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Made a start at coordination. I’m unsure whether it’s worth spending too long on the intricate accounts of Schlick and Reichenbach, and then of whose makes best sense of Einstein’s proposals. Then there’s plenty of recent literature on the subject.
For me, it would probably only be worth expanding if we could thrash out an account of what the nPOV has to say on the subject. Urs has suggested we look at Bohrification. That sounds like the best lead. Reading through the Bohr topos entry, however, it seemed to me that little is said there about how to integrate that with other parts of the synthetic QFT story. There’s the idea of the ’fifth axiom’, but shouldn’t we expect these quantum phase spaces to have appeared earlier as part of the quantization process. Or do we see it merely as way to interpret our way back from the weird quantum world to something as classical as possible so as to be able to relate theory to the recordings of our classical instruments?
Thanks! I have added some more hyperlinks into your text. Then I have added a brief paragraph on Bohr toposes.
These questions that you ask are good and deep questions that I don’t know the answer to! Maybe it’s wrong to think that there is an Independent 5th axiom to “synthetic QFT” which adds on a coordinating Bohr Topos. Maybe that coordination and/or the Bohr topos that mediates it are built in at a deeper stage of the axiomatics. I don’t know. That seems to be a very subtle question about ontology.
One thought, why the passage from a classical theory via quantization to a quantum one, only to return via Bohrification to something classical?
According to philosophers who talk about coordination, there’s already something to explain about with classical theories.
Scott Tanona in “Theory, coordination, and empirical meaning in modern physics”. In Discourse on a New Method, edited by M. Dickson and M. Domski, Open Court, 2010 writes at length at how to see the coordiation process going on in the Michelson-Morley experiment. He has the idea that principles such as the light principle allow us to relate theoretical derivations to what we experience in the pre-relativistic world of our laboratories. The thought is that we need some kind of independence between the new theoretical framework and some established empirical framework. But we shouldn’t imagine that the latter is just given to us. E.g., it might talk about interference patterns and light waves. It’s already an interpreted experience.
He also talks about the trickier case of quantum mechanics. I’ll have to read it again.
It would be interested to see a full derivation of a prediction from a theory to what is seen in the lab. E.g., what is a complete derivation from special relativity that my Michelson-Morley apparatus shows no sign of interference fringes moving?
What's the difference between coordination and interpretation (in the sense of ‘interpretation of quantum mechanics’)? It seems that this is all the measurement problem.
Of course, people don't realize that there's already a measurement problem in classical physics (particularly, but my no means only, in general relativity). So they treat ‘measurement problem’ as if it refers to something specially quantum, and talk about interpretations only of quantum physics. But it still seems to me to be all the same issue.
Isn’t the interpretation arising from a feeling such as:
I’ve got this very successful theory. It must tell me the truth about how the world is. I need therefore to be able to tell a satisfying story, interpreting the formalism, preferably one as similar as possible to my classical world view?
re #3: sure, the issue is not exclusively one of quantum physics. But the quantum aspect adds another subtlety. In the Bohr topos perspective you assume that you are already happy with understanding how probabilities to observe something in experiment map to probability distributions on a space called the phase space and wonder only about what further happens to that statement as the phase space is replaced by a non-commutative space ((or actually: a non-associative space, name the dual to a Jordan algebra)).
concerning this:
What’s the difference between coordination and interpretation (in the sense of ‘interpretation of quantum mechanics’)? It seems that this is all the measurement problem.
To me both are orthogonal to each other. Coordination alone in quantum mechanics is just a rule which for some experiment that you set up tells you which self-adjoint operator you are to pick from which C*-algebra. Then the theory gives probabilities for your measurement outcomes. No “interpretation of what is actually happening” is needed for this coordination.
OK, I think that I understand.
But maybe if we had a decent formalization of “coordination” that would also help with putting the discussion of “interpretation” into a context in which they can be analyzed more scientifically. Maybe something to think about. Or maybe with a good formalization of “coordination” the need for more “interpretation” just goes away. In any case, maybe something to think about.
