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    • CommentRowNumber1.
    • CommentAuthorsure
    • CommentTimeMar 22nd 2014
    • (edited Mar 22nd 2014)

    I'm a physics student who is really interested about foundational questions in physics. Obviously, studying foundations requires some philosophical (if not metaphysical) motivations and beliefs. I believe that physics is mainly algebraic (in the sense that it is relational: the object doesn't matter, only its relations with what surrounds it does), and that it is both non realist (quantum mechanics says so if you believe in locality) and requires the presence of an observer to make sense (relativity + QM). This is why, I started to study category theory. Yet, while studying the rudiments of CT through awodey's book, I noticed that some people (Urs Schreiber for example) were also studying foundational issues through higher category theory.

    I read the motivations of Urs Schreiber's big paper "Differential cohomology in a cohesive topos" in this respect, and even though I also think that the axiomatization of physics is a necessity, I'm not at all convinced that such approach fits what "physics" is. Let me explain.

    There is a huge difference between the questions a physicist is trying to answer and the ones a mathematican does. A physicist is a guy who is trying to build a theory that is, in some sense, a world view (a philosophical/metaphysical interpretation of the world, why the latter behaves as it does). A theory is an implicit framework, in which some explicit models are then incorporated. The goal of models is to make some (quantitative) predictions about some "real" (and non universal) physical phenomenon. The point is, in physics, a theory is plainly understood only when it is constructed from first principles. These principles must be simple, metaphysical, and expressible in natural language.

    Physics has basically three theoretical framework:
    1) general relativity,
    2) quantum mechanics,
    3) statistical mechanics.

    General relativity is totally understood from a physical perspective: it is build from purely metaphysical principles (strong equivalence principle, Galileo principle of relativity, and locality: there exists a maximum speed for information). Starting from those first principles, it is relatively easy to guess Einstein field equations by some though experiments.
    Quantum mechanics is not well understood yet: it is by no mean constructed from first principles, and this is why it lacks a metaphysical interpretation (ie, it lacks what physics is made for). Statistical mechanics is well constructed and well interpreted, even though there are some mathematical problems with its formulation.

    The point is, even though I totally agree that a mathematical and rigorous reformulation of well understood and motivated physical theories allows to go deeper in its understanding, the converse is totally false. Reformulating in a rigorous and deeper way the mathematical structures hidden in a non well understood physical theory (from a physical perspective) won't make you understand the previous theory better (again from a physical perspective). Let me give an example.

    Quantum field theory is a successful phenomenological framework that allows to predict many non trivial phenomenon with high precision. Still, predicting is not explaining.
    QFT lack the physical construction from first philosophical principles. It is not at all well understood, and there are non trivial problems to make sense of it (cf, the deep question about what is a field or a particle, considering that those depends on the type of frame you live in, the inequivalence of representations, ...).
    Yet, gauge theories were formulated, and reformulated in many deep and rigorous ways from many people through time. The same happens for all the quantization scheme (geometric/deformation) that are good explanation of the mathematical structures behind those theories. Still, can one claim that such reformulation helped our understanding of what is the meaning of quantum mechanics or field theory? Not at all ! From a physical perspective, those reformulation are totally useless.

    It really gives me the feeling that mathematical (re)formulations of physical theories won't help you to understand them better from a physical perspective (if, at first, they are not well understood).
    So, the question is simple. Why should I study the work of Urs and other guys if my aim is to understand the meaning of QM? How could such complicated mathematical structures
    help me in this journey? Moreover, don't you think that a physical breakthrough is, before all, coming from a revolutionary change of paradigm? A new paradigm, that as shown by history, always trivialized all the mathematical complications one had to deal with. Why should I therefore believe that going that far in the study of the structure of the current interpretation and formulation of QM and field theory worthes it? Wouldn't, all this work, be proven useless from a physical perspective, once a deeper meaning of what QM or field theory is eventually found?

