Bert Schroer has sent me a very long and interesting comment for posting here. I’ve put it into a separate web-page. It includes both a lot of history and many different ideas. Unfortunately I don’t have the time right now to write much about it in response, but just will make one point about the part that he explicitly addresses to me.
Schroer claims that geometric methods in QFT have so far only been useful in dealing with free fields in a fixed background gauge field or metric. This is largely, but not completely true. Most of our reasons for believing the standard model are based on perturbative quantization of gauge fields, and for this it’s true that geometrical methods are not strictly necessary. But for QCD, we need a non-perturbative quantization of the gauge fields, and here lattice QCD is the best we’ve got. It is based upon discretizing a geometrically formulated path integral, preserving as much of the gauge field geometry as possible. My own guess is that there is still a lot to be learned about non-perturbative quantization of gauge fields, based upon the geometrical formulation of the problem given by the path integral approach and I have been working on speculative ideas of how to do this (some of which even involve gerbes and algebraic geometry…). This is still work in progress, maybe I’ll someday find it really can’t work, but for now I’m quite optimistic.
There’s also a new survey paper on QFT from Fredenhagen, Rehren and Seiler. The authors discuss the current state of understanding of QFT, with some points of overlap with Schroer. Like Schroer, they also discuss string theory in detail, and are critical of the inability of the string theory research program to come up with precise statements about what the theory is supposed to be. They are however, much less forceful in their criticisms than Schroer.
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