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some basic text at hidden variable theory, just so as to satisfy some links…
Book by Sudbery (QM for mathematicians) has, in my memory (or did I read that elsewhere?), hidden variables of the 1st and second kind. The ones of the first kind can exist (for almost trivial reasons) and of the second can not at all, as in the entry, in that classification. The difference is somewhat subtle.
I added a stubby paragraph on Bohmian mechanics, linking to a potential article which maybe someday I will write.
Thanks.
But what do you have in mind with Bohmian mechanics being “well developed”?
It’s mainly a simple rewriting of Schrödinger’s equation, followed by an exclamation: “Oh, check out what it looks like now!” and then that’s pretty much it. What else has there been “developed”?
I mean that it makes sense, that you can actually do quantum mechanics with it, rather than just having a vague proposal. And there are people working on it; in particular, I think that there are some mathematical difficulties with quantum field theory (which is unsurprising, since that's not mathematically well defined) that are subjects of ongoing research. I am no expert, however, so I may be way off base.
The part of QFT that is not mathematically well defined has become rather small these days. Mostly it’s well defined only that practitioners like to stick to vague reasoning anyway because its still hard to compute.
What does it mean to “do quantum mechanics” with Bohmian mechanics? What I am aware of is that what is called “Bohmian mechanics” is an equivalent rewriting of the Schrödinger equation plus some philosophical sentiments. The first is trivially able to “do quantum mechanics”, about the second I don’t know what it has helped with.
But I maybe haven’t been following up on the discussion. If there is some new development, maybe you could point it out.
On the contrary, I don't know new developments, so I don't know how well the Bohmians have kept up with advances in QFT. (It's understandable, of course, if they're always a little behind, being in the minority.)
The problem is that Bohmian mechanics is more than just Schrödinger's equation and some philosphy. In the formalism, there are two aspects of reality: the wavefunction (governed by Schrödinger's equation) and a classical trajectory (governed nonlocally by the pilot-wave equation, not by the classical equations of motion). So if $X$ is configuration space, then a state of the universe (over all times) consists of a $\psi\colon X \times \mathbb{R} \to \mathbb{C}$ and a $q\colon \mathbb{R} \to X$, satisfying certain equations. For ordinary quantum mechanics, that works out just fine; but it's not immediately clear what to do when it's not legitimate to start with $X$. Wikipedia suggests that there are several approaches, and I don't know which if any really work.
In the many-worlds interpretation, the only aspect of reality is $\psi$; Bohm adds $q$, which amounts to picking out at all times which of the many worlds is the real world. The many-worlds interpretation takes them all to be real, which people sometimes have a problem with; even if they're all real in some sense, surely there is one world in which I actually live. The pilot-wave equation is the correct equation for tracing out a consistent history of one world, and Bohm says that only one such world is real (so that's the one that I live in).
Hey Toby, thanks, I know Bohmian mechanic. I still think it comes from a re-writing of Schrödinger’s equation, since that rewriting is what yields a continuity equation for a flow that satisfies a quantum-deformed Hamilton-Jacobi equation. It is this contiunuity equation which may be argued to justify looking at those “hidden” trajectories in the first place.
I think this has best been formalized by Edward Nelson, who constructed these “hidden” trajectories concretely as a stochastic process whose diffusion equation is equivalently described by the Schrödinger equation.
Or rather, one should amplify: all this is so away from the zero-locus of the wave function.
I am not asking about this standard formulation. I was asking about your assertian that Bohmian mechanics is “well developed”. Maybe I have the wrong association here, but for a “well developed” physical theory I expect that I can find textbooks on it that discuss non-trivial applications. But this is not so for Bohmian mechanics. The usual discussion stops having genuine physics content from the point where the Schrödinger equation is rewritten and realized – away from the zero locus of the wavefunction – as describing an exotic stochastic process. From there on all further discussion that I have seen is philosophical and concerned with the implications of this rewriting for the interpretation of reality. I doubt that there has been any development in Bohmian mechanics apart from this “philosophical sentiment”. That’s why I asked what you mean by Bohmian mechanics being “well developed”.
In any case, I have started an entry Bohmian mechanics now. You may want to add more stuff and in particular more references.
But since Bohmian mechanics is provably observationally equivalent to ordinary nonrelativistic QM, what possible sort of non-trivial applications could it have, if by “non-trivial” you mean “different from ordinary QM”?
I don’t know, that’s why I want to know what the developments would be.
To me, as I said above, what goes by “Bohmian mechanics” is an equivalent rewriting of the Schrödinger equation plus some chantiung that goes with it. The first is trivially equivalent to standard theory, of the second I have not yet seen a genuine use.
This is of course a common issue in all discussion of “interpretations” of quantum mechanics. As long as we all stare at the same equations, up to equivalence, and just feel inspired of different chants to sing over them, then why bother?
I think what is missing here is a more formalized framework for what the new “interpretation” would amount to. For instance the real point of a hidden variable theory should be that it predicts new phenomena at scales at which ordinary QM breaks down. This is the idea: hidden variable theory should mean that QM is just an effective theory at some scale which arises from coarse graining a more fundamental theory.
If one had such a statement, that would be interesting. Such as: Bohmian mechanics predicts that at scales about the Planck scale, the standard Schrödinger equation starts to differ from experimental observation and instead Bohmian mechanics is the only available accurate description.
However that won’t happen with what is done in Bohmian mechanics traditionally, because that is by definition equivalent to the standard Schrödinger equation and will succeed or fail jointly with it. Because it doesn’t differ from standard theory, up to a contractible space of philosophical sentiments.
I think.
A person from my former institute department in Zagreb is doing research on Bohm-like approaches to QM and QFT. He claimed that in relativistic situations there are some real differences:
Conventional relativistic quantum mechanics, based on the Klein-Gordon equation, does not possess a natural probabilistic interpretation in configuration space. The Bohmian interpretation, in which probabilities play a secondary role, provides a viable interpretation of relativistic quantum mechanics. We formulate the Bohmian interpretation of many-particle wave functions in a Lorentz-covariant way. In contrast with the nonrelativistic case, the relativistic Bohmian interpretation may lead to measurable predictions on particle positions even when the conventional interpretation does not lead to such predictions.
This is from his article
As long as we all stare at the same equations, up to equivalence, and just feel inspired of different chants to sing over them, then why bother?
Well, way back in the dark ages of the 20th century, people expected equations to actually mean something. No doubt we are all more enlightened now.
Yes. After the terrible experience of Everettian multiverse interpretations, we are more enlightened now.
@Zoran, thanks for the link, should be included in the entry. Every claim of an actual difference should be recorded, that’s what makes it worthwhile.
@Mike: Hm, you are being ironic now, I suppose, or sarcastic. (Correct me if not.) That will make online discussion really hard. If indeed it is sarcasm, then I am afraid that you have just a superficial impression of the issue at hand. I guess David C. in #13 is expressing the counter-sarcasm already. I think if there is interest, then we should discuss this without sarcasm, as the issue is not as clear-cut as you may think.
There are those famous historical debates from the first half of the 20th century where the big guys went through exactly this “the equations must mean what we have been used to all along”-trouble. After a while the general feeling changed. If an equation of physics doesn’t mean anything to you, then the problem is not necessarily in the equation. Every-day meaning of physics equations is the hallmark of 19th century physics. The revolutions in 20th century physics (relativity and quantum and their synthesis to QFT) threw all that out of the window.
I have lived with the Schrödinger equation for a good bit of my life now, and to me it is full of meaning, while the Bohmian rewriting distorts the meaning. This is how I feel about it. Now how are we to resolve this if you feel differently? I guess we cannot, and hence “meaning” of equations should be treated with a bit of care.
I think that it is easy to overestimate our ability to have intuition for fundamental physical reality. That feeling that physics must be about a point running around in a big confguration space may have that cozy touch to it that makes you say “it GOT to be this way”. But I think it’s the same “it GOT to be this way” that made Earth seem a disk neatly placed in the center of the Heavens.
To push this a bit further (which may be counterproductive in this dicussion here, but anyway) I believe that even the standard Schrödinger equation, weird as it may seem to the newcomer, is still closer to our human prejudices than to fundamental reality. Because I already know that it really comes from pull-push in twisted K-theory. And I already know that this is just the transgression of something that comes form pull-push in some higher chromatic cohomology theory. The deepest aspects of quantum physics which currently are understood are only understood at that level of abstraction. Looking back from here to Bohmian ideas makes them seem entirely out of place.
Again, that’s just a feeling of course. And since feelings here are many and varied, the only way to make progress is to look at what the math gives and not insist that it immediately coveys all its “meaning” to us.
By the way, over 11 years ago we had much the same discussion on spr, here is a trace. That was a little after I had that period as a student when I was all into Nelson’s stochastic mechanics interpretation of QM, the high-brow version of Bohm. (Hm, they say the internet does not forget, but my pdf review of Nelson’s stochastic mechanics seems to have disappeared? Sad.)
And in 11 years from now students will keep having this discussion over and again. Unless somebody manages to put an end to it and save people some lifetime. Well, probably this is one of the mistakes in life that everyone has to do himself ;-)
My comment about Everett was from someone who has had to sit through too many hours of multiverse-inspired philosophy, peaking (or its opposite) in a discussion of whether it is OK for me to kill you in a quantum situation (imagine a photon passing through some half-silvered mirror setting off a gun), since ’you’ survive in one branch of the multiverse.
More seriously, what matters in any ’philosophical’ thinking about some current physical theory is whether it leads to a good way forward for that theory. Bohmianism and Everettianism has never seemed to me the kind of thing that would do this. If Urs is right to lead us to “pull-push in some higher chromatic cohomology theory”, then (tautologically) whatever meta-scientific thinking got him there was of the right kind. I would suggest it’s a different kind of thinking, involving as one component the quest to apply the deepest of mathematical ideas. The track record suggests a range that have worked out very well: conic sections, the calculus, Fourier analysis, Riemannian geometry, Hilbert space theory. We should all be rather excited that the new one may be happening in our backyard.
I keep coming back to this, but in what kind of world could we be living, if Witten working on supersymmetric QFT could make sense of the geometric Langlands correspondence, and this not be giving us a very large pointer as to where we need to look for the next big physics breakthrough. I think there’d have to be a malicious demon teasing us.
@David, #16: yes, exactly.
