## April 15, 2008

### A Topos for Algebraic Quantum Theory (revised)

#### Posted by Urs Schreiber

[guest post by Bas Spitters]

Revised and vastly expanded versions of our work on: The Principle of General Tovariance and A Topos for Algebraic Quantum Theory are now available.

They were discussed before at the cafe here and here.

In this paper we relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces.

Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a $C^*$-algebra of observables $A$ induces a topos $T(A)$ in which the amalgamation of all of its commutative subalgebras comprises a single commutative $C^*$-algebra $A$. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum $\Sigma(A)$ in $T(A)$, which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a $C^*$-algebra (which is the noncommutative notion of a space).

In this setting, states on $A$ become probability measures (more precisely, valuations) on $\Sigma$, and self-adjoint elements of $A$ define continuous functions (more precisely, locale maps) from $\Sigma$ to Scott’s interval domain. Noting that open subsets of $\Sigma(A)$ correspond to propositions about the system, the pairing map that assigns a (generalised) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by $A$ is essentially turned into a classical physical theory, internal to the topos $T(A)$.

These results were partly inspired by the topos-theoretic approach to physics recently proposed by Doering and Isham.

Posted at April 15, 2008 9:32 PM UTC

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### Re: A Topos for Algebraic Quantum Theory (revised)

Is it possible to give a practical example in a attempt for a theory? For instance, how would that help Urs to view string theory in a new fashion

Posted by: Daniel de França MTd2 on April 16, 2008 2:13 PM | Permalink | Reply to this

### Re: A Topos for Algebraic Quantum Theory (revised)

Is it possible to give a practical example in a attempt for a theory?

It is a standard fact that much of the kinematics of quantum theory has to do with noncommutative $C^*$-algebras.

While the articles on natural toposophy tend to have “quantum” in their titles, strictly speaking what they do consider, to date, is “just” the theory of $C^*$-algebras:

Bas Spitters and his collaborators show that the theory of non-commutative $C^*$-algebras can be made to look entirely like the theory of commutative $C^*$-algebras by working internal to a suitable topos.

What this bare fact mainly achieves concerning quantum theory is that it indicates a way to better formalize the interface between a physical theory and the collection of “predictions” or “propositions” one can derive from it.

So far the very notion of “physical theory” has been rather vague. The toposopher’s results indicate that we might want to formalize a physical theory as something which comes in particular equipped with a pointed topos: a topos $T$ with a chosen object $\Sigma \in T$, called the “space of states”.

Given that data

- the formalization of the “collection of states” that the physical theory in question describes would be: “the generalized elements of $\Sigma$”, i.e. the morphisms $hom_T(1,\Sigma)$.

- the formalization of “a proposition about states” (such as “in this state the energy of the system described is within 5 and 50 Joule”) would be: a morphism $\Sigma \to \Omega$ from the chosen object $\Sigma$ to the subobject classifier $\Omega$.

- the formalization of “the prediction of the theory about proposition $P$ applied to state $\psi$” would be: the truth value represented by the morphism $1 \stackrel{\psi}{\to} \Sigma \stackrel{P}{\to} \Omega \,.$

The main insight, as far as quantum theory is concerned, of all this is: it shows that there is a unified formalization of “propositions about states in a physical theory” which applies to classical physics just as well as to quantum physics.

There is, at this state of the development, no physical insight to be gained from this, nor is it particularly useful to list examples: take any quantum mechanical system you like, then the toposophy work just guarantees you that the well-known and loved way to describe expectation values of observables fits into the above general systematics.

That’s all.

But I think it is pretty cool. It possibly sheds light on “what physical theory is”, which might come in handy one day and might help guide the way when it comes to building physical theories that go deeper than the ones we currently have. The enigma “quantum cosmology” has been pointed to in this context.

But you should be aware that there is, at this moment, no implication of this formalism which would affect the average working physicist. It seems that, currently, the effects on pure math are bigger: there is now a Gelfand-Naimark theorem for non-commutative $C^*$-algebras, for instance. That’s nice.

For instance, how would that help Urs to view string theory in a new fashion

Let’s say: “view quantum theories”.

Actually, I did think about how the toposopher’s point of view fits into the one I am talking about: I made a remark about the relation that i see in

The main observation there is this:

1) quantum theories of “general $\Sigma$-model type” have “spaces of states” which are the spaces of sections of certain $n$-bundles.

