## April 14, 2007

### Topos Theory in the New Scientist

#### Posted by John Baez

Our favorite science magazine has decided to take on Chris Isham and Andreas Döring’s work on topos theory and physics:

At the n-Category Café we serve only possible things for breakfast. But, many things are possible…

Over on the category theory mailing list, the renowned topos theorist Peter Johnstone writes:

Category theorists in general, and topos theorists in particular, may want to check out this week’s cover story in the New Scientist (www.newscientist.com). The author (Robert Matthews of Aston University in Birmingham) is clearly a fan of Chris Isham: it’s not clear to me whether he actually knows what a topos is, but he has committed himself to statements such as

“Topos theory could lead to a view of reality more astonishing and successful than quantum theory”

which is splashed all over page 32 of the magazine. Even if you don’t believe this (and I don’t think I do) it’s pleasant to see topos theory getting this sort of publicity.

The “impossible things for breakfast” line is, of course, borrowed from the logician Charles Lutwidge Dodgson:

“Alice laughed: “There’s no use trying,” she said; “one can’t believe impossible things.”

“I daresay you haven’t had much practice,” said the Queen. “When I was younger, I always did it for half an hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”

To someone steeped in classical logic, it may at first seem impossible to believe that the law of excluded middle:

$P or not(P) = true$

could fail, as it can in the intuitionistic logic described by topos theory. It may also seem impossible that the distributive law

$P and (Q or R) = (P and Q) or (P and R)$

could fail, as it can in quantum logic! But after serious thought, it seems eminently possible — especially upon examining the results of experiments in quantum mechanics.

Could the laws of logic have an empirical aspect to them? Could they be subject to revision and refinement as we carefully study the world around us?

Why not? Who ever promised us instant direct access to immutable truth? Even if logic is a matter of convention, some conventions are better than others — so we may want to change them.

Posted at April 14, 2007 2:55 AM UTC

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### SUBSCRIPTION REQUIRED; Re: Topos Theory in the New Scientist

I have a New Scientist subscription. I got it for my son, and recently renewed for another 3 years, even if the magazine is sometimes weirdly fact-checked (as Greg Egan and John Baez have pointed out). I cannot get their online edition to believe that I have a subscription, hence all that I (and some other of your readers) can see is:

Article Preview
Impossible things for breakfast, at the Logic Café

* 14 April 2007
* Robert Matthews
* Magazine issue 2599

Our rigid notions of true and false just don’t work for a quantum world. It’s time to dish up a new logic

CHRIS ISHAM has a problem with truth. And he suspects his fellow physicists do too. It is not their honesty he doubts, but their approach to understanding the nature of the universe, the laws that govern it and reality itself. Together with a small band of allies, Isham is wrestling with questions that lie at the very core of physics. Indeed they run even deeper, to such basic concepts as logic, existence and truth. What do they mean? Are they immutable? What lies beyond them?

After years of effort, Isham and his colleagues at Imperial College London and elsewhere believe they can glimpse the answers to these profound questions. They didn’t set out to rethink such weighty issues. When they started nearly a decade ago, the researchers hoped to arrive at a quantum theory of the universe, an ambitious enough task in itself. Yet in the process they might have …

The complete article is 2421 words long.

Posted by: Jonathan Vos Post on April 14, 2007 6:59 PM | Permalink | Reply to this

### Re: Topos Theory in the New Scientist

Yes, that’s all I can see too! Even though the University of California has a subscription, we can only see the electronic version after a certain number of months have gone by. Welcome to the bad old world of corporate-run science journalism.

I don’t really care too much what the New Scientist wrote, though I’m curious. My point in posting this entry was not so much to get Café regulars to read the article, or subscribe to New Scientist, as to give them a heads-up: topos theory has entered the volatile pop science market, and old pros like Peter Johnstone have noticed.

Anyone who wants to learn what Isham and Döring really did is advised to read their papers. I know, it ain’t easy. But you ain’t gonna find out in the New Scientist, that’s for sure!

Posted by: John Baez on April 14, 2007 8:12 PM | Permalink | Reply to this

### Re: Topos Theory in the New Scientist

I was reminded of David’s recent questions: “Whatever happened to the categories?” when seeing that Max Tegmark is being cited like this:

“There’s no doubt that we need something radical”, says Max Tegmark, a theorist at the Massachusetts Institute of Technology. “Whether this is it is another question. In the end the real test is: does it get us anywhere?”

In particular, since the claim seems to be that topos theory might help with issues in quantum cosmology.

My impression was that what Döring-Isham really do is to find a topos in which noncommutative algebras may be regarded as algebras of functions, i.e. to find a topos in which noncommutative geometry looks, formally, like ordinary geometry.

I can imagine that one aspect in which this could “get us anywhere” is that it could suggest certain constructions previously not thought of: simply by internalizing well known concepts related to function algebras into this new topos.

I’d be interested to learn in which sense Döring-Isham imagine applying this to quantum cosmology.

Maybe it is just too long ago that I seriously thought about the problems of quantum cosmology…

Posted by: urs on April 15, 2007 9:28 PM | Permalink | Reply to this

### Re: Topos Theory in the New Scientist

“My impression was that what Döring-Isham really do is to find a topos in which noncommutative algebras may be regarded as algebras of functions, i.e. to find a topos in which noncommutative geometry looks, formally, like ordinary geometry.”

