What categorical mathematical structure(s) best describe the space of “localized events” in “relational quantum mechanics”?

Question

In a recent (and to me, very enlightening) paper, entitled "Relational EPR", Smerlak and Rovelli present a way of thinking about EPR which relies upon Rovelli's previously published work on relational quantum mechanics (see http://arxiv.org/abs/quant-ph/9609002 ). In relational quantum mechanics, there is no non-locality, but the definition of when an event occurs is weakened from Einstein's strict definition and instead is localized to each observer-measurement apparatus, including subsequent observers. There are (informal) coherence assumptions to ensure the consistency of reports from different subsequent observers (all possible friends of Wigner).

All of this seems very similar to various results in modern categorical mathematics. Is there a standard mathematical structure which well describes the structure of the space of localized measurements which Rovelli has envisioned? I know of Isham's work on topos theory and quantum mechanics, but I think he is aiming at something a little different.

PS I first asked this on mathunderflow, then on mathoverflow but to no avail, I am therefore reposting it here.

Answer 1

To respond to Marek's comment: as a reader familiar with physics and categories, I can say why I am unable to even approach a response. The question cites some papers, but is not readable on its own (not "self-contained"). We only learn is that there is some nonlocal theory out there which obeys some consistency conditions. I am sure that this would seriously impair the possibility of help from Math Overflow.

Rather than trying to guess at the theories being asked about, then, maybe it's a good idea to say a word about Segal's axioms and S-matrix theory. Picture a quantum field theory on (IN)xR, where IN is a spatial manifold. Imagine that the incoming spacelike hypersurface of spacetime is a disjoint union of IN_i. To each component IN_i we assign a Hilbert space H_i, so an incoming state is in a tensor product of the H_i. The scattering of these states leads to an outgoing state in OUT_j, where OUT_j are the components of the outgoing spacelike hypersurface. This map from IN states to OUT state is called the S-matrix. "Factorizability" of the S-matrix says that when spacetime is made from "gluing" together two spacetimes along a common OUT/IN, the S-matrix map is the appropriate tensor product of maps. (This does correspond to a locality assumption, roughly that the path integral has a continuous integrand called the action.)

In two dimensions, where compact one-manifolds are very simple (always circles), this business simplifies greatly -- especially if your quantum field theory is independent of the metric on spactime -- and the "pair-of-pants" surface provides a multiplication map on the Hilbert space (two go in, one comes out), the notion of an algebra. Allowing four one-manifolds with boundary leads to the notion of a category (an algebra is a category with one object, for what it's worth).

Answer 2

I have been attacking this question in a series of papers over many years, My last 4-5 papers on gr-qc are relevant. Pardon me for not repeating them here.

Louis Crane

Answer 3

There exists a notion of locality in Physics, broadly given by a WKB-style approximation done to the Path Integral: if you do a steepest-descent approximation to the Path Integral, the critical points of the Action will give you the regions that mostly contribute to the Path Integral.

Now, it's possible to generalize this notion to gauge theories… and the name of the game is equivariant localization.

It is possible to categorify this notion of localization à la AQFT from n-functorial QFT.

I'm not sure this answers your question, in the sense that i don't know how much these concepts are related to "relational quantum mechanics". But, maybe this can help…