On the Jones polynomial via Chern-Simons theory:
On non-perturbative quantum field theory:
On supersymmetric quantum mechanics from the point of view of spectral geometry (“noncommutative geometry”):
and with an eye towards the superstring via 2-spectral triples:
Another survey is
On quantum probability theory and the operator algebra-foundations for quantum physics:
Jürg Fröhlich, B. Schubnel, Quantum Probability Theory and the Foundations of Quantum Mechanics. In: Blanchard P., Fröhlich J. (eds.) The Message of Quantum Science. Lecture Notes in Physics, vol 899. Springer 2015 (arXiv:1310.1484, doi:10.1007/978-3-662-46422-9_7)
Jürg Fröhlich, The structure of quantum theory, Chapter 6 in The quest for laws and structure, EMS 2016 (doi, doi:10.4171/164-1/8)
From Fröhlich 92, p. 11, on laying foundations for perturbative string theory via rigorous formulation of 2d SCFT in terms of conformal nets/2-spectral triples or similar:
I still have hopes, perhaps romantic ones, that string theory, or something inspired by it, will come back to life again. I believe it is interesting to attempt to formulate string theory in an “invariant” way, quite like it is useful to formulate geometry in a coordinate-independent way. One might, for example, start with a family $\mathcal{F}$, of hyperfinite type $III_1$ von Neumann algebras – to be a little technical – indexed by intervals of the circle with non-empty complement (or of the super-circle). It may pay to formulate the starting point using the language of sheaves. $[...]$ This structure determines a braided monoidal C*-category with unit, …; briefly, a quantum theory. From a combination of such tensor categories (left and right movers) one would attempt to reconstruct (symmetries of) physical space-time. String amplitudes would correspond to arrows (intertwiners) of the tensor category. $[...]$ it would provide a general way of thinking about string theory that does not presuppose knowing the target space-time of the theory.
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