By the way, we need not focus on my pet theories. If you take any actual research level formulation of quantum field theory (be it Whightman axioms, Haag-Kastler axioms, factorization algebras, various flavors of FQFT, etc. ) then none of them relates in any useful or way to that Bohmian suggestion. Nor does any of these formulations suffer from a problem that the point of view of Bohmian mechanics could conceivably solve.
I keep coming back to this, but in what kind of world could we be living, if Witten working on supersymmetric QFT could make sense of the geometric Langlands correspondence, and this not be giving us a very large pointer as to where we need to look for the next big physics breakthrough. I think there’d have to be a malicious demon teasing us.
That’s exactly how I feel, and how many people feel about this kind of stuff. Not everbody though, for sure.
Would be good to have a name for this, like those other names for philosophical standpoints. Maybe “mathematical realism” or the like.
Sorry, I was in a hurry and I left off my emoticon. (-: My intent was not to express a sarcastic or combative opinion of my own, but to say what I thought the big deal about Bohmian mechanics is (or was) to the people who believe(d) in it. Your comment about “why bother” seemed to me to be missing this historical context. I’m sure you are aware of the historical context, probably moreso than I am; it just didn’t come through to me in the comment.
I do feel like there is a larger issue surrounding what it means for an equation to “mean something”. Traditionally I think in physics, the equations were considered to refer to, or represent, some physical objects or reality, and that could be considered their “meaning”. Your descriptions of what the Schrodinger equation means to you doesn’t sound to me like this kind of “meaning”. My understanding is that this is one of the changes that took place in the 20th century, that physicists sort of gave up on having equations that actually refer directly to physical things and were satisfied with equations that could be used to predict observable phenomena.
Personally, I find that less satisfying, even if justififiable on practical/scientific grounds, so I can understand the attraction of Bohmian mechanics. It’s particularly nice that it gives such a simple meaning to the process of measurement; I still haven’t heard an explanation of measurement that makes much sense from any other perspective.
And now I have to run again, so I will leave this comment also not maximally edited, but I’ll put on lots of emoticons so you know I don’t mean to be argumentative. (-: (-: (-:
Would be good to have a name for this, like those other names for philosophical standpoints. Maybe “mathematical realism” or the like.
The trouble is this is taken already by the kind of position that holds that all mathematical entities exist, rather than the ’nominalist’ who believe none exist. So Penelope Maddy once held that she could actually see the set of eggs in her fridge. The trouble is there’s no discrimination as to the reality of profound mathematics.
But before we try to name it, isn’t there a difference between some unity of maths and physics as convergent (actually someone, Omnes, did give it a name in Converging Realities: Toward a common philosophy of physics and mathematics, which I see there’s a review of by me) and there being a deep mathematical reality, one face of which is relevant for physics?
A couple of sentences from your papers
in 4d topological Yang-Mills, this is the way Wess-Zumino-Witten theory and and Wilson loop actions appears as a codimension-2 corner theory and as codimension-3 defects, respectively.
Wilson loops in an ambient gauge theory are 1d topological prequantum systems that mathematically are the content of Kirillov’s orbit method
This suggests that geometric representation theorists should take the latter as one part of a larger connected whole. And given they’re in the game of forging links between geometric and arithmetic Langlands (see geometric representation theory), there ought to be some manifestation of this whole in the arithmetic flavour.
But back to the earlier question, is there some ur-structure whose avatars are seen in geometric and arithmetic situations, and consequently in physical geometry? So Witten is good for pure maths as through the physics avatar he glimpses this ur-structure.
There it is – physism. Omnes writes:
There are basic axioms for logic and mathematics. These axioms are laws of physics. They are recognized through two inseparable criteria: their fecundity in the construction of mathematics and their necessity for a statement of the law of physics. Their fecundity can be explained in view of the universality, subtlety, and richness of the laws: the basic axioms must be fecund enough to allow a statement of the laws in the language of mathematics.
Conversely, they generate every possible field of mathematics. New laws, new axioms, new fields, are possible and they may be discovered by further research. Consistency is equally necessary in mathematics and in the laws of physics, which are inseparable. Consistency cannot be explained, but it stands as one of the two criteria of truth. The other one is experimental falsification of a mathematical proposition purporting to express a law of nature. (p. 215)
If that wasn’t clear in #19, if there’s an orbit method coming from Wilson loops, shouldn’t there be other ’methods’ relating to other codimensions?
I still think it comes from a re-writing of Schrödinger’s equation, since that rewriting is what yields a continuity equation for a flow that satisfies a quantum-deformed Hamilton-Jacobi equation.
I'm not sure what you mean by this, since the Schrödinger equation itself is used in Bohmian mechanics as I know it, only there is also this extra stuff (which admittedly can be motivated from the Schrödinger equation too). But I will take a look at what you've written at Bohmian mechanics and discuss anything about that at it's own thread. I don't think that this gets at the heart of what we're discussing here.
This does:
for a “well developed” physical theory I expect that I can find textbooks on it that discuss non-trivial applications
OK, but we're not talking about a physical theory here, but an interpretation. Or rather, the physical theory simply is quantum mechanics, which Bohm interprets as a hidden-variable theory. Since people have shown how it works in a variety of (what might otherwise be) tricky physical situations, I think of it as well developed. (The many-worlds interpretation is also well developed. The Copenhagen interpretation is so well developed that people have come to realize that it doesn't really work! The Bayesian interpretation is not well developed, even though I personally favour it … so this is not a judgement about correctness.)
That said, …
This is the idea: hidden variable theory should mean that QM is just an effective theory at some scale which arises from coarse graining a more fundamental theory. If one had such a statement, that would be interesting. Such as: Bohmian mechanics predicts that at scales about the Planck scale, the standard Schrödinger equation starts to differ from experimental observation and instead Bohmian mechanics is the only available accurate description.
If a hidden-variable theory is required to have different predictions from standard QM, then standard Bohmian mechanics is not a hidden-variable theory. However, it does have variants (in which $q$ back-reacts on $\psi$) that make nonstandard predictions (or so Wikipedia says). To my mind, this is the value of having lots of interpretations around (kind of like the value of having lots of formulations of general relativity): they suggest variations that (while most likely wrong) are testable in principle and ought to be tested when possible (strengthening our confidence in the standard theory if the results come out as expected). The MWI also generates these, with variations in which interaction (beyond interference) is possible between the worlds.
David, Urs, I don’t understand what philosophical position you’re trying to name. You quoted
in what kind of world could we be living, if Witten working on supersymmetric QFT could make sense of the geometric Langlands correspondence, and this not be giving us a very large pointer as to where we need to look for the next big physics breakthrough.
but that doesn’t sound to me like a philosophical position, just an observation about the relationship of mathematics and physics.
A digression back a few steps: an example of a (theoretical) physicist discussing the physical reality corresponding to equations, or at least mathematical objects:
On-shell diagrams: physical quantities obtained by gluing together three-point amplitudes.
(slide 7 here)
Mike, yes a position has not been stated, only perhaps intimated.
The issue of what to make of mathematics finding application in physics is not generally treated in a terribly interesting fashion to my way of thinking, due to the avoidance of specificity. From this perspective one may as well boil the issue down to, say, an application of arithmetic to the sharing of sweets, as a representative of all forms of application.
The Witten case, on the other hand, may provoke us to wonder why a deep investigation of the best way to drive QFT forward turns up a particular 6d theory, which when compactified, etc., etc., yields via S-duality one form of the Langlands correspondence. The latter in turn derives from some attempt (lots of details needed) to provide a higher form of class field theory, the pinnacle of early 20th century algebraic number theory. Weil’s observations on the three parallel languages of what he calls the Rosetta stone then prompt the translation of the original Langlands conjecture into a geometric form, where we meet Witten.
So then, what to make of a case like this? Some options:
a) Nothing. There’s no more to explain than arithmetic used for sweet-sharing.
b) Not a big deal. From time to time, of course physics will come across complex structures which mathematicians have also reached by other means.
c) There are particularly ’deep’ structures involved here. These are few and far between. It’s not surprising that physicists and mathematicians converge, especially when you consider the history of their interaction, since physicists require deep structures.
d) Physism, or something like it, where mathematics is, or is a part of, physics (Arnold, Omnes).
Then we could dream of responses in tune with Urs (with my own little fantasies added):
e) Schreiberism: cohesive, and more so infinitesimally cohesive, $(\infty, 1)$-toposes support an extraordinary range of structure. Meanwhile, the space of cohesiveness has hardly been explored. At the moment it comes largely in plain and super-versions. There could well be fuller homotopic (less truncated) cohesiveness out there, as well as others of an arithmetic nature (e.g., rigid geometric nature). Witten has uncovered structure in the super-flavour of cohesiveness, which reveals more than the plain version. Some of this structure will manifest itself in an arithmetic flavour as pertaining to arithmetic Langlands correspondence.
The naturalness of Witten-like structures found in the super-world is accounted for by Urs’ axioms for synthetic QFT, explained as:
first axiom: not really an axiom at all, it just says to use foundational logic;
second axiom: is canonical in the following sense: given the foundational logic, the remaining freedom is adding modalities. A strong version of these is adjoint modalities. About the strongest of these that still admits interesting models is adjoint triples of modalities. There are precisely two choices for such: the yin-triple: monad-comonad-monad and the yang-triple: comonad-monad-comonad. So take them both. That’s the axiom of differential cohesion.
third axiom: not really an axiom either, rather the advice: use the canonical structures induced by the previous axiom: consider the homotopy fibers of the units of the comonads and slice over them (equivalently: make other types dependent on them). And consider relations in these slices. That’s synthetic prequantum field theory.
fourth axiom: that’s the one where there is certainly still the most room to understand how it is fully “god-given”. This is why I was after understanding the deep universal meaning of “motivic” stuff lately. But even at the not-yet-super-deep-level at which I have it currently, I think it’s looking pretty canonical. It says: linearize the above relations in the slice and sum them up. Eventually I hope this will be a canonical universal left adjoint construction on “relations dependent on $E_\infty$-ring types”.
These descriptions are independent of any particular form of cohesiveness.
What to make of such a position is not so clear to me. It could be taken as a version of (c), filling in part of the story of what is ’deep’.
One concern, value judgements are being made about the depth of mathematical theories. As has cropped up in discussion before, what then should be make of work not naturally expressed category theoretically, let alone in the cohesive HoTT framework, e.g., Hungarian style combinatorics? I guess we won’t be following Arnold who seems happy to write off a whole bunch which doesn’t fit his criteria as “ugly scholastic pseudo-mathematics”.
