After last month’s posting at Cosmic Variance about how String Theory is Losing the Public Debate, Sean Carroll seems to have decided to go on the offensive (or defensive…), with a piece in New Scientist entitled String theory: it’s not dead yet, which he reproduces and has a posting about here.

I can’t really disagree with Sean about either title. Yes, string theory is losing the public debate, and no, it’s not dead yet. Some of Sean’s claims in the New Scientist piece are descriptive claims about the behavior of theoretical physicists:

*String theorists are still being hired by universities in substantial numbers; new graduate students are still flocking to string theory to do their Ph.D. work…*

*Ideas about higher-dimensional branes have re-invigorated model-building in more conventional particle physics… Cosmologists thinking about the early universe increasingly turn to ideas from string theory.*

All of these are true enough (although the word “re-invigorated” might not be the most appropriate one), but don’t address the value judgment of whether any of this activity is a good thing or not. One could also come up with other evidence for continuing activity in string theory, such as the large number of press releases being issued claiming to have found new ways to “test string theory”, but the fact that these have all been bogus is relevant to evaluating whether this activity is a good thing or not.

Sean’s positive case for string theory is mostly about its role as a quantum gravity theory, acknowledging that the Landscape is a problem, and that progress has slowed since the mid-90s (although more accurate would be “come to a dead halt, now moving backwards..”). He describes that period as “it seemed as if there was a revolution every month”, displaying the predilection for over-the-top hype that has characterized much string theory salesmanship over the years. His claims about the achievements of string theory vary from relatively modest exaggerations (“The theory has provided numerous deep insights into pure mathematics”) to standard misleading propaganda:

“a promising new approach has connected string theory to the dynamics of the quark-gluon plasma observed at particle accelerators” (connected? wonder how strong the connection is…)

“it is compatible with everything we know about particle physics” (and also compatible with just about everything we know to not be true about particle physics…)

“Michael Green and John Schwarz demonstrated that string theory was a consistent framework” (there’s a lot more to consistency than canceling that anomaly…)

“It was realized that those five versions of the theory were different manifestations of a single underlying structure, M-theory” (would be nice if we knew what M-theory actually was…)

In the comment section Sean explains how string theorists have no intention of standing behind what used to be considered the main “prediction” of the theory, TeV-scale supersymmetry:

*If the LHC discovers supersymmetry, string theorists will be happy, but if it doesn’t there’s no reason to give up on string theory — the superpartners might just be too heavy.*

So, prospects for string theory remain bright, since with each new experiment the situation is: heads they win, tails doesn’t count.

Also at Cosmic Variance is the latest in an exchange between Joe Polchinski and Lee Smolin, entitled Science or Sociology? (some earlier parts of the exchange are here). I’m mostly resisting the impulse to get involved in various parts of that argument since Smolin doesn’t need my help: the points at issue don’t seem to me central to the claims of his book, and his positions and what he wrote in the book are perfectly defensible.

While I don’t see the point of arguing about things like how conjectural the AdS/CFT duality conjecture is (pretty damn conjectural I’d think though, since no one even knows what the definition of one side of the duality is…), it is interesting to see what it is that Polchinski finds most objectionable about Smolin’s criticisms. In the context of an argument about how much of a problem the positive CC was considered to be by string theorists in the late 90s, he strong objects to Smolin’s description of “a group of experts doing what they can to save a cherished theory in the face of data that seem to contradict it”, going on to describe the work on moduli stabilization that led to the landscape as “a major success” which Smolin is trying to paint as a “crisis”. Ignoring the argument about who thought what back then (although if you really care about this, for some relevant evidence, see the Witten quote), in a larger sense “a group of experts doing what they can to save a cherished theory in the face of data that seem to contradict it” describes precisely the behavior of Polchinski, Susskind, Arkani-Hamed, and many others in the face of the disastrous situation created by the “major success” of moduli stabilization.

The “anthropic landscape” philosophy is nothing more than an attempt to evade failure, and it is an failure of scientific ethics of a dramatic kind. Once one understands a speculative idea dear to one’s heart well enough to see that one can’t make any conventional scientific predictions using it, ethics demands that one admit failure. Instead we’ve seen scientists announcing a new way of doing science, even writing popular books and magazine articles promoting this. Most physicists (including even a sizable fraction of string theorists) are appalled by this behavior. If you don’t believe me, consult a random sampling of the faculty in your nearest physics department, or watch Susskind’s recent talk in Israel where he describes himself as at the center of a circular firing squad.