As far as I understand, this is not what they’re doing at all. In fact, they avoid working with noncommutative algerbas by working with their commutative subalgebras. Their topos is the topos of presheaves on the category of these commutative subalgerbas.

If there is a way to turn this approach around to look more like what you described, I would certainly be interested to hear about it!

What I find interesting about this approach is the possibility to describe field theory and gravity using the same formalism. In this case the topos would be sheaves over the site of manifolds with appropriate background structure (none, in case of gravity).

Isham himself commented on the later idea that it’s interesting but one would have to find a formal language associated with this topos in order to fit it into their “neo-realist” approach to physics. Haven’t thought about it so far.

Posted by: Squark on April 16, 2007 5:15 PM | Permalink | Reply to this

### Re: Topos Theory in the New Scientist

As far as I understand, this is not what they’re doing at all.

Okay, let’s see. The main point seemed to be that they find an object $\Sigma$ and an object $\mathcal{R}$ in their topos, such that every “quantum observable” (an operator, right?) yields a morphism $A : \Sigma \to \mathcal{R}$ in the topos, which can in some sense be interpreted as behaving like a function that maps “configuration space”, modelled by $\Sigma$, to the “space of real numbers”, whose role now is played by the object $\mathcal{R}$.

Posted by: urs on April 16, 2007 5:29 PM | Permalink | Reply to this

### Re: Topos Theory in the New Scientist

Yes, but I think it doesn’t work vice versa. That is, the algebra of observables cannot be recovered from the topos and the two objects alone. There are morphisms from Sigma to R that do not correspond to any quantum observable. Moreover, the noncommutative product on the observable algebra is not easily recovered by topos-theoretic means. In fact the noncommutative product doesn’t appear to play any direct role in their approach.

Posted by: Squark on April 17, 2007 7:36 PM | Permalink | Reply to this

### Re: Topos Theory in the New Scientist

Moreover, the noncommutative product on the observable algebra is not easily recovered by topos-theoretic means. In fact the noncommutative product doesn’t appear to play any direct role in their approach.

I see, thanks for emphasizing that.

While that makes my claim that their construction is about noncommutative geometry a little less than compelling – I admit – this would also – necessarily then – make the relation to quantum theory a little less than satisfactory. Or wouldn’t it? No commutators?

I must say that I gave up trying to read the paper when I failed to parse the equation (2.1) for daseinization.

Is there maybe a brief way to summarize the key steps in their construction, stripped off the more Heideggeresque contemplations, one that would fit into a blog comment?

Posted by: urs on April 17, 2007 8:28 PM | Permalink | Reply to this

### Re: Topos Theory in the New Scientist

Isham and Doring suggest a general framework for formulating physical theories using topos theory. The framework is very wide: so wide that it can encompass classical theory, quantum theory, (I think) local field theory, and who knows what else. The downside of this flexibility is the small amount of restrictions imposed on such a theory. In particular, it is not enough to specify the topos, the “state object” and the “value object”, one should also explicitely specify which morphisms count as observables.

The daseinization of a projection operator P is the following presheaf on the category of commutative subalgebras of our quantum obervable algerba A: given B a commutative subalgebra, the dasenization delta(P) at B is the smallest projection operator Q in B that is larger than P. Here, one projection operator P1 is said to be larger than another P2 when the image of P1 contains the image of P2.

Maybe I’ll attempt to summarize the paper later, too sleepy and have to get up for work tomorrow :)

Posted by: Squark on April 18, 2007 8:44 PM | Permalink | Reply to this

### Re: Topos Theory in the New Scientist

The daseinization of a projection operator P is the following presheaf […]

Thanks! This I understand.

How does that presheaf then give a morphism $\Sigma \to \mathcal{R}$ from the state object to the value object?

Posted by: urs on April 19, 2007 2:37 PM | Permalink | Reply to this

### Re: Topos Theory in the New Scientist

It doesn’t! delta(P) is a proposition, not an observable (which Isham and Doring prefer to call “quantity” due to their neo-realist convictions). A quantity is a morphism from the state object to value object, the classical analogue being a function on the phase space. A proposition is a sub-object of the state object, the classical analogue being a subset of the phase space. These sub-objects form a Heyting algebra (the intuitionist analogue of a boolean algebra).

Posted by: Squark on April 19, 2007 8:20 PM | Permalink | Reply to this

### Re: Topos Theory in the New Scientist

Okay, I get it. So then let me pose the correct question:

can you indicate how from an operator $a \in A$ we obtain a morphism $\Sigma \to \mathcal{R}$ ?

Posted by: urs on April 19, 2007 9:07 PM | Permalink | Reply to this

### Re: Topos Theory in the New Scientist

Isham and Döring suggest several alternatives for the “value object” $R$ that they denote $R_\geq$, $\R_\leq$ and $R_{\leftrightarrow}$.

$R_\geq$ is the so-called presheaf of order reversing real-valued functions over our category of commutative subalgebras. To any commutative subalgebra $B$ of our noncommutative algebra $A$, $R_\geq$ corresponds to the set of order-reversing real-valued functions on the set of subalgebras of $B$. Given $x$ an element of $A$, we can construct a morphism $\Sigma \to R_{\geq}$ as follows.