Thanks for trying to explain, David C. I think I understand (a) through (c), although those three seem more like a continuum to me than three discrete possibilities. But I don’t understand (d); what does it mean to say that mathematics “is, or is a part of” physics? Is there some Platonic philosophy of mathematics assumed, with the mathematical objects existing in the Platonic world being then objects of study by physics? And (e) seems to me rather like just a particular special case of the applicability of mathematics in physics, to which one could again react with philosophical interpretations such as (a) through (d), rather than itself a philosophical position.
Urs, I hope I didn’t offend you so badly that you gave up on this entire thread!
I’m not terribly sure what (d) means either.
I guess (e) would be quite a substantial working out of a case of (c), an elaborate illustration of what might be meant by ’deep’. If something general is to be taken from it, perhaps the push towards a freedom from caprice. (I borrow this from Collingwood’s analysis of the motivations for action in an ascending scale – utility, right, duty – where an element of capriciousness is removed at each step, the choice of use, the choice of law.)
So it might appear that QFT is a baroque construction, but really its a simple consequence of some basic decisions.
Okay, that makes sense to me; thanks! Let me know if you figure out (d).
Hi everyone,
sorry for the silence, I needed to concentrate on other things. Thanks a lot indeed for the ongoing nice discussion.
@Mike: okay, my fault, sorry. Thanks for sharing further thoughts.
@David: thanks for the pointer to Omnes and your review of it, I have quickly put that into a References-entry here: Converging Realities – Toward a common philosophy of physics and mathematics
@Toby: I don’t want to be a pain about “well developed”, I just feel that if the Bohmian perspective is the beginning of something interesting then it has never proceeded in an interesting way beyong the initial observation. It reminds me – if you have the patience to briefly follow me along this thought– of what Schellekens writes (here) about the idea that there might be a “landscape” of effective laws of nature: he points out that he was the first on record to say this publically (in a public talk), but that he didn’t consider making a publication about it because beyond that thought itself, there seems to be nothing much to say yet which would qualify as academic research. Maybe some day there is, but right now there is not. His implication of course is: all these people publishing all these articles on the multiverse are doing empty science. I feel it’s the same way with Bohmian mechanics. The original observation is worth a thought, but it has not “developed” into anything, to my mind. I hold the same view, I should say, about all the “interpretations” on quantum mechanics.
And, yes, I guess the main problem in the communication is or was that I would say that a “hidden variable theory” is more than just an “interpretation” of QM. But I can see your point about Bohmian mechanics being a hidden variable theory that happens to be just a (re-)interpretation.
Well, at least away from the 0-locus of the wavefunction, remember. ;-)
I’ll try to describe further that philosophical position for which I am demanding a name.
It is something like:
A fundamental theory of physics flows naturally out of its mathematics.
Or
A fundamental theory of physics will have a mathematical formulation that when you bring it to paper is more like a single entity with many facets, than lots of entities brought together.
Or
If in the study of your fundamental theory of physics you happen to find that the same kind of mathematics that you started to use naturally produces other theories (subtle variants) of physics than what you started with, then this is a sign that you should follow these other variants, that they get closer to the true fundamental nature of physics.
Maybe one can understand this position best through the people who are opposing it. For instance Bert Schroer, a big name in AQFT, has some rants on the arXiv (some of the articles with “String” in the title listed here and probably more, not sure now which of the many I had looked at) where he is attacking what he calls something like modern “dream”-physics or the like (I need to check which words exactly he uses). The dream-physics that he is attacking is that by people who follow mathematical hints in their fundamental physics research.
For instance in some online discussion I remember Schroer once denied that Chern-Simons theory is a quantum field theory. He just entirely rejected it as being a topic in physics at all.
People like this, hence holding a position opposite to the one that I am looking for a name of, see the task of mathematical physics to make something like a “photograph of the surface of what happens at accelerator experiments, taken on a photographic plate made of mathematics”. So they develop mathematics that accurately describes precisely what is seen. If some new effect is seen, some mathematics needed to describe it is added. If some effect turns out to have been due to error in experiment, then the math used for it is discarded and ignored from then on. You can see this in many article of one school in mathematical physics. They go in the following style (I’ll try to produce an impressionistic pciture):
“…let this here be the jet bundle over that vector space and say that a semi-Frechet section of it is a hyperlocal function on that which when restricted to A is a D, then we say that the Banach space completion of the algebra of microlocal distributions on E is the pre-Hilbert space of the field theory…”
And when the dust has settled indeed, there is the mathematical precise description of some field theory. What I am trying to say is: in this approach the math is not supposed to dance by itself. The math is kep on a very short leash. It is supposed to do the work that you want it to do, and not escape an inch to the left or right.
For instance in AQFT one assigns von Neumann algebras to open balls in spacetime. Now meanwhile the homotopy algebraists found that it is mathematically super interesting to assign instead $E_n$-algebras to open balls in spacetime. Allowing this leads to a flood of interesting things that follow. But much of what follows directly (and up to now) is what Schroer would probably call “dream physics”, notably it works best for topological field theories. Not only in fact. But still, the sociological phenomenon to be observed is: exactly nobody in AQFT pays any attention to factorization algebras, factorization homology, non-abelian Poincaré duality etc. at all. There is an entire silence. Because this is mathematics allowed to dance in its own way, and the people working on this are following more the maths more than the experimentally relevant physics, they don’t just stick to that photography of physics.
Another example is this whole supersymmetry business. On the one hand there are people who – like the old alchemists – put two harmless substances together: a few free boson on a 1+1 dimensioal worldsheet and then the same number of free fermions. Then adjust one relative coefficient in a way that seems natural. And – wham! – suddenly the mixture explodes into an enormous firework of structure. As if out of nothing there is worlsheet supersymmetry which implies target space supersymmetry which implies a cascade of physics-like theories starting in 11-dimensions and branching off to a plethora of structure, there is tons of math there and all the qualitative physics seen in experiments. Except for one little fact: all that physics is locally supersymmetric.
Now there are two reactions:
First, the people whose standpoint I am searching a name for say “Oh, wow, we have hit a gold mine, now we just need to follow the gold to see to which treasures of physics it leads us. It would be weird if this immense richness were not to lead us to the el-Dorado of fundamental physics that we are after”.
On the other hand, second, the other camp of mathematical physicists says: “Ah, bah, go away, there is local supersymmetry in all you are doing. This is not experimentally verified. Unless and until it is, there is absolutely no reason to waste time on chasing this dream. Let’s instead further refine those nets of microlocal differential operators on the jet bundle of .. etc. pp.”
Thanks, Urs, that’s a nice explanation. Can you point to historical examples that show the value of this approach to physics? Obviously it’s fruitful from a mathematical point of view, but how would you argue its worth to a physicist who cares, in the end, about describing physical reality? My outsider’s impression is that many successful physical theories were discovered first phenomenologically, and only then was the math invented or discovered to describe them, rather than by following the math where it leads naturally.
Well, at least away from the 0-locus of the wavefunction, remember. ;-)
Yeah, I don't actually understand that bit, so I suppose that I need to learn more about Bohmian mechanics.
Yes, thanks Urs. Your discussion is largely addressed to the question of how to go about doing mathematical physics. Dirac and Weyl could be mentioned as being on your side. Let’s have something by the former
It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. The Evolution of the Physicist’s Picture of Nature, Scientific American (1963)
But I’m hoping for a little more of an explanation. To do this I think one needs to think about the mathematics too. What is it for mathematics to “dance in its own way”? And what relation is there between physical intuition and mathematical intuition? Why could Atiyah and Witten talk so productively to one another, even though, as Atiyah confesses, he knew no physics?
I rather suspect we could learn a lot from Weyl, who Atiyah singles out as someone he most admired.
Can you point to historical examples that show the value of this approach to physics? Obviously it’s fruitful from a mathematical point of view, but how would you argue its worth to a physicist who cares, in the end, about describing physical reality? My outsider’s impression is that many successful physical theories were discovered first phenomenologically, and only then was the math invented or discovered to describe them, rather than by following the math where it leads naturally.
Right, so I’d say probably the central example where “mathematical dream physics” led to a dramatic success was Hilbert’s derivation of Einstein’s equations before Einstein even got them, by applying mathematical principles. I have started to collect some historical comments on that at Einstein-Hilbert action – History (more should be added there, but it’s a start). I think this is quite a remarkable story, if one looks at it.
And I would argue that there is a continuous connection from this insight by Hilbert to what is going on these days. Because a little later people took Hilbert’s successful principle – which is: write down as action functional the obvious terms that you can form from the given fields and which are invariant under the given symmetry – and found supergravity this way, by generalizing the symmetry a little in a “mathematically natural way”.
Or consider this: the effective action of string theory is a functional on pseudo-Riemannian geometries where not only the most obvious term with the required symmetry appears, the one that Hilbert used, but all terms with the required symmetry. Some other action principle (the “worldsheet action”) fixes the relative coefficients between all these terms in a subtle way, and the result turns out to fix a major theoretical defect of Einstein gravity: it is “renormalizable” (while agreeing with Einstein gravity at low energy). This is what in the “mathematical dream physics” of string theory is regarded as the strong indication that this is physically right.
On the other hand, various other teams in theoretical physics stick with that single Riemann scalar term that Hilbert used, reject the possibility that there might be any correction to that which becomes experimentally visible only at higher energy, and then fight for (and with) making quantum sense of this plain Einstein-Hilbert action. Either by dustifying the geoemtry (this is LQG) or by putting the theory on a dynamical lattice and hoping that all the unsolved questions about this will solve itself (this is CDT) or by hoping that the the plain Einstein-Hilbert action one day turns out to have a UV-fixed point.
One might argue that the approach of these latter teams accepts Hilberts insight, but not his method. Applying his method of exploring the space of mathematically natural theories of physics more widely leads away from plain Einstein gravity to variants which differ from it at high (undetecable) energy, and which feature all the rich structure that all those “mathematical dream physicists” these days are busy exploring (and often being accused for exploring).
Another example of “mathematical dreams be the truth in physics” is the decade-long history of geometric quantization.
