Is String Theory Testable?

I’ve been traveling in Italy for the past ten days, and gave talks in Rome and Pisa, on the topic “Is String Theory Testable?”. The slides from my talks are here (I’ll fix a few minor things about them in a few days when I’m back in New York, including adding credits to where some of the graphics were stolen from). It seemed to me that the talks went well, with fairly large audiences and good questions. In Pisa string theorist Massimo Porrati was there and made some extensive and quite reasonable comments afterwards, and this led to a bit of a discussion with some others in the audience.

I don’t think the points I was making in the talk were particularly controversial. It was an attempt to explain without too much editorializing the state of the effort to connect the idea of string-based unification of gravity and particle physics with the real world. This is something that has not worked out as people had hoped and I think it is important to acknowledge this and examine the reasons for it. In one part of the talk I go over a list of the many public claims made in recent years for some sort of “experimental tests” of string theory and explain what the problems with these are.

My conclusion, as you’d expect, is that string theory is not testable in any conventional scientific use of the term. The fundamental problem is that simple versions of the string theory unification idea, the ones often sold as “beautiful”, disagree with experiment for some basic reasons. Getting around these problems requires working with much more complicated versions, which have become so complicated that the framework becomes untestable as it can be made to agree with virtually anything one is likely to experimentally measure. This is a classic failure mode of a speculative framework: the rigid initial version doesn’t agree with experiment, making it less rigid to avoid this kills off its predictivity.

Some string theorists refuse to acknowledge that this is what has happened and that this has been a failure. Most I think just take the point of view that the structures uncovered are so rich that they are worth continuing to investigate despite this failure, especially given the lack of successful alternative ideas about unification of particle physics and gravity. Here we get into a very different kind of argument.

It was very interesting to talk to the particle physicists in Rome and Pisa. They are facing many of the same issues as elsewhere about what sort of research directions to support, with string theory often being pursued as an almost separate subject from the rest of particle theory, leading to conflict over resources and sometimes heated debates between them and the rest of the particle physics community. Many people were curious about how things were different in the US than in Europe, but I’m afraid I couldn’t enlighten them a great deal, mainly because I just don’t know as much about the European situation, although I’ve started to learn more about this on the trip. Several wondered if the phenomenon of theorists going to the press to make overhyped claims about string theory was an American phenomenon. I hadn’t really noticed this, but it does seem to be true. While the hype starts in the US, it does travel to Europe, with the US very influential in this aspect of culture as in many others. In the latest issue of the main Italian magazine about science, there’s an article explaining how certain US theorists have finally figured out how to test string theory with the new LHC…

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47 Responses to Is String Theory Testable?

  1. Levi says:

    “The fundamental problem is that simple versions of the string theory unification idea, the ones often sold as “beautiful”, disagree with experiment for some basic reasons.”

    This seems similar to the situation with Grand Unified Theories. I gather that SU(5) was the “beautiful” version, and when that version ran into problems much of the beauty went out of GUTs. It’s interesting to contrast this with cosmic inflation, where Guth’s original version didn’t quite work, but Linde and others found forms of inflation which worked better, and WMAP data gives a reality check.

    I should mention that I’m not a physicist, just a casual reader, so if I’m misinformed I hope somebody will point it out.

  2. Arun says:

    It would be nice to know what Porrati said, if at all possible.

  3. Joseph Smidt says:

    Great Post. I thought your comments on US/Europe string culture were interesting. Thanks for the slides.

  4. Irish physicist says:

    Off-topic – but congratulations on 3 years of blogging and Happy St. Patricks day too!

  5. woit says:

    Irish Physicist,

    Thanks! I hadn’t realized that the blog was started on a St. Patrick’s day. Surely some sort of homage to the Irish was unconsciously intended.

    Arun,

    I can’t recall exactly what Porrati’s points were, except that he said that he had five of them, and none of them were things that I really had a substantive disagreement with. Some of them were (from memory, and in loose translation, surely he would express these differently)

    1. String theory shouldn’t be thought of as a theory that leads to a unique, predictive model, but instead as a very general framework, like QFT, valuable for the different kinds of models it allows.

    2. He mentioned the “swampland” idea, that one could try and characterize those low energy theories that come from an ultraviolet completion like string theory.

    3. His main point I think was that as long as there was no alternative way to unify particle theory with quantum gravity, string theory would continue to be a main focus for people to pursue. Kind of the “only game in town argument”.

    4. He may also have mentioned the use of string theory in heavy-ion physics, in regimes where lattice gauge theory has trouble providing results.

    I guess I’m missing at least one…

  6. anon. says:

    Porrati’s 1st point is with all due respect exactly the argument that defended the use of epicycles by both Ptolemy and Copernicus: it seemed to be a very useful framework of ideas. (Ptolemy used epicycles in the earth-centred universe, c. 150AD. In 1543, Copernicus used epicycles in his final model of the solar system.)

    As a ‘general framework of ideas’, the false theory of epicycles was invaluable to Ptolemy, Copernicus and generations of physicists. But that useful approximate framework was really false, as Kepler eventually discovered. So in the end both the earth-centred universe and its general framework of ideas were discredited. Will the string theory framework of ideas similarly mislead generations?

