Friday, November 26, 2004

Category theory and physics

John Baez's "This Week's Finds in Mathematical Physics (Week 209)" is about one of John's favorite topics, namely category theory.

Incidentally, the previous week 208 was discussed here.

You know, category theory is the most abstract and "universal" part of rigorous mathematics, and it is also known as the (generalized) abstract nonsense. Actually the supporters of category theory don't view this name as an insult but rather as a cool compliment. Many of us have had a period in which the mathematical rigor was almost irresistable.

Category theory involves various theorems about sets, categories, functors, homomorphisms, and other abstract objects. Instead of saying "we perform an analogous procedure with a different kind of theory", category theorists give you a framework that translates vague words such as "analogous" into a mathematically rigorous structure.

This approach is meant to unify all of mathematics and all of rational thinking. You would think that the result must be very deep and hard to understand. But the actual results and tools of category theory are often closer to ordinary diagrams (combinatorial graphs) with arrows in between the objects - similar diagrams like those used by the people from the commercial sector in their PowerPoint presentations.

The mathematically oriented part of the string theory community is mostly excited by category theory. The so-called "derived categories (of coherent sheaves)" have been proposed as a generalization of K-theory to describe D-branes of various dimensions. Unlike K-theory that only knows about the allowed conserved charges of D-branes in various backgrounds, category theory should also know - and perhaps knows - something about their dynamics, for example the points in the moduli space at which the D-branes become (un)stable or (un)bound.

If you want to see some deep papers in this direction, see e.g.

The role of category theory in physics can therefore be described as a "progressive direction" within string theory. That's why at sci.physics.research, Aaron Bergman and Urs Schreiber replied to John Baez's week 209 that "he should be careful because he is secretly starting to work on string theory". ;-)


Nevertheless I think it is still fair to say that the dominant, physically oriented part of the string community is skeptical about the importance of the methods of category theory in physics - and, of course, for most of us the reason is simply that we don't understand where this approach came from. I've asked the same elementary questions to many people who've been trying to explain me derived categories - some of them with some success, most of them with no success whatsoever:
  • Are these notions and statements of category theory something that you can prove - or at least check in many situations - to be valid for string theory as we know it, or is it just an unproven conjecture that derived categories describe D-branes?
  • Or is category theory just a different language to describe the same physical insights that we know anyway, using other means?
  • Or is this categorical approach meant to redefine string theory? How do you prove that such a "new" string theory makes any sense?
The answers I've received were always a bit mixed. Some of the friends of category theory replied that category theory was just a different language to describe physics that we knew anyway - but "not quite". Such uncertain messages potentially make no sense, and it is hard to understand their meaning. In physics, we propose different conjectures about the real world, and it is important that we're not guaranteed that these conjectures will be true.

Probably the most serious conjecture of current high-energy theoretical physics is that string theory, once understood completely, describes the whole Universe around us exactly. Again, it's important that it is just a conjecture, albeit a very attractive one and one with a lot of circumstantial evidence. String theory itself is not just a conjecture, but rather a seemingly consistent mathematical framework.

Once we accept string theory as an objectively existing mathematical structure, a structure that we treat as a part of "generalized physics" - which is of course what all string theorists are doing every day - we can ask a lot of questions about its properties. But it's important that even in this case, the statements about string theory are just conjectures, and they need to be proved or supported by evidence, otherwise they're irrelevant and "wrong", in the physical sense.

I always feel very uneasy if the mathematically oriented people present their conjectures about physics, quantum gravity, or string theory as some sort of "obvious facts". One of the fascinating features of string theory is that its objects and investigations, even though they've been partially disconnected from the daily exchanges with the experimentalists, remained extremely physical in character. All of the objects that we deal with are analogous to some objects in well-known working physical theories, to say the least.

You may imagine that after 20 years and 15,000 papers, theoretical physicists who don't interact with the new experiments too much will end up with a completely abstract structure that does not resemble observable physics too much. It's not the case of string theory - which is a deep framework to unify all good features and facts about gauge theory, gravity, elementary particles, Higgs mechanism, confinement, and so forth. String theory is the most conservative among all the extensions of quantum field theories that go beyond the standard framework of QFT.

Well, category theory is exactly the type of science that would invalidate the previous paragraph.

Don't get me wrong: I do believe that string theory will eventually be formulated with very abstract rules - rules that will imply the features of the well-known systems and backgrounds with a geometric interpretation as an important special case. But I just don't see that this particular approach, diagrams with arrows from category theory, are getting closer to the right answer. If something is abstract, it does not have to be correct, and this particular thing just does not smell correct.

