### A Topos for Algebraic Quantum Theory

#### Posted by Urs Schreiber

I’ll try to summarize, very briefly, some key points of

Chris Heunen, Bas Spitters
*A Topos for Algebraic Quantum Theory*

arXiv:0709.4364v1 [quant-ph]

following up on our discussion of The Principle of General Tovariance.

(John Baez talked about Heunen&Spitters’ work in week 257.)

*Local abelianness*: a site for noncommutative things

The crucial underlying idea is the one featuring prominently already in Chris Isham’s work:

roughly speaking, given any noncommutative structure $A$, we may try to “cover” it by abelian things. This is supposed to mean that we form the poset $C(A) = \{ V \hookrightarrow W | V,W \subset A commutative sub-things\} \,.$ Objects are all commutative subthings of $A$. Morphisms are inclusions.

In the cases of interest in the context of the work by Baas, Döring, Isham, Landsman, Spitters, the things here are

a) either $C^*$-agebras

b) or von Neumann algebras.

I am not sure what the general condition is that would ensure that $C(A)$ actually is a site, and it does not matter much for the following, but it is good to *think* of $C(A)$ as defining something like a site of open subsets of the thing $A$.

**A noncommutative thing becomes abelian internal to its own site**

The main construction now is actually more or less just a sophisticated tautology (which is good!):

Given any $A$ (as far as $A$ is a set with extra structure), there is a canonical presheaf on $C(A)$ representing the thing $A$ in its own site: the tautological functor $\bar A : C(A) \to \mathrm{Set}$ which maps each commutative sub-thing $V$ of $A$ to the set underlying it: $\bar A : V \mapsto V \,.$

(There is a slighty annoying issue here with whether or not we are using $C(A)$ or $C(A)^{\mathrm{op}}$. Heunen and Spitters argue that $C(A)$ is more natural. Which means that you may want to think of it in many of the following formulas as $(C(A)^{\mathrm{op}})^{\mathrm{op}}$.)

If the “things” we are talking about are sufficiently well behaved such as to qualify as what are called “geometric theories”, a beautiful statement of abstract nonsense (corollary D.1.2.14 in Johnstone’s *Sketches of an elephant*) says that

Sheaves of things are these things internal to sheaves.

More precisely:

**Fact.** Let $T$ be a geometric theory and $\mathbf{C}$ any category. Denote by $\mathbf{Mod}(T,\mathbf{T})$ the category of $T$-models in the topos $\mathbf{T}$ with $T$-homomorphisms. There is an isomorphism
$\mathbf{Mod}(T,\mathcal{S}^{\mathbf{C}})
\simeq
\mathbf{Mod}(T,\mathcal{S})^{\mathbf{C}}
\,.$

Here a theory is essentially just a certain category, and a model is essentially just a functor from that category to somewhere else (an image of the abstract idea in some concrete context). $\mathcal{S}$ is any chosen ambient topos, which you are encouraged to think of as $\mathcal{S} = Set$, to get started.

$C^*$- and von Neumann algebras do satisfy the required assumptions here (at least after slighty tuning their definitions to adopt them to a more general setup then usually bothered with). So we find

The tautological functor $\bar A : C(A) \to Set$ is hence (mapped by the above isomorphism to) an abelian thing internal to sheaves on $C(A)^{\mathrm{op}}$.

This way the nonabelian thing $A$ becomes an abelian thing internal to its topos $Presheaves(C(A))$.

**The spectral presheaf is the internal spectrum of the internal algebra**

Isham and Döring prominently considered the presheaf $\Sigma : C(A)^{\mathrm{op}} \to Set$ which sends each abelian subalgebra to its Gelfand spectrum $\Sigma : V \mapsto Spec(V) = \mathrm{Hom}_{algebras}(V,ground field) \,.$ (See for instance this discussion.)

Now, Heunen and Spitters observe that the spectral presheaf is exactly what it should be: with $\bar A$ regarded as an internal commutative algebra, $\Sigma$ is the *internal* Gelfand spectrum.

That this is true is easy to see. Let $\bar K$ be the constant presheaf which sends everything to the set underlying the ground field. This is an internal “ground field object”. Then $\Sigma$ is the collection of algebra homomorphisms from $\bar A$ to $\bar K$, all thought of internally.

(This is corollary 10 in Heunen & Spitters. They are being somewhat more sophisticated then I want to be here, in that they consider the spectrum as a *locale*.)

**States on noncommutative algebras are internal distributions**

Chris Heunen and Bas Spitters keep going this way, pointing out how important structures on the nonabelian algebra $A$ are nothing but the corresponding structures on commutative algebras, internalized to presheaves on $C(A)$.

If you have an abelian $C^*$-algebra $A$, hence an algebra of continuous functions on some topological space by the Gelfand-Naimark theorem, you obtain good *states* on this, namely linear functionals
$\rho : A \to K$
(don’t confuse this with elements of the spectrum: these were also maps from the algebra to the ground field, but required to be algebra homomorphisms)

by integrating functions against a measure

$\rho_\mu : C(X)\ni f \mapsto \int_X f d\mu \,.$

After adopting a cool useful general formulation of the notion of integration abstracting from Lesbesgue integrals and others, called *algebraic integration*, Heunen and Spitters show that

*States on noncommutative $C^*$-algebras $A$ are nothing but internal integrals as above, on $\bar A$.*

That’s their theorem 15.

**Self adjoint operators are internal real numbers**

It keeps going this way. Everything works out just as one would wish.

We were all taught that self-adjoint operators *play the role* of real numbers. Heunen and Spitters make this precise:

The real numbers *internal* to $Presheaves(C(A))$ are the self-adjoint elements of $A$.

(I must admit I follow the details of this until they state their lemma 17, which is again a quote from Johnstone’s *Sketches of an elephant*. At this point Heunen&Spitter’s discussion seems to become rather less than self-contained. I’ll re-read this again tomorrow morning, when I am more awake…

**Summarizing slogan**

The theory of noncommutative algebras $A$ is that of commutative algebras

internalto the topos of presheaves on the site of open subalgebras of $A$.

## Re: A Topos for Algebraic Quantum Theory

Is this perspective then at odds with Connes’, who seems to enjoy noncommutative structures for what they are?

For example, in A View of Mathematics (p. 20) he says

And so,

Why are we so eager to abelianize everything?