    What is so interesting is that it seems to be disconnected from reality not just with regard to its failure to make testable predictions, but also at the input end. Instead of having solid input, everything which has been put into string theory is completely speculative. It is less testable than either of the epicycle theories, and has less solid evidence.

    People now laugh at the idea that a theory was once constructed in which the stars and planets were carried around the earth while imbedded in closed crystalline shells. At least that false model was an attempt to interpret data. Perhaps people will cry with pity in the future, reading how physicists defended 10/11 dimensional M-theory in the 21st century, without providing any evidence at all.

  7. Chris W. says:

    The pre-Copernican astronomers could be excused on the basis of epistemological naivete; their successors largely invented the understanding of science that is now being invoked in discussions of string theory.

    String theorists can’t be so excused. They should have known better, and should know better now. Certainly ‘general frameworks of ideas’ are important; they set the context for formulating problems. This is why metaphysics is important in science, even though most metaphysics ultimately proves worthless.

    The questions that must be asked now with respect to quantum gravity and unification concern the problem formulation. (Shiing-Shen Chern, who discussed the matter with Einstein in the 1940s, recognized this as the essential work of the physicist.) The alternatives to string theory in quantum gravity challenge the received wisdom in this regard, and for this reason alone are important. In this context Porrati’s main point (as stated by Peter, and echoed by many of Porrati’s colleagues) strikes me as a complete crock. The string theorists who adopt this attitude are the least likely to arrive at the crucial insights into the problem. One can hope they’ll at least have the good sense and simple honesty to recognize those insights when they appear, although I’m less and less optimistic about that.

  8. Vijay Shankar says:

    There seems to be many differences in opinions unfortunately based on nationality. What people would want Physicists to come up with is a theory that holds in all frames or an experimental method that would help us test all the theories. Until then, we can’t stop someone crying foul whenever there is a news about ‘revolutionary’ theories.

  9. tomj says:

    My question is: what is the difference between no theory and one which cannot be tested?

    I cannot figure out why string theory is a theory. It barely ranks as a hypothesis, and a poor one, very close to what my teenager would come up with. It is 100% mental.

    A theory, at minimum, should cover all the facts known, but as Einstein once said (something like): a theory should be as simple as possible, _but_ no simpler. The implication is that there has to be a careful balance, and the theory _must_ track data. How else could the complexity of theory be measured? Yes, you can predict new facts, but first you have to account for known facts. We have to start with the abilities of the observer. And the first ability is that of objectivity, and objectivity begins with the repudiation of belief.

    If a theory cannot be any simpler than necessary, how … really … how can a theory be more complex than necessary? If over simplification is a sin, complexity is beyond sin. A ‘theory’ (or set of words and math) which can ‘explain’ everything ‘after the fact’ is useless. Can someone please explain to me this: do physicists really believe that it is possible to formulate a complete description of the universe which will be testable? Because one possible reality is that we are incapable of this. We have thousands of years of data to suggest this conclusion, and only wishful thinking to suggest otherwise.

    I like the name of the book, it is important to echo prior thinking. But it might have been even more valid to call it ‘Beyond Reason’. Everyone seems to think that they have reason, that they think logically. And as long as we can avoid testing our reason and logic, we can continue to ‘think’ and ‘believe’ whatever we want. And if we become dogmatic in these untested beliefs, what is this? Science is not belief. Science is experiment. And experiment is based upon question, the antithesis of belief. Science is not an answer, science is a method.

  10. Ptolemy says:

    ‘I cannot figure out why string theory is a theory.’ – tomj

    Gerard ‘t Hooft:

    ‘Actually, I would not even be prepared to call string theory a “theory” – rather a model or not even that: just a hunch. After all, a theory should come together with instructions on how to deal with it to identify the things one wishes to describe, in our case the elementary particles, and one should, at least in principle, be able to formulate the rules for calculating the properties of these particles, and how to make new predictions for them. Imagine that I give you a chair, while explaining that the legs are still missing, and that the seat, back and armrest will perhaps be delivered soon; whatever I did give you, can I still call it a chair?’

    - http://www.math.columbia.edu/~woit/wordpress/?p=258#comment-5030

    Peter Woit’s argument of why a non-predictive framework is not science can be found on p211 of Not Even Wrong (UK ed.):

    ‘An explanation that allows one to predict successfully in detail what will happen when one goes out and performs a feasible experiment that has never been done before is the sort of explanation that most clearly can be labelled ‘scientific’. Explanations that are grounded in … systems of belief and which cannot be used to predict what will happen are the sort of thing that clearly does not deserve this label. This is also true of … wishful thinking or ideology, where the source of belief … is something other than rational thought.’