Polchinski ends by claiming that Smolin’s case for “group-think” and for a “sociological” problem with string theory is “quite weak”. This problem is obviously hard to quantify and a matter of perspective. While I don’t doubt that Polchinski sees himself as not suffering from “group-think”, if he were, he obviously wouldn’t think so. One thing I think is undeniable about the “sociology” of all this is that the blog phenomenon has put a lot of evidence out there for any unbiased observer to judge for themselves, and this is one of the main reasons for what even a fervent string theory proponent like Sean Carroll has noticed: string theorists are losing this debate.

Anyone who regularly follows the most well-known blogs run by string theorists pretty soon becomes convinced that they have a real problem. Lubos Motl is the Id of string theory on uncensored display. The fact that his colleagues promoted him and show signs of only having a problem with his politics, not his behavior as a scientist (if they have any problem with his calls for my death or other attacks on me, I’ve never seen evidence of it) is truly remarkable. Two out of three recent string theory textbooks prominently carry his endorsement. All another prominent string theorist blogger, Clifford Johnson, has to say about Lubos is “I thank him for his physics contributions and for widening the discussion.” This was in the context of an eight-part personal attack on Lee Smolin and me for having written books that Clifford steadfastly refuses to read. The other of the three prominent string theory bloggers is renowned for his sneering attacks on the competence of anyone who dares to criticize string theory, issues press releases claiming tests for string theory that other physicists describe as “hilarious”, while misusing his position of responsibility at the arXiv to stop links to criticism of string theory articles from appearing there. Among those string theorists without their own blogs who choose to participate in the comment sections of others, a surprising number seem to think that it is an ethical thing to do to post often personal attacks on string theory critics from behind the cover of anonymity. Less anonymously, a large group of string theorists at the KITP seem to have thought it was an intelligent idea to act like a bunch of jeering baboons, on video, for distribution on the web.

This kind of public behavior and the lack of any condemnation of it by other string theorists is what has convinced many physicists and others that, yes, string theory does have a “sociological” problem. I have to confess that my experience over the last couple years has caused me to come to the conclusion that the string theory community has a much greater problem with personal and professional ethics than I thought when I wrote my book. The fact that so many string theorists have decided to respond to my book and Smolin’s not with scientific arguments, but with unprofessional behavior I think speaks volumes for the strength of their scientific case, and this has been noticed by their colleagues, science journalists, and the general public. While I applaud Polchinski for behaving professionally in his response to the two books, I suggest that he should take a look at the behavior of many of his colleagues and ask himself again whether or not there might be a sociological problem here.

Urs,

It is not true that path integrals are not well-defined, except after the

fact. This is a comon misconception. In regularized QFT’s, there is absolutely no problem with the path integral. They are often studied numerically (that’s what lattice Monte Carlo people do) and, in some cases, even evaluated analytically!

As I said, defining field theories on arbitrary manifolds depends on

the problem under scrutiny. Standard analytic methods,

1. perturbation theory, 2. strong-coupling methods, 3. semiclassical approximations, 4. exact S-matrices and form factors, 5. the Bethe Ansatz, and 6. conformal FT methods using the trace anomaly, all of which have applicability in at least some field theories do not require any but flat manifolds.

This is a starting point for TQFT, not for QFT in general.TQFT is a particularly simple example of this sort of axiomatization, but it is not hard to extend to QFT in general. Just choose add more and more structure to the objects in your cobordism category. (I’m beginning to become somewhat skeptical of this particular axiomatization beyond the 1-categorical structure, but that’s just me.)

In regularized QFT’s, there is absolutely no problem with the path integral.That’s true, but I think it can be quite hard to regularize a QFT (nonpertubatively!). Even then, to really talk about the full QFT, one needs to prove the existence of the continuum limit (presumably varying the couplings as you decrease the lattice spacing.) This is, needless to say, rather hard.

Aaron,

I am not sure to what extent we disagree on this matter (perhaps not

at all).

As I tried to explain above, path integrals can in some cases be

evaluated analytically. No one who works with them seriously

questions their utility. Proving the existence of the continuum limit

is always the issue, but, generally speaking, path integrals have a better track record on this than other methods do. A stunningly

better track record!

For example, mathematical physicists (Nelson, Glimm, Jaffee, Simon, Frohlich, Aizenmann, Balaban) getting nowhere with the Wightman axioms had some real success after switching to Euclidean path-integral methods in the 70’s. Furthermore, semiclassical methods

(DHN techniques in 1+1, confinement in compact QED) are best

understood with the path integral.

