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This here is to mention something that I would eventually like to understand, not something that I have thought through. I am mentioning it in case that somebody else might find the question interesting, too, and maybe enjoy sharing thoughts.
One thing that is nice about Bohrification is that it makes the following statement true: “quantum states on a quantum algebra $A$ are precisely classical states internal to the Bohrified ringed topos corresponding to $A$”.
This is essentially a direct re-interpretation of Gleason’s theorem: this theorem says that quantum states on $A$ are already fixed by demanding them to be maps on $A$ that are (positive, normed) linear functions on all commutative subalgebras of $A$. Now, the immediate formalization of a map $A \to \mathbb{C}$ that is required to preserve certain structure on all commutative subalgebras is a fully structure-preserving function, but internal to the presheaf topos over the comutative subalgebras. That presheaf topos is the “Bohrification” of $A$, since Bohr said things that can be interpreted as being formalized by this process.
If one accepts this, the natural next question is: how are the quantum observables analogously re-formulated? As far as I am aware, the existing literature on this topic does not provide a really satisfying answer to this question yet.
But here is an obvious suggestion, that ought to be true if “Bohrification” is a good idea:
if I write $B(A)$ for the Bohrification of $A$ (a ringed topos that comes from a ringed topological space), which we are to think of as the generalized-space-incarnation of the quantum phase space encoded by $A$, then an observable should simply be a morphism $B(A) \to \mathbb{R}$ – a real valued function on that quantum phase space.
To make sense of this, we need to think of the real line here as living in the same context as $B(A)$. We might think of it as a ringed topos in the standard way. But this does not seem to me to yield something useful. The next obvious idea is to regard the algebra of complex functions $C(\mathbb{R})$ as the analog of $A$, and hence form the ringed topos $B(C(\mathbb{R}))$: presheaves on the poset of subalgebras of $C(\mathbb{R})$. So then observables should correspond to geometric morphisms $B(A) \to B(C(\mathbb{R}))$.
To check if this makes sense, we need to check if this reproduces the standard notion of quantum observables associated with $A$. Up to some flexibility in what exactly one wants to have, the observables should simply be the self-adjoint elements of $A$.
So the first question is: is there a natural injection from the self-adjoint elements of $A$ into the geometric morphisms $B(A) \to B(C(\mathbb{R}))$.
I have been thinking about this while hiking all day through sun and rain. No paper. No pen. So take this with a grain of salt. But I was thinking: functional calculus should provide this for us.
Given any self-adjoint element $a \in A \subset B(H)$ we get an algebra homomorphism $C(\mathbb{R}) \to A$ by $f \mapsto f(a)$. This induces a functor of posets of (commutative) subalgebras. And this is, I think, a morphism of sites for the Bohrified toposes. The resulting geometric morphism directly and evidently lifts to a morphism of ringed toposes.
I hope that’s right. I need to check this in detail. Could be that I am making a simple mistake somewhere. If true, the next question is: to which degree is this assignment onto? Which geometric morphisms $B(A) \to B(C(\mathbb{R}))$ do not arise from functional calculus this way? Can we find a simple extra condition on the geometric morphisms such that we characterize precisely those that do come from self-adjoint elements $a \in A$ by functional calculus? If we could, we’d have the desired statement that quantum observables on $A$ are precisely the suitably regular functions on the Bohrification of $A$.
I think now the answer to my question is: the extra condition on the morphisms of toposes is that they have an extra right adjoint.
So: For $A$ a $C^\ast$-algebra, the self-adjoint elements of $A$ are in bijection to the morphisms of Bohr toposes $Bohr(A) \to Bohr(C(\mathbb{R}))$ whose underlying geometric morphism has an extra right adjoint.
I think. Unless I am messing up some functional caluclus details.
I have added the discussion to Bohr topos.
If we could, we’d have the desired statement that quantum observables on A are precisely the suitably regular functions on the Bohrification of A.
How do you get impossibility of simultaneous mesaurement of noncommuting observables this way ? The purpose of foundations is eventually to get the quantum theory of measurement and what makes sense as a measured history. The functional calculus does a homomorphism for just one $a\in A$ and it does not say what happens with a next $b\in A$ measured after it. The quantum books have the notion of “reduction” of wave packet during measurement, but it does not answer questions like, if this happens instantaneously or gradually. There are alternatives like the Feynman’s approach to measurement theory via histories. Good theory of foundations should resolve these unsolved questions, otherwise it is just a reformulation of known science.
The Nijmengen school claims they can get probability law (from wave function) from their model of measurement, but the last paper which I looked at promised the derivation in a future article. Urs, do you know where it is and what kind of reduction this is compatible with ?