Before I come to this, I just noticed: in the other thread on the KK-mechanism, mathematician Bruce B. reviews the Lorentz force from the point of view of KK-reduction and concludes his (very nice!) slides with the exclamation:
Conclusion: electrically charged particles are moving along geodesics in 5-dimensional space! Similarly, charged particles in the standard model with gauge group $U(1) \times SU(2) \times SU(3)$ are moving along geodesics in 4 + 1 + 3 + 8 = 16 dimensions!
Interesting to reflect on this statement in the light of the present discussion. Becausee the Kaluza-Klein mechanism is mathematical dream physics.
Einstein got immensely excited about this in the latter part of his life (and image: he is the single mind who in his lifetime found the fundamental description of both electromagnetism and gravity, the only two fundamental forces known at that time, and then suddenly the math seems to be telling him that the former is moreover just a projection shadow of the latter.)
But as often with mathematical dream physics, it reproduces the known physics but also predicts a bit more. The KK-mechanics predicts the existence of “moduli” and was hence abandoned by non-dreamy physicists. (Notice that it is these moduli fields that drive these days, amlost a century later, that humongous discussion about the landscape.)
But the thing is, if one looks at the KK-mechanism with the unprejudiced mathematically trained eye, it is hard to not think that one sees a physical reality and conclude without doubt and with exclamation mark that
electrically charged particles are moving along geodesics in 5-dimensional space! Similarly, charged particles in the standard model with gauge group $U(1) \times SU(2) \times SU(3)$ are moving along geodesics in 4 + 1 + 3 + 8 = 16 dimensions!
Here is the other major example about “mathematical dream physics come true” which I wanted to mention: geometric quantization.
This ends up being another hare-and-hedgehog game between physicists and mathematicians, lasting several decades. In the role of Hilbert we now see Bott.
So physicists discover the need for quantization. First on $\mathbb{R}^{2n}$, which they split into “canonical coordinates and momenta”. Then later they (the mathematical physicists among them) refine to symplectic manifolds equipped with polarizations and consider polarized sections of the prequantum line bundle. Then that turns out not to always work and various “corrections” are added. Finally all these corections are subsumed in the notion of “metaplectic corrections”. Then later it is noticed that the metaplectically corrected polarized sections are just those counted by the index of the Dolbeault-operator. Then one notices that this in turn is just the index of the induced $Spin^c$-Dirac operator. And then, many decades after the chase began, one notices (maybe Bott was the first to do so explicitly) that all this long, long story ends up being nothing but one single statement in mathematics: geometric quantization is push-forward in KU.
(!)
So this story is historically not a case where mathematical dream physics made a prediction. It is rather a case of us all slapping our foreheads for wasting decades with fiddling around and not instead asking what might be a mathematically natural way to turn a complex line bundle into a single vector space. If you had asked any mathematician this question, they would have told you to push it to the point in KU.
Thanks for those examples, Urs. Maybe I misunderstand, though, but only Hilbert’s discovery of Einstein’s equation really seems to me to be relevant as an argument directed at a physicist with his/her eye on the “real world”. Finding new and beautiful mathematical ways to describe physical theories after the physicists have come up with them is great, but it doesn’t argue for the value of pursuing the mathematics where it leads in advance of the physics.
but it doesn’t argue for the value of pursuing the mathematics where it leads in advance of the physics.
Is it really so clear how to separate these? It’s not obvious to me what you would take as evidence. Do we need a mathematician picking up on some physics and driving forward the mathematics so as to later help the physics? But mathematical physicists complicate things. How do we describe Yang on gauge theory as opposed to Chern on fibre bundles, doing similar things simultaneously? How do we describe Weyl’s work on representation theory?
Surely all Urs is urging is that it generally pays off to frame physics in the most elegant mathematics available, whoever does it. And when this happens both mathematicians and physicists should pay attention.
I don’t think the point has anything to do with who is doing the work. I do think there’s a difference between (1) framing existing physics using more elegant mathematics and (2) starting from elegant mathematics that describes existing physics, extrapolating to new and even more elegant mathematics, and hoping/expecting the result to have implications for physics. Obviously sometimes that hope/expectation is valid, but it seems to happen rarely.
There are three examples in #34, which I will oversimplifyingly call (in order) Hilbert gravity, super gravity, and string gravity. I understand the Hilbert gravity example, and agree with Urs about it. I don't really understand the string gravity example, which runs into my ignorance of string theory; taking what Urs says at face value, however, it seems very sensible. But the super gravity example doesn't work for me. In the case of Hilber gravity and string gravity, we have a physical intuition for the way things should be, look for the best way to express this mathematically, and see where that leads us. In the case of super gravity, I don't understand why people are looking at that one at all, except with purely mathematical motives.
Mike, it may not matter who does the work, but I’m not so convinced that it’s easy to separate extrapolating mathematics from extrapolating physics, given that the intuitions will have been thoroughly intermingled. On one reading you could even describe Witten as doing your (2), the thing you think rarely works. But maybe you want this description. And, not to beat about the bush, how could your (2) not include what Urs is trying to do? Taking QFT and finding an elegant formulation in HoTT surely must count.
We need to see lots of examples of failing (2) activity. How about Weyl’s original gauge theory, where he drops length preservation? It didn’t work as planned,
Weyl’s uniﬁed theory, though ﬂawed, was not utterly doomed. It was to ﬁnd salvation during the development of Quantum Theory. (Quigley)
Then, Urs gave us Bott as filling the role of Hilbert for geometric quantization. How to rate this work?
On top of this should be added the thought that very few mathematicians or physicists add anything of lasting value, whatever they try to do. Is the success rate (however success is defined) of (2) activity worse than than for the kind of theoretical physicist who doesn’t push the mathematics in an elegant direction?
Two comments on Mike’s comments:
The order and path in which humans discover the thruth can hardly matter for this discussion. If I write down ten lengthy proofs in linear algebra using bases and matrices, and each time discover only at the end that there is a short proof using abstract vector spaces, then that’s my fault, and not that of the abstract theory of linear algebra. Even if I can give the next proof in matrix algebra again, I would be better advized to study the abstract theory.
But I am aware that the common idea that the mathematicians should just sit there and wait for physics to develop and then later come and clean up a bit after them. This is how one large sector of mathematical physics works, and many people share the sentiment that this is how things ought to be. This is why there are professors who do physics but don’t understand maths, and those who do maths but don’t understand physics and nothing much in between.
I am doubting that this is the way it should optimally be and am thinking that this is a state of affairs based on intellectual inertia.
Whether it happens “rarely” that fundamental physics ends up following the flow of the maths is maybe a matter of subjective perspective. I think it happens all over the place. But I am not arguing that there is proof that mathematical dream physics is bound to become true. Everybody is free to disagree. I’d just want a better word for it to be better able to communicate this standpoint.
@Toby, re #40:
The motivation for supergravity from physics is this:
A widely underappreciated fact is that the mere existence of fermions implies that physics is described not by manifolds, but by super-manifolds. A fermion field in field theory is not just a section of a spinor bundle. It is crucially a section of an odd-graded spinor bundle. So super-geometry (to be well-distinguished form super-symmetry) is experimentally verified ever since Stern-Gerlach.
Now given that Einstein gravity without fermions is the theory of Cartan-connections for the inclusion $SO(d-1,1) \hookrightarrow Iso(d-1,1)$, where on the right we have the isometry group of Minkowski spacetime, the natural question is what happens to this statement as one consideres instead Minkowski spacetime with an odd spinor bundle over it. The answer is that $Iso(d-1,1)$ gets promoted to the supersymmetry group and the corresponding Cartan geometry is supergravity.
This is not a proof of anything. But a natural motivation.
Another parallel motivation goes like this:
The worldline theory of a relativistic fermion is automatically locally supersymmetric (see here). The background theory can still be anything.
Also the worldsheet theory of a spinning string is automatically locally supersymmetric, even if one does not put this in. (This is how supersymmetry was first discovered in the West.) Now a miracle happens: the target space theory for a spinning string is necessarily locally supersymmetric.
This is how local supersymmetry is “predicted” by string theory. Namely, to summarize again:
experimentally we observe fermions;
for making gravity renormalizable we speculate that there are strings;
to make strings give fermions we consider spinning strings;
spinning strings automatically have worldsheet local supersymmetry and imply target space local supersymmetry.
Again, not a proof of anything. But certainly a physically well motivated conclusion. (Which might turn out all wrong, but it’s not far-fetched.)
I’d just want a better word for it to be better able to communicate this standpoint.
You might say it’s a form of ’rationalism’, but this has a very wide range of connotations. You could also give names, such as Dirac and Weyl. Atiyah writes:
The past 25 years have seen the rise of gauge theories–Kaluza-Klein models of high dimensions, string theories, and now M-theory, as physicists grapple with the challenge of combining all the basic forces of nature into one all embracing theory. This requires sophisticated mathematics involving Lie groups, manifolds, differential operators, all of which are part of Weyl’s inheritance. There is no doubt that he would have been an enthusiastic supporter and admirer of this fusion of mathematics and physics. No other mathematician could claim to have initiated more of the theories that are now being explored. His vision has stood the test of time.
and here
[Weyl’s] contemporaries are long since gone and only a few personal reminiscences survive. On the other hand the passage of time makes it easier to assess the long-term significance of Weyl’s work, to see how his ideas have influenced his successors and helped to shape mathematics and physics in the second half of the twentieth century. In fact, the last fifty years have seen a remarkable blossoming of just those areas that Weyl initiated. In retrospect one might almost say that he defined the agenda and provided the proper framework for what followed. He made fundamental contributions to most branches of mathematics, and he also took a serious interest in theoretical physics.
David,
thanks a lot for all those historical citations that you provide! I’d like to collect them all in some $n$Lab entry, either one on Weyl or one on mathematical physics or the like. But right now I don’t have the time to do so…
OK, I started a section at mathematical physics, called ’Attitudes towards mathematics’ (though the US audience doesn’t have that ’s’ in ’towards, does it?)
I found Dirac saying something rather similar to you.
The trend of mathematics and physics towards unification provides the physicist with a powerful new method of research into the foundations of his subject, a method which has not yet been applied successfully, but which I feel confident will prove its value in the future. The method is to begin by choosing that branch of mathematics which one thinks will form the basis of the new theory. One should be influenced very much in this choice by considerations of mathematical beauty. It would probably be a good thing also to give a preference to those branches of mathematics that have an interesting group of transformations underlying them, since transformations play an important role in modern physical theory, both relativity and quantum theory seeming to show that transformations are of more fundamental importance than equations. Having decided on the branch of mathematics, one should proceed to develop it along suitable lines, at the same time looking for that way in which it appears to lend itself naturally to physical interpretation. (The Relation between Mathematics and Physics)
This is 1939, so given that he says that it has “not yet been applied successfully”, it sounds like he’s excluding Weyl’s work to that date.
