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Hi,
If you haven’t looked at these yet, please have a look. They are calling it the biggest discovery in quantum mechanics for decades.
The reason I am posting this is that I would like to begin a discussion focusing on classical versus constructive (lack of excluded middle) mathematics as a means of dissecting the issue of realism in quantum mechanics. The articles listed are probing a very deep problem of realism in quantum mechanics. Does the wavefunction reflect our knowledge of a system or does it reflect reality itself. They use the language of ontic versus epistemic states and I will explain those.
Ontic states are variables which we think of as concerning an underlying reality and epistemic states are like probability distributions which reflect the information we have about a system. Were I to flip a coin and cover it and ask you to tell me about this coin, you would say it is 50 % heads and 50% tails. In my case, I peaked at the coin and the state that I would use to describe the coin is 100% heads, so the probability distribution just reflects the information we have been given and does not reflect reality. I could also have told you that the coin has landed on its side and is in the “Quails” state. You would then update your probability distribution to include a third state and this process begins again when I reflip the coin and ask you again what you think the state is.
The reason I am posting this here, is that this community seems familiar with the following research in the foundations of mathematics. In it I discovered the following interesting point:
Bifurcation of notions
On the other hand, differences in axiomatization or definition that make no difference classically can result in actual differences in behavior constructively. Therefore, classically equivalent notions often “bifurcate” (or “trifurcate” or worse) into multiple inequivalent constructive ones. This tends to happen whenever a concept involves negation and, to a lesser degree, disjunction and existential quantification.
I would like to offer for discussion the notion that the ontic states (bases and classical states, states of reality) and the epistemic states (quantum states, wavefunctions, states of knowledge) have been consistently treated to different axiomatizations in physics, perhaps even in the same paper/textbook without the author realizing it. Furthermore, I am offering that the multiple overlaping epistemic states which one can have for the same ontic state in quantum mechanics derives from the bifurcation between the classical mathematics one uses on the ontic states and the constructive mathematics on the epistemic states. It comes from the use of the existential quantification which one passes up from the ontic realm up to the epistemic realm when one talks about the elements of the Hilbert space or the space of quantum states. Further, we should find in the articles listed above that they represent an attempt to eliminate the constructive mathematics at the epistemic level. In particular, we see in Hardy’s paper the central axiom is of “Ontic Indifference” and here it is stated:
“Ontic indiﬀerence. Any quantum transformation on a system which leaves unchanged any given pure state, $| \psi \rangle$, can be performed in such a way that it does not aﬀect the underlying ontic states, $\lambda \in \Lambda_{| \psi \rangle}$ in the ontic support of that pure state.”
What we should note is the presence of the existential quantifier being passed up from the ontic states to the epistemic states. It is here where we can begin with dissecting this recent work in terms of foundations. Below, we the definition of “ontic support”:
“By the ontic support of a given state, $|\psi \rangle$, we mean the set, $\Lambda_{|\psi \rangle}$, of ontic states, $\lambda$ which might be prepared when the given pure state is prepared (i.e. those ontic states that have a non-zero probability of being prepared when the given pure state is prepared)”
You forgot to link to the actual article outside of a paywall.
There are a group of people trying to use the constructive internal mathematics of a topos to describe quantum mechanics. They use words like “the Bohrification of quantum logic”. For example, there is a theorem that if you take a noncommutative von Neumann algebra and regard it as an object internal to the topos of presheaves on its poset of commutative subalgebras, then it becomes an internal commutative von Neumann algebra and therefore has a spectrum, which is an internal locale.
Personally, it doesn’t seem to me that “quantum logic” matches constructive logic very well; about the only similarity I can see is that they are both not classical logic. I am more inclined towards Bob Coecke’s approach to quantum logic, which also has a very nice categorical interpretation.
Mike, Thanks for he response! I have enjoyed the topos approach to foundations, though it is very hard for me to follow. It was enough for me to understand that a generalized point or element is a map from a terminal object! :). I have met one other person that does topos approaches to foundations and I pointed out this thread in the hopes that they might pop in here and give us a few words on these latest papers that might be of interest to those who know topos theory and ncat theory. Mike’s post is a great start and thanks again.