Consider the spectral family $e_\lambda$ of $x$: $e_\lambda$ is the projector corresponding to $x \lt \lambda$. On each commutative subalgebra $B'$ of $A$, $e_\lambda$ yields the spectral family $f_\lambda(B'): = \mathrm{inf}_{\mu \gt \lambda} \delta(e_\mu)_{B'}$

Here $\delta(...)_{B'}$ means daseinization at $B'$. Each such family yields an element $\delta(x)_{B'}$ of $B'$ and hence a function on $\mathrm{Spec} B'$. We have $\delta(x)_{B'} := \int \lambda df_{\lambda(B')}$ Now we are ready to construct the morphism. The morphism should assign to each point $p$ in $\mathrm{Spec} B$ an order-reversing real-valued function on the subalgebras $B'$ of $B$. We define this function to be the evaluation of $\delta(x)_{B'}$ at the image of $p$ in $\mathrm{Spec} B'$ (equivalently the evaluation of the image of $\delta(x)_{B'}$ in $B$ at $p$). The $R_{\leq}$ story is similar with the following differences:

1) $R_{\leq}$ is the presheaf of real-valued order-preserving functions.

2) In this case the definition of $f_{\lambda(B')}$ is simpler: $f_{\lambda(B')} := \delta^i(e_\lambda)_{B'}$ Here $\delta^i$ is the so-called inner daseinisation. The inner daseinisation of $P$ at $B'$ is the best approximation of $P$ from below by a projector in $B'$ just as the usual (outer) daseinisation is the best approximation of $P$ from above by a projector in $B'$. $R_{\leftrightarrow}$ is the product of the presheaves $R_{\geq}$ and $R_\leq$, the morphism $\Sigma \to R_{\leftrightarrow}$ being the product morphism. This allows symmetry between inner and outer daseinisation.

Isham and Döring also discuss so-called $k(R_{\geq})$ as a value object. It is obtained from $R_\geq$ by a procedure that transforms monoid objects into group objects (note that $R_{\geq}$ is merely a monoid object, not a group object).

Posted by: squark on April 22, 2007 7:57 PM | Permalink | Reply to this

### Re: Topos Theory in the New Scientist

Two stupid question to see if I am following along:

one thing you didn’t mention explicitly but which is implicit in what you are saying (please correct me if I say now is incorrect):

The “object of states” $\Sigma$ itself is the sheaf on commutative subalgebas of $A$ which sends a commutative subalgebra $B$ to its spectrum (a set) $\Sigma : B \mapsto \mathrm{Spec}(B) \,.$ Okay, nice. And $R_{\geq}$ assigns $R_{\geq} : B \mapsto \{ f : \mathrm{subalgebras}(B) \to \mathbb{R} | f \mathrm{decreasing} \}$

You write:

the evaluation of $\delta(x)_{B'}$ at the image of $p$ in $\mathrm{Spec} B'$

I am being dense. Could you unwrap that statement a little further, for dummies like me?

I might well be confused about various different notions of spectrum floating around…

Posted by: urs on April 23, 2007 8:40 PM | Permalink | Reply to this

### Re: Topos Theory in the New Scientist

Firstly, Sigma is a presheaf rather than a sheaf. Our topos is the topos of presheaves on the category of commutative subalgebras. Sheaves don’t even make sense in this context since commutative subalgebras form merely a category, not a site, at least a priori.

B’ is a subalgebra of B hence there is a morphism from B’ to B (the embedding). Therefore, there is a morphism from Spec B to Spec B’ (Spec is a contravariant functor). Explicitely, given a point q in Spec B, thought of as a homomorphism from B to the field of complex numbers C, its image q’ in Spec B’ is defined by
q’(y) = q(y)
Here y is an element of B’ considered as such on the left side and considered as an element of B on the right side. In other words, q’ is the restriction of q to B’, or the pullback of q to B’ by the homomorphism B’ -> B. So, we can send p to some point p’ in Spec B’. Now, delta(x)_B’ is an element of B’ hence a function on Spec B’. We can evaluate this function at p’. If p’ is thought of as a homomorphism from B’ to C, the evaluation is just
p’(delta(x)_B’)

Posted by: Squark on April 26, 2007 8:38 AM | Permalink | Reply to this

### Re: Topos Theory in the New Scientist

Good, thanks. I understand. Very nice.

Thanks for taking the time for explaining all this, and for extracting this information from Döring-Isham. Maybe if I find the leisure over the weekend, I’ll sit down and summarize what we have said (i.e. what I have asked and you have replied) in a clean way.

Should snugly fit on a single page so far, and still goes a long way towards giving detailed technical insight into what’s actually going on.

Okay, so now you have taught me what the object of states $\Sigma$ is, what the object of values $R_{\geq}$ is, and how an operator $a$ (a quantum observable in the ordinary sense) can now be regarded as a morphism $a : \Sigma \to R_{\geq} \,.$

All right, what next? What exactly now is the punchline? Or are we not at that point yet? What else should I ask you to get to that point?

Do we have an injection $\{a \in A\} \hookrightarrow \mathrm{Hom}(\Sigma,R_{\geq})$ ?

I do recall that you said before that this is certainly not surjective.

What do we want to do? I seem to have gotten idea that I should hand you a topos and two objects $\Sigma$ and $R_{\geq}$ in it, together with a subset $U \subset \mathrm{Hom}(\Sigma,R_{\geq})$ and you read off from that some “physical theory”?

How much information of $A$ may be read off from the image of the map$\{a \in A\} \to \mathrm{Hom}(\Sigma,R_{\geq})$?

Do Döring-Isham discuss a toy example of all this, like the quantum mechanics of the free particle on the line? What is it their topos-formalism gives us in this case?