  11. r hofmann says:

    `Several wondered if the phenomenon of theorists going to the press to make overhyped claims about string theory was an American phenomenon. I hadn’t really noticed this, but it does seem to be true. While the hype starts in the US, it does travel to Europe, with the US very influential in this aspect of culture as in many others.´

    There are many examples of theoretical physicists
    working in the US (foreigners and US citizens) who do extraordinarily good work but on the short run are screened by those that produce overhyped newspaper headlines. My general impression is that the US culture supports the go for extremes in generating scientific opinion, publicizing of `results´, and network formation. This may be helpful in projects where a focus of resources is needed (Cobe, WMAP,…). On the theoretical side, however, it may at times just produce entropy, a lack of well-fermented orginality, and thus no gain in robust knowledge.

  12. Stacy says:

    A note on inflation, inspired by Levi’s comment:
    Actually, the situation with Inflation is quite analogous to that with string theory. The original idea was beautiful, and made a simple prediction (the universe should be flat) which solved the coincidence problem (to do with the evolution of the density of the universe). These together were compelling and propelled the theory to the dominance it enjoys today. But it did suffer problems (like a graceful exit from inflating) which have not entirely been solved. Worse, the compelling aspect of the flatness prediction – confirmed by the WMAP satellite – was that the density parameter should be unity – all in mass – in order to solve the coincidence problem. But it isn’t all in mass – we have now to invoke dark energy. This makes the coincidence problem worse.

    In other words, the compelling part of Inflation that led us all to believe it not only doesn’t work, but has made worse the problem it originally seemed to solve. I can’t help wondering if future generations of sociologists will debate whether speculative theories like string theory and Inflation were ever distinguishable from some sort of mathematically motivated religion.

  13. matteoeo says:

    I agree with Stacy, I think cosmology suffers just the same problems as string theory. Cosmologists can produce potentials that would suit to any possible dynamics of inflation and produce the desired spectrum of cosmic background radiation, without actually deriving them from the properties of the known QFT particles. The worse, cosmology at the moment is a melting pot of the most un-scientifical theories and hypothesis in town: dark energy, cosmological costant, strings and GUTs (early universe), inflation, higgs boson, quintessence, supersimmetry. In cosmology it seems one could just say whatever he wants without too much care about scientific estabilished facts. I was impressed once reading some articles that showed that accelerated expansion of the universe could be explained without any reference to cosmological costant and dark energy, but just owing to some very peculiar relativistic effect (I can give references if any of you is interested). The point is: before inventing theories about the universe, shouldn’t we study general relativity a lot better? And, before unifying gravity and the quantum, shouldn’t we try to understand the basis of QFT and the geometrical structure of QM, and the very profound implications of GR itself?

  14. woit says:

    Please, cosmology is off-topic. I’m not a cosmologist and don’t want to moderate discussions about cosmology.

  15. Alex Nichols says:

    I don’t think the epicycles analogy is correct.
    That’s an example of an incorrect theory that was disproved by subsequent observation, rather like to the ether theory.

    The suggestion being made is that string theory is incapable of falsification because it can’t be tested.
    Possibly true, but there are compelling reasons for believing that extended entities that fluctuate are the only possible basis for observable space-time.
    This could include strings, loop quantum gravity, spin foams, spin networks etc…

    Were it found that the higgs boson is a fundamental particle by the the LHC, all of these would be disproved.
    But aren’t the problems of falisfiability at high energy (planck or horizon size) equally true for all the other theories?

    Perhaps all the effort shouldn’t be going into one avenue of research. When it comes to funding though, governments may simply decide that we need more effort in applied physics, such as energy production.

  16. Alex Nichols says:

    BTW, could this finding have any heterotic implications? :-

    http://web.mit.edu/newsoffice/2007/e8.html

  17. The problem with particle physics, if it is a problem, is that we don’t have any new particles, and the very good theory we have for those particles looks pretty much like a kludge – all those undefined parameters hovering there like epicycles – which were very highly predictive, by the way.

    String theory is a heroic attempt to go beyond the SM, but so far hasn’t proven predictive in a confirmable sense. My guess is that we might be stuck without more input from the Universe, which is why everybody is pinning their hopes on the LHC.

    Maybe it will provide some clue that makes it possible to turn ST into a predictive theory, maybe it will make it more unlikely that ST has any reality, and maybe it will be mute on ST and other subjects.

    Only the last would be a bad outcome.

  18. off topic says:

    sorry to be off-topic, but let me point out that simplest inflation predicts Omega_total = 1: it is what we measure, and the fact that the total involves some components we don’t understand has nothing to do with inflation. Simplest inflation models also naturally produce a spectrum of scalar adiabatic Gaussian cosmological perturbations with spectral index ns = 1 +- 1/60. Each word has a precise meaning, and it agrees with data. (The deviation of ns from 1 is not yet safely seen).

    People tried and try to invent alternative to inflation, but it is not easy because inflation turned out to be good succesful physics. For example alternative models based on “simple string cosmologies” suggested wrong kinds of perturbations (isoentropic, ns not close to 1, etc), and a significant amount of additional complications seems needed to get what inflation naturally does.

