Ted Jacobson has put on the arXiv a transcription of a 1947 Feynman letter about his efforts to better understand the Dirac equation, in order to find a path integral formulation of it. The letter also contains some fascinating comments by Feynman about mathematics and its relation to “understanding”. In particular I like this one:

the terrifying power of math. to make us say things which we don’t understand but are true.

In some other places in the text he elaborates:

The power of mathematics is terrifying – and too many physicists finding they have the correct equations without understanding them have been so terrified they give up trying to understand them. I want to go back & try to understand them. What do I mean by understanding? Nothing deep or accurate — just to be able to see some of the qualitative consequences of the equations by some method other than solving them in detail.

The Dirac equation is something wondrous and mystifying. If one tries, like Feynman, to find a simple understanding of it in conventional geometric terms, one is doomed to failure. It is expressing something about not the conventional geometry of vectors, but the deeper and much more poorly understood geometry of spinors.

In terms of Feynman’s goal of finding a path integral formulation, the best answer to this problem I know of is the supersymmetric path integral. For one place to read about this, see David Tong’s notes, in particular section 3.3.1. In this paper, Atiyah gives a closely related interpretation of the Dirac operator in terms of an integral over the loop space of a manifold, using a formal argument in terms of differential forms on the loop space. I don’t think either of these though are what Feynman was looking for.

In any case, what one really cares about is not a single-particle theory, but the quantum field theory of fields satisfying the Dirac equation. Here there’s a standard apparatus of how to calculate given in every quantum field theory textbook. These standard calculations involving Dirac gamma-matrices fit well with Feynman’s “physicists finding they have the correct equations without understanding them have been so terrified they give up trying to understand them”.

What Feynman calls “terrifying” is what Wigner calls “unreasonable.” Why does math work so well, and what does it actually tell us about the world? Good questions, with no definitive answers.

Interesting, thanks for the link.

Unrelatedly: there’s a piece in the Giardian about Grothendieck. https://www.theguardian.com/science/article/2024/aug/31/alexander-grothendieck-huawei-ai-artificial-intelligence

Apparently there was a 5 part podcast about Grothendieck on Radio France last month https://www.radiofrance.fr/franceculture/podcasts/serie-alexandre-grothendieck-legende-rebelle-des-mathematiques

And there’s a conference on Topos next week https://ctta.igrothendieck.org/

I’d be really interested in a longer explanation of what the last sentence of this post means to say.

tulpoeid,

I’m working on understanding details of how Wick rotation works for spinors, and one striking thing is how hard it is to find a clear, coordinate-invariant geometric description of how spinor fields work, even in Minkowski space. The geometry of spinors and the Clifford algebra is subtle, but the standard treatments in a QFT book don’t even try to use this, instead just write down formulas and show that they work.

Peter,

I’m not sure this is what you’re looking for, but the lovely book by Gregori L.Naber ” The Geometry of Minkowski Spacetime-2nd edition” has a sizeable amount of material regarding spinors on Minkowski…

Anon X,

That’s about spinors in Minkowski space-time, there’s lots of places where this is done. Also lots of places where spinors in Euclidean space-time are discussed, which is a significantly different story. What’s not well-known and much more subtle is the relation between these two (e.g. how does Wick rotation work for spinors?).