I had another thought on the way in to work this morning. Mike mentioned Coecke’s approach which has also heavily influenced me. I have mentioned the following paper by Vicary before and in it he creates a category of comonoids internal to a dagger compact category. This category of internal comonoids would be the natural place to “host” the ontic stats of reality. Each of these categories could also host, (in an internal language?) diagrams in which we could interpret existential quantification. He points out that the cat of internal comonoids is a lot like Set. The adjunction he constructs could be used to map existential quantification diagrams from the base cat (the arguably quantum dagger compact cat) to the classical cat of internal comoinds and back. That functor is an adjunction too. It would be interesting to see if this also expresses the idea that ontic states lift to “non-overlapping) epistemic states.
You might have noticed that the papers, and most other work on the issue, frame this debate in a dichotomous light. Either the wavefunction is about information OR it is about reality. In fact, neither is going to be the case. Instead, we will have to update more fundamental pictures of physics so that, for example, information itself becomes physical and mechanics becomes informational. See Keye Martin’s work and Panangaden’s work. The truth will lie in an interesting new and big theory which will present a good picture of reality where information is physical.
Also, at some point somebody will notice that reality is not described by quantum mechanics. But by quantum field theory.
In particular, in the Coecke program state spaces are finite dimensional. That’s good for quantum computation. But for a philosophical discussion about the general reality of physics it may fall a bit short.
Hey Urs,
If you feel comfortable, I think it would not be just myself who would want to hear more of your thoughts on this. :) I feel that the infinity of the universe is almost guaranteed by the fact that systems can be causally disjoint. There is just no saying what the whole system is, because you are cut off from it. Black holes are the example that people normally talk about.
Also, did you notice that in Vicary’s paper, which is linked to above, he develops the topological quantum field theory right in Hilb using the internal comonoids in place of the normal category of cobordisms? What do you think of this? I think it is really neat!
the infinity of the universe
This has nothing to do with the infinite extension, or not, of the universe!
The space of states already of a single particle on the interval $[0,1]$ is not finite dimensional.
The only finite dimensional state spaces are those of topological theories and tensor factors of infinite-dimensional spaces that reflect discrete properties. For instance the space of states of the electron on the interval is the above infinite dimensional space tensored with $\mathbb{C}^2$. For quantum computation one uses only that finite $\mathbb{C}^2$-factor and ignores the rest. This is good for computation, but not for reflecting the fundamental nature of reality.
hear more of your thoughts on this
I haven’t looked at the articles yet that you mention above. On the issue of the Coecke-school approach compared to the Bohr topos-approach: these are not alternatives, but are complementary. I am not really into quantum logic, so I can’t tell you which one you’d want if you are interested in that. I do know however that the Bohr-topos approach can describe actual quantum field theory, a student of mine discussed that here.
in the Coecke program state spaces are finite dimensional.
This is an interesting question! My impression has generally been that when physicists work with infinite dimensional spaces, they generally like to pretend that they are finite dimensional as much as possible, or at least to do things with them that are only formally justifiable for finite-dimensional spaces, like use delta-functions as an orthonormal basis or integrate over spaces of paths. So I had sort of internalized a model where physicists are formally using very large finite-dimensional spaces (maybe space and time are discrete at the planck scale), but approximate them by infinite-dimensional spaces because continuous calculus is a super convenient tool.
these are not alternatives, but are complementary
Can you expand on that?
Hi Mike,
what the Coecke-program exploits is, at its heart, the FQFT formulation of the Schrödinger picture of quantum mechanics, where we consider state spaces given by objects in a suitable tensor category and “evolution”, hence operations on them by morphisms there.
What the Bohr-topos program is based on is the dual AQFT formulation of the Heisenberg picture, where one considers algebras of observables instead.
There is a crucial structural difference between finite and infinite degrees of freedom:
In the FQFT picture this has been made most explicit maybe by Stolz-Teichner in their program. Their work is today probably still the only one that systemtically considers FQFT in the non-topological case, the one of actual physical interest. They find that the asymmetry introduced in the infinite-dimensional case, where there is an evaluation map $V \otimes V^* \to k$ but no longer a unit map $k \to V \otimes V^*$ is a genuine structural property and not something to be discussed away. More recently, following an observation (yet another observation) by Segal they have refined this further, finding that the formalism of rigged Hilbert spaces, which is the one relevant in physics (that’s where the Dirac deltas etc live) is actually induced from the representation theory of such cobordism categories without full duals.
In the dual AQFT picture one can similarly see in the 2d case nicely that the “finite dimensional” case of algebras of observables of type II von Neumann factors is uninteresting, with all the interesting physics only appearing in the “infinite dimensional” case of type III factors.
