So this is going slower than I thought it would. I’m on 4 and I’ve basically given a definition that was incomplete. Maybe I’ll quit this endeavor if this one doesn’t go any better than the previous. I’m now going to make the assumption that if you could read NCG 2, then you were bored and so to cover more ground I’m going to make more demands on the reader.
A good example of the overview in NCG 2 is in electromagentism. Let V be a manifold. Let be the algebra of functions on V. It turns out the group of unitary elements of is the local gauge group of electromagnetism. Let D be the covariant derivative associated to the potential.
We can then express D by . Where is a -module and D as a derivation satisfies the Leibniz rule: where and . Now we know that each component of the potential couples equally with the Dirac spinors.
This is tending to be a little physics-like, so I’ll take a little diversion here. If you look at the Dirac equation it requires a collection of four objects satisfying: , , and when . These have 4 by 4 matrix representation that I won’t type (too cumbersome, but you can look up anywhere if you are terribly curious). Using Einstein summation convention, we get a nice compact form of the Dirac equation: where .
So we have an interaction with the EM-field through the potential . Just replace to get the new Dirac equation where the state function is a column spinor .
OK. So back to what I was driving at. This means we can identify with . So if we define D by the rule along with the Leibniz rule. This leads to the operator D behaving exactly as I derived (no that wasn’t for nothing) above. After redefining some constants we get is the left side of the “new” Dirac equation.
And so now we have an electromagnetic theory for noncommutative geometry. Which is actually pretty important since like we said before, if we want the four forces in a unified theory we need them to work in a quantum/noncommutative framework. Hmm…sorry, since I feel like the vote that went for NCG was not a vote for physics examples.