This past semester I taught our graduate class on Lie groups and representations, and spent part of the course on the Heisenberg group and the oscillator representation. Since the end of the semester I’ve been trying to clean up and expand this part of my class notes. I’m posting the current version, working title From Quantum Mechanics to Number Theory via the Oscillator Representation. This is still a work-in-progress, but I’ve decided today to step away from it a little while, work on other things, and then come back later perhaps with a clearer perspective on what I’d like to do with these notes. In a few days I’m heading off for a ten-day vacation in northern California, and one thing I don’t want to be thinking about then is things like how to get formulas involving modular forms correct.

There’s nothing really new in these notes, but this is material I’ve always found both fascinating and challenging, so writing it up has clarified things for me, and I hope will be of use to others. The basic relationship between quantum mechanics and representation theory explained here is something that I’ve always felt deserves a lot more attention than it has gotten.

In the past I’ve often made claims about the deep unity of fundamental physics and mathematics, One goal of this document is to lay out precisely one aspect of what I mean when making these claims. There are other much less well understood aspects of this unity, but the topic here is something well-understood.

One thing that struck me when thinking about this and teaching the class is that this is a central topic in representation theory, but one that often doesn’t make it into the textbooks or courses. Typically mathematicians develop theories with an eye to classifying all structures of a given kind. This case is a very unusual example where there is effectively a unique structure. The classification theorem here is that there is basically only one representation, but it is one with an unusually rich structure.

When I get back from vacation, I plan to get back to work on the ideas about twistors and unification that I’m still very excited about, but have set to the side for quite a few months while I was teaching the class and writing these notes. More about that in the next few months…

This is great, thanks for this. I decided to (re)tackle Not Even Wrong again this summer. My graduate work in mathematics is some years behind me (and was applied math / engineering). I’m enjoying making this a summer reading project. As part, I picked up Weyl’s “Symmetry” and Hall’s “Lie Groups, Lie Algebras, and Representations” based on your suggestions in the readings. Also found a copy of Penrose’s “The Road to Reality” (that’s a summer project in itself…) I look forward to adding these notes to my reading list. Have a great vacation!

I look forward to the number theory section! Your notes remind me slightly of Mackey’s book

Unitary Group Representations in Physics, Probability, and Number Theory.John Baez,

The main reason I lost steam and haven’t yet finished the notes is that I want to spend the summer working on other things. But also relevant are the following:

1. The way the story of the oscillator representation and the theta correspondence is set up in detail over the real numbers goes over straightforwardly to finite fields, the p-adics, and over the rationals via the adeles. Beyond that statement, it’s tricky to write out details of this accessible to non-number theory experts without starting to write an introduction to a lot of modern number theory that would be a different and larger project (and not really within my expertise).

2. There’s a simple explanation of the theta function and its properties as matrix elements of the oscillator representation, which I wrote down. The details of the story of modular forms and how theta functions fit into it are quite complicated and this is where I realized I was in danger of getting into writing up something much more complex than I had energy for. Maybe someday on that…

3. There’s a lot of readable expository literature on automorphic representations of SL(2) and the Langland correspondence with Galois representations. But, not so much on what happens for the metaplectic double cover. Before trying to say anything about this, I’m hoping to consult with some local experts and get a better understanding of that story. The story of the Heisenberg group over the adeles is yet again something different that I haven’t found a detailed explanation of (that I can understand…).

In an ideal world, what I’d like to turn this into someday would be a document that works out in detail the simplest case of this whole story, including the number theory parts. I hope this would complement the usual math expositions, which emphasize generality, while making clear that this part of number theory and fundamental QM are about exactly the same thing (the oscillator representation).