(sorry, am having sever hardware problems here, I wanted to say:)
… not correct to consider linear combinations of observables $a + b$ of $a$ and $b$ do not commute with each other, hence not correct to demand that a state is a function $\rho : A \to \mathbb{C}$ on the observable algebra $A$ which is required to be linear on $A$. Instead, it ought to be a function which is required to be linear only on all commutative subalgebras of $A$.
The Bohr topos is precisely that context internal to which it is true that a map $A \to \mathbb{C}$ is externally such a function which is required to be linear on all commutative subalgebras.
(Of course Gleason’s theorem then says that for $A = B(H)$ with $dim H \gt 2$ every function linear on all commutative subalgebras is already entirely linear, thus showing that in these cases the “naive” definition is in fact “operationally correct”, if you wish.)
It seems you did not understand me. You are telling me what something is by your definitions before you described the world. My question is however, about the theory of measurement. So your machinery is an input. The output should be descriptions of reality. Reality involves some sort of sequence of events in real time. The events include thing which we call measurement. Now, saying that something is measureable or not measurable by definition of some algebra of measurables in any internal sense is what you give me as an answer. But this is circular. The success of the story is if one can define what are the meaningful histories (sequences of events) and then to see that the history in which one measures $a$ and then $b$ must give different results from the one in opposite order. So Nijmengen machinery should give description of reality. Then from the reality I should get the picture of the world, and then if this corresponds to known physics this will give interpretation. Then the interpretation tells what is the true observable. It better be what the theory started with, but with precise picture of the “reduction”, Born probability rule etc. I do not believe that a thing like Gleason theorem would tell me that as it does not tell in usual quantum mechanics what the theory of measurement should be. “co-probed by all possible collections of commuting observables” is an abstract classification within the formal system of the theory. I want to see the meaning of such detached statements from a concrete model of the measurement. Take the internal mechanics in Bohr topos and describe how the evolution there describes the reality, and then form the model of measurement in this picture. Then one should calculate from the model that one can observe exactly what the postulates said at the beginning.
That is why people went into quantum logic: to find the theory of measurements which would have interpretable language and the way how various facts fit together into more complex facts. Unfortunately it was not very developed as a logic and did not have incorporated the temporal aspect enough.
In other words in 3 I am talking about the relation between reality and the “context of Bohr topos”. The words in latter should be statements in such a logic whose statements have logical validity in a model of reality and should be interpretable hence in reality.
In particular, is the reduction of the quantum packet in the process of measurement in the Nijmengen model instantenous ?
The approach we are talking about is at best in its infancy. There is no specific theory of measurement here. All we have is the observation that some classical theorems on quantum foundations (Gleason, Kochen-Specker) talk about doing things commutative-subalgebra-wise and that this is naturally equivalently re-formulated as doing things in copresheaves over the poset of commutative subalgebras. Most of the work on this so far tries to identify some very basic facts about this ringed topos, some of which you might interpret as being suggestive of taking this ringed topos seriously as a context for looking at quantum mechanics.
I think what is needed is some nontrivial theorem about Bohr toposes that shows that some genuine advantage is gained by looking at them. Some statement about QM or QFT that would hardly be conceivable without the notion of Bohr topos. I am not sure if such a theorem has been found.
The question that Nuiten looked into was an attempt to guess and find such a statement: one observes that Bohrification turns local nets of observables into presheaves of ringed toposes. So to the extent that AQFT applies, every quantum field theory is encoded in a presheaf of ringed toposes over spacetime. An evident question one can ask for this and which is not easily asked without passing to Bohr toposes is: when is this a sheaf? Nuiten finds that it satisfies spatial descent by local geometric morphisms precisely if the original QFT is causally local (precisely if the net of observables is in fact a local net). I think this kind of statement is beginning to sound like something that would prove the Bohr-topos perspective to be genuinely useful. But even here, it is not quite clear to me how far this carries.
So now in this thread here I was trying to push further, trying to see what other aspects of QM might find a useful reformulation in terms of the Bohr topos. This is a trial-and-error process. I don’t know where it is leading and whether it is being successful. But it’s an open question I find intersting enough to think about it. In particular since an increasing number of people around me do ;-) So I want to gain an educated opinion on this matter.
There is – almost by definition – a nice way to understand quantum states in terms of the Bohr topos. So here I was trying to see if similarly there is a nice way to understand quantum observables in terms of the Bohr topos (because the existing proposals for such identifications in the literature are maybe not fully satisfactory).
So by the way, I have a question for you: apart from the topos-theoretic aspects about this thread here, there is a simple underlying issue in noncommutative geometry:
I find the following fact striking: in the context of $C^*$-algebraic noncommutative geometry, the central theorems of functional calculus simply say:
Fact For $A$ a $C^\ast$-algebra, functions $Spec A \to \mathbb{R}$ are precisely self-adjoint elements of $A$.
