December 6, 2007

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 internal to the topos of presheaves on the site of open subalgebras of $A$.

Posted at December 6, 2007 10:45 PM UTC

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Re: A Topos for Algebraic Quantum Theory

…given any noncommutative structure A, we may try to “cover” it by abelian things.

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

As mentioned above the notion of scheme is obtained by patching together the geometric counterpart of arbitrary commutative rings. Thus one might wonder at first why such patching is unnecessary in noncommutative geometry whose basic data is simply that of a noncommutative algebra. The main point there is that the noncommutativity present already in matrices allows one to perform this patching without exiting from the category of algebras.

And so,

Thus there is no need in noncommutative geometry to give a “gluing data” for a bunch of commutative algebras, instead one sticks to the “purest” algebraic objects by allowing simply noncommutative algebras on the algebraic side of the basic duality,

$Geometric Space | Non Commutative algebra$

Why are we so eager to abelianize everything?

Posted by: David Corfield on December 7, 2007 8:54 AM | Permalink | Reply to this

Re: A Topos for Algebraic Quantum Theory

I paraphrased one of the main ideas of the toposophers of quantum mechanics by saying:

…given any noncommutative structure A, we may try to “cover” it by abelian things.

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

At least the kind of “abelian cover” that I mentioned is a rather different concept compared to the abelian patching that Alain Connes is referring to in the paragraph that you quote.

It might be useful to think of it this way:

the nature of the “abelian cover” of $A$ that I mentioned, namely the precise nature of the category $C(A)$, is actually invisible. Or supposed to be. What we really want to say is that

The theory of noncommutative algebras in a topos $S$ is equivalent to the theory of commutative algebras in some topos $T$ internal to $S$.

That the topos $T$ in question can be modeled by considering presheaves on a category of abelian subalgebras of $A$ might be useful for explicit computations, but is actually not something that pertains to the intrinsic meaning of the above statement, I would say.

So the way I am thinking of it, I would actually regard the toposopher’s perspective as a strengthening of Alain Connes’ point of view:

they kind of say: yes, Connes is precisely right in that the noncommutative algebra $A$ doesn’t have to be thought of as being patched together from abelian things. And we can actually make this precise: because the noncommutative algebra is one single commutative algebra – when regarded in the right context.

Why are we so eager to abelianize everything?

Let’s qualify this question a little: why are the toposophers of quantum mechanics so eager to abelianize everything?

(Because in other areas people are equally eager to make everthing in sight non-abelian ;-)

And the answer is: because they think this is the right way to deal with the vexing measurement problem of quantum mechanics.

The non-abelianness of the quantum phase space implies that the connection between the corresponding physical model and its interpretation is subtle. The toposophers say:

this subtlety is precisely the need to interpret everything internal to a suitable topos. And the construction of this happens to involve, if you look at it externally, “local abelian covers” of the nonabelian situation.

Okay, that’s what I can say. Now I’ll think about how to say it better. And probably before I can do that John Baez will chime in and give a crystal clear account.

Posted by: Urs Schreiber on December 7, 2007 9:58 AM | Permalink | Reply to this

Re: A Topos for Algebraic Quantum Theory

David wrote:

Why are we so eager to abelianize everything?

I think it’s because people like situations where a state of a system simply consists of a state for each part. They like situations where we can freely duplicate and delete information. In other words, they like symmetric monoidal categories that are cartesian. The classic examples are the category of sets and various categories of ‘spaces’. There’s something very geometrical about cartesian categories.

Algebras form a symmetric monoidal category with their usual tensor product, but it’s not cartesian.

Commutative algebras also form a symmetric monoidal category with their usual tensor product, and this isn’t cartesian either. But, the opposite of this category is cartesian!

In other words, commutative algebras are dual to ‘spaces’ — or technically, ‘affine schemes’ — and affine schemes form a cartesian category. This is the idea behind algebraic geometry, which secretly goes back to Descartes.

Noncommutative algebras are just different. I’ve always thought we should just accept them for what they are. But, it’s interesting that we can often think of them as giving rise to commutative algebras not inside the category of sets, but inside some other cartesian category!

You can think of this trick as a clever way to keep ahold of the cartesianness we love so much.

Posted by: John Baez on December 8, 2007 8:02 AM | Permalink | Reply to this

Re: A Topos for Algebraic Quantum Theory

I think the oldest example of the idea of studying something non Abelian by looking at Abelian subthingies might be the theory of simple Lie algebras, where you use the Cartan subalgebra (invented by Killing, of course, just as the Killing form was invented by Cartan) to analyse the structure and representations. In this case all Abelian subthingies are conjugate, so probably there is no need for topoi. Did anybody look at non semisimple Lie algebras from this point of view? I think also for Kac-Moody algebras Cartans are not always conjugate, so that might be interesting, and connected to quantum field theory.