Or, if you feel that I am asking the wrong questions here: what would be the right next question to ask after we know what $a : \Sigma \to R_{\geq}$ is?

Posted by: urs on April 27, 2007 10:34 AM | Permalink | Reply to this

### Topos-theoretic Reformulation of Quantum Observable

I’ll try to summarize what Squark has taught me about how Döring-Isham phrase quantum observables $a$ (operators) as morphisms $a : \Sigma \to R_{\geq}$ in a certain topos.

The goal is to have a quick reference for the key points of the construction – and to find out to which degree I actually understand the construction (meaning: I’d be grateful for any corrections of the following).

To get started, we fix some algebra $A\,.$ I am not sure what conditions exactly we need to demand of $A$. Probably we are on the safe side if we imagine for a moment that $A = B(H)$ is the algebra of bounded operators on some given Hilbert space.

Given the algebra $A$, we form the category $\mathrm{Comm}(A)$ of commutative subalgebras of $A$. (This name is my invention, probably Döring-Isham have their own name for that.)

Objects here are commutative subalgebras $B \subset A$ and morphisms are inclusion relations of subalgebras.

The topos $T$ that we are talking about is that of Set-valued presheaves on $\mathrm{Comm}(A)$ $T = \mathrm{PSh}(\mathrm{Comm}(A)) = \mathrm{Hom}_{\mathrm{Cat}}( \mathrm{Comm}(A)^{\mathrm{op}} , \mathrm{Set} ) \,.$

A natural example of such a presheaf is that which sends each commutative algebra $B$ to its spectrum $\mathrm{Spec}(B)$, which is the set of algebra homomorphisms to the ground ring $\mathrm{Spec}(B) = \mathrm{Hom}(B,\mathbb{R}) \,.$ For every inclusion $i : B^\prime \hookrightarrow B$ of commutative subalgebras of $A$, we get a map $i^* : \mathrm{Spec}(B) \to \mathrm{Spec}(B^\prime)$ by precomposing a homomorphism from $B$ to $\mathbb{C}$ with $i$.

This presheaf of spectra is what we now want to call the object of states $\Sigma := \mathrm{Spec} \,.$

(And all this depends on the choice of $A$, so more precisely we might want to put a subindex ${}_A$ on all these symbols).

Another presheaf on $\mathrm{Comm}(B)$ is the object of values $R_{\geq} \in T$ which sends each commutative subalgebra $B$ of $A$ to the set of real-valued functions on subalgebras $B^\prime$ of $B$, whith these functions required to be decreasing with respect to the inclusion relation on subalgebras: $R_{\geq} : B \mapsto \{ f : \mathrm{subalgebras}(B) \to \mathbb{R} | f \;\mathrm{decreasing} \} \,.$ For any inclusion of algebras $i : B^\prime \hookrightarrow B$ we naturally get an inclusion of sets of subalgebras $\tilde i : \mathrm{subalgebras}(B^\prime) \to \mathrm{subalgebras}(B)$ with which we may pull back these functions. Hence $R_{\geq}$ is indeed a presheaf on $\mathrm{Comm}(A)$.

A morphism of presheaves $f : \Sigma \to R_{\geq}$ is a natural transformation of the corresponding functors, hence an assignment of morphisms of sets $B \mapsto ( f_B : \mathrm{Spec}(B) \to R_{\geq}(B) )$ for each commutative $B \subset A$ compatible with the restriction maps in the obvious way.

By the general logic of spectra, each element $b \in B$ of the algebra $B$ defines a function on the spectrum of $B$. Given an element $x : B \to \mathbb{R}$ of the spectrum, we simply set $b(x) := x(b) \,.$

Given any element $p \in A$ which is a projector, we get an element $\delta(p)_B \in B$ for each commutative subalgebra $B$ of $A$ defined to be the projector in $B$ with the smallest image that still containts the image of $p$.

(I notice I may not fully understand this yet: what if such a projector does not exists in $B$?)

This construction is then generalized from projectors to arbitrary operators by using functional calculus. This way every operator $a \in A$ defines for every commutative subalgebra $B \subset A$ an element $\delta(a)_B \in B \,.$

Now we are ready to define a morphism $a : \Sigma \to R_{\geq}$ for each element $a \in A$ of the original algebra.

For each subalgebra $B \subset A$ it is a function $a_{B}$ that sends a point $f$ in the spectrum of $B$ to a decreasing function $\mathrm{subalgebras}(B) \to \mathbb{R}$. Namely we set

$a_{B} : f \mapsto (B^\prime \mapsto f(\delta(a)_{B^\prime})) \,.$

I think this summarizes what Squark has been telling me above.

I still have little intuition for why exactly we want to consider precisely this construction.

Posted by: urs on April 29, 2007 4:48 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

Ok, thanks for this nice explanation. We’re all ready now to move onto the next step!

I’m a bit confused as to how to think geometrically about some of these things. In other words, I’m always trying to compare $Comm (A)$ to the category $Open(X)$ of open sets of a topological space. Consider the $\delta$ construction above. Should I think of that as the algebra analogue of thinking of a point $x \in U$ as $x \in X$? Or should I think of it in the other direction, as some kind of restriction map, like the map which sends a vector bundle over $X$ to a vector bundle over $U \subseteq X$?

(By the way, I think your $x$’s and $a$’s got mixed up in your last few paragraphs. Also there is this typesetting issue with primes; it seems prettier to write $B^'$ instead of $B'$.)

