Anti-Quill Part 10: Mathematics, Physics, and "Esoteric"

 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.

Anti-Quill Part 9: Quantization-2 | Geometric Quantization

 The aim now is to answer what really defines a quantum mechanical system or a quantum field theory, which in a sense is "field degrees of freedom" (quotes because this is very non-trivial physically) that we obtain from some classical mechanical system or a classical field theory. A classical theory consists of a phase space $T^{*}M$ that is a symplectic manifold $(\mathbb{R}^{2n}, \omega )$ and observables are functions $f\in C^{\infty }(T^{*}M)$ that are elements of a Poisson algebra $\mathcal{P}$. which comes with Poisson brackets $\{\cdot , \cdot \} : C^{\infty }\times C^{\infty }\to C^{\infty }(M)$. Quantum observables are then going to be operators on some $\mathcal{A}(\mathcal{H})$ with a commutator $[\cdot, \cdot ]$ on $\mathcal{H}$. Quantization is then just a function $Q: C^{\infty }(\mathbb{R}^{2n})\to L(\mathcal{H})$, space of linear operators on $\mathcal{H}$. In stronger operator algebraic terms we would actually be more careful about what we do with this (i.e. selecting $\mathcal{B}(\mathcal{H})$), but for now we can be more superficial. So the problem is really just going from this classical phase space to a quantum Hilbert space. Note that I specifically said "quantum Hilbert space", since it is usually easier to obtain a pre-quantum Hilbert space from pre-quantization, where we identify from our Poisson manifold a collection of a hermitian line bundle, a connection on this line bundle and a Hilbert space that is etymologically one but not the actual constrained Hilbert space -- in this abuse of terminology one can consider this pre-quantum Hilbert space the auxiliary Hilbert space, on which polarization is done. I will not get into what polarization actually does for now, but this is typically the actual quantization problem. After polarization, we will end up with the actual quantum Hilbert space, which would be the quantised theory. 

So what we did here is called geometric quantization, which is only one of many sorts of quantization. The aim here is of course to end up with a Hilbert space. In a similar fashion, you could ask what happens with field theories. A classical field theory, after all, is also equipped with Poisson algebras and eventually we want to quantize this. Since QFTs describe algebras of observables, you could define it as really a functor from categories of globally hyperbolic manifolds to those of $C^{*}$-algebras satisfying the Haag-Kastler axioms. However, the obvious issue is that field theories have infinite degrees of freedom and therefore the phase space is infinite dimensional. In such cases you would do the so-called refined algebraic quantization, which will be a different post eventually. This was in fact, touched upon in the Hilbert space of de Sitter quantum gravity paper by Suvrat et al, without realising the full technicalities of what they were doing! On the other hand, a more suitable general scheme is that of deformation quantization where the Poisson brackets are "deformed" into commutators. Then there is stochastic quantization and more interestingly, Berezin-Toeplitz quantization which has to do with compact symplectic manifolds and Kostant–Souriau operators, which are related to the Hamiltonian vector field in a certain condition, which I am not entirely sure about.

Anti-Quill Part 8: Quantization-1

  In physics a very natural construction is that of quantization, where we take a classical system and quantize it. This involves, to be very naive, taking a symplectic pair $(M, \omega )$ and turning the functions in $L^{2}(M)$ into operators, and the Poisson brackets $\{\cdot, \cdot \}$ into commutators $[\cdot, \cdot ]$. But getting to $L^{2}(M)$ is hard. In fact, prequantization is as far as we've gotten to here. The objective at the end of the day is to take this symplectic manifold we have and obtain a Hilbert space. Really, what you would work with would be subspaces and not $L^{2}$ spaces, etc etc. There's many sides to how you do this though. In fact, this is a very crucial part of physics -- or in broad terms mathematical physics, since a lot of the developments in this field come not from physicists but from mathematicians. E.g. deformation quantization was formulated by Maxim Kontsevich (the "t" comes before the "s", this comment is directed at my collaborator), a lot of geometric quantization was done by mathematicians just trying to do symplectic geometry and brane quantization, while initiated by Gukov and Witten and is of interest to string theorists primarily, is also of interest for differential geometry and representation theorists. The case with brane quantization is that there's boundary conditions involving Lagrangian and coisotropic A-branes, and stuff I don't fully understand yet. See Gukov-Witten and Gaiotto-Witten for more about this. Interestingly, the ordinary sort of quantization that physicists do is not fully meaningful either. A good example is the work by Suvrat et al on the Hilbert space of de Sitter quantum gravity, where you take the ordinary Hilbert space (obtained from Dirac quantization) and you have to solve the constraints next, for which you define the rigging map and do group-averaging. This forms the so-called refined algebraic quantization scheme, which is really just Dirac quantization extended. So even though quantization as a term seems trivial from undergrad physics a la canonical quantization, the actual mathematically clear way to do it is very nontrivial and they branch out into several formalisms. In the words of Ludwig Faddeev, "quantization is an art, not a science". More on this soon.