    PS: The questions I'm asking are serious, I'm not just freely provoking.
    PPS: Couldn't HoTT plays a deeper role in the understanding of physics?
    PPS2: I believe that category theory is deep and should be used in physics, but why higher category theory and all this complicated stuff?
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2014
    • (edited Mar 23rd 2014)

    While the claim of an axiomatics is making the bigger public impact, this is not what drives the development.

    What drives the development is understanding interacting full non-perturbative quantum field theory, problems of the kind that the Clay Millenium Problem asks for.

    Doing so requires dealing with global quantum anomaly cancellation. Deep problems of global quantum anomaly cancellation in low dimensional quantum field theory made Freed, Hopkins and others realize that (higher) gauge fields have to be described as cocycles in differential cohomology. But there remained open technical issues.

    Based on my PhD thesis work I had made the rather natural observation that a proper formulation of differential cohomology takes place in higher topos theory. A few years later, in his 2011 talk at the Simons center, Hopkins said that this is how his seminal article on differential cohomology with Singer “should have been written”. When applying this to open problems in field theory it dawned on me that actually there is a simple axiomatics in a higher topos which gives the theory of differential cohomology (thanks to hints from Richard Williamson, who is back on the nForum these days). By the way, regarding your PPS this is formalized in HoTT.

    In fact the axioms are better than I even realized. Recently Bunke-Nikolaus-Völkl observed that where I wrote “differential cohomology in a cohesive \infty-topos” one is entitled to write “differential cohomology IS cohesive \infty-topos theory”. Using this they now solved some remaining open technical issues in the quantum anomaly cancellation of the type II superstring (still unpublished, but let know if you want to hear details), following Distler-Freed-Moore.

    Regarding higher category theory in physics: it is a fact that the locality of nn-dimensional field theory is incarnated in monoidal nn-category theory. That’s how it comes about.

    Regarding the mysteries of quantization: what I have to offer on these is here.

    • CommentRowNumber3.
    • CommentAuthorsure
    • CommentTimeMar 23rd 2014
    Thanks for your fast answer. So basically, the questions you're asking yourself with such approaches are more mathematical questions rather than physical ones.
    Understanding the mathematical foundations of field theory/QM is clearly an interesting thing, but I'm afraid that I don't have the mathematical skill/knowledge required in order to see what this could bring to understand (from a physical perspective) the physical theories. Considering that there is not yet a "good" interpretation and formulation of QM/Field theory from first principles, don't you think that the mathematical understanding of the hidden structures in the present formulations of the latter theories won't help in their physical understanding? Again, it appears that the mathematical complexity involved in the formulation of physical theories always resulted from a badly understood paradigm (think about epicycles theory or aether one).

    PS: Don't make me say what I didn't. I believe that it is really important to explore the mathematical structures of a physical theory, but only when the latter is indeed well founded from a physical perspective. I really have the feeling that all this work is just some way to discover new mathematics rather than new physics. Surely, this is interesting in its own right, but as a physicist, I don't understand the motivations and it really looks like it doesn't answers any physics questions.
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2014

    I am not sure if we are communicating. Above I highlighted how this is all about understanding physics questions.

    • CommentRowNumber5.
    • CommentAuthorZhen Lin
    • CommentTimeMar 23rd 2014

    Perhaps the problem is in the phrase “physics questions”. I spoke to some colleagues in the fluids research group a while ago and got the impression that they didn’t think the Navier–Stokes millennium problem was relevant to their work – they regarded it as a question in PDE theory.

    • CommentRowNumber6.
    • CommentAuthorsure
    • CommentTimeMar 23rd 2014
    No its not. The process of quantization is not a physics question at the moment, it is a mathematical question.
    A physics question deals with "why is the world behaving as it is?" and is asking you to find the underlying philosophical principles that would eventually allows you to build a meaningful theory. From this theory, which is nothing else than a "point of view" on the world we're living in, some canonical models should be able to predicts basic non trivial phenomenon.