It would be good to dig up some sceptical opinions on the role of maths. There is
physics is to mathematics as sex is to masturbation,
dubiously attributed to Feynman, and maybe not so easy to interpret precisely :). Surely there are some hard-nosed physicists who like to keep their maths simple, and happy to say so.
though the US audience doesn’t have that ’s’ in ’towards, does it?
We do; it’s just that there’s a brand of US prescriptivism that wags its finger at it. For a while I dropped the ’s’, just to keep some people happy, but I never understood what was supposed to be wrong with it, as I grew up saying and writing ’towards’ before learning of this weird shibboleth.
Thanks, David, that quote in #46 by Dirac is excellent! Yes, that’s about what I was trying to express. I didn’t know that Dirac said something like this.
And thanks for starting the entry_mathematical physics_. That looks just like I was hoping for. As soon as I have a spare minute I may join in editing further. Right now its a bit hectic here…
Right now its a bit hectic here…
The Barcelona meeting. I wonder if you could get some more from Joyal on loci.
Pardon, Urs, 48 The entry mathematical physics has originally been started by me, in the moment when it was maybe less interesting: April 20, 2010 03:16:47 by Zoran Škoda, revision 1: http://ncatlab.org/nlab/revision/mathematical+physics/1 David added new recent material.
Right, sorry, you started the entry, David now kindly started to add there quotations on perspectives on mathematical physics.
Here’s Roger Penrose, not exactly someone who “likes to keep math simple”, from The Road to Reality on what he calls the “miracles” of string theory:
As the mathematician, Richard Thomas… remarked to me …
… To a mathematician these things cannot be coincidence, they must come from a higher reason….
Yet, it still may well be that this ’something’ is of purely mathematical interest, without there being any real reason to believe that it takes us any closer to Nature’s secrets.
and later
As far as I am aware, such mysterious relationships are not normally put forward to support proposals for scientific (as opposed to mathematical) theories.
and again
One thing is certain, however, and that is that such mathematical miracles cannot always be a sure guide. During the course of my own studies of twistor theory, I have come across several different pieces of encouragement that would seem to come under the heading of ’miracles’… I have no wish to make a comparison between the miracles of twistor theory with those of string theory. But they cannot both be unambiguous signposts, because… the two theories are, as they stand, incompatible with each other!
The way I read this objection, he’s saying that pure mathematics is already full of surprising connections between unrelated things. So the fact that some part of mathematics known to be relevant to physics is surprisingly connected to some other part of mathematics doesn’t mean we should expect that that other part of mathematics is also relevant to physics.
It’s difficult to tease apart what we’re talking about here. I wasn’t under the impression that Urs’ story of QGFT-in-CoHoTT was a proposal of any new physics; am I wrong? I am all in favor of formulating physical theories in the most mathematically elegant and appropriate way. I suppose there may be physicists who object even to that, but I think there is a separate objection to justifying a new physical theory by its mathematical elegance. Certainly it’s happened often that new physical theories turn out — after we understand them fully — to be more elegant than their predecessors, but there’s lots of elegance in mathematics that isn’t useful for describing reality (so far as we know). So (I think the question is) is it a good idea, in the search for new physics, to be led by mathematics?
If I understand correctly, even Weyl was not successful at first, and only later was it realized that the mathematics he developed was relevant to physics in a different way. So (to take the other side as strongly as possible) one might argue that his work was not a success of physics but rather a success of mathematics. We know separately that it happens often that some unsuspected part of mathematics turns out to be relevant to a new physical theory, and so what if it should happen that such a part of mathematics was originally developed by a physicist hoping to use it in a different way?
Thanks, Mike, for these comments. Some quick reactions:
That comment about twistors and string theory being incompatible is ironic in retrospect, because right about the time that Penrose wrote that book, Edward Witten formulated “twistor string theory” (which is strings propagating on the twistor space) and found that this yields a way to predict those “MHV” scattering amplitudes in Yang-Mills theory which ever since then has become a huge field of study (see the parallel thread on the “amplituhedron”). So in retrospect Penrose’s counter-example has been realized to instead be an archetypical example.
(Matthew Strassler on his blog is right now starting a series on this aspect of “string theory usefulness of the second kind”. On the nLab there are more comments on this at string theory results applied elsewhere.)
It is true that the QFT-in-HoTT is meant to capture the established, and experimentally verified physics (and the same statement holds for string theory, not to forget). But it is also true that QFT-in-HoTT automatically does so in a way that naturally supports the string theory perspective on physics. For instance in QFT-in-HoTT 1-dimensional quantum mechanics arises as the “boundary theory” of a 2-dimensional topological field theory. This is the “holographic” perspective of string theory and quite unlike how the non-string mainstream tries to put quantum mechanics on foundational footing. Generally, what QFT-in-HoTT spits out directly are higher topological field theories and then the non-topological field theories are found as secondary effects. Conversely, we are finding in HoTT rigorous formalizations of string theorist’s “holography”, just recently I pointed this out at off-shell Poisson bracket. And then in the context of the “Brane bouquet” I kept pointing out how remarkable it is (to my mind) that if you study QFT-in-HoTT in the supergeometric model, then just starting with the canonical line object and considering its homotopy extension theory, the entire broad structure of “M-theory” just so appears, out of no-where. (I keep saying this, and keep thinking that there are only really two reactions to this: either to doubt it and claim that I am making an error, or else to be rather amazed. I don’t want to go around advertizing my insights as “amazing”, but probably I didn’t quite convey how noteworthy this seems to be.)
Finally, yes, I certainly agree, if we just say “all elegant and beautiful mathematics must imply physics” then we end up in a sad mess, already because it seems that pretty much every matheamtician finds his or her subject “beautiful” and “elegant”. Therefore I am not going around trying to promote this. I do still think that there is something which maybe escapes our ability to phrase it in words and which if we could phrase it in something different than words would express what these words are trying to and failing to express. (If that makes any sense ;-). So I won’t be promoting any such principles. But it seems to me that just working on QFT-in-HoTT is a promising way to be silent whereof one cannot speak.
I wasn’t under the impression that Urs’ story of QGFT-in-CoHoTT was a proposal of any new physics
Doesn’t joining up the dots in a certain way result in more than the dots? And if the way of joining the dots is principled and hard to emulate, as with the brane bouquet, so much the better.
But let’s not forget that some of the original anti-dream mathematics proponents that Urs described above wouldn’t even have gone so far as to admit the worth of the branes for physics, let alone appreciate the way they can be tied together.
Now if something wonderful emerged for physics by moving to QFT-in-HoTT beyond the 2-truncated super-world, that would be a wonderful instance of Dirac’s methodology.
@Urs
Speaking of Americanisms, you seem to be hypercorrecting “-i/yse” to “-i/yze” (cf advertise, advise, analyse).
@David re @54:
right, and as an example I think the interpretation of the Hamilton-de Donder-Weyl equation of motion as a map from $\mathbf{B}\mathbb{R}^n$ to the automorphisms of some object in the cohesive slice topos over some $\mathbf{B}^n U(1)_{conn}$ counts as an genuine contribution of QFT-in-HoTT to current problems in field theory. This needs to be expanded on more to yield sociological impact, but I will be claiming that this solves some open issues in covariant field theory, see section 3.3 in Classical field theory via Cohesive homotopy types (schreiber).
@Zhen Lin: thanks for alerting me. I haven’t been thinking about that, it’s just my fingers. But I’ll try to correct that in the future. Thanks.
Maybe I’m just running up against my ignorance of modern physics here. I need a popsci exposition of all this stuff; most of the threads around here that you’re citing make my eyes glaze over.
some of the original anti-dream mathematics proponents that Urs described above wouldn’t even have gone so far as to admit the worth of the branes for physics
What are branes worth for physics? I was under the impression they were still hypothetical mathematics, like strings.
Maybe I should clarify some more. When I say “it seems to be rare” what I mean is that I personally haven’t heard and understood very many examples that convinced me. I would love to hear and understand more examples, but they need to be explained very very simply for me to understand them.
What I think I’m looking for is cases where “following the mathematics” has led to new physics which makes new predictions that are later validated by experiment. The case of Einstein and Hilbert seems to qualify: Hilbert followed the mathematics to derive Einstein’s equation, which of course makes tested predictions. I don’t see how Weyl qualifies; if I understand correctly, following the mathematics led him to more mathematics but not to useful physics, and then later on someone else was following some other physics and found that Weyl’s mathematics was what they needed. Are the other examples being cited of this nature? Or am I behind the times in expecting physics to be rooted in experiments?
Good points. I’ll reply incrementally as a find time.
Concerning branes: while as fundamental objects in the physics of the observable universe they are hypothetical, as an organizing tool of physics that sheds light on the physics of the observable universe they play a role.
Notably the way that branes appear in string theory connects the Yang-Mills gauge field theory that sits on the brane worldvolume with the gravity theory that governs the spacetime in which the branes themselves sit. This allows to study the Yang-Mills gauge field theory on the brane alternatively via the theory of gravity “in the bulk”. This is what the “AdS/CFT correspondence” is about. It leads to plenty of insights into the nature of Yang-Mills theory and of the observable world, if we consider branes of dimension 3+1. Several of such insights are listed at string theory results applied elsewhere.
Quite generally, this is how string theory with all its phenomena (string, branes, etc.) is affecting the understanding of the established physics of the observanle universe: it works like a huge directory of different field theories and their interrelations. Making use of these interrelations that are only understandable by thinking in terms of string theory increaes the insight into just Yang-Mills gauge theory as such.
Maybe the most striking example of this is the relation of AdS/CFT to condensed matter physics (see here for pointers): a while back people noticed that to compute a property called the “shear viscosity” of the quark-gluon plasma that was created at the RHIC collider, by AdS/CFT one can alternatively compute the Hawking temperature of a certain black hole in anti-deSitter spacetime and translate one computation into the other.
Have to run now. I’ll give more examples later.
