I've recently begun trying to understand a formulation of quantum field theory as a functor from a category of spacetimes-with-boundaries (bordisms) to a category of Hilbert spaces, as reviewed in [1]. Not sure if this formulation has an accepted name, but it's associated with Atiyah and Segal.

My interest is motivated by the promise of a nice perspective on anomalies [2]. I'm not a mathematician, though, and I'm struggling to clearly understand how local observables are encoded in this formulation. I understand that the formulation described in [1] and [2] includes topological QFTs, which don't have local observables, but it also includes traditional QFTs on pseudo-Riemannian manifolds. Section 2.2 in [1] discusses correlation functions and the state-operator correspondence in the context of conformal QFTs, and the conclusion of that section says

"It should also be possible to construct such a net [of observables] from a QFT in our sense.

I added the emphasis on the word "should," which suggests that maybe nobody has actually worked out the details yet. Or have they?

Question: In the formulation of QFT described in [1] and [2], how are local observables encoded for not-necessarily-conformal QFTs?

Here's what I've tried so far:

  • I have read (not fully understood, but read) the references listed at the end of this post, among others.

  • I have a guess about how local observables might be encoded, but I haven't found clear confirmation/clarification of my guess.

Here's my guess: Think of a correlation function expressed using the functional-integral language, between given initial and final states, schematically like this: $$ \langle\Psi_f| O_1(x_1) O_2(x_2)\cdots|\Psi_i\rangle \hskip4cm \\ \hskip1cm \sim \int [d\phi]\ \Psi_f^*[\phi]\, O_1(x_1) O_2(x_2)\cdots e^{iS[\phi]}\Psi_i[\phi] $$ where $\Psi_{i,f}$ depend only on the field variables $\phi$ at the initial and final times, respectively, and the $O_n$s in the integrand are expressed in terms of the field variables $\phi$. We can think of $\Psi_{i,f}$ as state-vectors in Hilbert spaces associated with the initial/final boundaries of a truncated spacetime (a bordism). Now, choose a neighborhood $N_n$ of each of the spacetime points $x_n$ where the observables are localized, and "integrate out" all of the field variables localized inside those neighborhoods. The resulting functional integral has a similar form except that each $O_n$ is replaced by a functional $\Psi_n[\phi]$ of the field variables $\phi$ on the boundary of the neighborhood $N_n$, like state-vectors in new Hilbert spaces associated with these new boundaries of a spacetime-with-holes (a new bordism).

These state-vectors $\Psi_n$ are uniquely determined by the $O_n$ in the context of the given correlation function, but does $O_n$ somehow determine $\Psi_n$ without that context? And does $\Psi_n$ somehow determine $O_n$? If the answers are "no," then is there still some other sense in which the formulation described in [1] and [2] encodes local observables? This isn't meant to be multiple questions; I'm just describing what I've thought about so far.

This feels related to the state-operator correspondence in CFT, but I'm still just a beginner when it comes to the special tools used in CFT, so I'm not sure yet if that's a helpful connection.

References cited:

Related references that I've consulted:

  • $\begingroup$ In a TQFT there’s no local observables. QFT with local observables don’t satisfy Atiyah-Siegel axioms. There’s a middle ground — in a background-independent theory with local ovservables, the vector spaces that correspond to spatial hypersurfaces are infinite dimensional. $\endgroup$ – Prof. Legolasov May 31 at 8:25
  • $\begingroup$ I’m having a hard time understanding your question. Could you clarify — why do you expect that Atiyah-Siegel holds for non-background-independent theories? $\endgroup$ – Prof. Legolasov Jun 3 at 2:23
  • $\begingroup$ @SolenodonParadoxus References [1] and [2] say that a QFT can be described as a functor from a bordism category to a category of Hilbert spaces, and the bordism category doesn't have to be a category of (merely) topological manifolds. Explicitly, [1] says: "A category of QFT exists for each fixed spacetime dimension d and a structure S on manifolds. Here, the structure S can be e.g. smooth structure, Riemannian metric, conformal structure, spin structure, etc." $\endgroup$ – Chiral Anomaly Jun 3 at 2:30
  • $\begingroup$ @SolenodonParadoxus ...and [2] says: "Quantum field theories typically require topological and geometric structures on the spacetimes on which they are defined. We will call these structures the field theory data of the quantum field theory under consideration. For instance, the theory of a free spin 1/2 field on oriented spacetime requires a spin stucture and a Riemannian metric. More generally, the field theory data can be abstracted as a sheaf over spacetime. In the following, ‘manifold’ will always mean ‘manifold endowed with the relevant field theory data’." $\endgroup$ – Chiral Anomaly Jun 3 at 2:30
  • $\begingroup$ @SolenodonParadoxus I was probably wrong to call this Atiyah-Segal. Maybe it's a generalization that was inspired by the Atiyah-Segal approach to topological QFTs. (Thank you for the feedback, by the way. I'm not confident in any of this; just starting to learn it.) $\endgroup$ – Chiral Anomaly Jun 3 at 2:31

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