Extract from This Week’s Finds in Mathematical Physics (Week 257):

Though they probably don’t think of it this way, you can think of their work as making precise Bohr’s ideas on seeing the quantum world through classical eyes. Instead of talking about all observables at once, they consider collections of observables that you can measure simultaneously without the uncertainty principle kicking in. These collections are called “commutative subalgebras”.

You can think of a commutative subalgebra as a classical snapshot of the full quantum reality. Each snapshot only shows part of the reality. One might show an electron’s position; another might show its momentum.

Some commutative subalgebras contain others, just like some open sets of a topological space contain others. The analogy is a good one, except there’s no one commutative subalgebra that contains all the others.

Topos theory is a kind of “local” version of logic, but where the concept of locality goes way beyond the ordinary notion from topology. In topology, we say a property makes sense “locally” if it makes sense for points in some particular open set. In the Doering-Isham setup, a property makes sense “locally” if it makes sense “within a particular classical snapshot of reality” - that is, relative to a particular commutative subalgebra.

[…]

Doering and Isham set up a whole program for doing physics “within a topos”, based on existing ideas on how to do math in a topos. You can do vast amounts of math inside any topos just as if you were in the ordinary world of set theory - but using intuitionistic logic instead of classical logic. Intuitionistic logic denies the principle of excluded middle, namely:

“For any statement P, either P is true or not(P) is true.”

In Doering and Isham’s setup, if you pick a commutative subalgebra that contains the position of an electron as one of its observables, it can’t contain the electron’s momentum. That’s because these observables don’t commute: you can’t measure them both simultaneously. So, working “locally” - that is, relative to this particular subalgebra - the statement

P = “the momentum of the electron is zero”

is neither true nor false! It’s just not defined. Their work has inspired this very nice paper:

15) Chris Heunen and Bas Spitters, A topos for algebraic quantum theory, available as arXiv:0709.4364.

so let me explain that too.

I said you can do a lot of math inside a topos. In particular, you can define an algebra of observables - or technically, a “C*-algebra”. By the Isham-Doering work I just sketched, any C*-algebra of observables gives a topos. Heunen and Spitters show that the original C*-algebra gives a C*-algebra in this topos, which is commutative even if the original one was noncommutative! That actually makes sense, since in this setup each “local view” of the full quantum reality is classical.

What’s really neat is that the Gelfand-Naimark theorem, saying commutative C*-algebras are always algebras of continuous functions on compact Hausdorff spaces, can be generalized to work within any topos. So, we get a space in our topos such that observables of the C*-algebra in the topos are just functions on this space.

I know this sounds technical if you’re not into this stuff [you don’t say - DF]. But it’s really quite wonderful. It basically means this: using topos logic, we can talk about a classical space of states for a quantum system! However, this space typically has “no global points” - that’s called the “Kochen-Specker theorem”. In other words, there’s no overall classical reality that matches all the classical snapshots.

Oddly enough, last night as I was trying to get back to sleep after the earthquake (!!!) I suddenly had a very clear mental image of the pullback square which shows the correspondence between a subset and a characteristic function (it’s the one on the cover of Goldblatt’s book on topos theory, which I’m still plodding through). Turning it over in my mind, I was convinced that I knew what was in each corner of the square and what the arrows between them were, and that I now understood how, in a topos, the sub-object classifier could provide a value for the “degree of identity” between two objects. Looking it up in the morning, I was delighted to see that I had indeed pictured the diagram correctly; whether my impression of having understood its import was correct remains to be seen…