    To predict is not to explain. The epicycle model, based on the geocentric theory of the universe (the earth is at the center of the universe), is totally capable of predicting the orbits of the planets and stars. Yet, the ways to build models inside the geocentric theory lack many constraints and justifications in order to be "canonical". There's nothing that explains why you should take circles instead of ellipses or parabola for example.
    The good point of view, or at least, a deeper one, is obviously based on the three Newton's laws:
    - Galileo principle of inertia: there's no privileged position in space, nor privileged frame, even though there are two classes of them (physics doesn't give a fuck about speed).
    - Second law, that tells you that acceleration is an absolute and is what makes the difference between "nothing happens" and "there's something non trivial happening".
    - Third law, that is simply conservation of impulsion.
    From this theory, one can then justify a canonical model for gravitation: the so called universal law (K/r^2). Based on this simple law, which is easily explained and justified from symmetry arguments (equivalent to conservation laws thanks to Noether's theorem), one can easily compute the different form of orbits of the planets. A single model, that is based on one justified formula is enough to predict everything in the sky. That's what a canonical model is, and this is why Newtonian mechanics is way deeper than the geocentric one, or Aristotle's F = mv one. The Newtonian theory is, I insist, a point of view on the whole universe, and we have many good reasons to believe it is a meaningful one because of the presence of such powerful predictions from so little.

    Now, if you were a mathematician back in the times, it would have certainly been really interesting, from a mathematical perspective, to study the hidden structures in the geocentric theory (principally based on the epicycle models). Surely, there have many interesting geometric things inside, and a rigorous (and complicated) formulation of it could have led to big mathematical breakthroughs. Yet, how could such study helped to find Newton's laws? From a physical perspective, all this work would have been totally useless.

    Let us now speak about quantum field theory, by keeping in our head what is a well founded physical theory, and what is a non well founded one.
    QFT is constructed "on" QM. Although, there are some "canonical" models in QM that lead to good predictions (think about the hydrogen atom, Young's double-slit experiment or the harmonic oscillator), QM is at the moment not well understood: there is no construction of it from first principles. For QFT, its even worse. We have the principle of gauge invariance to tell us "how" to build models. Right, fair enough, but we still need to chose one gauge group. Which one to choose? Why? Nobody knows: there are no justifications to take the usual gauge groups people takes. This lack of constraints is a lack in "canonicalicity", and is a clue that the underlying theory might not be the good one (ie, QFT is not satisfying from a physical perspective, which is also understood by many people who knows all the problem coming from the inequivalence of representations, ...).

    Moreover, there are two examples in the history of physics that perfectly shows that predicting is not explaining. The modified aether theory is perfectly able to predict everything special relativity does. There's no way to discriminate it from special relativity. Why did physicist choose special relativity, then? Simply because its explaining power is much deeper, and because it is perfectly constructed from first principles. Also, the computations of the relativistic phenomenon by special relativity are expressed with much less mathematical complexity than they are in the aether theory. Note that there is also no way to discriminate the usual interpretation of QM, with Bohm's pilote waves theory. It is also interesting to remark that physicists didn't necessarily do the right choice of theory in history. Let me just cite how Einstein-Cartan theory (torsion) is being ignored by most physicists, while it might be a better interpretation of gravity than general relativity (curvature).

    To conclude, I hope that you understand that, as a physicist, I have no reason to believe that revealing the hidden mathematical structures of a non well founded physical theory (as geocentric, aether, or QFT) is useful in order to understand a better "point of view" of the theory. Moreover, the mathematical complexity is, as shown by the geocentric or aether example, a consequence of the badly understood physics. I have no reason to believe that the mathematical structures involved in physics are that complicated, considering that when the phenomenon were eventually well interpreted (Newtonian mechanics, theory of relativity, ...) this complexity vanished instantly. Finally, let me recall once again that the lack of constraints and justifications in models, is to me a clue that the underlying theory is not at all well understood.