Posted by: Bruce Bartlett on April 30, 2007 12:04 AM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

(By the way, I think your $x$’s and $a$’s got mixed up in your last few paragraphs.

Thanks! Fixed.

Also there is this typesetting issue with primes; it seems prettier to write $B^\prime$ instead of $B'$.)

Yes, I am aware of that. I was stubbornly sticking to the prettier code, even though it has not the prettiest output at the moment…

Posted by: urs on April 30, 2007 6:46 AM | Permalink | Reply to this

### Primes (again)

Yes, I am aware of that. I was stubbornly sticking to the prettier code, even though it has not the prettiest output at the moment…

If you want something TeX-compatible, then type B', which produces the desired output ($B'$) with a minimum of fuss.

The Computer Modern fonts have an over-large, unnaturally-low “prime” glyph, which makes it possible to type B^\prime and produce something that looks right. There’s no corresponding glyph in the Unicode repertoire, so B^\prime is never going to look right. While it’s possible to type B\prime and get the desired output, I don’t really recommend it.

Posted by: Jacques Distler on April 30, 2007 8:11 AM | Permalink | PGP Sig | Reply to this

### Re: Primes (again)

Now I am slightly confused. Let me see:

the code I originally used was indeed B’. I think it was the output “$B'$” produced by that, which Bruce complained about. Now I have changed the code to B^\prime. That does indeed produce better-looking output “$B^\prime$” (the prime sits decently high) on my system.

I seem to recall that it wasn’t always this way. Unless I am mixed up, a while ago typing B’ did produce the expected output with the prime not too low.

Posted by: urs on April 30, 2007 1:11 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

We’re all ready now to move onto the next step!

Okay, good. Do I sense a certain impatience here? :-)

I’m a bit confused as to how to think geometrically about some of these things.

Yes. There ought to be a good way to think about it. Whether geometric or not. It would help me to see precisely which problem is solved by this construction. I understand that this construction is supposed to circumvent the Kochen-Specker no-go theorem.

That theorem, apparently, tells us that we cannot, in general, conistently identify all elements $b \in B \subset A$ of any commutative subalgebra with real numbers. This means, I guess, that in general $\mathrm{Spec}(B)$ may be empty.

So presumably the point of the exercise here is then not to look at a fixed commutative $B \subset A$ but at all of them at once. Clearly some of them do have nonempty spectrum (for instance all those generated from a single element of $A$).

Posted by: urs on April 30, 2007 1:26 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

Urs wrote:

I am not sure what conditions exactly we need to demand of A. Probably we are on the safe side if we imagine for a moment that A=B(H) is the algebra of bounded operators on some given Hilbert space.

In fact, Isham and Doring only consider the case B(H) but I tried to generalize everything to an arbitrary (technically we probably need it to be C*) algebra since we started from the possible connection to noncommutive geometry.

I notice I may not fully understand this yet: what if such a projector does not exist in B?

It does since 1 is in B :-)

That theorem, apparently, tells us that we cannot, in general, conistently identify all elements b∈B⊂A of any commutative subalgebra with real numbers. This means, I guess, that in general Spec(B) may be empty.

No. A commutative algebra is defined by its spectrum. In the present context we are considering C*-algerbas and so algebras and spectra are “the same” (the categories are anti-equivalent) by the Gelfand-Naimark theorem. Note I’ve been trying not to go into technical functional analytic details regarding the precise kind of algebras we ought to consider. Disregarding the precise mathematical context, ideologically, taking the spectrum of a commutative algebra is always an information preserving operation.

The Kochen-Specker theorem talks about non-commutative algebras, more precisely, about the algebra B(H) with dim H > 2. The way we avoid it is by working with commutative subalgebras and we need all of them since we don’t want to lose information.

Do we have an injection

Yes, but you correctly note it’s not a surjection

All right, what next?

Maybe the next thing to note is the way Doring and Isham treat quantum states.

Classically, each point in the phase space defines a global element of the state object. However in the quantum case global elements don’t exist (this is the Kochen-Specker theorem; which brings us back to the issue of avoiding it).

Luckily, there is an alternative approach: each point p in the phase space P defines a subset T(p) of the set of subsets 2P of the phase-space (no, it’s not a typo, I’m considering a set of sets here!) Namely, given p a point of the phase space P, a subset U of P is in T(p) when p is in U.

Analogously, each quantum state ψ (a vector in H where A = B(H)) defines a subobject T(ψ) of the presheaf of clopen subobjects of the state object (clopen meaning closed and open in the sense of the topology we have on the Gelfand-Naimark spectra). T(ψ) is constructed as follows. A clopen subset X of Spec(B) is in T(ψ) when ψ is in the image of the projector corresponding to X.

This allows us to assign truth value to propositions of the form “the value of quantity A is in the range Δ” given a quantum state ψ. The truth value is an element of the Heyting algebra of global elements of the subobject classifier of the topos (it’s an “intuitionistic” truth value). Later Isham and Doring explain how to do this even for more complicated propositions that involve quantifiers. Thus Isham and Doring claim to achieve the philosophical goal of being able of discussing properties quantum systems actually posses rather than only measurements that can be made with the quantum system as in the usual interpretation of quantum mechanics.

Posted by: Squark on April 30, 2007 9:43 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

Classically, each point in the phase space defines a global element of the state object. However in the quantum case global elements don’t exist (this is the Kochen-Specker theorem;

Ah, thanks for saying this!

