Dear Alan,
I enjoyed reading your essay, which I think is both well-motivated and well-explained. I have a couple of questions/remarks.
1. Like you, I tend to find certain aspects of quantum theory and quantum field theory less well-motivated than relativity, which is based on simple physical principles. In particular, even in ordinary quantum theory, I am not very fond of the Hilbert space/operator algebra view, which takes things for granted that are convenient mathematically but physically dubious. I prefer Feynman's sum-over-histories viewpoint, even though the math is harder, because the physical ideas are clear and refer only to entities with obvious physical meaning.
2. Have you read the essays by Torsten Asselmeyer-Maluga and Jerzy Krol in this contest? They together propose new methods of trying derive quantum gravity from smooth manifolds alone. I find their ideas interesting, but I will warn you that their approach is not something that can be fully understood the first time through, at least not for me. Anyway, since you focus on theories based on geometry, I thought you might find them interesting.
3. There are some interesting new ways in which Hopf algebras are showing up in quantum information theory these days, which is rather striking for something with essentially a geometric origin. There is a preprint about this by Sasakura, and I have written some about it myself, although unfortunately it isn't in any shape to publish yet.
Paradoxically, although my mathematical work involves mostly manifolds and algebraic varieties, my ideas about fundamental physics are quite different. I suspect that manifolds are "too good to be true" physically because of their special order-theoretic properties and the fact that they imply things like nonmeasurable sets. My view is that manifolds have dominated physics historically mostly because they are mathematically convenient, much like Hilbert spaces are mathematically convenient for quantum theory. However, a lot of the papers I have been reading lately are causing me to reconsider this, and I like to keep an open mind.
I prefer to try to build fundamental physics from more primitive notions like causal relations, which lend themselves to information-theoretic and even computational approaches. I note from your bio that you taught computer science; it's a bit ironic that a mathematician such as myself would try to think about physics in terms of information theory, while a computer scientist would think about it in terms of geometry!
Anyway, I enjoyed reading about your work! Take care,
Ben Dribus