Another comment on the idea that finite systems approximate infinite systems well enough:
I once had a discussion about this with Alain Connes, concerning the physical relevance of his result about outer automorphisms of type III factors. He amplified the following point, which I think is central:
Naively it looks as if the Schrödinger-Heisenberg evolution equation says that time propagation in QFT is an inner algebra automorphism, because one writes
$A(t) = \exp(i t H) A(0) \exp(-i t H) \,,$for $H$ the Hamiltonian. But a careful formalization for the case of quantum field theory shows that this is not so. Starting for instance with a lattice approximation of your quantum field, say a $n \times n$ lattice as for the Ising model, where a field operator $A$ as above acts on the Hilbert space $\oplus_{0 \leq i, j \leq n } \mathcal{H}_0$ for $\mathcal{H}_0$ the (finite dimensional) space of states at one one lattice site, the Hamiltonian is in typical cases of the form
$H = \sum_{i,j} H_{i,j} + interaction terms$being the sum of Hamiltonians acting on each lattice site, and similarly some interaction terms relating neighbouring lattice sites.
So for finite $n$ here, time evolution is indeed an inner automorphism.
However, as we scale this to the limit of infinite size, this changes: the correct algebra to take in the limit has as elements arbitrary but finite formal sums of operators on the lattice sites. So in the limit the Hamiltonian is no longer in fact an element of the algebra. And hence the Schrödinger/Heisenberg time evolution equation becomes an outer automorphism and as such structurally quite different from the finite-dimensional case.
What is the physical meaning of infinite dimensionality? I believe that the mathematics of infinite dimensionality is structurally different from that of finite dimensions, but can you say anything to convince me that it isn’t just an artifact of the mathematics we’ve chosen, that there are real observed physical phenomena that can’t be explained by a finite-dimensional model?
I see what you are after. Yes, phase transitions, universal behaviour, scaling laws and such phenomena are not exhibited at any finite lattice stage of the approximation, but only in what technically is called “the thermodynamical limit”.
This is a good question to have a decent nLab entry about. I’ll try to look into it. Not right now, but maybe in a few days.
I have started thermodynamic limit – a stub to serve as a reminder for me when I have more time to come back to this.
Hi,
is the comment directed at me or at Mike? I am not sure I understand.
The set of integers, $\mathbb{Z}$, say appearing in a physical system given by a 1-dimensional lattice, do you count that as a “truly infinite”?
What would be a solution to your probleme in this physical 1-dimensional lattice case?
Let’s try to gauge what we are talking about: my comments above were meant as a start of supporting my claim (more details have to wait) that a formalism that only sees finite dimensional Hilbert spaces misses certain physical phenomena which either have no finite approximation, or else for which one needs to take the “thermodynamic limit” of the finite approximations.
The discussion of non-trivial 2d QFTs coming from local nets of type $III$ vN factors (the “infinite” factors) via quantum lattice systems and their uniformly hyperfinite algebras are a genuinely quantum mechanical (as opposed to statistical) example.
All of this deserves to be expanded on, but this is what I was trying to indicate.
I’m not really interested in a philosophical discussion about whether anything in mathematics is “truly infinite”. I work with infinite sets all the time and prove existence results about them that have nothing to do with finiteness. My question was whether the infinite-dimensional parts of mathematics really have importance in physics.
This MO question seems vaguely related to the question of importance of infinite-dimensionality in physics. At least, I think infinite-dimensional things tend to be even harder to work with constructively than they are classically.
I am now getting the impression that all three of us are each talking about a diferent question. :-)
Concerning constructive mathematics in non-finite physics: there is for instance the constructive Gelfand duality theorem and together with the Bohr-topos perspective it allows to speak about all those non-finite QFT systems constructively.
Urs 11:
as we scale this to the limit of infinite size, this changes: the correct algebra to take in the limit has as elements arbitrary but finite formal sums of operators on the lattice sites
What does it mean “correct” ? What is wrong with taking some “completion” ? I mean physically – do we really get new observables if we make it bigger ?
Here are my naive thoughts: In any stage of measurement on such limit observable one has the same meaning and result as one would have from some finite approximation; only infinitely many measurements could see a difference. So I could work with a system with more operators, and for a theoretical physicist thinking on one or another would be the same. I know that from teh point of view of classification etc. one likes to be in a smaller world of allowed models (algebras), but I do not see that this preference in the choice has anything to do with physical requirements.
What does it mean “correct” ?
Correct observable means: local observable. Or rather: observable that is approximated to any given accuracy by a local observable.
