M Theory Lesson 147
First let us consider the relation on matrices defined by $A \simeq B$ if $[A,B] = \lambda I$ for a scalar $\lambda$. It is not necessarily transitive, except for triples satisfying rules of the form
$[A,B] + [B,C] - [A,C] = 0$
but it is reflexive (since $[A,A] = 0$) and symmetric (since $[A,B] = -[B,A]$). In the world of categories we think of transitivity as a triangle of arrows, but we might weaken this triangle by allowing 2-arrows, or even higher dimensional structure.
Under this equivalence, the usual Heisenberg rule $[X,P] = i \hbar$ is a kind of equivalence between position and momentum. If we exponentiate this expression we find that
$\textrm{exp}(XP)=\textrm{exp}(i \hbar) \textrm{exp}(PX)$
which naturally reminds us of the Weyl rule for the discrete Fourier transform underlying the mass matrices. Now we see that $\hbar$ naturally defines a root of unity, and there is no reason to assume it takes on a fixed value. Moreover, when the root of unity is specified by the dimension of the matrix, as is the case for the Fourier transform, the value of $\hbar$ is specified.
$[A,B] + [B,C] - [A,C] = 0$
but it is reflexive (since $[A,A] = 0$) and symmetric (since $[A,B] = -[B,A]$). In the world of categories we think of transitivity as a triangle of arrows, but we might weaken this triangle by allowing 2-arrows, or even higher dimensional structure.
Under this equivalence, the usual Heisenberg rule $[X,P] = i \hbar$ is a kind of equivalence between position and momentum. If we exponentiate this expression we find that
$\textrm{exp}(XP)=\textrm{exp}(i \hbar) \textrm{exp}(PX)$
which naturally reminds us of the Weyl rule for the discrete Fourier transform underlying the mass matrices. Now we see that $\hbar$ naturally defines a root of unity, and there is no reason to assume it takes on a fixed value. Moreover, when the root of unity is specified by the dimension of the matrix, as is the case for the Fourier transform, the value of $\hbar$ is specified.
posted by Kea | 9:18 PM
2 Comments:
I'm guessing that if were in your presence I could get you to explain this. But it's rather terse. What's a "triangle of arrows" mean? Maybe you could draw it. And I followed the link but didn't see any mention of "Weyl" so that's mysterious too.
Part of the reason I'm complaining here is I've been watching seminars given at the Perimeter Institute and they are very easy to understand. I had gotten too much used to people making short, difficult to understand lectures, and writing short, terse, papers that don't explain things very well and I'd forgotten how easy it is to understand a nice long 3 hour lecture.
Carl, the main problem with some of these posts is that I'm really not sure where they are going! But you are right that lectures are the best way to learn something. I watched a few half talks, but I'm at a bit of a disadvantage being here in NZ, because the streaming video often cuts out (and that's the best format for me to use), and then I'm only too tempted to stop watching when the talk gets boring. So far I've chosen a few string theory seminars.
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