Maybe we can play a bit with these ideas. (Next Wednesday I am member of a defense committee of a PhD defense on the topic of Bohr toposes. So I’ll be in Bohr topos spirit for the next days anyway…)
So as was mentioned above, maybe it is good to play with “coordination” in the classical/prequantum context first. Not sure, but here is one observation:
If we are given a phase space in form of a symplectic manifold , then there is the classical/prequantum analog of the Bohr topos, namely the presheaf topos over the site of Poisson-commuting subalgebras of the corresponding Poisson algebra.
If we regard the phase space as an object of the slice of the ambient topos over the moduli space of closed 2-forms, and if we consider a prequantization to an object of , then the “prequantum Bohr topos” could be taken to be the presheaf topos whose objects are abelian groups equipped with a homomorphism
(so I am making use of the details discussed at prequantized Lagrangian correspondence).
Hence the site is a sub-category of on the objects of the form , for an abelian group object.
Phrased this way, we immediately have a generalization of Bohr toposes in mechanics to a notion of prequantum Bohr -topos in -dimensional field theory: by the discussion at Classical field theory via Cohesive homotopy types (schreiber) this is just given by replacing in the above by and allowing now to be an abelian infinity-group object.
So this might be something to explore for exploring “coordination”: this evades the full quantum aspect and concentrates on the classical/prequantum aspect, but generalized from mechanics to field theory, which may have a much better chance to allow for interesting statements.
(The idea in much of the Bohr topos discussion and similar philosophizing over fundamentals of physics that from just mechanics instead of genuine field theory one can indeed extrapolate to the actual questions (such as in Isham’s work “quantum cosmology”) is generally a bit, say, dangerous).
Hm, so what can we say about ?
Hm…
Re 9,
Or maybe with a good formalization of “coordination” the need for more “interpretation” just goes away.
That’s Tanona’s opinion. But he does everything descriptively, without any formalism.
Re 10, I’ll see if I can understand this later, but one thought for now, Tanona makes a big deal of special (and general) relativity needing coordination, before moving on briefly to what he admits is the trickier quantum case. Is anything like this appearing here in a difference between non-relativistic and relativistic classical theories? Is it that coordination is needed for both, but extra consideration should be given to the passage from the theory to the specificity of the reference frame of the laboratory?
Thanks for the pointer to Tanona. I have now added a remark on that here. Please check if the citation is correct. I didn’t manage to find a free copy of the article.
Concerning relativity: I think once formalized, the difference of the coordination process is not different here. In both cases we need to find some phase space of the system to be described, reduce the constraint symmetries, and then coordinate functions on the reduced phase space with “actual physical observables”. This is a process that works (to the extent that it does work) for any local action functional and some of them may have Lorentzian symmetry, some may have Galileian symmetry, some may have AdS-symmetry, some may have weird other symmetries.
To my mind the coordination process is about the same in all cases. The fact that with Galileian symmetry there seemed, historically, less coordination necessary is that the necessary coordination was more obvious to our human minds. But “for the formalism” which has no biological prejudices, Galileian coordination is typically actually harder than Einsteinian coordination. There was once a post by John Huerta, I think, on teaching GR, which ended in marvelling that making the Galileian picture fully systematic is actually harder than doing it for GR, because it is more convoluted, mathematically less simple.
re #10:
After posting this yesterday I came to think: maybe I am just arguing for the image of the stabiiization under the dependent product map .
re #13: that’s a bit too enigmatic for me – you’re looking to stabilize in the classical world? Is there some sense in which you’re aiming for a commutative square – quantization and coordination commute? Would a prequantum Bohr (n,1)-topos be taken to a Bohr (n,1)-topos?
re #12: Here’s an extract from Tanona on QM:
on this view we do not need to explain the apparent classicality of quantum systems because quantum mechanics requires and relies on that apparent classicality—indeed, in a sense, the theory is about what happens in that classical world. On this view, then, the characterization of collapse as a separate physical process is misguided because the phenomenon which collapse is supposed to address concerns not an actual process within quantum mechanical theory but rather the coordination between empirical measurements and representations of quantum systems. Until we first get clear on this relationship, it is premature to propose new processes to account for features of that relationship.
