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.

## Re: A Topos Foundation for Theories of Physics

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