What I can imagine is that some insights of category theory will be useful to understand some things about D-branes and M5-branes - categories are possibly relevant for the "gerbes" - but it remains an open question whether category theory as we know it today will ever become a useful language for physics. No doubt, the mathematicians among us will try to impose their culture on the rest of the community. But I feel that this is the influence that would move the string theorists away from physics.

It's great if some people study string theory because of purely mathematical motivations, and I don't want to question the meaning of mathematics. What I want to question is the self-confidence of many mathematicians with which they want to use their mathematical ideas directly in physics. String theory is first of all a physical theory, and it should be studied because of physical motivations - the primary physical motivation is to locate the right ideas and equations that describe the real world.

Category theory has been used by many to achieve completely wrong physical conclusions - for example, by considering the "pompously foolish" quantization functor, many people have claimed that everything that happens in a classical theory has a counterpart in the "corresponding" quantum theory. This is just a lousy idea that we know to be wrong because of hundreds of different reasons - yet, the people who have a religious respect to the diagrams in category theory view this incorrect map as a "self-obvious dogma". Incidentally, this dogma is one of the reasons that makes the category theory community and the loop quantum gravity community correlated. It may be nice to be rigorous, but it's always more important to be correct: if the specific kind of rigor leads us to stupid conclusions in physics, we should avoid it.

Categories and "continuous vs. discrete"

Urs Schreiber on sci.physics.research describes "categorization" as "stringification" - which is the step from "quantum mechanics" to "conformal field theory". In other words, "categorization" is nothing else that adding dimensions to your auxiliary "worldvolume". Well, adding dimensions is not such a mysterious thing. Moreover, we know that sometimes it can be done and sometimes it cannot. Even if it can be done, some things are preserved, but others are completely changed.

Category theory often resembles linguistics (or even postmodern literary criticism): it is a science about arrows between different objects and about creating new objects from these arrows, but it does not really care too much whether the objects exist and what are their real properties. Although "categorization" is making things "more continuous" (e.g. by adding dimensions; making quantization), the actual diagrams that category theorists eventually talk about are extremely discrete objects.

If a complete discrete diagram tries to encode an extremely continuous, infinite-dimensional object, we should always carefully ask whether we know that the non-trivial, infinite-dimensional, highly continuous object really exists. Its existence is never guaranteed and drawing simple-minded diagrams from otherwise "rigorous and deep" category theory can't help us to decide about the "hard questions" associated with quantum field theory, string theory, or any other objects that require some sort of "infinite-dimensional continuous" structure. If we create physical theories as complex as quantum field theory or more, the difficult part is always the task to make the highly continuous, infinite-dimensional objects and expressions meaningful. On the other hand, one can invent zillions of different discrete rules - such as the rules for n-categories - and it is simply not guaranteed that a set of combinatorial, discrete rules will be relevant for anything in physics (which includes string theory).

In my opinion - and not only mine - it is important for all of us to realize that we have not yet solved the fundamental physical goal of string theory - to predict and confirm new phenomena and facts about the real world. And category theory does not seem to help us in this goal, I think. Do you disagree?


  1. Don't let your dislike of individuals cloud your judgement about the subjects they work on.

  2. Thanks for your advice. Incidentally, I like the individuals who believe(d) it (those in string theory) - Paul Aspinwall, Dave Morrison, Albion Lawrence etc. - this is a purely scientific question. Ockham's razor is clearly much more important for me than for other people. Scientific terms and structures should not be reproduced unless it's necessary.

  3. Category theory may have little or nothing to do with physics as yet, but surely it's too early to predict its future? Even if it looks like abstract nonsense now, it may turn out to be quite useful. Think of differential geometry. Or the zeta function.

    Anyway, looking at the discussion over at the string coffee table and the papers they discuss, it looks like categories may even be useful for string theory, in more ways than one. Looks like one way of generalizing the notion of `symmetry groups for particles' is to have `symmetry categories for strings'.

    It is true that most of the rigor in mathematics turns out to be quite unnecesary for physical applications. But that doesn't make the math irrelevant.


  4. Lubos said:

    What I can imagine is that some insights of category theory will be useful to understand some things about D-branes and M5-branes - categories are possibly relevant for the "gerbes" - but it remains an open question whether category theory as we know it today will ever become a useful language for physics.

    I say:

    On a related note, are you convinced that "gerbes" are relevant for a physical theory? Frankly, I am not. It is a very interesting question what the effective world volume theory is that lives on M5 branes when M2s end on them. The M2 ends are self-dual strings which become massless in the limit where the M5 branes between which they are stretched become coincident. One can then imagine that this gives you something one might want to call "gauge strings", massless strings that tranform in a gauge group very much analogous to YM theory constructed from branes.