  19. matteoeo says:

    I’m sorry I went off topic and I will retain from writing again, but nevertheless I think it’s interesting to see how the scientific method has been mistreated and pseudo-scientific claims are made in almost any field of natural sciences and humanistic “sciences”.
    Or do you think that this bad string theory story is just an occasional mistake soon to be recovered?
    My question was: do we know enough of the physics of the 20th century before adventuring in the physics of the 21st? I don’t think this question is off-topic.

  20. Peter Woit says:

    matteoeo,

    There are all sorts of problematic claims made in different sciences. I just don’t want this blog turned into a discussion forum about all of them, but want to keep it focused on things I know about and am willing to moderate discussions of. The question of the evidence for inflation is an interesting one, and “off-topic” makes to-the-point comments, but I’m not an expert on this, and there are good blogs out there run by people who are, so that’s where the discussion should really take place.

    My point of view is certainly that the Standard Model QFT remains poorly understood in many ways, and that problem deserves more attention. There are lots of other issues in physics that aren’t well-understood, but again, I don’t want to moderate discussions of issues I don’t know much about.

  21. Robert says:

    If your words were as reasonable as your slides congratulations for this nice presentation. For the philosophy of science section of the German Physics Societey I had indended to give a talk with a very similar subject (but of course slightly different conclusions). Unfortunately for personal reasons I could not attend the conference.

    Just a minor point of nitpicking (and we have discussed this before): When you say there is no clear cut experimental prediction I would qualify that with “to be performed with currently available experimental technology”. Otherwise I strongly believe your claim is wrong, at least if a weakly coupled description exisists (that is there is — possibly after a duality — a stringy description with g

  22. Robert says:

    Sorry for the sudden end of the previous comment. I wanted to say g less then less than (i.e. \ll in TeX) 1 but typing that froze my firefox (probably the script that does the preview. Luckily, I did not lose the post as after a few minutes it popped up a box asking me if I wanted to cancel a script. So I could still press the submit button. But there seems to be a bug either in the script or in firefox…

  23. Robert Musil says:

    Peter,

    Please correct me if I mistake your views, but I believe you have several times made clear that while you harbor skepticism over many aspects of string theory as physics, you believe that much extraordinary and important mathematics has resulted from string theory. The “Mirror Conjecture” is one such example. Admiration for string theory mathematics spin-offs is widely shared by many of the worlds leading mathematicians.

    But there are some very troubling aspects to even this, very real, admiration for string theory inspired mathematics – at least to my eye. It’s trivial to formulate the Mirror Conjecture: Just flip the Hodge array on the diagonal and ask for a variety. But nobody bothered to ask the question before M-theory was posited. Moreover, the first few examples of the Mirror Conjecture are not hard to prove (although the entire conjecture is), yet nobody bothered to investigate them before M-Theory was posited. One main (or at least common) example that supposedly demonstrates the mathematical importance of the Mirror Conjecture – finding those curves – was being pursued (apparently) by exactly two Norwegians on a computer before the Mirror Conjecture came up. Yet the Mirror Conjecture is supposed to be ultra-important mathematics. There is something very strange here.

    Perhaps what is strange here is reflected (oops! and unintentional pun) in the constant references to physics in all mathematical programs regarding the Mirror Conjecture (or at least the ones with which I am familiar). “Golly,” the mathematicians seem to say, “What I’m noodling over has relevance to the real world! It must be important mathematics!” But if it turns out that string theory is not important physics, I believe it would be a first if the associated mathematics were really all that important – regardless of the level of enthusiasm it has inspired. After all, string theory inspired quite a lot of ill-considered, unchallenged enthusiasm as physics for quite a while.

    In other words, I can’t shake the sense that the enthusiasm over the Mirror Conjecture (for example) has itself a hall-of-mirrors aspect: Mathematicians (even very good ones) love it supposedly because it is “intrinsically” wonderful mathematics. But it’s a strange kind of intrinsically wonderful mathematics that nobody gave a dam about before the physics came along in the form of string theory – even though it’s wonderful mathematics whose formulation is trivial and whose first few examples are easy and whose supposedly important applications nobody cared about enough to work on but two Norwegians (not that I have anything against Norwegians, mind you).

    Of course, on the other side of the hall of mirrors we find the string theorists reassuring themselves that their theory must be important (or even correct) because the mathematics is so wonderful. Bing, bing, bing goes the wonderful image across the hall – each time a little more distorted as it recedes.

    Personally, I find this hall of mirrors aspect of things disturbing, perhaps because I associate halls of mirrors with lower-budget hotel lobbies trying to look bigger than they are. Somehow I get a similar feeling from the mathematical spin-offs of string theory.

    Do you have anything to say on this?

  24. r hofmann says:

    Dear Robert Musil,

    although I have no idea about the Mirror Conjecture what you say about it and its embedding into the modern relationship between physics and mathematics strikes me as an intelligent observation. Thanks for the info.

  25. David B. says:

    Dear Robert:

    Mirror symmetry is not just about “flipping the Hodge diamond”.

    When you say
    “But it’s a strange kind of intrinsically wonderful mathematics that nobody gave a dam about before the physics came along in the form of string theory.”

