This interesting-looking preprint has just appeared on ArXiv:
Theo Johnson-Freyd, Heisenberg-picture quantum field theory, arXiv:1508.05908
It argues for a different category-theoretical formalism for TQFT than the `Schroedinger-picture‘ Atiyah-Segal-type axiomatization that we are used to. The `Heisenberg-picture‘ functor it proposes has as its target a category whose top level is pointed vector spaces instead of numbers, and whose second to top level is associative algebras instead of vector spaces. The preprint argues that this formalism is better physically motivated, and one might dream that it is better-suited to analyze “semiclassical limit” conjectures such as the AJ conjecture and its variants.
I’m very happy to see this sort of playing-around with the foundations of TQFT, which I am happy to believe are too rigid. I expect there should be a useful Dirac picture also, and that there are other alternative axiomatizations also. Let’s see where this all leads!