2) A section of an $n$-bundle is naturally represented by a morphism from the terminal $n$-bundle to the given one.

3) Therefore a state in this context very naturally comes as a morphism of the form $\psi : 1 \to \Sigma$.

4) Therefore one should try to see if this morphism of $n$-bundles could be usefully regarded as a corresponding morphism in a topos category. That would then be a natural realization of the toposopher’s formalization of quantum theory. And in particular, it would naturally indicate what it would mean to make a proposition about a state in a “generalized $\Sigma$-model” quantum theory.

Posted by: Urs Schreiber on April 16, 2008 6:32 PM | Permalink | Reply to this

### Re: A Topos for Algebraic Quantum Theory (revised)

Hi Urs.

Great blog. Great Math.

I’m a Msc. Math student in Argentina. I’m getting close to the end of my curricula and I’ve been looking for a nice subject to study and prepare my thesis for some time now. It so happens that a friend of mine showed me a few papers by Alain Connes and Matilde Marcolli - they work in the Noncommutative Geometry paradigm.

For what I’ve read, they are very much aware of the role played by theoretical and quantum physics in the develpment of mathematical ideas - a view which I agree with to a large extent: the notion of Geometry being the bridge between reality and mathematics.

They take up Heisenberg’s position towards Quantum Theory: the theory of states and the behaviour of the spectral lines put in evidence the need of a NON-COMMUTATIVE concept of space. I believe this somewhat differs from Bohr’s criterion…

Actually, they choose not to make use of Grothendieck notion of topos (as a generalization of a topological space) but to deliberately attach a NON-COMMUTATIVE algebra to a geometric object X, and make it play the role of the usual algebra of continuos functions over X. Then, they make use of the whole arsenal of K-theory and operator theory on that algebra and say stuff about X.

Now my question is, to what extent the Connes-Marcolli approach is equivalent to the one via topos theory?

I’m a real beginmer with all this, but so far, it’s great.

keep it up!

manu

Posted by: manuel puebla on May 3, 2008 10:14 PM | Permalink | Reply to this

### Re: A Topos for Algebraic Quantum Theory (revised)

Actually, they choose not to make use of Grothendieck notion of topos (as a generalization of a topological space) but to deliberately attach a NON-COMMUTATIVE algebra to a geometric object X, and make it play the role of the usual algebra of continuos functions over X.

Yes, so the difference is that in Connes’ NCG one does not consider non-commutative algebras locally and then tries to glue them to global structures.

In other contexts one does that, though, to some extent or other. The “local nets of operator algebras” that are used in “algebraic quantum field theory” (AQFT) are co-presheaves of noncommutative algebras: there is the noncommutative algebra (“of observables”) associated to a Minkowski space as a whole, but there is also a rule for how to pick sub-algebras of that for each open subset of the big space.

A more ambitious approach to merge noncommutative geometry with the ideas going back to Grothendieck is that by Kontsevich and others, such as Rosenberg. See for instance Kontsevich & Rosenberg, Noncommutative smooth spaces and many more recent developments.

That said, though, I want to hasten to add that I feel that reducing Alain Connes’ approach to geometry to the question of non-commutative algebras is missing the point: the term “Non-commutative geometry” for his approach is not very accurately descriptive. A more fitting term would be: “(not necessarily commutative) spectral geometry”:

because the point is not to generalize the usual local, patchwise description of spaces to the noncommutative setup and then follow the usual local construction of metric data and all that derives from this. Rather, the point is the observation that there is an elegant “global” description of all that in terms of operator algebra: the notion of a spectral triple.

So the important lesson from quantum mechanics here is: the energy eigenvalues of the quantum particle propagating on compact space (a 1-dimensional “$\sigma$“-model) know a lot about the Riemannian geometry of that space.

Not all quantum mechanical systems come from a geometric $\sigma$-model though. But once we know how to read off the geometric data of “target space” from the spectrum of the particle, we can give something like a geometric interpretation of those “non-geometrical phases”, too. And that’s what Alain Connes’s notion of spectral triple formalizes:

a spectral triple is one way of formalizing what “quantum mechanical system” means – plus a toolbox for extracting “effective target space geometry” from that data.