I’ll add a part of this to coordination.
Thanks for the further details on Tanona!
And yes, sorry for being so brief on that suggestion about slice toposes. Let me explain:
so by discussion that we recently already had elsewhere, given a prequantized phase space exhibited by a map , a -tuple of commuting Hamiltonians is a concrete group homomorphism
hence commuting 1-parameter subgroups of automorphisms of , regarded in the slice over .
For the site of the Bohr toposes consists of all such Poisson-commutative subgroups, hence the site of the prequantum Bohr topos consists of all concrete maps of the form
as varies. Looking at it this way, I was now searching for the most natural general version of this statement. First, it seems that once we look at things globally, we could and should allow all abelian smooth groups, not just copies of the additive group of real numbers. So then we’d be looking at all concrete maps of the form
for abelian, and hence for an abelian infinity-group, i suppose.
Notice that the abelianness here is a reflection of the “Bohr classicality” which encodes that we are looking at tuples of observables that all (Poisson-)commute with each other, hence that can all be unambiguously measured simultaneously.
Phrased this way it siggests that in the perspective of higher geometry the Bohr classicality is really about stabilization. So we might want to see if we can just replace in the above abelian -groups with spectra.
To that end, observe thatby construction we have an inclusion
which sends the unique point to itself and group elements to automorphisms of that.
So therefore I was thinking now that maybe what this is really telling us is that we are just looking generally at stable objects in the slice topos. That would seem to capture the information of what should be the “Bohr infinity-topos” while at the same time being an object that deserves to be considered on purely formal grounds.
But this is just a wild idea so far. I’ll need to think about it more. But yesterday I began thinking that this might be interesting.
I have changed the second paragraph at coordination. It used to say:
For example, to connect Einstein’s equations about the curvature of space-time to predicted observations of the direction of light coming from distant stars and passing close to the sun, general relativity postulates that light follows a null geodesic.
But this is a little problematic, since of course with the usual action functional for point particles on pseudo-Riemannian spacetimes, one derives (instead of postulating) that they follow geodesics, and null geodesics if they are massless.
In any case, it seems to me the first example to be given at this point of the entry is something more basic. I put in this now:
For example an abstract differential equation on the sections of some abstract fiber bundle does not constitute a theory of physics unless one specifies which kind of physical field these sections are meant to represent.
The originators of the notion were thinking also about how the physics translates into things I can see. There’s nothing more to a theory than to this translation process, and this involves not just the mathematical physical theory, but also how it is specified that that theory gives rise to measurements.
They were impressed by Poincaré’s conventionalism. Perhaps you can see the first pages of Friedman’s chapter [here]https://books.google.co.uk/books?id=e9TjZc9wNUAC&pg=PA71). (Actually he goes on to say there’s something more subtle in Poincaré’s views than this conventionalism.) There may be two different theories with their respective coordination principles generating equally accurate predictions. Choice then is down to convenience.
So as well as specifying what kind of physical field is in play, they would also want to know how field values translate into measurements.
So as well as specifying what kind of physical field is in play, they would also want to know how field values translate into measurements.
That I took as being implied. To reflect this better, I have now expanded the paragraph, to read as follows:
For example an abstract differential equation on the sections of some abstract fiber bundle does not constitute a theory of physics unless one specifies which kind of physical field these sections are meant to represent.
The nature of the field however determines a rule for how to measure it: If we are told that the theory is meant to be about the electromagnetic field, say electromagnetic waves, we test it by measuring, say, Lorentz forces on electrons; while if we are told that the theory is meant to be about gravity, say gravitational waves, we test it by entirely different measurements (e.g. via the LIGO experiment).
But this distinction may not be reflected in the plain mathematics that constitutes the physical, theory: In the previous example both electromagnetic waves and gravitational waves may be described after gauge fixing by the same kind of wave equation. Hence coordination is involved in specifying what aspect of observable reality this equation is meant to be referring to.
Perhaps you can see the first pages of Friedman’s chapter
I have added the pointer to the entry as: Friedman 99
The Poincare heated disk example is nicely given here. People still differ as to how convincing that is that we could retain Euclidean geometry come what may.