    So let me ask my question once again: do you think that your work can lead to physics breakthrough?
    • CommentRowNumber7.
    • CommentAuthorigor
    • CommentTimeMar 23rd 2014

    @sure, let me weigh in, as another physicist. It seems to me that you have already answered your question, in the negative. From what I understand, your view (which I do not necessarily agree with) is that QM and QFT are not “well founded” and are likely to be replaced with something different, just like the geocentric model was replaced by the heliocentric one. Your question then sounds to me as follows: can mathematical investigations into the structure of QFT help one find this eventual replacement? Of course, nobody can answer that with a certain Yes or No, simply because nobody knows whether any such replacement exists and, if it does, how it is to be found. However, most active mathematical investigations of QFT, including the directions that Urs alluded to earlier, do not have the overthrow of QFT as a primary goal.

    So, if you are indeed focused on finding alternatives to QFT, then you’d have to look elsewhere for answers. That is, unless, you’d be satisfied by stumbling onto such answers by accident. But in that case, of course, it won’t really matter what you choose to study.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2014
    • (edited Mar 23rd 2014)

    re the last question in#6:

    Yes. Of course to discuss this you need to recognize a modern fundamental physics breakthrough. Your emphasis of Newton’s laws above reminds me of my theoretical physics profs back then who would teach us classical mechanics from Goldstein’s book and not know about symplectic geometry. They were still leaders in their respective fields and would not see the need to modern more fine-grained insights into the laws of physics.

    But when it comes to fundamental high energy physics there has been some progress in the centuries since, and the modern questions range deeper. That one needs the language of mathematics to talk about them, and not philosophical principles as you suggest, is an empirical fact of life which you won’t have much luck with fighting against.

    The big open question in modern fundamental physics is the UV-completion of the standard model plus gravity. (Any remaining doubts about this were disspelled just a few days back by bicep2.) The main contender here is string theory. Building a model here means building a string vacuum, which in turn means to build a 2d SCFT of central charge 15, well defined on all genera.

    The subtlety of accomplishing this is widely underappreciated in the physics community, due to the wide-spread lure of those philosophical principles that you appeal to.

    I know the kind of attitude well which you seem to be displaying. Back when I was in Hamburg and my boss would speak to the local string theorists about his classification of rational 2d CFT they (some) would yawn and suggest that they knew all this 20 years back. But they didn’t, they just followed the custom in theoretical physics to declare any widely held believe after a few years a fact. The actual fact is that the full classification revealed that some of the modular invariant CFTs in the literature extended in more than one way to a full 2d CFT, while other’s didn’t at all. Given that each of these is a potential part of a stringy UV completion of the standard model, this matters.

    Now most string vavua are built not even algebraically like this but geometrically, via sigma-models. This is even more dangerous. Sometimes it is good to remember that nobody has much of an idea if the notorious string vacua landscape actually exists the way commonly thought, since very little is known for sure about which of its conjectured points actually exist as full CFTs.

    The reason that this is subtle is that there are, as I mentioned above, famous and famously subtle global quantum anomaly cancallation conditions obstructing the existing of what naively may look like a consistent superstring background. Staggeringly, an argument for what actually the full consistency condition on a type II background really is was advanced only in 2009, by Distler-Freed-Moore.

    (I said this and the following already in my first message, but I suppose the point was lost. I urge you to actually look at this material if you are really interested in a discussion. See their very last remark, for emphasis.)

    Now this quantum anomaly cancellation requires the string’s background gauge fields to be cocycles in certain twisted versions of differential cohomology. But it wasn’t entirely clear what the right concept of twisted differential cohomology here really is. This missing ingredient was recently provided by Bunke-Nikolaus-Völkl, based on axiomatic cohesion.

    One can actually nicely motivate the axioms of cohesion from deep philosophical principles, which you say physics is about. But once one has the axioms formulated mathematically, one is to study the mathematics to find out what a consistent superstring background is, which then has a chance of providing a consisten UV completion of the standard model plus gravity.