A very nice discussion along these lines is in

At the end of these notes the authors indicate how to move from spectral triples describing 1-dimensional quantum theory to “2-spectral triples” describing 2-dimensional quantum theory. More detailed studies in this direction are currently being undertaken by Yan Soibelman (who on request might send you detailed unpublished notes), I have talked about that in Spectral Triples and Graph Field Theory.

Now my question is, to what extent the Connes-Marcolli approach is equivalent to the one via topos theory?

Maybe it is important to note that it can be ambiguous to talk about THE approach via topos theory. Topos categories can be used in different ways here.

In particular, there are always two sides to a topos: spaces on one hand and logic on the other.

The Kontsevich-Rosenberg approach mentioned above is more along the space-like sheaf-theoretic use of toposes. Whether their noncommutative algebraic geometry is equivalent to Connes’ spectral geometry i do not know.

The use of toposos in the context of noncommutative geometry which we have been talking about here, however is, at least superficially, different:

the work by Döring, Heunen, Isham, Landsman, Spitters shows, I think, that

one can regard a given non-commutative (spectral) geometry with non-commutative algebra $A$ as a given commutative (spectral) geometry internal to a suitable category $C(A)$

Here $C(A)$ is the category of contravariant functors from the poset of commutative subalgebras of $A$ to the category of sets.

I haven’t even mentioned the word topos in saying this, just to make the main point: their approach says something about “internalization”: interpreting one mathematical structure (commutative $C^*$ algebras in this case) in another category $C(A)$.

That this category $C(A)$ here happens to be a topos is, from this point of view so far, secondary. But, yes, it is a topos. And that it is a topos then allows one to see how the “internal logic” of this topos might help straighten out the conceptual puzzlements induced by looking at the non-commutativity of “noncommutative spaces”. These puzzlements are known, in quantum physics, as the measurement problem or the like, and have been the main motivation for Chris Isham to think about all this.

Posted by: Urs Schreiber on May 4, 2008 10:26 AM | Permalink | Reply to this

### Re: A Topos for Algebraic Quantum Theory (revised)

So I am thinking of the toposopher’s work as providing an “interface” or “back-end” to both quantum mechanics and Connes’ spectral geometry (which I regard as essentially the same thing, anyway): it may provide the right point of view for “connecting theory with observation”.

For you might ask the question: okay, so suppose all these speculations turn out to be right, and the spacetime around us is, indeed, if one looks closely enough, a non-commutative space of sorts (in particular the Kaluza-Klein reduction of a 10-dimensional space to a 4+ small-6 dimensional non-commutative space the way Alain Connes reads the standard model), then: what does that really mean operationally?

In a way, the question is: what is a physical theory? You can talk about “wavefunctions” and “Hamilton operators” and all that all day, but how do I know the symbolism you describe with these words qualifies as a theory of physics and is not just some recreational exercise in esoteric language?

If you tell me that you have found out that space “really” is described by this and that algebraic structure, how do I know how this statement about mathematical symbols relates to what I’d like to think of as “reality”.

I think of the toposopher’s work as a step towards axiomatizing what “physical theory” means.

And their answer is this: whatever formalism you have that you want to regard as a physical theory, it must provide you, in the end, with a pointed topos:

a topos $T$ together with an object $H \in Obj(T)$.

This data will then encode what the “predictions” of your physical theory are, given that the world is in a certain “state”:

the “states” are the elements of $H$ $\psi : 1 \to H$ and the “predictions” (the proposotions) about states are the morphisms $P : H \to \Omega \,,$ where “1” denotes the terminal object of the topos and $\Omega$ the “subobject classifier”, which is such that morphisms $1 \to \Omega$ are precisely the “truth values” as seen in the topos.

So if your physicsal theory comes with a pointed topos $H \in Obj(T)$ as its back-end, I know what it means to pick a state $1 \stackrel{\psi}{\to} H$ described by the theory, and choose a “prediction” $H \stackrel{P}{\to} \Omega$ and then find out “how true” this prediction is on this given state by forming the morphism $P(\psi) : 1 \to \Omega$ defined by $P(\psi) : 1 \stackrel{\psi}{\to} H \stackrel{P}{\to} \Omega \,.$

Of course that’s still just symbols, and you might wonder whether this really helps with connecting theory with “reality”, in particular since these “truth values” $1 \to \Omega$ can be, in general, rather weird-looking things.