Regarding detection, e.g., in LIGO, the point is that there needs to be further stipulations as to how what is recorded there is taken as a measurement of a wave stretching the length of an arm. Of course, the instrument is designed with that intention in mind of displaying a reading of the wave strength. But this in turn must rely on assumptions about the behaviour of the measuring device under changing conditions (as in the Poincare case).
Michelson-Morley usually comes up here as an experiment which was read in two ways: there is no aether or lengths contract along the direction of motion through the aether.
Of course, in that case other factors are at play:
Although the estimated difference between these two times is exceedingly small, Michelson and Morley performed an experiment involving interference in which this difference should have been clearly detectable. But the experiment gave a negative result — a fact very perplexing to physicists. Lorentz and FitzGerald rescued the theory from this difficulty by assuming that the motion of the body relative to the æther produces a contraction of the body in the direction of motion, the amount of contraction being just sufficient to compensate for the difference in time mentioned above. Comparison with the discussion in Section 11 shows that also from the standpoint of the theory of relativity this solution of the difficulty was the right one. But on the basis of the theory of relativity the method of interpretation is incomparably more satisfactory. According to this theory there is no such thing as a “specially favoured” (unique) co-ordinate system to occasion the introduction of the æther-idea, and hence there can be no æther-drift, nor any experiment with which to demonstrate it. Here the contraction of moving bodies follows from the two fundamental principles of the theory, without the introduction of particular hypotheses; and as the prime factor involved in this contraction we find, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point. Thus for a co-ordinate system moving with the earth the mirror system of Michelson and Morley is not shortened, but it is shortened for a co-ordinate system which is at rest relatively to the sun. (Einstein 1916)
David, much of what you write here would fit nicely into the entry, you might go ahead and put them in!
By the way, I came back to this when reading Tim Maudlin’s recent comment on Peter Woit’s blog, the comment here:
To become a physical theory, or rather be part of a physical theory, a mathematical formalism has to be interpreted, or, as I say, accompanied by a commentary.
I tried to point there to the nLab entry on coordination, but Peter Woit is deleting my comments.
Busy at the moment, but later.
Good heavens. Barred by Woit. Is that for your support of string theory?
I regularly hear from people whose comments he deletes. Everything that doesn’t support his agenda.
Ok, I’ve added something along the lines of #20.
Thanks!
I have tried again to point Tim Maudlin to this here.
Maudlin would surely know of this. The interesting issue is the one Tanona raises towards the end of our page in the context of QM:
…the characterization of collapse as a separate physical process is misguided because the phenomenon which collapse is supposed to address concerns not an actual process within quantum mechanical theory but rather the coordination between empirical measurements and representations of quantum systems. Until we first get clear on this relationship, it is premature to propose new processes to account for features of that relationship.
Then coordination can be seen as more fundamental than interpretation, in the sense of interpretations of QM.
Maudlin would surely know of this.
Curious then how he advertizes this as his personal way of thinking about the situation. His case might be helped by pointing to the established history of the concept.
The interesting issue is the one Tanona raises
Not sure. At this point I feel natural language is beyond its range of applicability. I’d rather we phrase the coordination business more formally, say as in the paragraph on Bohr toposes, maybe with some linear logic thrown in (as we have discussed elsewhere) and then see what really remains of the measurement problem.
Why assume that this only involves natural language? Why can’t “Until we first get clear on this relationship” include the use of Bohr toposes?
True. I need to find time to think about this with more leisure.
Peter W has Urs’ comment visible now but then dissuades any further discussion in the following comment.
I thought the example someone gave about electric and magnetic fields ( and , or equivalently the Maxwell tensor , a curvature) vs potentials (for instance the magnetic potential/connection ) quaint, since we have experiments (i.e. the Aharonov–Bohm effect) that can detect in the presence of zero . It seems people either have to accept in a classical theory the reality of the potentials, or that in QM there is a real effect (at least, that’s how I read the Wikipedia article, not great I know…)
There is never a decent discussion to be had on Woit’s blog, the best one can achieve is sneak in a link to some place that does make sense, in the hope that at least some readers eventually see the light.
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