    There are more stories like this. Maybe just very briefly: a big discussion in fundamental physics these days concerns the quantum gravitational behaviour of black holes. These arguments stack vague ideas on top of conjectures on top of vague ideas based on the AdS/CFT duality conjecture. At the same time almost nothing precisely is known about how holography actually works in this case. Now, using the extended local QFTs (the ones I mentioned require higher category theory for their description) one may precisely capture at least the higher CS/WZW variants of holographys and study these. Quite a few of the widely held claims about holography turn out to at least receive qualification in these precisely understood examples. I hope it is clear that by running this backwards it has direct impact on those current questions of black hole physics.

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeMar 29th 2014

    The epicycle model, based on the geocentric theory of the universe (the earth is at the center of the universe), is totally capable of predicting the orbits of the planets and stars.

    I don't agree with this. Far from being totally capable of these predictions, any actually existing epicycle model could only predict orbits up to the level of precision of the observations that went into that model. The Newtonian theory, besides having all of the good points that you identify, is also superior on the purely phenomenological level of making predictions (and this much is already true with Kepler). There were few refinements to Newtonian models before the advent of Einstein's theory, and each one led to the successful prediction of the existence of a physical object that could also be observed in completely different ways (such as by sighting the reflection of sunlight off of it). Refinements to epicycle models did not do this.

    The modified aether theory is perfectly able to predict everything special relativity does. There’s no way to discriminate it from special relativity.

    This I agree with. But here, the superiority of special relativity is perfectly clear from a purely mathematical standpoint. The aether theory features a quantity, the reference frame of the aether, that is necessarily unobservable; special relativity lacks this and is much more elegant. Sure, it is superior for the other reasons that you mention, but mathematics alone can also identify it as better.

    Surely, [epicycle models] have many interesting geometric things inside, and a rigorous (and complicated) formulation of it could have led to big mathematical breakthroughs. Yet, how could such study helped to find Newton’s laws? From a physical perspective, all this work would have been totally useless.

    Epicycles are of no interest mathematically; they are completely ad hoc. (Similarly, the precise layout of the planets and large nearby planetoids in our solar system, which refine Newtonian models, are of no interest mathematically; of course, Kepler tried to relate them to the Platonic solids, but this didn't pan out.)

    But now imagine a world without Copernicus and Kepler, but with Brahe and Galileo; in this world, Brahe uses his own data to develop highly sophisticated geocentric models, and Galileo discovers the mathematical regularity underlying these models. Only at this point, where we have a law to predict more and more precise epicycles, can a geocentric model actually match the predictive capabilities of our world's heliocentric models. But at this point, we can also observe the superiority of the heliocentric models from a purely mathematical perspective. We only await the stroke of insight (perhaps due to Galileo himself) that suggests this mathematical simplification, and the history of science rejoins our world. (Newton follows, and so on.)

    This never really happened. Even with special relativity, this is not how it happened (despite the work of people like Poincaré who arguably could have done it this way). But with general relativity, it did —not by Einstein but by Hilbert, at about the same time. So it can be done.

    I don't think that anybody is happy with QFT, or even string theory, as we have it now. But research into its mathematical structure could well lead to the discovery of the same sort of insight. Arguably, it already has, but we have to work out the consequences of this new mathematical framework to be sure that it's right.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMar 29th 2014
    • (edited Mar 29th 2014)

    Thanks, Toby. I didn’t react to the “all this math looks epicyclically complicated” in the above, since I thought we should first try to clarify what it does, before we enter discussion of whether it’s beautiful.

    Of course as regulars here are all too tired to hear me repeat, part of the point I like to advertrize is that the math is actually shockingly elegant, simple, and far less epicyclic than traditional formulations.

    It often happens that people mistake “simple/easy/elegant” for “familiar” and “complicated/epicyclic” for “unfamiliar”. I once illustrated it this way: most people on the street will think they perfectly understand how gravity works while they think the inner working of a nuclear reactor is complicated. But what they really mean is that they are very familiar with stuff dropping to the ground and very unfamiliar with nuclear dynamics; while on absolute grounds, scientifically, fundamentally understanding nuclear dynamics turns out to be actually simpler than fundamentally understanding gravity.

    Something like this is happening here: the derivation of classical field theory and at least good bits of quantum field theory from just homotopy type theory with some axioms added is fundamentally much simpler than anything traditionally done, but of course it looks more unfamiliar at the moment.