But at least now these symbols are guaranteed to organize themselves consistently in a manner that allows you to consistently think of them in terms of truth values, etc. That’s already progress, since it’s at least less murky than the situation in quantum mechanics before the advent of the toposophers.

Posted by: Urs Schreiber on May 4, 2008 10:54 AM | Permalink | Reply to this

### Re: A Topos for Algebraic Quantum Theory (revised)

hi urs.

thank you very much for your quick response!

What really impressed me of Connes’s work was the fact that also many other non-physics-related examples such as tilings, discrete groups, non commutative measure theory, non commutative differential geometry, non commutative number theory (all very different in nature) were displayed by the same “unified philosophy”: attaching an operator algebra and attacking with k-theoric weapons, cohomology and functional analysis tools…

i wonder still that one actually can get from one view to another, somehow. at this level, mathematics is just like being in a disney movie… so i believe (or wish to believe) that in the end, Kontsevich, Isham and your approach will be related somehow, to some extent.

i have to say that i’m overwhelmed by the whole lot of math that’s out there… sometimes it feels like no matter for how long i study, i’ll always be caught in the “Preliminaries”…

right now, i have some sort of vague understanding of some stuff i read here and elswhere - locally, i find the meaning of every mathematical term, but gluing all those patches togteher is still complicated.

any recommendation? where to start? what to read first?

{right now i’m reading commutative algebra stuff (atiyah- macdonald), in order to see what those DIABOLIC schemes are… and was about to pick up a C*-algebra / k theory book. - to see what that DIABOLIC index theory is}

congrats and keep it up!

cheers

manu

Posted by: manuel on May 4, 2008 3:33 PM | Permalink | Reply to this

### Re: A Topos for Algebraic Quantum Theory (revised)

i’m overwhelmed by the whole lot of math that’s out there… sometimes it feels like no matter for how long i study, i’ll always be caught in the “Preliminaries”…

This feeling I know all too well. I am always hoping that one fine day I will get to the point that I see the full big picture of the relevant things in math and physics. Sometimes I feel like I am making progress. On other days it seems rather as if the net advance is negative.

On the other hand, I have the impression that there is room for making lots of unification and structural simplification in major concepts in math and physics. (I think here at this blog a general tendency of the discussions is interest in such unification.)

I should maybe admit that I am mostly thinking here about theoretical physics and the math related to that (which does span a fair bit of all the math that is out there, though). Here, my general feeling is that we are currently experiencing, after the beginning of the 20th century moved us out of the closed Newtonian world, an age of rapid discovery of new territory, with the mapmakers lagging sadly behind.

One day we’ll look back at noncommutative geometry as we do today on Riemannian differential geometry: as on a well-sorted collection of concepts with clear relation to other areas. But not yet. Now is the time to figure out how all things quantum and geometry hang together.

in the end, Kontsevich, Isham and your approach will be related somehow, to some extent.

I am hoping that in the end we will be able to tell a single coherent story which has one concise big main storyline into which a myriad of smaller sub-plots organize themselves to a grand epic. This should be such that whenever you wonder: “but why that definition?, why does that concept turn out to be so central?” there should be a reply: “because it fits into the big story in such and such a way”.

That big story somehow revolves around “quantization”. The task of our age is to figure this out.

any recommendation? where to start? what to read first?

Hm, that depends on what you really need. You’re into math, not physics, I gather. And beginning to look into algebra and geometry?

Posted by: Urs Schreiber on May 4, 2008 4:16 PM | Permalink | Reply to this

### Re: A Topos for Algebraic Quantum Theory (revised)

Manuel wrote:

i have to say that i’m overwhelmed by the whole lot of math that’s out there… sometimes it feels like no matter for how long i study, i’ll always be caught in the “Preliminaries”…

Your feeling is correct. Since mathematics is infinite in extent, you and I — and indeed our whole civilization, and every civilization in the Universe — will always be caught in the “Preliminaries”. There will always be stuff we don’t quite understand yet, which is much cooler than anything we understand so far.

So, the main thing is to relax and accept this. Keep studying math, and it will keep getting more and more interesting, and keep making more and more sense.