Another example of mathematical reasomning predicting physics that is later experimentally verified:
Pauli’s prediction of the existence of the neutrino: in 1930 the weak beta decay seemed to violate conservation of momentum, and hence the invariance of the laws of nature under the spacetime isometry group. Bohr then suggested to ignore the math and postulate that conservation laws hold in some other way, to fit experiment. Pauli instead insisted that the symmetry group should act, hence by Noether’s theorem that momentum had to be conserved. Accordingly he postulated that the missing momentum has to be carried by a new fundamental particle not detected yet. Back then, postulating a new unobserved particle was regarded as a major thing, not lightly taken and generally frowned upon. But that hypothetical neutrino was then later indeed found.
More examples later…
One more example of mathematical reasoning predicting physics that is later experimentally verified:
Maxwell’s introduction of the “displacement current” and the prediction of electromagnetic waves.
The original form of what is now called Maxwell’s equations for electromagnetism were lacking one term, later called the “displacement current”. Only with this term added to Maxwell’s equations do they obtain the very symmetrical form that today we summarize in exterior calculus by just writing the six symbols $\mathbf{d} \ast \mathbf{d} A = J$. Without that term the equations are mathematically much less natural.
How exactly Maxwell argued for the missing term seems to be a matter of historical debate. But he mentions “mathematical analogy” and, as Wikipedia has it, is generally thought to have been motivated by the desire to perfect the symmetry of the field equation and, maybe more importantly, to obtain consistency with the continuity equation for the electric current (which is otherwise violated).
So I think this is an example of where the mathematics guided the (major) prediction of new physical phenomena that are later experimentally confirmed.
I should maybe add that by playing with these exterior calculus equations of the kind of Maxwell’s equations but next for higher degree forms leads to much of the lore about higher dimensional charged branes and the higher gauge fields that they are charged under. This arguably comes out of the same mathematical flow which made Maxwell arrive at the prediction of electromagnetic waves.
More examples later…
What I think I’m looking for is cases where “following the mathematics” has led to new physics which makes new predictions that are later validated by experiment.
1) Had Hilbert’s work been delayed by five years (or even one year), would this count merely as neatening up Einsteinian theory? Does the fact that it was nearly simultaneous make a difference? Wasn’t the legacy of Hilbert’s reformulation the more important thing? As Urs says above,
And I would argue that there is a continuous connection from this insight by Hilbert to what is going on these days. Because a little later people took Hilbert’s successful principle – which is: write down as action functional the obvious terms that you can form from the given fields and which are invariant under the given symmetry – and found supergravity this way, by generalizing the symmetry a little in a “mathematically natural way”.
2) Had Weyl guessed right by opting for a U(1) gauge theory rather than one based on changing lengths, would we count this as an instance of following the mathematics? He was very well-versed in physics. I dare say the physicists would have claimed him as one of their own. As it was, they were able to pass it off as the work of a mathematician. At what point as we pass from a Hilbert to a Weyl to a Witten to a Feynman are we able to say someone was following the physics rather than following the mathematics? Why is Hilbert’s “write down as action functional the obvious terms that you can form from the given fields and which are invariant under the given symmetry” to be taken as mathematics rather than physics? Were Witten to come up with a principle of this kind, would you call it mathematics?
3) Is there such a difference between the kind of thing Weyl was trying to do and what Dirac succeeded in doing, leading to predictions of the positron? One was trying to unify electromagnetism and general relativity, the other quantum theory and special relativity. Can’t I see these as two mathematical physicists, one not quite succeeding, but developing an idea which when modified would be crucial for physics, the other rather more straightforwardly succeeding?
Although Dirac did not at first fully appreciate what his own equation was telling him, his resolute faith in the logic of mathematics as a means to physical reasoning, his explanation of spin as a consequence of the union of quantum mechanics and relativity, and the eventual discovery of the positron, represents one of the great triumphs of theoretical physics, fully on a par with the work of Newton, Maxwell, and Einstein before him. Wikipedia
In view of his work on quantum mechanics, it’s not difficult to imagine a counterfactual history where Weyl achieves this. After all he does have the Weyl equation for massless spin-1/2 particles named after him, where the Dirac equation deals with massive spin-1/2 particles.
Re Urs @ 53, Strassler also explains the application of Witten’s twistor string theory here (near the beginning).
Thanks for these! I had thought of the displacement current late last night, but not the neutrino or the positron. Now that I’m thinking about particles, would you count the Higgs as well? Wasn’t it originally suggested by looking for a mathematical way to unify electromagnetism with the nuclear forces?
Had Hilbert’s work been delayed by five years (or even one year), would this count merely as neatening up Einsteinian theory? Does the fact that it was nearly simultaneous make a difference?
I think the thing that makes a difference to me is that Hilbert didn’t already know the answer when doing it. If Hilbert’s work had been delayed by five years, but Hilbert hadn’t read Einstein’s work before doing his, then I think it would be equally convincing. The question (or at least a question, the one I’m asking right now) is how to argue that “following the mathematics” is a justifiable way to look for physics when you don’t already know the answer. I’m not saying that it would be totally irrelevant if Hilbert had known of Einstein’s work first, but I think it wouldn’t support the position as strongly. This is also why I’m looking for examples of actual successes rather than almost-successes: even if in hindsight the distinction is due only to historical accident, being able to point to an actual success is a stronger argument against someone who claims that following the mathematics never leads anywhere new.
I also agree that there isn’t a perfectly clear dividing line.
Concerning the Higgs field: I had thought about mentioning the Higgs field as an example, but then I thought you’d have too easy a time to shoot it down :-)
Because with the Higgs field “following the mathematics” very much constrains what is possible, but does not completely identify it.
So the problem is to add masses to fermions in a field theory without giving up the mathematical principle (if you wish) of local gauge invariance.
One solution to that is is the Higgs mechanism, and that seems to have turned out to be realized in nature. But mathematically also “technicolor” is a possible solution. Until recently the people who promoted technicolor had lots of good arguments for that. Now with the latest LHC measurement it seems that the room for arguing that technicolor is realized in nature has become rather small, though. But mathematically technicolor is maybe actually a bit nicer, because it is “more pure”: there the Higgs field is not a new kind of field, but just an effective bound state of the kind of fields already present in the theory.
Concerning the contribution of Hilbert in comparison to that of Einstein:
I think the crucial aspect is that Einstein’s considerations are necessary to connect the mathematics to the observations.
Without Einstein, Hilbert might have been able to speculate that there is a physical field modeled by pseudo-Riemannian metrics which extremize the integrated Ricci scalar, but maybe he would never have been able to shown that the physical theory thus defined is approximated by Newtonian gravity in a suitable limit. This is what Einstein dealt with.
This is maybe the most subtle aspect of foundational mathematical physics, how to map the mathmatics to, in the end, our actual concious sensations.
I had had this discussed with Davic C. elsewhere a while back. To illustrate the problem: there are people out there who claim or are close to claiming that some functor from cobordisms to somewhere which they constructed is “quantum gravity”. But with just some such functor given and no further information, it seems to be hard to decide what it even means for such a claim to be true or false. How would such a functor be used to predict out observations?
For classical gravity, this is what Einstein answered, by explaining in which way for instance geodesics on a pseudo-Riemannian manifold can locally be interpreted as particles subject to gravitational field. This is a piece of information that does not just follow from Hilbertian considerations.
This topic is known as ’coordination’. Logical positivists/empiricists were very concerned with this issue of tying mathematical frameworks to empirical observations. For instance, you can see something on Reichenbach’s version here.
Thanks, that’s very helpful. The “coordination” issue seems like perhaps it ought to be emphasized more.
Thanks for that link, David!
Yes, this sentence which I find there
Particular principles, $[...]$ are supposed to establish a correspondence between something in experience and our representation of it in the form of a mathematically axiomatized scientific theory (à la Hilbert).
seems to be about what I just said (or the other way round ;-)
Also, I agree very much with what you say in #62. Mike is of course right that it is more impressive to have the theory before having the measurements it predicts, but in the end this is not the question, I’d say. If all theoretical physicists on the planet would declare to go out of business for 500 years and only then come back to formulate the theory that explains all the experiments done until then, it will in the end have to be the same theory after all that they would come up by theoretizing all along.
Anyway, I have to quit now. But I do hope we can eventually collect more of the good bits of this thread in the nLab entry on mathematical physics.
it will in the end have to be the same theory after all that they would come up by theoretizing all along.
I wonder. For one thing, would all the same experiments have been done in that intervening 500 years, if the theoreticians weren’t around proposing wacky ideas that needed to be (dis)proven?
(Further responses may be slow, yesterday my computer received a shot of water while I was sitting outside on the lawn. Now the keyboard is acting the way I would if I had received the same amount of Whiskey…)
Concerning that point you make, Mike: I think I certainly see what you mean. Certainly, the idea that fundamental physicsis is in some way modeled on mathematics in a way “natural” for mathematics is impressively highlighted if using this guiding principle we are able to predict phenomena not yet predicted.
I am just wary that while this is impressive when it happens, much like a good magicians trick, this cannot really be the ultimate criterion by which we estimate the value of that idea. On the one hand, the idea might be right and we might still be not good enough to see what predictions it leads to. Or it might be wrong, and we may spectacularly luck out with some predictions which then would be misleading.
Rather, I think in the end what counts as an objective check of the idea is whether one can give a formalization of fundamental physics in some mathematical foundation such that it is “short” ort “not overly long” by some measure (for instance by lines of Coq code).
To give more illustration of what I mean: on various discussion forums on the internet one can find the following argument rehashed:
contributor A states that string theory is interesting because it “predicts gravity and gauge theory” (referring to the fact that assuming that fundamental particles are fundamental string excitations automatically implies that the effective field theory which describes phsics is of the form of Einstein-Yang-Mills-Dirac theory, only the parameters of that are not uniquely specified.)
contributor B objects that this is not interesting, because we already knew that there is gravity before string theory was found, hence that it is not a prediction but a “postdiction” .
I think that in these kinds of discussions contributor B is missing an important point. It is true that it would have been impressive for our human souls if string theorists had predicted that apples fall to the ground before this had been noticed in experiment; but for the theoretical understanding of the world the order in which we understand its aspects is really not fundamentally relevant.
For instance if I had been born with a built-in understanding of all of electromagnetism (much like we are born with a built-in understanding of 3-dimensional Euclidean geometry) then that would not make Maxwell’s equations any less interesting or noteworthy.
