The n-Category Café

For the following purpose, we will describe quantum field theory in AQFT terms, which focuses on the “algebras of observables”.

Interlude: Schrödinger and Segal versus Heisenberg and Haag-Kastler

We have mostly been talking about functorial descriptions of quantum field theory here, in which QFTs are conceived as representations of cobordism categories $$U : n\mathrm{Cob} \to \mathrm{Hilb} \,.$$ Objects $X$ of $n\mathrm{Cob}$ are sent to spaces of states $$V_X \,,$$ while morphisms $X \stackrel{\Sigma}{\to} Y$ are sent to operations on states $$V_X \stackrel{U(\Sigma)}{\to} V_Y \,.$$

In quantum physics textbooks (which usually know nothing about functors), this perspective on quantum field theory is called the Schrödinger picture

There is another perspective on quantum field theory, known as the Heisenberg picture.

This is obtained, roughly, by applying the endomorphism functor $$\mathrm{Ad} : \mathrm{Hilb} \to \mathrm{Bim}$$ to the Schrödinger picture. This functor sends an isomorphism of Hilbert spaces to an isomorphism of the corresponding endomorphism algebras $$\mathrm{Ad} : (V \stackrel{f}{\to} W) \mapsto (\mathrm{End}(V) \stackrel{\mathrm{Ad}(f)}{\to} \mathrm{End}(W)) \,.$$ Where a vector in $V$ is called a state, an element in $\mathrm{End}(V)$ is called an observable.

We all know that, at some point, the above Schrödinger picture is too coarse. Instead of 1-functors that propagate states globally over $n$-cobordisms, we want to see $n$-functors that propagate $n$-states over pieces of $n$-cobordisms #.

In particular, if our cobordisms are equipped with a Lorentzian structure, we might want to propagate states over little causal diamonds: $$\array{ & \nearrow \searrow^{\gamma} \\ x &\;\Downarrow^\Sigma& y \\ & \searrow \nearrow_{\gamma'} \,. }$$

This Schrödinger picture of extended QFT should have a corresponding extended Heisenberg picture, too. But nobody has yet tried to make this connection explicit.

Instead, people have directly guessed (proposed) one way to talk about the “extended Heisenberg picture”: the result is known as the theory of nets of local observables and usually addressed as (local) algebraic quantum field theory.

Notice that nobody except me ever talks about the “extended Heisenberg picture”. But they should! :-) The above picture

Schrödinger picture : $n$-functorial QFT :: Heisenberg picture : AQFT

ought to be drawn in full detail, eventually. Here it serves me just to put the various concepts that I consider in the following in perspective.

(I should add that somewhat parallel to the AQFT crowd with their pre-co-sheaves of local algebras (to be described in a moment), there are people working, in the context of 2-dimensional conformal QFT, on sheaves and gerbes of vertex operator algebras, which is based on a very similar idea, but differs in detail.)

End of interlude, beginning of introduction: nets of local observables

It should now be at least plausible that the following definition has chances to capture some aspects of quantum field theory on Lorentzian domains:

Definition (Haag-Kastler nets). For a given Lorentzian manifold $M$, denote by $O$ the category whose objects are its causal diamonds (the union of all time-like curves between a given pair of points in $M$) and whose morphisms are inclusions of these. A net of local observables is a certain pre-cosheaf of $C^*$-algebras on $O$.

(I say “certain” pre-cosheaf because there are some funny details about this. On the one hand, a sheaf condition is curiously missing, on the other hand it is usually demanded that all co-restriction maps of algebras are by injective morphisms. This looks unnatural to me. I suspect that we will eventually find a a more natural formulation of this.)

But anyway. The only other axiom of real relevance for the following discussion is this:

Locality property: We demand that whenever the causal diamonds $O_1 \subset O_0$ and $O_2 \subset O_0$ are spacelike separated, then the sub-algebras associated to them mutually commute.

Finally, we assume that the inductive limit of all these “local” algebras over all causal diamonds does exist. The resulting algebra (“of observables on $M$”) shall be called $$A \,.$$ This is the object of interest now.

End of introduction, finally the main point: the arrow theory of localized endomorphisms.

As every algebra, $A$ lives in $$\mathrm{Bim} \,,$$ the 2-category whose objects are algebras, whose morphisms are bimodules between these and whose 2-morphisms are bimodule intertwiners.

