### M Theory Lesson 293

Last time we considered chord flips on a square. Recall that a chorded square labels a point at the source or target of an associator edge.

A basic three leaf tree is what labels the associator edge. In other words, an associator edge is like two four point diagrams connected by a single three point diagram. This is backwards (or rather, Poincare dual) to the situation of interest in twistor QFT, which is where a four point vertex may factorize into pairs of three point vertices.

As it happens, there is such a thing as a dual square, where points become edges and edges points. In this context, roughly speaking, the square would be called a Hodges diagram. The vertices on a Hodges diagram stand for twistor variables in a massless field theory. Poincare duality is a simple enough idea, so massless particle physics really does see associahedra.

That's category theory, folks. You can keep saying that category theory has nothing to do with physics. But Nature does not agree with you.

A basic three leaf tree is what labels the associator edge. In other words, an associator edge is like two four point diagrams connected by a single three point diagram. This is backwards (or rather, Poincare dual) to the situation of interest in twistor QFT, which is where a four point vertex may factorize into pairs of three point vertices.

As it happens, there is such a thing as a dual square, where points become edges and edges points. In this context, roughly speaking, the square would be called a Hodges diagram. The vertices on a Hodges diagram stand for twistor variables in a massless field theory. Poincare duality is a simple enough idea, so massless particle physics really does see associahedra.

That's category theory, folks. You can keep saying that category theory has nothing to do with physics. But Nature does not agree with you.

## 8 Comments:

Just a random thought which popped in my mind during morning walk yesterday.

Baez et al are playing with very formal looking formal structures by replacing vectors of Hilbert space with Hilbert spaces and obtain results which make them happy. This looks to a physicist who has learned his Feynman diagrams more or less a waste of time. Just a pearl game.

I however realized that this might make sense in quantum measurement theory with finite measurement resolution described in terms of inclusions of hyperfinite factors of type II_1.

The included algebra represents measurement resolution and this means that the infinite-D sub-Hilbert spaces obtained by the action of this algebra replace the rays. Sub-factor takes the role of complex numbers in generalized QM so that you get non-commutative quantum mechanics.

For instance, quantum entanglement for two systems of this kind would not be between rays but between infinite-D subspaces corresponding to subfactors.

One could build a generalization of QM by replacing rays with sub-spaces and I think that quantum group concept does more or less this: the states in representations of quantum groups could be seen as infinite-dimensional Hilbert spaces.

What seems unavoidable is infinite dimension of Hilbert spaces replacing rays of Hilbert space - very natural in realistic physical theories in which the physics below length scale cutoff involves infinite number of degrees of freedom.

Hi Matti. Of course I agree that the random categorification game is not the way to think about new mathematics for physics. However, I agree with your insights here. One can think of the study of the hierarchy of n-Hilbert spaces (which is n-categorical) as a way to probe the mystical category of Motives - by which I mean the as yet undefined one, the one of which Grothendieck dreamed.

But the constructive arithmetic is missing - and who cares about Hilbert spaces, per se. This is a point that I believe you appreciate, and it is why I am interested in categories with prime objects, and constructive p-adic MUB structures and so on.

It's good that it deals with simple diagrams that correspond to vertices in physics. Do you have any example of category theory can be used to help make a checkable prediction, or does that require a lot more research and development?

Anonymous, the categorical version of the standard model, which I refer to in this post, is unfortunately not something that many people work on - you can probably count Matti, Carl and me, but I can't think of anyone else.

Now this version of the SM

does not agreewith all the predictions of the SM. For instance, if we could understand the higher dimensional operad polytope 'tilings' a little better, then in principle we could compute amplitudes for multijet processes at the LHC that would conflict with SM predictions.Outside SM type scattering amplitudes, there are plenty of predictions asociated to this new physics. Now one could always argue that these things are not connected to category theory, since the case is far less clear than for scattering amplitudes, but all a physicist can do is try to make quantitive predictions within a consistent framework. Whether or not it is possible to convince others that a given framework is consistent is not a question of mathematics (category theory or otherwise) but a question of physical principle.

I found the This Week's Finds explaining the idea of 2-vector space.

The 2-vector space defined as category of vector spaces (one forms n-tuples of vector spaces as analog for points of vector space) with tensor product and direct sum defining algebraic operations does not look interesting although it occurs in representation theory of groups.

The (to me rather) fuzzy notion of fractal quantum-dimensional factor space of HFF obtained by dividing HFF by included HFF might however give rise to much more interesting 2-structure. Fractal 2-operator algebras and 2-state spaces would be obtained.

This category would have direct physical interpretation as what one obtains by forming all possible many particle states with single parton corresponding to a given inclusion of HFF characterizing the particular measurement resolution. In zero energy ontology situation comes even more interesting.

What would make the structure interesting is that the tensor product would not be trivial anymore since the action of sub-factors involved should be effectively complex multiplication just as in ordinary quantum theory since it does not change change the physical state in the measurement resolution considered.

If I have not totally misunderstood the basic notions, this would force irreducible entanglement describing the interaction and expressible in terms of Connes tensor product. The category in question would describe a hierarchy of interacting systems with Connes tensor product coding for the interactions.

The M-matrices as generalizations of S-matrices - defined in terms of Connes tensor product and characterized by measurement resolutions- would also relate naturally to a 2-structure with respect to tensor product. Connes tensor products of M-matrices would characterize the multiple Connes tensor products of zero energy state spaces.

Yes, Matti, I think that is basically right, although I have my own confused way of thinking about it, and I suspect that Connes' tensor product should become something much richer, in categorical terms.

Do you actually mean Poincare duality? If so, in which space?

Rhys, I mean Poincare duality in the 1D space defined by the graph. So a square is just a square.

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