Similarly I think that, coming back to one of my examples above, if with hindsight, after many twists and turns, we finally understand that geometric quantization is simply push-forward in K-theory, then this does not make me value this insight less than had somebody suggested it much earlier. The lesson I take out of this is that a very fundamental aspect of nature (quantization) is effectively identical to a very fundamental aspect of mathematics (natural operation in stable homotopy theory) and that seems to be noteworthy irrespective of the – usually chaotic – path along which we humans gather this understanding.
But I do hope we can eventually collect more of the good bits of this thread in the nLab entry on mathematical physics.
Right, but not the easiest task to boil it down to its essentials. On the one hand, there’s a line of thought, being pushed by you and me, that there’s more to contributing to physics that devising equations leading to predictions, and that a lot of work in mathematical physics is to be praised for making best mathematical sense of existing physics. On the other, there’s a line that wants something more, represented by MIke, and presumably also by Dirac if he thinks the methodology he proposes hasn’t been applied at the time he said it in 1939 (so not even Hilbert with regard to Einstein), where a development made for as purely mathematical reasons as possible leads to advances in physics.
One flippant remark: perhaps Mike is now uniquely well placed to do this having worked with Urs on cohesive homotopy type theory for QFT, without (I hope this is fair) having much grounding in the physics. Or imagine if I could work out what less homotopically-truncated cohesion looks like and synthetic QFT interpreted there generated something important.
More seriously, the issue of coordination came up, and that needs to be elaborated. One might say that the work of coordination is some of the deepest in physics. Emphasis on it in recent years comes from a philosopher of science I admire, Michael Friedman. See, e.g., from
But can this beguiling form…
on p. 177 of Kant, Kuhn and the Rationality of Science. Under his scheme, we can see different tasks to be done: formulation of new mathematics, allowing the formation of new physical principles, which when coordinated allow the expression of laws of empirical regularities.
There are developments of his views about what precisely it is one coordinates to that I’ve wanted to get around to understanding.
A comment from someone firmly in the anti-dream camp.
There is an obvious, grounding observation to be made here: not all mathematics can be right (in the sense of physics). I use this, rather vague, statement to label a direct conclusion of the empirical fact that many (most?) physical theories have turned out to be wrong, despite having been formulated in mathematical terms that have seemed very natural and elegant (perhaps both physically and mathematically) to their proponents. And I don’t mean wrong-with-a-kernel-of-truth wrong (like Newton vs Einstein), but wrong wrong (like vortex knots in the ether vs atoms, or caloric vs kinetic).
So, it seems inescapable to me that in any discussion of physics, the value and correctness of any role played by mathematics in physics is to be judged by the usual scientific criteria: tying together known data, predicting new data, Occam’s razor. The question of naturalness or elegance I think is strictly secondary and will take care of itself. I am often reminded of the (probably mangled) quote (whose attribution I am not sure of) “Whatever the right physical theory turns out to be, it will be considered beautiful.”, which addresses precisely that issue. Whenever I affirm my anti-dream position, I am mostly reacting to someone trying to relax the above judgment criteria and allow special exceptions for mathematics.
Finally, to address Urs’ desire for giving a name to what appears to be an implicitly held position by many people, including some famous ones, on the relation between mathematics and physics. Unfortunately, I can’t contribute usefully to that particular slice of the discussion. Ironically, that is because my position mirror’s Urs’ position on Bohmian mechanics vs Schroedinger equation. In particular, I already see an excellent (idealized) relationship between the two. Namely, among other things, mathematics all logical implications, generalizations, abstractions, specializations, dualities, isomorphisms, etc. that the human mind can imagine, and all of that with extremely permissive motivation restrictions. Mostly, it is sufficient that the work is interesting to the practitioner. On the other hand, physics ruthlessly selects the best available mathematical tools for its own purposes, whose relevance is judged by the above mentioned scientific criteria, and discards the rest. Mind you, what is discarded or ignored is not to be never looked at again. It simply goes back into the same mathematical pool of ideas that are called on the next time it is necessary.
From my perspective, all of the historical anecdotes that have been brought up in this thread already fall into one or the other category of work. Occasionally, however, some people (including Dirac, for instance) try to add some chant, or dream-like aspiration, to this routine that attempts to elevate mathematics to a more special status than is allowed by scientific principles. And as a response to that, I firmly stand anti-dream.
Urs: (Further responses may be slow, yesterday my computer received a shot of water while I was sitting outside on the lawn. Now the keyboard is acting the way I would if I had received the same amount of Whiskey…)
Is it possible to dry with pure alcochol or something else what sucks water quickly ? Maybe low vacuum pump as it is in any chemistry lab ?
Igor: your user entry hides your real personality. Is this intentional ?
Good to hear from the other side!
Concerning your two “wrong wrong” examples:
This was surely a case of religious motivation providing the force to persist with some outlandish speculation. Smoke rings, and other spinning things, show a certain stability. Atoms are stable and come in many different kinds. Knotted circles come in different kinds. Therefore, atoms are knotted vortices of ether, and elements correspond to knot types.
Thomson tries to make sense of stable vortices of knotted ether, and doesn’t get very far; Tait begins the classification of knots, and makes no connection from them to the elements.
Verdict: fairly wild speculation.
There’s also a position called structural realism. One kind of structural realist wants to say, “Sure the nature of the substance was wrong”, but there was enough structurally correct in the caloric theory to give us the heat equation. This is in defense against the anti-realist who points to many successful theories, where now major ingredients have been given up. The response is that it was the structure which did the work in any success there was. This tends to get preserved through dramatic reinterpretations of theory. See, e.g., Votsis’ paper.
@Zoran, definitely not. it was a surprise to me that I have a user entry to think about.
@David, very interesting! Let me make sure, though, that my point is not drowned in historical detail. Whatever the contribution to mathematics of these ideas, it would have been difficult to correctly judge their relevance to physics during their development if one were to be guided purely by the sense elegance and naturalness experienced by the practitioners.
Great, the user entry http://nforum.mathforge.org/account/296 changed to the informative one :) thanx
Hi Igor, thanks for joining in!
I don’t think we actually disagree on your first paragraphs in #74, re the last paragraph in my #53 above.
By the way, right now I have trouble just parsing your paragraph in #74 that starts with “Ironically,..”. Can you try to just say this again? Currently I am not sure what the message of that paragraph precisely is. Once I understand better what it is saying, I can try to react. Sorry. (Maybe I am just too exhausted, I should get some rest now…)
@David #73: Nice summary. In case it wasn’t clear, personally I do want to agree with the point that you and Urs are pushing, and which Urs expressed well in #72. But I think that the “other side” has a valid point as well, relating to the “coordination” issue, which should be respected and addressed, not just dismissed. There probably is an intrinsic value in finding more elegant formulations of known physical theories even if they do not make any different new predictions, at least within reason. But since the ultimate goal of physics is to understand reality, I think it’s entirely reasonable for people to be initially suspicious of new theories which don’t seem to contribute to such understanding.
In the end it may boil down to a question of what “understand” means. Do we understand something when we can predict its behavior? Or only when we have an “explanation” of that behavior? What is an explanation anyway, and what is it that makes one explanation better than another? Is there any objective standard? I know this is something that philosophers of science have thought about for a long time. (I even took a course on the philosophy of scientific explanation as an undergrad, but unfortunately I don’t remember much of it.)
perhaps Mike is now uniquely well placed to do this having worked with Urs on cohesive homotopy type theory for QFT, without (I hope this is fair) having much grounding in the physics.
Yeah, I’d been having a sinking feeling that it might end up like that. Maybe in my copious spare time. d-:
By the way – even though I hesitate to say it because for some reason or other the following keyword is known to raise negative reactions among some of you – but the idea of Isham and others, which I like to think of as “Bohr toposes” or similar, is just that: to formalize that “coordination” issue.
Namely the idea is that whenever you claim to have produced a physical theory, in the end you should provide a topos (or similar) and a map from the propositions about a speicifed object in that topos to propositions about physical observation.
A while back I had expressed it to Joost Nuiten over coffee like this:
I said that maybe in the end we must add a fifth axiom to the four (or so) axioms for “synthetic QFT” in cohesive HoTT. Currenty the last axiom leaves us with the statement that a quantum field theory sends correspondences in slice toposes to linear maps of $E$-infinity-modules, for an $\infty$-ring $E$.
Now for $E = KU$ and also actually to some extent for $E$ an elliptic ring or $tmf$ , we happen to now how to “coordinate” this with physical observation.
But next suppose we manage (since there is indication that this should be true) to quantize this way generally certain n-dimensional field theories with the “space of phases” $E$ being a (integral etc.) Morava K-theory, or similar.
The mathematics seems to suggest that. But even if we can get it to work, in the end we’d be running into a coordination issue: currently I don’t know how to interpret a map between $K(n)$-module spectra as the propagator of a field theory.
So I was beginning to think that in the end there must be a fifth axiom. Something that tells us how to associate with a given $E$ etc. some (“Bohr”-) topos or similar such that the propositions in its internal logic “are” the physical/observational statements about the corresponding “physics”.
I am not claiming that the specific “Bohr toposes” as currently conceived necessarily live up to this task. But I think the motivation of Isham and others to consider something like them in the first place is to deal with the “coordination” issue in a formalized way.
And that seems to me to have something to it.
That’s an interesting idea and a nice way to think about the motivation. Can you say anything about measurement from such a point of view?
Can you say anything about measurement from such a point of view?
There are some comments on how to express in Bohr toposes the notion of “expectation value of a quantum observable” at Bohr topos.
But right now the nLab is down, and since my personal computer is broken, I cannot restart the server. So I can’t check the page to remind me about how much it says.
@Urs, I was simply referring to the irony that my disagreement with one of your positions uses the same arguments as you used to support a different position of yours.
Of course, if your position were summed up in the last paragraph of #53, we would have no disagreement. However, you seem to be advocating a position (unless I am misreading your intent) where direction for research in physics should be taken from cues of elegance of certain mathematical structures. Moreover, you seem to be looking for a name for this tendency. I call this tendency pure and simple mathematics. Moreover, the fact that some famous people, in famous anecdotes from the history of science, have called such work “physics” and then happened to be right in the sense that physics did indeed adopt some of their work, I call an accident. So, I disagree with the adoption of these kinds of methods and motivations in what is mainstream physics. I think it is this position on which we likely disagree. Unfortunately, I might be in the minority in this opinion.