    You are trivializing the contribution from physicists and mathematicians. The truth is that mathematicians had not suspected that the problem of counting curves in Calabi-Yau manifolds (a typical problem of enumerative geometry) could be related to the theory of deformations of the complex structure on the mirror geometry.

    You are also trivializing the problem by making statements like “even though it’s wonderful mathematics whose formulation is trivial and whose first few examples are easy”.

    The formulation is not trivial at all, and it took quite a while before someone produced a complete mathematical proof of the first few examples.

    I don’t like these misinformed statements about the relationship between research in string theory and mathematics. They seem to be crafted for purposefully misleading the public at large.

    Many profesionals use simple statements like “flipping the hodge diamond” when giving presentations in order to explain the simplest aspects of mirror symmetry to an uninformed audience, and to try to give them something they might relate to. In this way they can share the excitement of the subject. Don’t mistake those statements for the research that is done in the subject.

  26. Peter Woit says:

    Robert (non-Musil),

    The slides pretty accurately reflect what I said. In this talk I wanted to just as clearly as possible state the facts of the matter and avoid any editorializing.

    One thing that I should have put in the slides was a comment about the issue you raise, the claim that the testability problem for string theory only arises at low energy, that if we could do Planck scale experiments, it would be testable. I think we’ve probably discussed this before, but I would claim that the string theory framework continues to be not testable even at that scale. As you acknowledge, even a qualitative prediction of the kind I assume you have in mind (standard distinctive aspects of perturbative string spectra or scattering amplitudes) rely on the string coupling being small enough for the perturbation approximation to be good. Such a prediction is not falsifiable, since it could be evaded simply by saying “well, maybe the string coupling really is not small enough”.

    In practice, it is true that if we could do experiments at arbitrarily high scales, we’d presumably see what the structure of quantum gravitational effects is, and would see whether this looked at all like anything that had ever shown up in studies of string theory.

    Robert (Musil)

    David B. is right. The “Mirror Conjecture” and the associated mathematics it has generated go far, far beyond what you mention and are much deeper than “flipping the Hodge diamond”. As an example of this, next week at the IAS they’ll be an important mathematics workshop on “Homological Mirror Symmetry”, focusing on relations to the geometric Langlands program. This is a very active and important area in matthematics. It has pretty much nothing to do with attempts to unify physics via string theory, but it’s great mathematics, and maybe someday it will turn around and inspire some physics.

  27. Robert Musil says:

    Peter,

    Thank you for your as-always thoughtful response. David B. seems a very intelligent and knowledgeable (if somewhat excitable) fellow, but he is certainly not right in mischaracterizing me as asserting that the Mirror Conjecture ends with the Hodge Diamond formulations. Indeed, I’m not aware of any comprehensive formulation of the Mirror Conjecture. Manifolds with mirror-symmetric Hodge tables are called geometrical mirrors. My point in this regard is (and was) that the Hodge Diamond formulation is trivial to state and notice and that nobody had bothered to do either prior to the positing of M Theory. Yet now that very formulation is deemed to be inherently wonderful mathematics. Of course, this is not an argument for dismissing or downgrading the significance of any version of the Mirror Conjecture. But to start the discussion it does help to get the question right.

    Nor is David B.’s assertion that there are no easy examples of Mirror Symmetry right. Indeed, it is not that hard to find references to this fact in papers by central practitioners in the field. Of course, some of the known examples were by no means easy.

    As for the geometric Langlands program, I’m not knowledgable in the area of mathematics. I realize that geometric Langlands is an active area of research considered promising by many very smart people. But promise and “rich” structure alone didn’t make string theory great – or even important – physics. I’m not sure if I see why one can already conclude that Geometric Langlands is great mathematics – and evaluating the importance of Mirror Conjecture relationships to GL is another step after that.

  28. Peter Shor says:

    If somebody comes up tomorrow with a beautiful new theory which unifies gravity and QM and is much simpler than string theory, and if the LHC produces results that agree with its predictions, I assume that nearly all the string theorists will drop their current research and jump on the bandwagon.

    The real question is (a) without any hints of an alternative, are any of them going to abandon string theory research, no matter how unpromising it looks and (b) whether a hint of a promising alternative is enough, or whether it takes a fully formed theory. For instance, if the LHC produces a Higgs mass close to that predicted by Connes, are any of the string theorists going to take this as a hint that maybe they’re on the wrong track, and Connes on the right one?

    Any wagers on this?

  29. Kea says:

    Any wagers on this?

    What are we betting on? How long it will take the String theorists to figure out what’s going on? Actual experimental outcomes at the LHC? Oooohhh, this is fun.

  30. A.J. says:

    Several comments for Robert Musil:

    1) The relationship between Hodge diamonds predates M-theory by several years. It’s part of the story physicists like to tell about M-theory and a simple example of a mirror phenomenon, but I don’t think it’s of deep importance. More of a decorative note.

    2) I’ve never heard anyone claim that the existence of manifolds with mirror hodge diamonds was the important or deep part of the story. Complaining that others are calling it “inherently wonderful mathematics” seems like a bit of a straw man. Who exactly has said this?