Sometimes other methods work better for specific cases

(integrable QFT’s in 1+1 for example) , but for all-around utility

the path integral has no serious rivals.

I’m all for path integrals. There are, I’m led to believe, some reasonably tought no-go theorems about the nonexistence of interesting measures on spaces of functions, however, so it remains a very interesting question as to what exactly a path integral is. I sometimes can’t help but feel that the lack of a proper definition after all these years might mean that we’re missing something important. But, then, the success of lattice gauge theories certainly speaks against that.

In some sense the axiomatizations of Atiyah and Segal and co. are an attempt to do an endrun around the lack of a proper definition of the continuum path integral. What it is an attempt to capture, I think, is the local nature of the path integral, that it must obey gluing and the existence of states in a Hilbert space to express that gluing. The structures expressed in these axioms clearly are present, but I doubt capture the whole story. One possible extension is the idea of a hierarchy of n-categorical structure, but the lack of a proper definition of the objects involved makes it tough to figure out how to go from usual physics to these rather abstract notions.

(To quote Dan Freed: “I define a mathematician’s category number to be the largest n such that he/she can think about n-categories without getting a migraine.”)

“I define a mathematician’s category number to be the largest n such that he/she can think about n-categories without getting a migraine.”LOL! Cute. Guess for physicists we can reduce n to, say, n – 3.

Aaron,

Forget all the no-go theorems (or rather, take them with a grain of salt).

Lattice path integrals are rigorously defined. The mathematicians who

have problems with them are geometers, not analysts, and the real

basis of path integrals is analysis, not geometry. The only issue is how to renormalize the answer. If there is a critical point in the space of couplings, there is a continuum limit (and then one has to address issues

of triviality).

I’m not sure we’re arguing, although I’m under the impression that there are still issues with chiral fermions and global symmetries. I’d just really, really like to see a proof of the existence of the continuum limit in an interesting QFT — the triviality issues, as I understand them, generally reflect the existence of a Landau pole, right? I’m not sure it’s fair to say that its only the geometers that have problems — no-go theorems in measure theory count as analysis in my book, and analysts like to prove things just as much as any mathematician.

Aaron,

I repeat that path integrals are healthy, and the best way to

look at quantum field theory in most cases.

There are of course subtleties, but the issues you are raising

are not fundamental difficulties with the path integral. There

are problems with putting the standard model on a lattice,

for example, but the origin of these is related to the

Nielsen-Ninomya theorem, which is also a problem in any regularization. Luescher may have solved this problem.

The Landau pole only suggests triviality of some QFT’s. It’s

less meaningful than generally believed. There are better

arguments for triviality (Ginzburg’s criterion, epsilon

expansions, 1/n expansions, polymer methods) for

scalar field theories.

There are no measure-theoretic no-go theorems, at least none

which are meaningful. Look at constructive field theory for

example (I listed a bunch of names earlier). The

measure-theoretic issues for Euclidean path integrals

were first addressed by Mark Kac a long time ago. Edward

Nelson also worked on the foundations of this subject.

Take a look at Simon’s book on Functional Integrals.

Look, I agree that path integrals are the way to go, but if there were no difficulties, there would be existence proofs for interesting QFTs in four dimensions. There aren’t. I’d love to believe this is just a result of the technical complexities of the limit, but until someone actual proves something who knows?

Aaron,

Yes, of course there aren’t existence proofs of four-dimensional theories (though there are existence proofs of two and three dimensional theories).

Anyway, I don’t think rigorous existence proofs are the first step towards understanding the continuum limit (I say this as someone

who has spent some time trying to see how rigorous methods in

lattice field theory work). What we want is a good way to calculate physical quantities in that limit, using some sort of controlled approximation scheme. This is where perturbation theory fails for theories like QCD. If that can first be done, and it is a big if, then

we can worry about cleaning up the arguments mathematically.

I’d like both thank you very much. I can say that because I’m not actually working on the subject :).

“This is a starting point for TQFT, not for QFT in general.”TQFT is a particularly simple example of this sort of axiomatization, but it is not hard to extend to QFT in general. Just choose add more and more structure to the objects in your cobordism category.Is this so? My impression, admittedly based on what I saw 15 years ago, is that the point with topological theories is that they are locally trivial – there are no local dofs. With only finitely many global dofs, it is hardly surprising if you don’t encounter the really difficult problems of QFT, which have to do with infinitely many local dofs.

Thomas,

Actually there is a sense in which some topological field

theories are nontrivial. As I understand it, their correlators are

purely topological, but they do have nontrivial degrees

of freedom. The original models of Floer and of Witten

(not Chern-Simons) are examples.