That point of view on the Kochen-Specker theorem certainly does motivate a topos-theoretic approach now: we can read KS as a statement about the presheaf $\Sigma$ of spectra of commutative subalgebras of $A$. This, then, certainly motivates further study of $\Sigma$.

It was motivations like that, which make all these constructions we were talking about look more compelling, that I was looking for.

Also, I was confused, for a while, about what precisely the Kochen-Specker theorem is actually saying, hence some of the nonsense I had been saying here. But now things are getting clearer.

This allows us to assign truth value to propositions of the form “the value of quantity A is in the range $\Delta$

That’s interesting.

But now that you brought me to this point of understanding Isham-Döring, I am wondering why you reacted so negatively to my assertion that they seem to have a way to regard noncommutative geometry as ordinary geometry internal to a funny topos.

It’s true that the product of elements of $a$ plays no direct role in the topos description (though it is of course indirectly present in the spectra of the subalgebras), but still it seems that all this machinery is a way to make elements of a noncommutative algebra look like functions.

And moreover it seems that of all of quantum mechanics actually only the noncommutative aspect (namely of the “quantum phase space”) is relevant here.

Posted by: urs on May 1, 2007 10:46 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

The reasons I don’t think this approach has great kinship with noncommutative geometry are:

1) Addition and multiplication of noncommuting observables cannot be defined in topos-theoretic terms (as far as I can see). It is true that they are used in the construction, but that just goes to say the construction actually depends on the algebra and not on something less. On the other hand, addition and multiplication are essentially the only structure on the set of “functions” the noncommutative geometry guys use.

2) Each algebra requires its own topos, there is no single topos which allows describing all noncommutative geometries.

In particular, I find point 1 rather disappointing from a physical perspective, and this is why. Once we can formulate statements about the quantum system in some formal language and assign truth value to these statements, the next thing one would like is some sort of an axiom system. I’m not sure how axiom systems work in multi-valued logic but I’m sure Isham and Doring know. :-) In particular, these axioms would represent familiar laws of physics like the second law of Newton for a point particle. However, this law

mx”(t) = F(x(t))

involves a 2nd derivative which is essentially a linear combination and it’s not clear how to describe it topos-theoretically. It appears to me this is an impediment to constructiing a reasonable axiom system and thus to actually understanding physics using the Isham-Doring approach.

Posted by: Squark on May 3, 2007 10:11 AM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

Addition and multiplication of noncommuting observables cannot be defined in topos-theoretic terms (as far as I can see).

Maybe that’s related to the fact that the map from operators to morphisms $\Sigma \to R_{\geq}$ is not surjective?

Because if we do know that two such morphisms come from two operators, then we would know what it should mean to add and multiply them: we would define these operations in terms of the corresponding operators.

So maybe one would need an additional axiom which encodes this property abstractly and cuts down the admissable morphisms $\Sigma \to R_{\geq}$ to exactly those for which there is an $a \in A$ such that it induces this morphism.

Could it be that there is some regularity condition on morphisms $\Sigma \to R_{\geq}$ which would ensure this?

Even classically, for instance, in order to be able to implement Newton’s law we need a regularity condition on all maps from states to numbers: these need to be sufficiently differentiable.

Posted by: urs on May 4, 2007 11:20 AM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

Isham and Döring are indeed trying to circumvent the Kochen–Specker theorem. I mentioned this back in week135 And here’s a puzzle I raised — my kind of puzzle, not a puzzle about 100 prisoners who each have a colored dot on their forehead and each flip a coin and need to guess something…

While I’m talking about quantum logic, let me raise a puzzle concerning the Kochen-Specker theorem. Remember what this says: if you have a Hilbert space $H$ with dimension more than 2, there’s no map $F$ from bounded self-adjoint operators on $H$ to real numbers with the following properties:

a) For any self-adjoint operator $A$, $F(A)$ lies in the spectrum of $A$,

and

b) For any continuous $f: \mathbb{R} → \mathbb{R}$, $f(F(A)) = F(f(A))$.

This means there’s no sensible consistent way of thinking of all observables as simultaneously having values in a quantum system!

Okay, the puzzle is: what happens if the dimension of $H$ equals 2?

Posted by: John Baez on May 1, 2007 1:30 AM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

SPOILER WARNING: A solution (I hope) to John’s puzzle coming up.

Consider H of dimension 2. For each set of 2 1-dimensional orthogonal subspaces {V,W} in H, choose one of them (V or W). Given A a self-adjoint operator, there are 2 options:

1) A = λ Id and we set F(A) = λ.

2) A has two eigenspaces, V and W. They form a set {V,W} as above. Therefore one of them is “the chosen”, say V. Suppose the corresponding eigenvalue is λ. We set F(A) = λ.

The reason this doesn’t work for dim H > 2 is that more complicated degeneracies can occur.

Posted by: Squark on May 1, 2007 4:41 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

a) F(A + B) = F(A) + F(B)

or the weaker requirement

b) F is continuous

Then we get a Kochen-Specker-like theorem form dim H = 2 as well.

Posted by: Squark on May 1, 2007 4:58 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

But without specifying how V is chosen, you may not satisfy the second condition. For example, say I always set F(A) to be the larger eigenvalue. Then F(f(A))=f(F(A)) will fail whenever f is decreasing.

Posted by: Mark on May 1, 2007 7:54 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

Mark wrote:

But without specifying how V is chosen, you may not satisfy the second condition

Yes I will. Note that the choice does not depend on the eigenvalues, it only depends on the eigenspaces. Your “example” is not a special case of my procedure.

