This post is a slight detour from the quantization discussions, and is meant to be a note of caution to theoretical physicists about what "mathematics" is.
Here is the average theoretical physics (especially hep-th) scenario. Average Joe theoretical physicist says let $F_{n}$ be this quantity we want to compute, but not only (1) can they not compute it in any sense other than a toy model in $n\to 0$ or $n\to \infty $, (2) $F_{n}$ is not even well-defined. A very popular example is the path integral, in which case not only do physicists not know how to compute them generally, the $\mathcal{D}\Phi $ or whatever term is not even a well-defined measure. A mathematical physicist, then, looks at $F_{n}$ and says huh, this does not make sense. Let me formalise this. And then uses stuff like cohomology theory, differential graded algebras and other formal tools to make a meaningful interpretation of this, which in the path integrals case would lead to the Balatin-Vilkovisky formalism and supermanifolds. Average Joe then says this doesn't make sense. Why are you doing all this when you don't get anything computable from it? As if he could compute it in the first place. Mathematical physicist then says hm, let me formalize QFTs then, since clearly these path integrals and partition functions are still superficial. And then uses category theory to say that a QFT is a functor from some category $\mathbf{Cat}_{A}$ to $\mathbf{Cat}_{B}$. Average Joe does not know what a category is, and says this is all just esoteric nonsense! The smarter theoretical physicist, on the other hand, asks what these $\mathbf{Cat}_{A}$ and $\mathbf{Cat}_{B}$ are. Mathematical physicist then says these could be categories of globally hyperbolic manifolds and categories of $C^{*}$-algebras, as in the case of locally covariant QFTs, or for $n$-bordisms, like for topological QFTs like Chern-Simons theories. Average Joe does not care. He goes there is still nothing to compute, as if he were able to compute anything. However, what you get from this are: (1) a proper program for gauge theories from BV formalism and (2) a formal notion of, eg, algebras of observables and Hilbert spaces associated to not just free field theories in Minkowski spacetime, but also on curved spacetimes (with TQFTs you get a background independent formalism of course which has its own formulations). Average Joe does not find this amusing. He says, where does this even come from?! This comes from picking a Hadamard 2-point function and a formal $\star$-product and you end up with propagators and correlators as usual.
Then there is the scenario that Urs Schreiber described in this post, which is an excellent observation that circles back to what I am saying. In that example, the topic of discussion is the AdS/CFT duality, which for Average Joe is just the GKPW dictionary combined with entanglement wedge reconstruction. Interestingly, at the time that post was published (2007), the Lewkowycz-Faulkner-Jafferic-Maldacena-Suh works were not published, and therefore notes on modular flows were not a part of that Joe description. In any case, for a mathematical physicist, it does not have to be this precise statement about the original Maldacena duality. In fact, I wrote an essay with Aayush Verma to this effect back in 2024. However, attempts like algebraic holography by Rehren were dismissed very quickly since they did not reflect the physicist approach. Indeed, at the time, algebras of observables were not very well understood, and it would take over a decade to get the Liu-Leutheusser-Witten papers on such constructions of von Neumann algebras and their automorphism groups. Therefore, modular theory was not incorporated into these criticisms of Rehren duality and I take it that incorporating Tomita-Takesaki theory into the picture resolves some issues. In any case, one of the main criticisms at the time was that not only do you not get the exact AdS/CFT correspondence, you just get duality of states and not a state-correlator correspondence! Which in my opinion is interesting, because back then you could not even make sense of reconstruction in a form other than the GKPW dictionary. This takes virtually nothing away from Rehren's duality, and in fact shows that in the two sets of approaches that Schreiber mentions (that of Atiyah-Segal and that of Haag-Kastler, which is synonymous here to perturbative AQFTs), the pAqfts approach does much to axiomatize a general notion of holography in terms of algebras of observables. Which have worked to an extent already in a somewhat more lenient fashion from Liu-Leutheusser works. Now, if a functorial formalism for this is more meaningful I cannot say (yet, as I am working on this precise issue), but the outcome of this situation will still be the same: a formal notion of holography in terms of categories and operator algebras and deformation theory will be found, while Average Joe will still say but why categories?!
What you should take away from this post is that even though a lot in hep-th can be "computed" in some limit without rigorous formalization, the works that mathematical physicists do is not esoteric and is the work that Average Joe was supposed to have done decades back. While it might seem as if getting a categorification of some theory is too unnecessary or irrelevant to what you are computing, if that axiomatization does not make sense in the first place, your computations have been either a very successful mistake or a very irrelevant computation in the first place. In short: no, mathematical physics is not pedantic.
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