Notice that we could regard algebras as 1-object $\mathrm{Vect}$-enriched categories. This would suggest an obvious 2-category structure on them, too. But this 2-category naturally embeds into $\mathrm{Bim}$. The latter is like the category of profunctors between 1-object $\mathrm{Vect}$-enriched categories. Keeping this in mind, I’ll be talking about $\mathrm{Bim}$.

We noticed before that $\mathrm{Bim}$ is monoidal since $\mathrm{Vect}$ is braided. The tensor unit in $\mathrm{Bim}$ is $$1 := \mathbb{C}$$ the ground field $\mathbb{C}$ itself (That’s the ground field used in quantum field theory applications.)

Now, we may regard $A$ in a trivial way as a $1$-$A$-bimodule. The corresponding morphism in $\mathrm{Bim}$ we write $$1 \stackrel{A}{\to} A \,.$$

Since the left algebra is sort of trivial, we think of this bimodule as just being a right module for $A$.

For our physics needs, this is called the vacuum representation of our algebra of observables $A$.

The real entity of interest is the monoidal category $$\mathrm{End}_\mathrm{Bim}(A)$$ of $A$-$A$-bimodules. To make contact with the QFT literature, we would restrict this to those bimodules and their homomorphisms that come from functors between algabras and natural transformations bertween these.

More importantly, we want to restrict to localized algebra endomorphisms. This are all those that act nontrivially only on a sub-algebra of $A$ corresponding to some causal diamond $O$.

So we get a monoidal sub-category $$\mathrm{End}^\mathrm{loca}_\mathrm{Bim}(A) \subset \mathrm{End}_\mathrm{Bim}(A)$$ of local endomorphisms.

By composing any endomorphism of $A$ $$A \stackrel{f}{\to} A$$ with the “vacucum representation” $$1 \stackrel{A}{\to} A$$ we get another representation $$1 \stackrel{A}{\to} A \stackrel{f}{\to} A \,.$$ The representations obtained from $\mathrm{End}(A)$ this way are called the Doplicher-Haag-Roberts (DHR) representations.

Of course, for us this passage from endomorphisms to “representations” simply amounts to forgetting the left $A$-action on an $A$-$A$ bimodule.

The interesting point of all this is that, due to this locality requirement, the monoidal category $\mathrm{End}^\mathrm{loca}_\mathrm{Bim}(A)$ is braided!

Namely, using unitary operators $U_1$ and $U_2$ in $A$ with spacelike separated support, we may translate any two localized endomorphisms $f_i$ to endomorphisms $\tilde f_i$ with spacelike separated support $$\array{ & \nearrow \searrow^{f_i} \\ A &\;\Downarrow^{U_i}& A \\ & \searrow \nearrow_{\tilde f_i} }$$ which therefore commute $$(A \stackrel{\tilde f_1}{\to} A \stackrel{\tilde f_2}{\to} A) = (A \stackrel{\tilde f_2}{\to} A \stackrel{\tilde f_1}{\to} A) \,.$$ Transporting the commuted $\tilde f_i$ back along $U_i$ defines a braiding on $\mathrm{End}^\mathrm{loc}(A)$.

One shows that this braiding

- is symmetric whenever the dimension of the Lorentzian manifold is greater than 2

- is nontrivial in dimension 2.

(Compare all this with section 2 of the above review)

What does all this have to do with amplimorphisms?

Nothing directly yet. But they come in as follows:

In physics applications, we will be thinking of the equivalence classes in $\mathrm{End}^\mathrm{loc}(A)$ as the superselection sectors of the theory, and of equivalence classes of simple objects (irreps) as ‘elementary (particle) excitations’ of the theory.

The Doplicher-Roberts reconstruction theorem then makes us want to understand $\mathrm{End}^\mathrm{loc}(A)$ as the category of representations of the “gauge symmetry” of the theory.

Since $\mathrm{End}^\mathrm{loc}(A)$ is braided, this requires a braided version of the original Doplicher-Roberts result.

It turns out that we may find a certain Hopf algebra, and a certain notion of representations of it, based on amplimorphisms, such that it is equivalent to $\mathrm{End}^\mathrm{loc}(A)$.

This I will maybe describe elsewhere.