Hi I just wanted to thank everyone for participating in this thread. I wish I had lots to add but I am clearly deferring to the folks here as the experts.
I guess the only two cents that I would throw in here is to point out my other threads on what I called structural inductionand Bayesian update. If we have an abstraction of structural induction over categories we are allowing for unbounded structure just via the inductive type proof. My approach to physics is to replace spaces (Hilbert spaces, manifolds and sets)with categories themselves. The finite physics approach of Vicary and anyone doing quantum gravity is reflected in my obsession with tiny categories and how they build into other categories. This has inadvertently also solved another problem which I wanted to solve which was to provide an alternate presentation of the theory of categories. Structural inductive type proofs, taken together form a presentation of the theory of categories. I think that some neo realists might object to a vision of physics where the background is so relative: you can only know structure which your apparatus will reflect in terms of IT’S structure preserving maps. Perhaps an objection is that it skirts the issue. Sometimes I think that only causality is real. Then I think of the causally neutral work of Hardy and Spekkens and then feel that causation and correlation are just post hoc conclusions…….but of what?
I am committed to the idea that foundational problems in physics cannot be addressed without addressing the foundations of mathematics. So, while it may look the same: replace Hilbert spaces with categories and start your thinking at the point of tiny structure, I am actually cleaning out the foundation where, in the other case of Hilbert spaces, sets can still generally interfere with problems of interpretation. It works like this: how am I presenting physics?…with Hilbert spaces. How am I presenting Hilbert spaces? With the category of Hilbert spaces. How are you presenting the theory of categories? In set.
Ben, what you write above sounds pretty wild. Some keywords that you mention remind me of things that I am interested in and others here are interested in, but the way you state the above paragraphs sounds weird.
There is plenty of discussion here of type theory, physics, foundations, categorfication, foundations of physics, categorification in physics etc. So from the point of view of your inclinations it might be interesting.
If this is to be continued: can we find a way to make this into a more systematic discussion? The first thing would maybe be to break down all these ideas into little bite-sized pieces and see if we can see what it actually is that you are talking about (I currently can’t see it!).
Let’s see, maybe we start with this sentence of yours
Structural inductive type proofs, taken together form a presentation of the theory of categories.
This does not quite parse for me, although very similar-sounding sentences might parse. Maybe we can try to just sort out this piece of your above message, for a start.
So: what do you actually want to say with this sentence?
Hi Urs,
Some keywords that you mention remind me of things that I am interested in and others here are interested in, but the way you state the above paragraphs sounds weird.
ya, I get that a lot. Again, your patience is appreciated.
I’m visiting Toronto this weekend, but I am mainly looking forward to sitting in cafe’s and doing math! I have Topooses and Local Set Theories with me and Awodey’s small text on Category theory. What a vacation! I just hung a white board in my den (previously a spare bedroom). I want to collect all these ideas that I have been asking about and put them together into something concrete. Let me see if Toronto will inspire me to at least elaborate on my posts.
Hi Urs,
The whole structural induction idea is very new to me and hard to express, but here goes. Mike has suggested in the thread I offered that we can use categorical cell complexes to host something like a partially ordered set of categories. Suppose we have a method of induction-style proof where we prove statements about a structure and then have an induction hypothesis that states that if it is true on $x$ and $x R y$ then it is true for $y$. Next we prove the statement for the minimal elements (…and so on, but it is a bit beyond me so I am at a loss). Thus, we assume we have the partially ordered collection of categories, we have the induction proof method over the partial order, and finally we discover that our collection of partially ordered categories is actually all categories. Maybe it is CAT, the cat of all cats. I don’t know, as I don’t know what categorical cell complexes are. I still have to decide if that is what I am thinking about. Mike seems to have agreed that we might both be thinking of a collection of cats that is, indeed, all cats.
We start there.
Next, we must understand that any statement proved with the inductive proof method is actually a statement about the kinds of structures on which we are basing the partially ordered collection. As we have seen, though, the structures are just categories and nothing more specific or general. Thus, any inductive-type proof (as described) is actually a proof about categories. We then start writing out each of these proofs. After a long time we will start to consider the collection of all such proofs, or statements using that proof method. This collection is, therefore, all statements one might prove about categories. Hence, we are considering a way to present the theory of categories.
FWIW, there are no partial orders in the solution I presented.
Edit: Well, maybe there are. I guess the partial order in question would have, as elements, categories equipped with a cell complex decomposition, and as relation the notion “is a subcomplex of”.
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