    3) What is important, as David B. more or less pointed out, is that we can relate moduli spaces of complex structures to moduli spaces of symplectic structures. This is incredibly non-trivial, and potentially very useful.

    4) While I agree that “This shows up in string theory” isn’t necessarily a good rationale for a mathematical research problem, I think it’s a poor reason to dislike good mathematical ideas. And the notion that there’s a topological field theory which carries information about the space of curves and maps to a fixed target has proven to be a fertile source of algebro-geometric ideas.

  31. A.J. says:

    Peter (Shor):

    If the Connes et al prediction comes out right, I imagine that some people will take the hint and start working on it. On the other hand, I’ve also seen some stringy speculation around the fact that the noncommutative space in Connes, Marcolli, & Chamseddine has KO dimension 6.

  32. Robert Musil says:

    A.J.,

    You are quite right in that current interest in mirror manifolds is due to the idea is that along with the equality h1,1(X) = h2,1(Y ) of moduli numbers of Kahler structures on X and of complex structures on Y, the whole symplectic topology on X is equivalent to complex geometry on Y, and vice versa. In that sense perhaps I should have been more explicit about the means of establishing the Hodge equivalences. But the first examples of this equivalence are not hard, nobody was looking at them, etc.

    I’m not sure what you mean by “The relationship between Hodge diamonds predates M-theory by several years,” unless you are referring to the earlier computer results. I’m not aware of any general Hodge diamond conjecture that predated the positing of M Theory.

    All that being said, I don’t see why my points don’t still stand. For example, while I don’t mean to be snide or obtuse, neither do I see why the assertion that something is “a fertile source of algebro-geometric ideas” is a very good basis for concluding that those ideas or their source are important. The argument seems to completely assume its conclusion. Am I missing part of your point?

    There is clearly great and broad enthusiasm for some mathematics derived from (or spun off from) string theory – much of it among very smart and accomplished people. But there was (and is) just such enthusiasm for string theory itself – an enthusiasm only recently seriously challenged. That challenge has been made from one redoubt: In physics one at least has the check on the products of such enthusiasms that at some point or other those products must be EXPERIMENTALLY TESTABLE (although, as this blog cogently points out, some string theory practitioners are struggling mightily to avoid even that check). There is no such check in mathematics. So how do we know that the mathematics spun off from string theory are not just empty enthusiasms? It’s just silly to deny that a lot (as Peter points out, perhaps not all) of the enthusiasm is derived directly or indirectly from string theory itself. To make matters worse, some of the best mathematicians speaking to the public about mathematics spun off from string theory often make claims for its importance that are absurdly over the top (Michael Atiyah, for example). Certainly just asserting that one thing or another is “great mathematics” or the like doesn’t advance matters, does it? What does?

  33. A.J. says:

    Robert:

    OK first, most of the ideas of mirror symmetry predate M-theory by several years. The former is part of the body of evidence for the latter. If you want more direct evidence: Kontsevich’s homological mirror symmetry lecture is from the summer of 94; Witten’s M-theory announcement from the fall of 1995.

    Second, why are we still talking about Hodge equivalences? This is a hint that there’s something interesting going on, not the end goal of any major research efforts.

    I don’t understand what your metric for “importance” is. But it seems to me that algebraic geometers have judged Gromov-Witten theory to be important and interesting because of the ideas it’s brought into their field, not because it’s connected in some way to a much larger program in a different field. So, yes, by this standard, it’s important. If you mean important in some other sense, I really don’t have anything to say to you.

    My point basically is this: You have a reasonable abstract point about a potential relationship between relative levels of enthusiasm about physics and mathematics, and a cute metaphor about hotel mirrors to go with it. But I think you’re quite wrong to single out mirror symmetry as an example of the phenomenon you’re talking about.

    And I suspect you will have a hard time finding actual examples. It’s true that some mathematicians like to talk and daydream about important physics connections, but I think you’ll find that the physics-derived ideas which mathematicians have really taken the time to develop intensely have been those which are useful and interesting as as mathematics.

  34. Robert Musil says:

    A.J.,

    You mention “I don’t understand what your metric for “importance” is.” Well, let’s take that seriously. Terry Tao advanced a set of criteria for “bad mathematics” that I believe were discussed in this blog a while back:

    • A field which becomes increasingly ornate and baroque, in which individual results are generalised and refined for their own sake, but the subject as a whole drifts aimlessly without any definite direction or sense of progress; or

    • A field which becomes filled with many astounding conjectures, but with no hope of rigorous progress on any of them; or

    • A field which now consists primarily of using ad hoc methods to solve a collection of unrelated problems, which have no unifying theme, connections, or purpose; or

    • A field which has become overly dry and theoretical, continually recasting and unifying previous results in increasingly technical formal frameworks, but not generating any exciting new breakthroughs as a consequence; or

    • A field which reveres classical results, and continually presents shorter, simpler, and more elegant proofs of these results, but which does not generate any truly original and new results beyond the classical literature.

    Is it clear that the mathematics spun off from string theory has avoided each of these? It seems at least arguable that one, perhaps more, of these criteria fit uncomfortably well. Not that an answer to this would end the discussion, of course.

