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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 refs 1, 2, and 3. The formulation was originally crafted for topological QFTs (which don't have local observables) and conformal QFTs, but it can also be used for traditional QFTs on pseudo-Riemannian manifolds, as explained in this excerpt from ref 1:

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 $S$.

...and from ref 3:

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. ... Depending on the structure we obtain different types of quantum field theories: topological, `spin,' conformal, etc.

Question: How are local observables encoded in this formulation of QFT?

Section 2.2 in ref 1 answers the question for conformal QFTs, using the state-operator correspondence, but I don't know how to generalize this to not-necessarily-conformal QFTs. Ref 4 seems to partly address question, but only indirectly (and I'm having trouble parsing the higher-category-theoretic math).

References cited:

  1. Tachikawa (2017), "On 'categories' of quantum field theories" (https://arxiv.org/abs/1712.09456)

  2. Monnier (2019), "A modern point of view on anomalies" (https://arxiv.org/abs/1903.02828)

  3. Dijkgraaf and Witten (1990), "Topological gauge theories and group cohomology" (https://projecteuclid.org/euclid.cmp/1104180750)

  4. Schreiber (2008), "AQFT from n-functorial QFT" (https://arxiv.org/abs/0806.1079)

Related references:

Chiral Anomaly
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    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. – Prof. Legolasov May 31 '19 at 08:25

1 Answers1

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This is discussed briefly in Sec. 3 of

Let us explain briefly how to get from this functorial picture to the traditional language of local observables and vacuum expectation values. For a point $x$ of a $d$-manifold $M$, we define the vector space $\mathcal{O}_x$ of observables at $x$ as follows. We consider the family of all closed discs $D$ smoothly embedded in $M$ which contain $x$ in the interior $\mathring{D}$. If $D'\subset\mathring{D}$ then $D\setminus\mathring{D'}$ is a cobordism $\partial D'\rightsquigarrow \partial D$ and gives us a trace-class map $E_{\partial D'}\to E_{\partial D}$. We therefore have an inverse system $\{E_{\partial D}\}$ indexed by the discs $D$, and we define $\mathcal{O}_x$ as its inverse-limit.

Now suppose that $M$ is closed and that $x_1,\dots,x_k$ are distinct points of $M$. Let $D_1, \dots, D_k$ be disjoint discs in $M$ with $x_i\in\mathring{D}_i$. Then, $M' = M\setminus\bigcup \mathring{D}_i$ is a cobordism from $\bigsqcup \partial D_i$ to the empty $(d-1)$-manifold $\emptyset$, and defines $Z_{M'} : E_{\sqcup\partial D_i}\to E_\emptyset = \mathbb{C}$. Using the tensoring property, we can write this $$ Z_{M'}:\bigotimes E_{\partial D_i}\to\mathbb{C},$$ and then we can pass to the inverse-limits to get the expectation-value map $$\bigotimes\mathcal{O}_{x_i}\to\mathbb{C}.$$

PonderingPan
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  • I'll let a few days pass before I accept the answer (recommended practice), but I think this is exactly what I needed. Thank you! – Chiral Anomaly Apr 20 '23 at 00:29