    Now, many people have tried to promote Wilson lines to "Wilson surfaces" to understand how to construct an effective theory of such gauge strings. I think the attempt is worthwhile, but the conceptual problems (no sense of ordering on a surface, etc.) with this straightforward approach are so massive that in hindsight I would call it naive and failed. We should all realize: There are things we simply do not know how to do. We cannot write down the action for self dual gauge fields. We cannot write down a gauge theory with a global symmetry described by an exceptional group. (Such theories are known implicitly to exist from brane constructions -- analogous to our knowledge that self-dual strings live on M5 branes).

    I like to see it this way: The implicit knowledge that theories exist for which we can't write down the Lagrangian is evidence that we lack some technology to fully understand all aspects of string theory. In the case of self-dual gauge fields we have a nice patch -- we supplement some action by a physical state condition and everything is nice and dandy. Maybe people should try to understand how to do something along these lines for theories with global exceptional symmetries and hopefully learn something about how to approach the much harder problem of what I called "gauge strings"?

    The reason I write all this is that it is my perspective on Lubos' criticism of too abstract procedures -- with which I very much agree. Progress in these areas is more likely to come from physical insight than from abstract mathematics. Once understood it will be nice to construct the rigorous mathematical framework for such new theories and see if we could have gotten it directly this way. ;))


    PS: Lubos, did you mean to say in your comment that Paul, Dave and Albion believe in category theory? I don't think that's true. All of them are darn good physicists...

  5. Hey Michael! Thanks for your interesting and sophisticated ideas. Yes, I agree with your assessment of the gerbes and the straightforward ways to extend Wilson lines to surfaces, as well some other points - like the sentence that the progress will come from physical intuition.

    By saying that Paul, Dave, and Albion believe in category theory, I meant that they not only "believe" :-), but they have written advanced papers where they argue that derived categories etc. are essential for the right description of D-branes etc. Two of them are mentioned in the updated version of this article (reload). Dave Morrison, when he was here weeks ago, was very positive about the approach, calling it "absolutely correct", or something like that ;-)

    Cheers, Lubos

  6. OK, I see. But these papers, if very technical, are serious, are they not? Also, jusged by citations they contain new important ideas (~100 and ~50 citations, respectively). They may not be the most ground breaking papers, but I think it's unfair to even compare them to the loop quantum gravity bla bla (that usually gets 0 citations). Also, once again, look at the accomplishments of the authors. They *are* good!

    Thanks, Michael

  7. OK, I see. But these papers, if very technical, are serious, are they not? Also, jusged by citations they contain new important ideas (~100 and ~50 citations, respectively). They may not be the most ground breaking papers, but I think it's unfair to even compare them to the loop quantum gravity bla bla (that usually gets 0 citations). Also, once again, look at the accomplishments of the authors. They *are* good!

    Thanks, Michael

  8. They are definitely good! I did not want to compare it with loop quantum gravity, of course! ;-) It was just a comment on whether this formalism should become an unavoidable part of our toolkit.

  9. Hi Lubos and Michael -

    I knew that Lubos was going to be sceptical of the 'categorification is stringification' claim
    ( ;-)

    Let me just add a couple of coments:

    Regarding Michaels comment, it is not true that we don't know how to construct a sensible Wilson surface. The ordering problem can be dealt with in a systematical way, and it is among other things category theoretic reasoning which guides one what this way should be. Of course you can choose other tools than categories to arrive there, if you like, like loop space differential geometry.

    I would like to emphasize that the idea of categories is one very close to the heart of physics. It's all about gauge and duality equivalence, if you wish. Categories are there to make the notion of natural transformation work, which is nothing but the most general way so formulate a similarity transformation:

    So in physics we know that two things are equal if they only differ by a unitary transformation

    F = U G U^+

    (a gauge transformaiton)

    or more generally by a similarity transformation

    F = A G A^-1.

    In the cases where A is not invertible this still makes sense if we write it as

    F A = A G,

    called an intertwining relation.

    Since F and G here are hence to ways to look at the same map, we should in general think of them as images of a given third map h

    F(h) A = A G(h).

    This is the most general useful way to say that F(h) and G(h) are 'the same up to generalized gauge
    transformation'. And this is nothing but the definition of a natural transformation between functors F and G.

    Category theory is nothing but a notational framework to make this general notion of 'similarity' work.