  35. Robert Musil says:

    A.J.,

    I first want to be very clear that I appreciate your thoughtfulness and intelligent comments. I also want to apologize in advance for popping in this second post before you have a chance to respond to or digest the first.

    With respect to Kontsevich’s seminal address at ICM, Zurich 1994, it is worth keeping in mind that Kontsevich’s himself characterized what he was doing as follows (I quote from his address):

    “Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Calabi-Yau manifolds V, W of dimension n (not necessarily equal to 3) one has
    dim Hp(V,q) = dim Hn−p(W,
    q). ….

    “We describe here a not yet completely constructed theory which has potentially wider domain of applications than mirror symmetry. It is based on pioneering ideas of M. Gromov on the role of ∂-equations in symplectic geometry, and certain physical intuition proposed by E. Witten.”

    The relevant references to Witten’s “intuitions” are to two papers: Topological sigma models, Commun. Math. Phys. 118 (1988), 411-449 and Two-dimensional gravity and intersection theory on moduli space, Surveys in Di . Geom. 1 (1991), 243–310.

    I believe these quotes address several questions and concerns expressed in your posts above (why we are talking about Hodge numbers, for example). I also believe these passages support my points.

  36. A.J. says:

    Robert:

    Tao didn’t give that list as criteria for bad mathematics. It’s just a list of dangers (somewhat exaggerated as Tao admits) which might have detrimental effects on the development of a field. I think it’s misleading to treat it as checklist for identifying “bad mathematics”.

    That said, the only danger I see being remotely applicable is is the 2nd one. But I don’t think it’s a particularly great danger. For one thing, judicious borrowing of physical intuition has a pretty good track record. (Donaldson theory, Chern-Simons, knot polynomials, mirror symmetry, Seiberg-Witten theory, and so on.) And for another, mathematicians have a habit of concentrating on problems they think are solvable. No one is butting heads with 4d Yang-Mills theory right now, because it’s probably out of reach. But there’s lots of motion in the Gromov-Witten theory of orbifolds right now; people are getting things done.

  37. A.J. says:

    Robert:

    I don’t see how the Kontsevich quotes support your points. Perhaps you’d care to explain? You’ll probably have to take some care to spell out carefully what you mean, since we seem to be talking at angles.

    Some of the confusion may stem from the term mirror symmetry. The symmetry gets its name from the duality of the hodge diamonds, but it’s just a name. The actual set of ideas involved is considerably richer than the name implies. Most of it has been developed in the years since Kontsevich’s lecture.

  38. David Williams says:

    Hello
    Please have mercy on an old scocial science Phd.

    I have had, basically, only a pragmatic and professional education except for a couple of biology courses and a stint as a biology teaching assistant (where I first encountered the scientific method) but I have indulged my interest in popularized science writing. I use this information in debating the champions of religion.

    In debate the basic successful argument is that Science is not based on belief but on questioning and testing. Recently, String theory has become widely accepted in physics. I love the idea in the sense that it tells us that the universe is a symphony. HOWEVER, String theory appears to arrive at the position of a Unified Field Theory only by relying upon
    a. mathematical solutions b. solutions that require positing multiple universes.

    May I ask you these questions.
    In your opinion are mathematical solutions the equivalent of an empirical test? Although I’m told (and I simply have to accept or not – at the level of my math and science skills) that M theory will offer an opportunity to empirically test String Theory. I cannot, to my satisfaction, imagine an empirical test for multiple universes.

    And, If empirical tests are not available by the very nature of String Theory is this idea no better than religious belief?

    I am inclined, therefore, to simply leave String theory to it’s own devises and conclude that it lacks scientific credibility and that we are stuck with the contradictions between General Relativity and Quantum Dynamics. We would otherwise be as lacking in evidence as the religious. Why have physicists so departed from scientific standards?

    Hope you will be able to spare the time answer this query.

    David P. Williams, PhD
    3181 Micmac St. Halifax
    Nova Scotia Canada B3L 3W3
    (902) 454

  39. tomj says:

    David,

    I have similar concerns. String theory is hyped or hoped beyond belief. This is serious, because if a scientist is supposed to be exact and careful about their theory, their work, etc., why doesn’t this carry over into their public descriptions?

    I somehow stumbled across this site a few weeks ago. But for several years I have firmly believed that there was something not right about the ‘theory of everything’ crowd.

    At that point I was trying to track down some more concrete details about these theories. But nothing concrete ever appeared. Instead, I ran across some made up cafeteria dialog between a string theorist and (I guess) a LQG theorist. I think the point of the dialog was to highlight the lack of evidence for either theory, but more important for me was another principle: science is about the unknown, not the known or the unknowable.

    If science is expanded to cover the unknowable, you forfeit the ability to apply Occam’s razor. Occam’s razor isn’t a theory, it isn’t a law of nature, it is a check on logic: it requires experiment. If a theory has no experimental results, how can you compare it to one that does? If a theory predicts unknowables like multiple universes, how can this win out over a theory that predicts only the one we experience?