Igor, can we look at the example of the prediction of the Neutrino in a bit more detail? (maybe as an archetype of the prediction of an infinitude of particles such as KK-modes etc.)
I still need to find a source that gives a sufficiently detailed discussion of the history. But my broad understanding is roughly this:
Energy/momentum conservation is observed to be violated in weak beta decay.
Now Bohr takes the anti-dream position and tries to change the math used for describing the physics to match what is being observed. He proposes that the conservation laws must be changed to capture the observation. Nothing unseen is being predicted, no dream-land seems to be entered.
Pauli instead insists that the mathematics of symmetry must be what applies unchanged, and that instead it must be due to our restricted observation of the world that there is an apparent discrepancy. He enters mathematical dream land which leads hom to predict an unseen particle. Many physicists are unhappy with such a dreamy suggestion. Until it is eventually verified.
Is that roughly right? Even if that doesn’t convince anyone of anything, would it not be fair to say that it is at least one example?
Pauli wrote his own history of events – Zur älteren und neueren Geschichte des Neutrinos.
Urs, to refresh my memory on the subject, I’m looking at Sec 14(d) of Abraham Pais’ Inward Bound. If I were in Pauli’s shoes, I would point to the fact that energy conservation is not observed anywhere else first, before appealing to symmetries. Moreover, the escape of a neutrino particle, would have left a particular signature in the energy spectrum of the beta-particle, which is more constrained than just saying “energy is not conserved”.
But I should now check what Pais has to say about that. From what I know about Pauli, he was one of the ardent anti-dream physicists of the time. After all, he was the one who almost shut Yang up in an early presentation of the Yang-Mills theory with a question about the mass of the vector bosons: too low and it should have been seen by then!
much like we are born with a built-in understanding of 3-dimensional Euclidean geometry
Actually, we develop this understanding through extensive (albeit scientifically naïve) empirical observation in the first several months after we are born.
This makes the human brain all the more amazing, in my opinion. It suggests that if a human were born into a space with different geometry, then they would probably be able to work it out just as easily!
Re Toby @ 88, already in the late nineteenth century, some had come to the conclusion that experience must play a role in the geometry we come to see with (e.g., Helmholtz). On the other hand, there were some general restrictions, e.g., we must see in a geometry of constant curvature to allow the free mobility of objects without shape changes.
Rereading Pais’ account of events is very illuminating, especially his chronology of ideas about beta-decay in the early ’30s. It appears to me that purely mathematical considerations, such as Noether’s theorem, did not have a dominant influence on Pauli or other people at that time. (An exception might actually be Dirac, who in 1936 reversed his position and considered that energy non-conservation could be a way to avoid the purely mathematical problems he saw with divergences in QED.)
It seems that both proposals, Bohr’s energy non-conservation and Pauli’s new particle, were met with strong initial skepticism. Though I don’t see that the opposition to Bohr’s idea came primarily from Noether-style arguments. Though it was definitely known that usual classical and quantum mechanical laws (which were formulated in a time translation invariant way) required energy conservation. However, I would attribute the inertia behind keeping that framework to the strong available evidence that energy is indeed conserved in other processes (going back to the thermodynamics of the 19th century, but also including contemporary experiments, like cloud chambers, which established energy conservation in individual microscopic processes). On the other hand, the skepticism in Pauli’s idea seemed to have dissipated quickly (within the ’30s, only a few years after Pauli’s proposal, and much earlier than direct detection of neutrinos) because concrete scattering/decay models were promptly constructed (culminating in Fermi’s model) and favorably compared to experiment.
So, in conclusion, I would be hesitant to draw very close parallels between the neutrino and KK proposals.
Hm, right it seems that Pauli kept saying and crucially said so in his famous letter that his goal is to “save the energy conservation theorem” but that he never ever mentions, it seems, that energy conservation is not just something we observe, but is a mathematical consequence that follows unless the whole foundations of field theory would be changed.
This is weird! But it means that you are right and mathematical arguments did not play a role in the prediction of the Neutrino. They could have considerably strengthened Pauli’s point, but apparenty were not.
Can this be, that in this whole discussion around the 1930s nobody ever mentioned Noether’s name? Nobody points out that giving up energy conservation (in flat spacetime, as the case is) is tantamount to giving up the entire known foundations of physics?
Okay, these are question that I’d be interested in knowing the answer to, but they are a bit tangential to the discussion here.
So let me try instead my second bullet: the displacement current. Again, I’d need to look up the details of the history, but above I am claiming that this is an example where physics was predicted by taking a partial matehmatical theory (Maxwell’s equations without the “displacement current”) and observing that it would “much more naturally” want to have a further term added.
Coming back to how much Pauli’s argument drew on Noether’s theorem:
This document here:
does start out with amplifying Pauli’s appreciation of symmetry and conservation laws. However the citations remain a bit weak. p. 5 cites Pauli in 1953 as saying
I am very much in favour of the general principle to bring empirical conservation laws and invariance properties in connection with mathematical groups of transformations of the laws of nature.
That sounds a bit like “I am very much in favor of Noether’s theorem”, which is not the strongest kind of statement that one would hope for here. Not sure what to make of this.
On p. 6 the author of the above document says
Pauli’s belief in the absolute credibility of symmetry principles led him to defend conservation laws even when at that time the empirical evidence was doubtful. His prediction of the neutrino is a great example.
This is indeed what I’d expect to hear. But again, the actual quote by Pauli right after that on the same page is much weaker:
am myself fairly convinced … that Bohr with his corresponding deliberations concerning a violation of energy conservation is entirely on the wrong track! …. The idea of a violation of the conservation of energy in β-decay is and remains, in my opinion, cheap and very clumsy philosophy.”
If he had really been relying on symmetry and the Noether theorem, he could have said “provably wrong” instead of just “cheap and clumsy”. Especially since “clumsy” suggests “possible, even if not enjoyable”, where instead he should have exclaimed: “impossible!”.
So my appreciation for Pauli is just diminishing a bit, sadly. On the other hand, even if he never says so, it still remains true that mathematics provided the strongest argument for Pauli’s position, even if he maybe didn’t fully realize it. Maybe if some Hilbert had been following this dispute over the beta-decay, he might have pointed it out.
Interesting for our discussion then also the next quote in the above pdf from Pauli concerning the Neutrino, which is this:
I’ve done a terrible thing today, something which no theoretical physicist should ever do. I have suggested something that can never be verified experimentally.
;-)
You see now why the philosopher of science, Imre Lakatos, was so tempted by ’rational reconstructions’, that one could tell the tale of what was the most rational path to a new position, taking into account the theoretical resources of the time. Of course, one acknowledged that history seldom panned out this way, but one shouldn’t fault theoretical resources, just because humans fail to realise its potential.
He’s always severely criticised for this, but I think there is a point.
Maybe if some Hilbert had been following this dispute […]
Maybe Hilbert himself! After all, Noether had been working closely with him when she proved her theorem. According to Wikipedia, she was led to this by considering a problem that had been troubling Hilbert: the nature of energy conservation in general relativity!
David C. writes:
You see now why the philosopher of science, Imre Lakatos, was so tempted by ’rational reconstructions’, that one could tell the tale of what was the most rational path to a new position, taking into account the theoretical resources of the time. Of course, one acknowledged that history seldom panned out this way, but one shouldn’t fault theoretical resources, just because humans fail to realise its potential.
He’s always severely criticised for this, but I think there is a point.
While I didn’t know that Lakatos made this point, but I sure think now that the prediction Neutrino is an example.
Toby reacts:
Maybe if some Hilbert had been following this dispute […]
Maybe Hilbert himself! After all, Noether had been working closely with him when she proved her theorem. According to Wikipedia, she was led to this by considering a problem that had been troubling Hilbert: the nature of energy conservation in general relativity!
So I suppose Hilbert was involved back in the late 1910s maybe 1920s with these general considerations? But he or somebody like him didn’t seem to be involved in the issue of the apparent violation of energy conservation in weak beta decay measured around 1930?
I don’t know the history. Probably both Nother and Hilbert were no longer actively following science at that late stage of their lifes? Or maybe they were, but not in what might have been a debate more among experimentally inclined physicists than among theoreticians? Or maybe it’s a sign of the general phenomenon that Noether’s contribution took a long time to be fully appreciated.
It seems that happens more often than one would hope. That the answer to what keeps the scientific community busy is known all along, but not appreciated.
I have now added some of the above story on Pauli and the neutrino to neutrino.
It seems that happens more often than one would hope. That the answer to what keeps the scientific community busy is known all along, but not appreciated.
Hear! Hear!
I wonder why Pauli believed neutrino detection to be permanently out of reach.
I’ve done a terrible thing today, something which no theoretical physicist should ever do. I have suggested something that can never be verified experimentally.
Presumably he knew we were bathed in neutrinos, so it must have been just a question of detection. Seems odd that less than 30 years of technological improvement allowed something thought impossible.
While I’m on this, what exactly is being implied by
Notice that back then, predicting unobserved and possibly practically unobservable fundamental particles was not taken as lightly as in some circles it is these days (e.g. in supersymmetry and/or string theory)?
Is this praising Pauli for being worried about predicting something then out of reach, whereas people aren’t sufficiently careful today? It’s odd though if Pauli had been exaggerating the difficulty. On the other hand, the Planck scale is seriously out of reach.
Now we are overlapping a bit with the discussion in the neutrino thread. There I have replied to the first part of the above comment, so here is now a reply to the second:
I think to appreciate the postulation of the neutrino one needs to imagine back those days before the entire zoo of particles discovered throughout the 20th century was known. Back then the world had seemed to consist of only very few fundamental ingredients, and the idea that these were just the tip of an iceberg had not been around.
There is some mental development until then 50 years later people get so used to the idea of new particles waiting to be discovered, that some start postulating infinite towers of unobservable massive particles, namely higher KK-modes and higher string modes.
In fact in Einstein’s times even the single lowest Kaluza-Klein-mode, the dilaton, made people reject the KK-unification of gravity and the gauge forces. The idea then lies dorman through the middle of the 20th century, only to be revived much later when meanwhile much of the particle zoo of the standard model had been found. The basic idea of KK-theory hadn’t changed through these decades, only (some) physicist’s feeling about it had.
Ah, that’s where #98 went. I thought it had been lost, so rewrote it for the other thread.