    And there is a nice 'physical' way to look at this, too: Two images of a given point can either coincide or not. But two images of a given arrow/string can be 'congruent' without being 'equal'. There can be a map A translating one image to the other such that they coincide.

    This is simple but useful. For instance regarding strings as morphisms and an association of categorified group obects to them as their 'image' in that group in a theory of 'gauge strings' one finds that the above mentioned notion of natural transformations describes precisely how gauge transformation in such a 'gauge string' theory generalize with respect to the ordinary notion - it gives you the cocycle description of a nonabelian gerbe with only a couple of lines work. That's nice I think.

    Finally, regarding Luboš's concern that categories are somehow 'discrete' because there are arrows between objects: They can be perfectly continuous, both the object and the morphism spaces can be continuous, smooth, even manifolds if you like. The simplest example is path space, the configuration space of an open string. It can be regarded as a category whose arrows are the strings over the objects which are the points of spacetime. Categorifying superQM in that category gives you the super CFTs of the string. That's nice I think.

  10. Hi Urs! On the contrary, I am enthusiastic about your explanation that categorification is stringification! :-) It's a nice confirmation that these things are not much more than a fancy renaming of the concepts that we know.

    Concerning the continuity of categories. I understand that the categories can represent very continuous objects. But that's not enough. The tools used to show that a given category exists or not are very formal and discrete - not continuously calculational - and therefore these formal proofs don't mean much.

    You know, if we talk about the existence of a physical theory or a physical object, we never mean some abstract "existence theorems" based on some subtle details of set theory. We're talking about a very specific calculational framework that is formally based on infinite-dimensional objects, but they are such that they admit a calculation of physically interpretable finite numerical answers to well-defined questions.

    This is by no means guaranteed by the simple arrow-graphs used to argue in category theory. The games with the arrow have eventually very little in common with the actual physical questions.

  11. Hi Urs!

    Concerning the generalized similarity relations - I will probably never understand this kind of thinking - it is much like groupoids, monoids, and their sibblings.

    If two systems of operators are equivalent in physics, there must exist an honest similarity transformation - and if the scalar products in both groups were already chosen and chosen identically, then the similarity map must be unitary.

    I don't understand why you think that (or, in what sense) generalizing the similarity map to the non-invertible case with the f(h) F = G g(h) or whatever you wrote is "useful". Could you be more specific what is it useful for?

    One can generalize things in various ways, and weaken various conditions - but in physics, we usually need most of these conditions for the things to be consistent. In gauge theory, we need a gauge group, not a gauge semigroup or gauge groupoid, and similarity transformations must be honest, and not rigged.

    I understand that in mathematics one can make weaker assumptions to prove *some* theorems because not all the structure is necessary. But I totally agree with Feynman's comment about exactly this question that it is a completely irrelevant task to try to be efficient in this mathematical sense - it never hurts if you say an equivalent thing twice, or if you make an assumption that is not necessary for a specific problem, but it is necessary for the most general problem.

    For the most general problem in physics, things must be "full", and crippling the objects in physics - unitarity equivalence, group, etc. - in various ways - groupoid, generalized categorial equivalence, monoid, semigroup - does not seem to be too useful, as far as I can say.


  12. Hi -

    finally I have some leisure to chat on the web again: I have now posted a reply to your comments to the String Coffee Table: href="

  13. I think you misunderstand category theory (CT) if you think its diagrams are like those found in powerpoint presentations. Unlike most of mathematics, the diagrams of CT are used in the very act of reasoning, rather than being mere illustrations of reasoning. In this, CT diagrams are like the geometrical diagrams of Euclid, and Euclidean geometry is the only other major branch of mathematics to use diagrams in this way.

  14. To support my comment on the role of diagrams in category theory as reasoning engines, I quote some statements from a standard CT text:

    M. Barr and C. Wells [1999]: Category Theory for Computing Science. Montreal: Les Publications CRM, 3rd edition.

    "When the target graph of a diagram is the underlying graph of a category some new possibilities arise, in particular the concept of commutative diagram, which is the categorist's way of expressing equations." (page 93)

    "This point of view provides a pictorial proof that the composite of two graph homomorphisms is a graph homomorphism. . . . . . The verification process just described is called "chasing the diagram". Of course, one can verify the required fact by writing the equations (4.14) and (4.15) down, but these equations hide the source and target information given in Diagram (4.13) and thus provide a possibility of writing an impossible composite down. For many people, Diagram (4.13) is much easier to remember than equations (4.14) and (4.15). However, diagrams are more than informal aids; they are formally-defined mathematical objects just like automata and categories." (page 96)