Because dually this are $C^\ast$-algebra homomorphisms $C(\mathbb{R}) \to A$.(Where by “$Spec$” here I mean the algebraic-geomety Spec, not the commutative $C^\ast$-spec: just the formal dual.)
My question is: do you know any place in the literature where this nc-geometric interpretation (evident and tautological as it may be) is made explicit? That self-adjoint elements of $A$ are precisely the real functions on the $C^\ast$-algebraic geometry spectrum of $A$?
8 Nice post Urs! I think that one of the Nijmengen papers does claim to DERIVE the Born probability rule. So at least there is one such a theorem. Do not take me for bad, I know that it is long way to go, and I do like the approach a lot. It looks very natural mathematically. But once one makes some statements about observables it is natural to ask about their supposed role in full story. I do not know what is envisioned by now by the authors. One should be careful when using intuitive terms like measurement and observable, what is just the name for an abstract notion and when a statement with those has actual interpretable content in the original sense. I think we should continue to ask questions and look if the things are possibly nearly answered, or at least make sense in the proposed outline of a new language.
I do not know the answer to your question at the end of 8. (maybe it was rethorical question?)
(maybe it was rethorical question?)
No, not a rethorical question. I am genuinely wondering: did people not realize this? Is it too obvious? Is it not obviuous enough?
It seems, while very simple, to be a conceptually important statement to make explicit.
impossibility of simultaneous mesaurement of noncommuting observables
So maybe I should phrase it this way: I understand that such a simultaneous combination is not in the notion of observable in this context by the definition of the topos. However even in classical physics absolutely simultaneous measurements are anyway not feasible to check; so the question is weather the limit in time of the consecutive measurements exists. By the dependence on order it should not. But this depends on the measurement theory to be developed. Regarding that the guys can get Born rule out of the theory of measurement (the promised result in the survey in their recent Commun. Math. Phys.), I suspect some elements of the picture do exist.
Hi Zoran,
I thought about your question about measurement. Here is an observation, which might be relevant:
suppose that given a QFT (say 1-dimensional, for quantum mechanics) we measure some observable $x_{t}$ (an observable in the algebra of observables corresponding to some time slice $t$). Then it is natural to pass from the Bohr topos $[\mathcal{C}(A), Set]$ to the slice topos
$[\mathcal{C}(A), Set]/\langle x_t\rangle \simeq [\langle x_t\rangle/\mathcal{C}(A), Set] \,,$where $\langle x_t \rangle$ denotes the ($C^\ast$-)algebra generated by $x_t$.
What is this like? Like this: the new cosite $\langle x_t\rangle/\mathcal{C}(A)$ is simply the poset of those commutative subalgebras of $A$ that contain the element $x_t$. So the new slice topos is the context in which everything is coprobed by consistent observations that are consistent with the observation of $x_t$.
We can now make the next measurement of some $x_{t_2}$. And pass to the next slice topos
$[\langle x_t\rangle/\mathcal{C}(A), Set]/ \langle x_{t_2}\rangle_{x_t} \simeq [\langle x_t, x_{t_2}\rangle/\mathcal{C}(A), Set] \,.$This is now the context where every consistent set of observations must contain $x_t$ and $x_{t_2}$. And so on.
So we get a kind of “classically consistent histories” of abservations naturally reflected in internal topos-logic.
Maybe that’s useful to make explicit. I need to think about it.
Maybe there is a possibility of establishing some sort of semantics of time sequences of events coming from the chain of evauations at different slice topoi ?
Yes, I guess so. But for really getting the standard notion of consistent histories one needs to do something else: because by successively passing to slice toposes we build a sequence of commuting observables, only.
I am not sure what this is good for. The slice of the Bohr topos over a given observable is something like the context of all quantum kinematics that is classically compatible with that observable.
But wait a second, I do not understand quite what do you mean by classically consistent histories. In the next stage, at later time, you can measure any observable without any limitations. The only thing is that because of the reduction in the process of measurement of the first observable, the observable which you measure now can not be coming by unperturbed time evolution from an observable at the first time o measurement which did not commute in first place. So anything, even noncommuting one can be measured at later time, just the constraint is that it is not related in a predictable way from a noncommuting one in previous time. I hope you get my idea (it is hard to say).
Yes, so that’s why I am saying that iterated slicings of the Bohr topos – even though it looks a bit like forming a “consistent history of measurements” is much more restrictive than what is usually understood by this technical term.
So: I don’t have a good answer to your question yet. The observation about the meaning of slicing the Bohr topos was just an idea that I thought I’d mention.
It is certainly a worthy idea! I like it very much, just trying to make sure that we did not get into blunder. Maybe there is a more sophisticated way to use it eventually. :)
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