If you feel overwhelmed by the diversity of material, perhaps it’s good to focus more tightly for a while. Get ahold of a couple of good books and read them over and over, doing calculations and proving theorems on the side, until you thoroughly understand them. This is especially good to do when you’re on vacation in the countryside, or stuck on a desert island, or in prison.

When you’re less isolated, the opposite strategy can be helpful: get some friends and talk to them about math every day, perhaps while drinking coffee. Develop a regular routine. Working on problems together is much better than superficial chats, though a bit of superficial chat is good too. These days I learn the most math while working and talking with James Dolan at various local cafés. Often I let him explain things, sometimes I explain things to him, but the most fun is when we invent new mathematics together.

And then, of course, there’s looking at books and papers in libraries and online. But it sounds like you’re already doing this. This can be a bit demoralizing if you don’t balance it with the other activities I just mentioned, since it’s easy to find things you need to learn faster than you can actually learn them.

Posted by: John Baez on May 5, 2008 1:58 AM | Permalink | Reply to this

### Re: A Topos for Algebraic Quantum Theory (revised)

hi again urs.

yes, i’m a struggling mathematician wannabe. i studied electronic engineering for 4 years and the quit school for a year. then i switched to pure mathematics. been doing pure math for 3 years now.

i have 4 more classes to take before i get my masters degree. here in my country there’s no minor-major division, or bachelor-graduate division: it’s all a single 6 year chunk of compulsory classes, spanning most basic areas, but covering the content of most grad school programs elsewhere (i.e. grad school in the US), plus a few elective classes. In the end, i guess it’s equivalent to a Msc…

what i’ve already seen/learned:

abstract algebra (linear algebra, groups, rings, modules, galois theory)

analysis (real analysis-measure theory, complex analysis, functional analysis)

classical projective geometry.

i still have to take differential geometry, a class on PDEs and a few more.

i took a short seminar on category theory and right now, a commutative algebra course (aimed to algebraic geometry).

i’m really on the geometry kind of vibe for mathematics, as i said in an earlier comment, i find geometry as the bridge between reality and mathematics (or i should also say between physics and mathematics…) - Geometry as the motor of mathematics through history.

really curious about grothendieck.

homology theory, k theory, categories? geometry and number theory via schemes?

uuuuuuuuu it’s so easy to get lost!

where/ with what did you start off?

best regards
manu

Posted by: manuel on May 4, 2008 6:38 PM | Permalink | Reply to this

### Re: A Topos for Algebraic Quantum Theory (revised)

where/ with what did you start off?

You should know that I was raised as a physicist. I could name some books I found influential. But if I think about it, what really “started me off” and kept me going is: listening to and then talking to people.

Back when I was a student I was reading a fabulous newsgroup that no longer exists as such (though a group by that name still exists): “sci.physics.research”.

We learned more from a three minute spr post than we ever learned in school. #

Here “learning” mostly means: getting pointers to the really cool stuff.

That group was originally run by John Baez, of course. Rumour has it that he now instead runs a weblog… ;-)

Back then I had a bunch of files which contained printouts of John Baez’s TWFs. If you haven’t seen them yet, have a look right now!

Sometimes people ask me: “so you post all this stuff to the web and still find time to do research?” Then I tell them that this is a misunderstanding. It’s not that I have conversation on the web and do my work separately from that. The conversation on the web is an important source of intellectual exchange for me. I always felt I needed more discussion than I could get in person around me.

(This recently changed a bit when I had the big luck to get Danny Stevenson as a colleague for a few months. But our paths have already separated again…)

Sorry, you were asking about books. When I was a student I had the biggest fun in my life with Theodore Frankel, The geometry of physics. (Should be followed/accompanied by Bott and Tu.)

Posted by: Urs Schreiber on May 4, 2008 8:06 PM | Permalink | Reply to this

### Re: A Topos for Algebraic Quantum Theory (revised)

Well, I should point you to John’s How to learn math and physics!

Posted by: Urs Schreiber on May 4, 2008 8:10 PM | Permalink | Reply to this
Read the post Integrals and Valuations using Geometric Logic
Weblog: The n-Category Café
Excerpt: Bas Spitters on the theory of integration in a topos.
Tracked: May 19, 2008 7:55 AM
Read the post Where is the Philosophy of Physics?
Weblog: The n-Category Café
Excerpt: Should we devote more time to the philosophy of physics?
Tracked: May 19, 2009 2:19 PM

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