Posted by: Maarten Bergvelt on December 7, 2007 5:35 PM | Permalink | Reply to this

Re: A Topos for Algebraic Quantum Theory

Maarten wrote:

I think the oldest example of the idea of studying something non Abelian by looking at Abelian subthingies might be the theory of simple Lie algebras, where you use the Cartan subalgebra […] to analyse the structure and representations. In this case all Abelian subthingies are conjugate, so probably there is no need for topoi.

But it’s still possible that topoi could help clarify what’s really going on, at least for people who understand topoi.

When, shockingly late in life, I first seriously tried to understand simple Lie algebras and their representations, I spent some time trying to figure out the deep inner meaning of the usual Cartan subalgebra / Weyl group / weight lattice baloney.

I call it “baloney” because while it’s great, everyone seems to blindly accept it, without asking why it really works. No matter how good a piece of math is, we need to keep rethinking it, or it becomes stale.

I didn’t get too far, but I had hopes. I was thinking about Lie groups rather than Lie algebras, but that shouldn’t matter much. I was trying to see a compact simple Lie group $G$ as somehow ‘built from’ the action of the Weyl group $W$ on a maximal torus $T$. And, I wanted to use this to describe the usual category $Rep(G)$ of finite-dimensional representations of $G$ in terms of the action of $W$ on $Rep(T)$. $Rep(T)$ is equivalent to the category of vector bundles on the dual group $T^*$, also known as the “weight lattice”. So, I hoped this stuff would give a nice conceptual approach to the usual “highest weight” approach to constructing irreducible guys in $Rep(G)$.

I didn’t make much progress, so I posted a question about it to sci.math.research:

I’m enjoying Adams’ book on Lie groups, where he downplays the semisimple Lie theory and emphasizes compact Lie groups in comparison to standard treatments. One of the basic results is as follows.

Suppose $G$ is a compact Lie group, $T$ a maximal torus, and $W$ the Weyl group (the normalizer of $T$ mod the centralizer of $T$). Then the following two algebras are isomorphic: the algebra of class functions on $G$, and the algebra of $W$-invariant class functions on $T$, where W acts as automorphisms of $T$ in the obvious way. The isomorphism is given by restricting class functions on $G$ to class functions on $T$.

Now the algebra of class functions on $G$ is also called the representation ring of $G$, for good reasons. The category of representations of $G$ is a monoidal category with a notion of direct sums, and from any such category one can extract an algebra whose elements are formal linear combinations of objects; direct sums in the category correspond to addition in the algebra, and tensor products in the category to products in the algebra, in a well-known way. (This is the Grothendieck ring construction.) Each representation of G yields an element of the representation ring, and concretely speaking, the latter is just the character of the representation.

So I suspect the following. Let $C$ be the category of finite-dimensional representations of $G$. Then there is some category $C^'$ of representations of $T$ equipped with some extra bit of structure involving $W$, such that $C$ and $C^'$ are equivalent as monoidal categories with direct sums. (To be more formal, instead of “with direct sums” I could talk about abelian categories, but it’s not those category-theoretic niceties that are the issue here, it’s the group theory.)

To get from $C$ to $C^'$, we first of all simply restrict any representation of $G$ to a representation of $T$. But then we need to retain some extra bit of structure involving $W$ to make sure we don’t lose any information (so that $C$ and $C^'$ are equivalent). What is it? Whatever it is, it’s reflected in the fact that when we take the character of a representation of $G$ and restrict it to $T$, we get something $W$-invariant.

The process of getting back from $C^'$ to $C$ is closely related in spirit to the “highest weight representation” construction. But I’d like to talk about it in a way that doesn’t use any arbitrary choices (like a choice of Weyl chamber).

I got a reply from Gavin Wraith which suggested thinking of $T$ as a group in the topos of $W$-sets, and trying to use this as a kind of stand-in for $G$. It’s easy to see that $Rep(T)$ becomes a category in the topos of $W$-sets. But, I don’t think we got as far as we wanted, namely to describe $Rep(G)$ in this language.

However, what’s nice is that we see a way to get from a compact connected Lie group to a compact connected abelian Lie group in a certain topos.

This is almost exactly analogous to what Heunen and Spitters do for C*-algebras!

The main difference is that Wraith considers a topos of presheaves on a certain groupoid, while Heunen and Spitters consider a topos of presheaves on a certain poset. Wraith’s groupoid consists of maximal abelian subgroups of the given Lie group, with ‘conjugations’ as isomorphisms. Heunen and Spitters’ poset consists of all abelian C*-subalgebras of the given C*-algebra, with inclusions as morphisms.