Posted by: Squark on May 1, 2007 8:02 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

Of course you’re right. That’s what I get for skipping a sentence at the beginning (pointing out that you chose one of each pair before even considering A).

Posted by: Mark on May 2, 2007 12:24 AM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

Squark gave a solution to my Kochen–Specker puzzle. It sounds right… but it doesn’t sound very ‘physical’.

I could be wrong, but I think there’s a way to make the solution sound more like physics: the physics of spin-1/2 particles.

Any self-adjoint operator $A$ on $\mathbb{C}^2$ can be written as

$A = a_0 1 + a_x \sigma_x + a_y \sigma_y + a_z \sigma_z$

in a unique way, where the $a_i$ are real and $\sigma_x, \sigma_y, \sigma_z$ are the Pauli matrices. Write $\vec{a} = (a_x,a_y,a_z)$.

So, a self-adjoint operator $A$ on $\mathbb{C}^2$ is just a real number $a$ and a vector $\vec a$. We can think of it as measuring the spin of a spin-1/2 particle in the $\vec a$ direction and adding $a_0$.

I think there’s a way to use $a_0$ and $\vec a$ to write formulas for functions $F$ satisfying the Kochen–Specker conditions:

a) $F(A)$ lies in the spectrum of $A$;

b) For any continuous $f: \mathbb{R} \to \mathbb{R}$, $F(f(A)) = f(F(A))$.

But, I forget how these formulas go, if I ever knew!

Squark also mentions:

a) $F(A + B) = F(A) + F(B)$

or the weaker requirement

b) $F$ is continuous

then we get a Kochen–Specker-like theorem for $dim H = 2$ as well.

The continuity condition suggests that any formula for $F(A)$ requires an arbitrary choice that we can’t make in a continuous way. I don’t remember hearing about this before, but clearly it’s happening in Squark’s own solution to the puzzle: to define $F(A)$ he needs to pick a 1-dimensional eigenspace of $A$!

Sub-puzzle: what bundle are we finding a section of when we choose a 1-dimensional eigenspace of $A$ for each self-adjoint operator $A$, and why is there no continuous section? What (presumably famous) topological obstruction prevents us?

The other condition Squark mentions,

$F(A + B) = F(A) + F(B)$

goes completely against the grain of the Kochen–Specker philosophy. If $A$ and $B$ don’t commute, there’s no way to measure $A$, $B$, and $A+B$ simultaneously, get numbers $F(A)$, $F(B)$ and $F(A+B)$, and check that $F(A + B) = F(A) + F(B)$. The supposedly startling thing about the Kochen–Specker theorem is that the condition

$F(f(A)) = f(F(A))$

only involves commuting observables, namely $A$ and $f(A)$ — yet it’s still too strong to get solutions!

Posted by: John Baez on May 1, 2007 8:03 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

Let me prove my remark about F’s continuity. Assume to the contrary that F is continuous and satisfies the Kochen-Specker conditions. Given P an orthogonal projector of rank 1, F(P) is either 0 or 1. However, the space of such projectors is just the 1-dimensional complex projective space (the Riemann sphere) and in particular connected. Hence either all projectors get 1 or all of them get 0. However, the KS conditions imply that F(1-P) = 1-F(P) and this is a contradiction.

Posted by: Squark on May 1, 2007 8:37 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

Sub-puzzle: what bundle are we finding a section of when we choose a 1-dimensional eigenspace of A for each self-adjoint operator A, and why is there no continuous section? What (presumably famous) topological obstruction prevents us?

Does it perhaps have something to do with the fact that $SU(2)$ is a double cover of $SO(3)$? When I see your Pauli spin matrices notation, I think ‘Clifford algebra’ and ‘spin stuff’ and hence $SU(2)$, while if I think of the fact that the two eigenspaces are orthogonal (since $A$ is selfadjoint) I think ‘rotations’ so ‘$SO(3)$’. But I can’t put these thoughts together in a precise way .

Posted by: Bruce Bartlett on May 1, 2007 8:56 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

Does it perhaps have something to do with the fact that SU(2) is a double cover of SO(3)?

I’d think so.

Consider John’s formula from above $A = a_0 1 + a_x \sigma x + a_y \sigma_y + a_z \sigma_z \,.$ Take $a_0 = 0$ to get the trace-free self-adjoint operators.

$\mathrm{SU}(2)$ is the exponentiation of that: $\mathrm{SU}(2) = \{\exp i(a_x \sigma_x + a_y \sigma_y + a_z \sigma_z)\}$ Write $\vec a = \alpha \vec e_a$ with $\vec e_a$ the unit vector along $\vec a$ and $\alpha$ the length (norm) of $\vec a$.

Then both $\exp i\alpha(\vec e_a \cdot \vec \sigma)$ as well as $-\exp i\alpha(\vec e_a \cdot \vec \sigma)$ map to the element of $\mathrm{SO}(3)$ given by rotation by angle $2\alpha$ around $\vec e_a$.

Both these operators have the same two eigenspaces, and such pair of eigenspaces uniquely characterize pairs of $\mathrm{SU}(2)$-elements that map to the same element of $\mathrm{SO}(3)$.

