The n-Category Café

Peristalithic does not seem to be in the OED. It could be a typo for peristalitic meaning having to do with peristalsis, the process by which food is moved through the digestive tract. Or, it could be a modification of peristalith, “a ring or row of standing stones surrounding a burial mound, cairn, etc.”.

On part III, p.34, last paragraph, where it says

Of particular interest [are] tools for constructing theories that go beyond quantum theory and which do not use Hilbert spaces, path integrals, or any of the other familiar tools in which the continuum real and complex numbers play a fundamental role.

Few days ago I had a stimulating discussion with somebody working in general relativity.

He vividly recounted how he was trying to sensitise his students to the fact that in the history of physics, people have again and again taken various things to be obviously linear, only to later find that what looked obviously true was just the first approximation to a curved situation. The flatness of earth, the flatness of space, that of spacetime.

His conclusion was that assuming quantum mechanics to be evidently linear, no matter how closely we inspect quantum mechanical systems, is just one more fallacy in this series of linear fallacies.

Certainly he is not the first to speculate about non-linear QM, but his perspective was a rather noteworthy one, I found.

Since he also revealed himself as secretly being a “structuralist”, as far as theoretical physics is concerned, I offered him the following perspective:

Quantum theory is, apparently, the theory of representations of cobordism categories.

This perspective empowers us to deal with any potential nonlinearities in a robust way: we simply adapt the codomain of the representation and consider functors from cobordisms to categories that don’t necessarily have a forgetful functor to $\mathrm{Vect}$.

The point being, that this allows us to consistently determine what will and what will not change should we ever discover that the Hilbert space of states of a fundamental particle is, beyond some energy threshold, say, no longer conceivable as a linear space over the complex numbers.

For instance consider the body of work on correlators in topological and rational conformal 2-dimensional field theories using state sum models involving Frobenius algebras internal to certain monoidal categories.

Due to the way these are formulated internally to some context, the bulk of this work remains rather unaffected by the precise nature of this context, provided some collection of assumptions is satisfied.

As a result, there is a pretty much entirely diagrammatic way to conceive all of 2-dimensional topological and rational conformal theory, and the only point where an assumption on the linear nature of the context is used is when you want to translate such a diagrammatic correlator into an actual number.

Should there ever be evidence that, say, the quantum mechanics of the fundamental string needs to be conceived as being non-linear in some sense, we would make the necessary adjustment only at this point where we realize our diagrams by internalizing them appropriately. Most of the desireable structural aspects the quantum theory (as the sewing constraints) would carry over undisturbed, as it should.

So far there is no indication that we need to do this, so nobody has seriously looked into it.

But similar remarks actually apply also to the domain of our representation-theoretic problem: maybe we want to tamper with the nature of the category of cobordisms. This is not a linear category. Which is the reason why it can be relatively hard to deal with.

But maybe we want to reverse the above reasoning, then, and intentionally approximate something by a linear model which we know is nonlinear.

Again, this can be done by internalization, without affecting the basic tenets of what quantum theory is:

a good example is maybe what is called “topological conformal field theory”. There the category of conformal cobordisms (the domain of our representation functor) is replaced by the category which has the same objects, but where the Hom-spaces (the moduli spaces of conformal cobordisms with given boundaries) are replaced by the their homology complexes! This way the domain becomes a category enriched in complexes, $V$, and hence linear in this sense.

The entire resulting “topological conformal” quantum theory now is entirely about $V$-enriched representation theory. Remarkably, it turns out that this linear approximation still knows a lot about the true non-linear physics that we are really interested in.

In general, I believe that this is one of the powers of fully employing category-theoretic reasoning in the context of physics: it allows us to disentangle physical structures (arrow theory) from physical implementations (internalization).

Therefore it should befit us to think closely about what the arrow theory behind our physical theories really is, in particular behind quantum theory.

In search for a paved way into this body of ideas I was scanning the list of references, looking for Chris Isham’s older work on “quantization on a category” # which expresses an idea that I am comfortable with. Couldn’t find such a reference. I am wondering what the reason for that is.

I don’t understanding what you’re looking for and not finding, Urs. “Older work which expresses an idea you are comfortable with”? It sounds like you’re getting tired of radical new ideas.

I imagine you know about this trilogy:

• Chris Isham, A new approach to quantising space-time: I. Quantising on a general category.
• Chris Isham, A new approach to quantising space-time: II. Quantising on a category of sets.
• Chris Isham, A new approach to quantising space-time: III. State vectors as functions on arrows.

There’s also this related paper:

• Chris Isham, Quantising on a category.

We would certainly benefit from somebody going through the effort of providing us with a summary of a summary of Döring-Isham’s work.

I spent about an hour looking at the four parts in order to get a glimpse of the main ideas.

Certainly, in that one hour I must have missed many, many important points. My impression was, that the main idea revolved around a way to explain the difference between the concept of classical and quantum observables by internalization in different topoi.

The authors consider an object $$\Sigma$$ in the topos, which is to be thought of as the object of states. In the classical case this is a symplectic space, in the quantum case it is an algebra of operators (or rather the respective sheaves represented by these).

Then they consider an object $$\mathcal{R}$$ in which observables take their values. In the classical case this is just the real numbers. I forget what it is in the quantum case.

Then, a physical observable is modeled by a morphism $$A : \Sigma \to \mathcal{R}$$ in the topos. This is now something that models the assignment of values to states.

In the classical setup this is literally nothing but a real-valued function on the phase space of the system.

My impression was that much of the work (especially part II and III) revolves around understanding how quantum observables can be understood as morphisms $$A : \Sigma \to \mathcal{R}$$ too, when regarded in a suitable topos.

From a Numb3rs rerun last Friday:
a topos theory for analyzing Bach’s sonatas
or words to that effect

and better

(personal) relationships as described in a graded tensor category

jim

Very interesting. I find it intriguing that in the description of quantum
mechanics using a presheaf topos, the restriction maps of a presheaf move
in the direction of less information; whereas in the intuitionist
perspective on topos theory, the restriction maps generally move in the
direction of more information (e.g. as you restrict to a smaller
open set, you know more about where a hypothetical point may be located).

There seems to be at least one thing conspicuously missing, though:
where do the probabilistic predictions of experiments come in? The
daseinisation of a self-adjoint operator, as defined in section 3 of
paper III, seems only to capture the minimum and maximum values which
that observable might take on. But what about the actual probability
distribution of results? Am I missing something?

Mike

That reminds me of C.F.v.Weizsäcker’s “Kreisgang” (sort of hermeneutic circle) philosophy of Physics (Aufbau der Physik 1985, engl. The Structure of Physics 2006).

I never seen him mentioned around here (or e.g. his last student Holger Lyre). Any comment on his work?

I just started reading A topos foundation for theories of physics. I was pleased to see intuitionist logic showing up, at least partly because I’m a constructivist myself. In particular, I remember reading a thesis long ago in which someone (I forget who) analyzed the potential of using constructive math instead of classical for the conventional formulation of quantum mechanics. The thesis found situations in which the time-evolution of the wave function was not constructive; but I noticed that these cases happened to be covered by the uncertainty principle, so in once sense it didn’t matter.