The n-Category Café

I received some very useful feedback. One open question for me is/was this:

I describe how 2-functors on 2-paths in Minkowski give rise to their “endomorphism co-presheaves” on causal subsets, which are necessarily local nets satisfying the time slice axiom.

The open question is: how does this work the other way around? given a local net, can we construct the 2-functor that it is the endomorphism co-presheaf of?

One helpful suggestion from the audience was this:

assume the local net $A$ has the following two properties:

- the algebra $A(O)$ assigned to any double cone $O$ is maximal: with $W^O_L$ the wedge region left of $O$ and $W^O_R$ the wedge region right of $O$, the inclusion $$A(O) \hookrightarrow A(W_L^O)' \cap A(W_R^O')$$ (prime denotes commutant, as usual), expressing the locality of the net is actually an equality $$A(O) = A(W_L^O)' \cap A(W_R^O)' \,.$$ Similarly then for $W^x_L$ and $W^x_R$ the left and right wedge at any point $x$, $$A(W^x_L) = A(W^x_R)' \,.$$

- the net satisfies a split property which says that for $y$ spacelike right of $x$ the inclusion of wedge algebras $$W^y_R \subset W^x_R$$ always factors through a type I factor $N$ as $$W^y_R \subset N \subset W^x_R \,.$$

Under some extra assumption which is apparently discussed in

S. J. Summers, On the independence of local algebras in quantum field theory, Rev. Math. Phys. 2 (1990), 201-247, this implies that then $$W^x_R \vee (W^y_R)' \simeq W^x_R \otimes (W^y_R)'$$ (where $A \vee B$ is the vN algebra generated by $A$ and $B$, as usual).

This can be found disucssed at the beginning of section 2.2 of

G. Lechner, On the construction of quantum field theories with Factorizing S-matrices

p. 20-21.

So, assuming the net $A$ satisfies all this we have for $O = O_{x,y}$ the causal subset with left corner at $x$ and right corner at $y$ the identity $$\begin{aligned} A(O) &= A(W^x_L)' \cap A(X^y_R)' \\ &= (A(W^x_L) \vee A(X^y_R))' \\ &= (A(W^x_R)' \vee A(X^y_R))' \\ &= (A(W^x_R)' \otimes A(X^y_R))' \,. \end{aligned}$$ Now take $H_{x,y}$ to be the total Hilbert space but regarded as a module just for $A(W^x_R)' \otimes A(X^y_R)$. Then the module endomorphisms should be $$End(H_{x,y}) = (A(W^x_R)' \otimes A(X^y_R))' = A(O) \,.$$

So the idea is that the 2-functor $Z$ of which $A$ might be the endomorphism co-presheaf assigns to paths $x \to y$ this $H_{x,y}$. To a point $x$ it might assign the vN algebra $A(W^x_R)$.

I am not sure yet exactly if I can see the full 2-categorical structure this may be hinting at, but it does look suggestive.

Another things I was being pointed to are some technical subtleties concerning the implementation of the “Schrödinger picture” of QFT in dimension $d \gt 2$.

There is

C. G. Torre, M. Varadarajan,

Quantum Fields at Any Time

which shows that everything is fine in dimension $d = 2$

and

Functional Evolution of Free Quantum Fields

which demonstrates in a simple example some subtleties for higher dimensions.

The problem arises, as summarized on p. 14, for time evolution between Cauchy sur^faces which cannot be transformed into each other by an isometry of the ambient spacetime (which is assumed to be flat Minkowski space in their computations).

In section B, starting on p. 6, the authors amplify that no such problem arises in the Heisenberg=AQFT pciture.

I started thinking about formalizing more the structure that is used here to turn a transport 2-functor, i.e. a differential nonabelian cocycle, into a local net. So about formalizing the role played by the Minkowski structure on base space in this construction.

Maybe, just as one can consider differential cocycles equivariant with respect to a groupoid, here we may have to talk about “poset equivariance”, encoding the lightcone structure of the underlying Minkowski manifold in a poset structure.

I am not sure yet what exactly to do, but started playing around with something like this:

our Minkowski (or globally hyperbolic Lorentzian) space naturally inherits the structure of a poset, as we know, by taking $x \leq y$ precisely if the point $y$ is in the future of the point $x$, equivalently, if and only if $x$ is in the past of $y$ (meaning that there exists smooth curve connecting $x$ with $y$ the Minkowski norm of whose tangent is everywhere non-negative).

So let me write $\mathbf{X}$ for the smooth poset we have, which can be thought of as a category enriched in (-1)-categories, i.e. enriched in the monoid $(\{true, false\}, \otimes = and)$.

Then $$IsFuture(x) := \mathbf{X}(x,-) : \mathbf{X}_0 \to \{true, false\}$$ is the subobject classifyer for the future of $x$, in that the future $Future(x)$ is the subset arising as the pullback $$\array{ Future(x) &\to& \{true\} \\ \downarrow && \downarrow \\ \mathbf{X}_0 &\stackrel{\mathbf{X}(x,-)}{\to}& \{true, false\} }$$ and $$IsPast(y) := \mathbf{X}(-,y) : \mathbf{X}_0 \to \{true, false\}$$ similarly is the subobject classifier for the past of $y$.

The crucial structures in the business of local nets are those causal subsets $O_{x,y}$: these are precisely the intersections of the future of one point $x$ with the past of another point $y$. So their subobject classifier is $$(a \mapsto \mathbf{X}(a,y)\otimes\mathbf{X}(x,a) ) : \mathbf{X}_0 \to \{true, false\}$$ so that the causal subset $O_{x,y}$ is the pullback $$\array{ O_{x,y} &\to& \{true\} \\ \downarrow && \downarrow \\ \mathbf{X}_0 &\stackrel{\mathbf{X}(-,y)\otimes \mathbf{X}(x,-)}{\to}& \{true, false\} } \,.$$

So suppose we start with something like a poset-covariant 2-functor on 2-paths $P_2(X)$, maybe a 2-functor from $$Codesc(N(\mathbf{X}), P_2) = \int^{[n]\in \Delta} P_2(N(X)_n)\otimes O(\Delta^n) \,,$$ where $N(\mathbf{X})$ denotes the nerve of the poset $\mathbf{X}$ and otherwise I am following the notation used here.

Then somehow the task is to naturally obtain from that a co-presheaf on the poset of $O_{x,y}$s using just abstract nonsense as above.

Okay, let’s see. What’s the natural expression for the poset structure on the causal subsets $O_{x,y}$ themselves. We simply have $$(O_{x',y'} \subset O_{x,y}) \Leftrightarrow ( (y' \leq y) and (x \leq x') ) \Leftrightarrow (\mathbf{X}(-_2,y)\otimes \mathbf{X}(x,-_1))(x',y')$$ assuming that both $O_{x,y}$ and $O_{x',y'}$ are non-empty, in that $x \leq y$ and $x'\leq y'$.

Hm, so what now? Can anyone see how to proceed from here along the abstract-nonsense route?