Maybe we should try to get the benefits of both approaches, by working with presheaves on a category where the morphisms are ‘inner monomorphisms’ — that is, conjugations followed by inclusions.

Posted by: John Baez on December 7, 2007 7:22 PM | Permalink | Reply to this

Re: A Topos for Algebraic Quantum Theory

John,
Are you asking for a Barr-Beck type description of reps of G in terms of reps of T? (ie coming from the adjoint
paper restriction-induction?) If we replace Rep T by Vect we get that Rep G are the same
as modules over the algebra C[G] (regular
rep) in (ind) Vect, which I presume
is what Barr-Beck says here. So for
Rep T shouldn’t we be saying Rep G are
objects in Rep T equipped with an action
of the algebra object C[G] in (ind of)
Rep T? Now I agree this is a purely
is how to think of this algebra object
(the regular representation) in terms of
T and W.. not sure how to answer that.

OTOH, I’m not sure I like this point
of view so well — as stacks G/G (adjoint action) and T/W are quite different, they just have the same functions on them –
once we look at more subtle invariants
we can feel the difference more.
Of course for noncompact semisimple groups
this becomes more serious: how do you
think of the collection of (conjugacy
classes of) maximal tori modulo
the corresponding Weyl group as a model for G/G? On the level of characters again this works, this is one of Harish-Chandra’s great contributions, but I would love to understand how it might apply to categories of representations!

David

Posted by: David Ben-Zvi on December 8, 2007 4:48 PM | Permalink | Reply to this

Re: A Topos for Algebraic Quantum Theory

John wrote:

I was trying to see a compact simple Lie group $G$ as somehow ‘built from’ the action of the Weyl group $W$ on a maximal torus $T$. And, I wanted to use this to describe the usual category Rep($G$) of finite-dimensional representations of $G$ in terms of the action of $W$ on Rep($T$).

For finite groups, there is an intriguing statement of this kind which I call “Kirillov’s method”. You can find it on page four of this paper.

Suppose we have a short exact sequence of groups,

(1)$1 \rightarrow N \rightarrow M \rightarrow G \rightarrow 1$

By picking a set-theoretic section, we can think of this as a homomorphism of 2-groups

(2)$G \rightarrow AUT(N).$

This means that Rep($N$) carries a 2-representation of $G$. In Urs’s language, this is the canonical 2-representation .

Kirillov says that we can reconstruct Rep($M$) as the category of $G$-invariant objects in Rep($N$),

(3)$Rep(M) \cong (Rep(N))^G.$

I find that remarkable but I have to admit that I don’t understand it! (Even though my phd is about 2-representations). I’m not even sure if it’s meant to be an equivalence of monoidal categories, or just an equivalence of categories (or possibly abelian categories). Unfortunately Kirillov’s paper is a bit confusing in places.

I don’t know if the trio consisting of the Weyl group, the maximal torus and the Lie group $G$ can be cast into this formalism.

Posted by: Bruce Bartlett on December 9, 2007 11:28 PM | Permalink | Reply to this

Re: A Topos for Algebraic Quantum Theory

I think also for Kac-Moody algebras Cartans are not always conjugate

KM algebras have at least unique Dynkin diagrams, so Cartan algebras are isomorphic. In contrast, most infinite-dimensional Lie algebras, e.g. Virasoro, don’t have Dynkin diagrams and Cartan matrices; twice a root can still be a root. For Lie superalgebras the situation is even more complex even in the finite-dimensional case: some have a unique Dynkin, some have several, and some have none, e.g. diff algebras in (0|m) dimensions.

Re: A Topos for Algebraic Quantum Theory

Might it help to ask the question in even more generality?

It seems what we are talking about is something like local stabilization or the like.

The algebra $A$ is a set (in particular), which is 1-tuply monoidal. Hence we can globally think of it as a 1-object category.

But subsets of $A$ may actually be doubly monoidal, the abelian subalgebras. These we can promote to one-object one-1-morphism 2-categories.

Maybe we should think about how we’d handle nonabelian 2-algebras in a 2-topos context.

These are 1-tuply monoidal categories. I suppose toposophers would want to look at them as being “locally triply monoidal”, namely look at all their symmetric braided monoidal subcategories.

Or would we maybe proceed in two steps: first look at all braided monoidal subcategories and then at all symmetric monoidal subcategories of all of these?

This looks like a more difficult question than the original one. But sometimes it helps to ask the more general question in order to lift accidental degeneracies of low dimensions that may be misleading.

Posted by: Urs Schreiber on December 10, 2007 9:24 AM | Permalink | Reply to this
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