    • CommentRowNumber11.
    • CommentAuthorsure
    • CommentTimeMar 30th 2014
    • (edited Mar 30th 2014)
    Toby: I would disagree that Hilbert found general relativity alone. Basically, the hardest part to find general relativity is to understand the strong equivalence principle, and then that acceleration can be seen as curvature of space-time (cf Einstein's disk/Ehrenfest paradox). Once this interpretation of the world has been found, it is, as I said, relatively easy to find Einstein field equations if you're familiar with differential geometry (which allow you to code that simply by adding "generalized" Galileo principle of relativity, ie, diffeomorphism invariance). Moreover, as far as I know, Hilbert himself attributed his success to Einstein, and never claimed to been the founder of general relativity.

    This is why, I don't think that a mathematical reformulation of a complicated and non well understood physical theory could lead to any physical breakthrough.

    Ur: Your approach might indeed by simpler and much more elegant from a mathematical perspective than usual QFT. Yet, you're not finding something else than QFT as an underlying structure, you're finding QFT again. In some sense, the interpretation of the world (ie, the physical theory) is closed under those mathematical reformulations. I really believe that physicists are rushing too much for 50 years. Quantum mechanics is still not well understood from a physical perspective, field theory is known to have many strange paradoxes (not in the sense of antinomy) that say a lot about how our perception of nature has to be changed.

    For example, it is well known that the very concept of particle is relative to the observer (rindler observer vs inertial) and doesn't necessarily makes sense! How could then one construct a theory based on the concept of particle? It is also well known that the same happens for fields. There are many "ontological" problems with field theory that strongly indicates that it might be better not to rush, but to stop and start thinking about what QM "really" is. So actually, I would totally agree with your argument about how people sometimes mistake familiarity with understanding. The point is, most physicists are familiar with quantum mechanics, but few are the ones who have deep insight into what it means. Still, they are the ones who rush into QFT, supersymmetric version of it, and eventually into string theory and so on. I believe that the previous foundational questions I mentioned are not going to be solved by a headlong rush. There are also two big foundational problems in general relativity that needs to be solved and that could really help our understanding of these previous issues: to understand clearly the difference between linear acceleration and spinning/rotating, and to understand the charge paradox (both of these problems are related to the fact that a physical object might not need to exist for everyone).

    Physics is in this respect totally different from mathematics. In pure maths, it is totally legitimate to "rush" and construct a high building with the same foundations (essentially ZFC + LK). This is so because high level abstraction as done in mathematics can always lead to new language, results, and encompass understanding of other fields.
    Yet, it is also legitimate to ask, even in pure maths, if ZFC + LK is indeed a good foundation to describe what mathematics "truly" is. Indeed, there are many problems in this axiomatic scheme: one could think about the field F_1, or about non commutative geometry. One could also bring all the computability problems, the fact that constructibility seems to be a better paradigm than "truth", or the ontological problems coming from ZF (everything being a set, even the number pi. ETCS seems to be a deeper foundation in this respect) and so on. This search for better/deeper foundations is of course, even in maths, not at all free from philosophical principles.

    The analogy with physics is evident. In physics, what we are trying to do is precisely to always find a deeper formulation/theory of the physical world, as a logician might try to find a deeper formulation of the mathematical world. Yet, most pure maths are totally useless in this respect. The same happens in physics: most physics stuff that is done in a given theoretical framework is useless in this respect. Even worse, a fundamental physicist is contrary to the pure mathematician, much more constrained by physical reality. There's no point constructing a 150 floors building with same foundations: this is the goal of engineering.
    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2014

    Hey “sure”,

    I still don’t see that we are communicating.

    Not sure how to usefully continue this exchange. If you do want some answer from me (maybe you don’t) I suggest to focus on one concrete aspect, for currently i see too many things being mixed up in too random a fashion.

    • CommentRowNumber13.
    • CommentAuthorsanath
    • CommentTimeApr 30th 2014