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?

Clarification in response to comments:

The original version of this question used the name "Atiyah-Segal," but I'm not sure that's the right name for the formulation described in the question. This clarification was requested in comments:

In a TQFT there’s no local observables. QFT with local observables don’t satisfy Atiyah-Siegel axioms. ...why do you expect that Atiyah-Siegel holds for non-background-independent theories?

The following excerpts clarify which formulation the question is referring to. References [1], [2], [3] 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, regarding the formulation described in the question, [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. ... They are supposed to satisfy the standard axioms of Atiyah and Segal..., properly modified for the structure $S$. ...a QFT Q determines a functor from a suitable bordism category to the category of vector spaces. Traditionally, a QFT for smooth manifolds is called a topological QFT (despite the fact that smooth manifolds and topological manifolds can have interesting differences), a QFT for Riemannian manifolds are simply called a QFT (without adjective), a QFT for conformal structure is called a conformal field theory (CFT), etc.

...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’. ... The Atiyah–Segal axioms are a convenient way of formalizing the notion of quantum field theory into a mathematical object. In this framework, a $d$-dimensional quantum field theory $\mathcal{Q}$ is a symmetric monoidal functor from a bordism category $\mathcal{B}^d$ to the category of Hilbert spaces $\mathcal{H}$. $\mathcal{B}^d$ should be the bordism category of manifolds endowed with the field theory data required by $\mathcal{Q}$.

...and [3] says:

In a convenient axiomatization of quantum field theory..., the structure of a $d +1$ dimensional quantum field theory includes... a functor $\Phi$ from the category of closed $d$-manifolds [more precisely: *-manifolds] into the category of Hilbert spaces. ... Here ∗ can be any extra structure: for example an orientation, a spin structure, a complex structure, or a metric. Two manifolds are isomorphic if there exists a diffeomorphism that preserves the structure. Depending on the structure we obtain different types of quantum field theories: topological, `spin,' conformal, etc.

Here's what I've tried:

  • 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.

References cited:

Related references that I've consulted:

  • 1
    $\begingroup$ 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 '19 at 8:25
  • 1
    $\begingroup$ 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 '19 at 2:23
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    $\begingroup$ Several months after posting this question, I found this paper by @UrsSchreiber - "AQFT from n-functorial QFT" (arxiv.org/abs/0806.1079). The abstract says, "There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. ... In this note we present an observation about how these two axiom systems are naturally related..." Digesting this paper may take me a long time, though (I'm a slow learner), so I'd still welcome answers! $\endgroup$ – Chiral Anomaly Oct 26 '19 at 19:30
  • $\begingroup$ Update: The paper arxiv.org/abs/0806.1079 linked in my previous comment uses the name "FQFT" for the formulation I described in my question, and it focuses specifically on the case with "pseudo-Riemannian structures, and with flat Lorentzian structure (Minkowski structure) in particular." However, while the paper proposes how to construct a net of local observables from a given FQFT, it focuses mostly on 2-dimensional spacetime, and it doesn't say much about how to go the other direction. So, while this paper is interesting, it doesn't fully answer the question that I'm asking here. $\endgroup$ – Chiral Anomaly Mar 15 at 23:41

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