I was visiting the math department at Dartmouth the past couple days, and gave a colloquium talk there. It’s now available online.

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I’m using the term “weight” in the precise sense it is used in representation theory, the label for a representation of T. More generally it can label reps of some Abelian subgroup, e.g. multiplicative groups. If the group is the real numbers, the weights are given by arbitrary numbers, not necessarily integers like in the U(1) case.

Why specifically is the word “weight” used? Weyl was very precise with language so by weight he would have meant the exponent on a multiplicative factor, in whatever sense (in the talk, the context is projective representations of spin groups).

In the sense of Weyl geometry, the weight of the Dirac spinor field is imaginary, ie. The weight of a Yang-Mills field should be something like iek Tk as a kind of dyadic.

Hi Danny,

The “weights” I’m talking about are for compact Lie groups G and come about as follows:

1. For G of rank r, this means there is a subgroup T of r copies of U(1). For any representation of G, you can restrict attention to T and think of it as a representation of T. If you start with an irreducible representation of G, it will break up as several irreducible representations of T (all one-dimensional since T is Abelian). These representations of T are the “weights” of the original G representation.

2. More explicitly, an irreducible representation of U(1) is labeled by an integer n, and e^{i\theta} acts by multiplication by e^{in\theta}. An irreducible representation of T is labeled by r integers.

Things are different and much more complicated for non-compact groups (like the conformal group), then you typically have copies of R (the real numbers) in the maximal Abelian subgroup. Representations of R are labeled not by integers but by real numbers.

RE Peter-Weyl and Borel-Weil theorems –

One thing I have been thinking about is extending the idea of conformal weight in an attempt to understand the relation of the covariant derivative

(d/dm + ie Tk Akm)

in field theory, to

(d/dm + N Am)

in Weyl conformal geometry – where N is the Weyl conformal weight. Am I right in saying that the “weights” mentioned in your talk are some kind of generalized phase which will turn out to correspond in gauge theory to charges?

I used the latex macro package called “prosper”. For now the latex source file is at

http://www.math.columbia.edu/~woit/dartmouth.tex

I love the interview with Dirac – that’s priceless!

Once a friend drove Dirac from the airport to the hotel in New Orleans. It had recently rained a great deal and snakes had emerged from the swamp, which Dirac noticed with a short comment like “snakes are interesting”. My friend was on the organizing committee for the conference, and part of his job was to deliver Dirac and find something entertaining for him to do in New Orleans. So, he picks the zoo and the herpatarium in particular and asks Dirac if he’d like to see it “since you like snakes”. Dirac said “I did not say I liked snakes. I said they were interesting.”

I felt really good about both Dirac and Weyl after reading that story. On his part, Weyl was the first person to really understand the significance of the Dirac equation. Weyl was very generous with praise when it was deserved.

Aside, what tools did you use to create this?