    My problem with the proponents of string theory is that their ideas fall into the category of ‘known’ or ‘unknowable’. That is, their statements lead me to believe that they know something (strings are the basic building blocks of everything) or their theory covers stuff we can’t know (multiple universes, etc.). In the first case, they are lying, or using language in a very sloppy way. If they are sloppy with English, why should I think that they are not sloppy in their math or logic?

    What I don’t understand is that if a scientist makes wild statements that they ‘know’ something or that their theory implies ‘unknowable’ realities, why shouldn’t I remember their unscientific approach? Either put up, or shut up.

    Known = technology
    Unknown = science
    Unknowable = fantasy

  40. r hofmann says:

    A.J.,

    it’s of undeniable educational value to follow the debate between Robert Musil and yourself.

    `No one is butting heads with 4d Yang-Mills theory right now, because it’s probably out of reach.’

    This statement is, however, outright false and confirms
    pretty much the relevance of Terry Tao’s above quoted criteria.

    Best, RH

  41. Ralf says:

    David and tomj,

    String theory/M-theory is a speculative research program which therefore would not be covered at all in the popularized science press if the latter were responsible. Officially, string theory is “accepted” exactly as that. In practice this doesn’t stand in the way of string theorists taking over high energy physics, in part because the field has been short of new ideas for three decades. In such cases the subjective criteria for what can be regarded “reasonable” ideas become rather flexible. The “scientific method” exists only in the imagination of philosophers of science. “Occam’s razor” cannot be “applied” like a theoretical analog of a lab test.

    People can be honestly deluded about things that are crucial to their identity, like their love life, their social life or their professional world. In addition, string theorists view themselves as intellectually superior to everybody else, which automatically degrades any objections brought up by those.

    Finally, in reality there is simply nothing about string theory that can be related to laypeople. Anybody who writes about it is only heaping nonsense on a foundation of nonsense which again rests on a foundation of nonsense. It is utter intellectual dishonesty to pretend otherwise. Supersymmetry, by itself, cannot possibly be assessed by a non-physicist. Every account makes it appear much more reasonable than it is. Grand Unification sounds almost like a no-brainer if one doesn’t know the details. The technical details on which string theory is built—and which are never even mentioned in the popular press—render it, in my opinion, deranged and demented. And it is exactly this wide gap between actual physics and string theory that—perversely—facilitates the public’s susceptibility for it. The public never registered anything from the Schrödinger equation onward because they don’t like the absence of visualizability.
    That is why they prefer the faux visualizability of General Relativity—and of string theory, of course. Physics is not the Riemannian geometry of the 19th century. It was Einstein, after all, who commented that one should explain everything as simply as possible—but not simpler.

  42. a says:

    dear David,

    let me try to give an answer to your “Why have physicists so departed from scientific standards?”. It is oversimplified and caricatural, but I think it captures a relevant aspect of the question. Do you believe that an average rational human being would choose option A or B?

    Option A is what is happening now.

    Option B is “I spent my life working on strings, but, contrarily to what press said, initial hopes mostly disappeared. Maybe I could start doing some other physics, but I only have expertise in strings, that is a highly specialized topic: so I resign from my academic job”

  43. A.J. says:

    R. Hofmann:

    Sorry about that. I was not expressing myself clearly. (Why can’t you people just read my mind?!) A precise formulation: “Few if any mathematicians are attempting to construct 4d Yang-Mills theory in the sense required by the Clay Millenium prizes.” Obviously plenty of people are thinking about 4d Yang-Mills in a non-rigorous fashion, or trying to work out various facts about its topological analogues. But no one’s managed to do anything interesting as far as construction & mass gap goes.

  44. r hofmann says:

    A.J.,

    I see … That problem was formulated by E. Witten and a famous Harvard mathematical physicist, right?

    Best, RH

  45. A.J. says:

    Yes,

    I don’t know anything about how the Clay Foundation works, but at the least the problem description was written by Edward Witten and Arthur Jaffe.

  46. Ari Heikkinen says:

    Peter,

    Just a couple of questions, when you say:

    ‘The fundamental problem is that simple versions of the string theory unification idea, the ones often sold as “beautiful”, disagree with experiment for some basic reasons.’

    Do you mean by this that what’s in Greene’s book (that “beautiful idea”) of particles being tiny vibrating strings of which amplitude and wavelength corresponds to different masses and force charges of them and that those “extra” dimension are curled up in Calabi-Yau shapes are what disagree with experiment?

    ‘Getting around these problems requires working with much more complicated versions, which have become so complicated that the framework becomes untestable as it can be made to agree with virtually anything one is likely to experimentally measure.’

    And by this something that Greene’s book don’t mention?

  47. Peter Woit says:

    Ari,

    One of the main problems is that you have to do something to fix the size and shape of the Calabi-Yaus, and the only ways people have found to do this involve introducing a lot of complex, ad hoc structure. This is the “moduli problem”, and I don’t remember what Brian says about it in his book. His book was written now quite a few years ago, before people had any solution at all to the problem. Back then I suspect there was a lot more optimism that a simple solution could be found.