, and the remaining square commutes. But this is exactly the top face of the cube. So, the proof is done.

2010-01-11