So I guess we can regard the space of all these pairs of eigenspaces, over each $(\alpha,\vec e_a)$, as the bundle $\mathrm{SU}(2) \to \mathrm{SO}(3)$ itself. And if so, there will be a more sophisticated way of saying this…

Posted by: urs on May 1, 2007 10:25 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

Both these operators have the same two eigenspaces, and such pair of eigenspaces uniquely characterize pairs of $SU(2)$-elements that map to the same element of $SO(3)$.

I should maybe add for clarity that $\mathrm{exp} i (\vec a \cdot \vec \sigma) = \mathrm{cos}(\alpha) 1 + i \mathrm{sin}(\alpha) \vec e_a \cdot \vec \sigma$ (using the standard, though somewhat improper vector notation, which we already used above).

Posted by: urs on May 3, 2007 2:13 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

I spoke to Andrew Stacey about this one, and he came up with the following, which I’m going to rephrase slightly into the language we’ve been using.

We need to associate to each self adjoint 2x2 matrix

(1)$A=a_0 1 +a_x \sigma_x + a_y \sigma_y + a_z \sigma_z$

one if it’s eigenvalues, in a natural way. To say this more geometrically, it’s better to say that we’re trying to associate to each $A$ one of it’s eigenspaces in a natural way:

(2)$A \mapsto eigenspace of A.$

Ok. Now the identity term in $A$ plays no role, and neither does the overall scale factor (they don’t change the eigenspaces), so we can assume we’re dealing with matrices of the form

(3)$A = a_x \sigma_x + a_y \sigma_y + a_z \sigma_z$

where $a_x^2 + a_y^2 + a_z^2 = 1$. So our matrices are parametrized by $S^2$.

On the other hand, their two eigenspaces are orthogonal. So once you’ve fixed one eigenspace, you’ve fixed the other. In other words, the ‘space of orthogonal pairs of eigenspaces’ is $\mathbb{C}P \cong S^2$.

Here’s the caveat. Since our map must be natural, i.e. $F(f(A)) = f(F(A))$, we need to associate to $A$ and to $-A$ the same eigenspace . So we’re really trying to find a map

(4)$F : S^2 / \mathbb{Z}_2 \rightarrow S^2$

And this means we’re trying to find a section $F$ of the bundle

(5)$S^2 \rightarrow \mathbb{R}P^2$

which is not possible.

I don’t know how this relates to our discussion in terms of $SO(3)$ and $SU(2)$… I’m sure it does, in some way!

Posted by: Bruce Bartlett on May 3, 2007 3:12 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

Almost. As Bruce explains, the problem for dimension $2$ condenses down to finding a section of the projection $S^2 \to \mathbb{R}P^2$. This can be done, but cannot be done continuously.

Posted by: Andrew Stacey on May 3, 2007 4:42 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

Now the identity term in $A$ plays no role

Hm, what do you mean by that? Certainly the eigenspaces may change when you add something proportional to the identity.

On the other hand, using that exponential formula it is possible to express terms that contain the identity operator by terms that don’t.

Posted by: urs on May 3, 2007 5:05 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

When I say that the identity term doesn’t change the eigenspace, I just mean that if $Av = \lambda v$, then

(1)$[A + a_0 1] v = (\lambda + a_0) v$

so that $v$ remains an eigenvector of $A + a_0 1$… though it’s eigenvalue has been shifted by $a_0$.

Posted by: Bruce Bartlett on May 3, 2007 5:24 PM | Permalink | Reply to this

### Re: Topos-theoretic Reformulation of Quantum Observable

I don’t know how this relates to our discussion in terms of $\mathrm{SO}(3)$ and $\mathrm{SU}(2)$

$\mathbb{R}P^2$ can be though of as the subset of $\mathrm{SO}(3)$ of rotations by $\pm \pi$ about arbitrary axes. $S^2 \to \mathbb{R}P^2$ is the double cover of that. So it looks as if this just needs to be extended to rotations by arbitrary angles.

I think the restriction to $\pm \pi$ comes from your normalization of $\vec a$. So if instead arbitrary $\vec a$ are allowed, this should give the double cover of all of $\mathrm{SO}(3)$. I guess.

Posted by: urs on May 3, 2007 6:02 PM | Permalink | Reply to this
Read the post Physical Systems as Topoi, Part I
Weblog: The n-Category Café
Excerpt: Doering on his work with Isham on topos theoretic models of quantum theory and general theories of physics.
Tracked: July 22, 2007 8:43 PM
Read the post Physical Systems as Topoi, Part III
Weblog: The n-Category Café
Excerpt: The third part of the talk.
Tracked: July 23, 2007 8:32 PM

### Re: Topos Theory in the New Scientist

Here’s a new FQXI article on Isham’s work to reformulate Quantum theory using topos, contains t’Hooft’s comments on it:
http://www.fqxi.org/community/data/articles/Isham_Christopher.pdf?PHPSESSID=ddf4dbe965a99d67bfdddf0266c69aa9

Posted by: tytung on October 1, 2007 6:17 AM | Permalink | Reply to this
Read the post The Principle of General Tovariance
Weblog: The n-Category Café
Excerpt: Landsmann proposes that physical laws should be formulated such that they may be internalized into any topos.
Tracked: December 5, 2007 7:10 PM
Read the post The Concept of a Space of States, and the Space of States of the Charged n-Particle
Weblog: The n-Category Café
Excerpt: On the notion of topos-theoretic quantum state objects, the proposed definition by Isham and Doering and a proposal for a simplified modification for the class of theories given by charged n-particle sigma-models.
Tracked: January 9, 2008 10:26 PM

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