The functorial definition of QFT is not restricted to topological theories. Put any structure S you like (conformal, Riemannian, etc) on your cobordisms, and you’ll get a corresponding S QFT (conformal, etc).

TQFTs are just the best understood cases. But 2-d CFT is now also pretty well understood. Full rational 2-d CFT is pretty much completely understood this way. And, actually, only this way. (See this for more.)

And, as I said, this is nothing but making sense of the concept of a path integral. If you happen to have a path integral which already makes sense by itself, all the better. Then you have a way to construct such a functor using functional integrals. But if your path integral happens to be ill-defined, you might still be able to construct your functor otherwise.

At our workshop taking place currently the main focus is on understanding functorial Riemannian 2-d QFT.

(That’s because Stephan Stolz and Peter Teichner noticed recently that to get elliptic cohomology over points (knows as modular forms) from the landscape of 2-dimensional QFTs, one doesn’t necessarily need to restrict attention to superconformal ones. Supersymmetry alone implies that the torus partition function is a modular form (see this preprint for more).

In most QFT cases which are physically interesting, there is absolutely no clue what the measure on the space of fields should be which we would like to integrate over.

We hadn’t really mentioned extended QFT yet.

But it is true, that the idea is that in the end n-dimensional QFT is really an n-functor on extended n-cobordisms instead of just a 1-functor on n-cobordisms.

This, too, though, is closely related to the path integral: namely to evaluating the path integral with certain boundary data kept fixed. The idea goes back to Dan Freed, at least, who does present it in path integral language in his papers from the beginning of the 90s (see the references given here.)

Hopkins and Freed are proposing, at least in talks, refinements of this idea to infinity-functors. As mentioned here.

“In most QFT cases which are physically interesting, there is absolutely no clue what the measure on the space of fields should be which we would like to integrate over.”

False.

Urs,

I don’t want to be a churl, but I am a bit sensitive about this

issue, since I have worked on it for most of my career, and

often hear statements such as yours. So let me just mention

that I gave examples above where the measure is perfectly

fine. The problem with the field theories we’d like to understand

better is not in defining the path integral (which can be

regularized), but in removing the cut-off.

With respect, Peter, that’s like saying that taking the limit isn’t important in the Riemann integral. The whole point of defining the path integral rigorously is to remove the cutoff.

Aaron,

I don’t see where Peter is saying that taking the limit isn’t important. But there are physically interesting theories (e.g. Yang-Mills) where conjecturally we think we know how to take the limit, and there is a huge amount of numerical evidence to back up this conjecture.

Just to add to what Peter W. said:

There are examples above where the limit CAN be taken. In particle

scalar field theories in 3 Euclidean dimensions. One can also

prove that the 5 dimensional case has no interacting limit (and

argue the same in 4d, though not yet rigorously).

Since no one has solved QCD, we don’t yet know what the limit

is, and it has not proved to be an interacting theory. Nobody

who works on this problem doubts that the limit exists, however.

Formal renormalizability on the lattice (just perturbation theory

with a lattice cut-off) certainly indicates that the limit

exists, though it is not a proof. In any case, if you are theoretical

physicist, you want a calculational scheme more than a formal

proof.

I have some experience with this sort of problem, so I think

I have some feeling for what doesn’t work (which is most

of what people have tried). Perhaps the formal ideas Urs is

discussing are mathematically interesting, but they do not

shed light on the real problems of four-dimensional theories.

If you believe Borcherds’s notes on QFT, there does not exist a Lebesgue measure on the relevant spaces, so assuming that such a limit exists, I’d like to understand precisely what this is.

The reason why this would be useful is that there are field theories that don’t admit nice Lagrangian descriptions. It seems reasonable to me that there are plenty of “measures” that aren’t simply exponentials of integrals of functions involving the fields and their derivatives. I’d like to know what they are.

Aaron,

I don’t seem to able to get my view through to you or Urs. That

may have to do with the choice of problems we work on. Unlike

the two of you, I actually work on non-perturbative aspects of gauge theories an other field theories, and have for some time. That’s why I have been very insistent that much of the issues you raise are not the right issues.

I am aware that there are field theories which do not (at least yet)

have Lagrangian descriptions. But…

The field theories we NEED to know about for the purpose

of understanding experiments in high-energy or condensed-matter

physics (possibly with the exception of some special CFT’s) admit

a Lagrangian description.

The measure-theoretic issues Borcherds worries about may be important in certain approaches to QFT. They may give insights of one sort or another. But they may not be relevant to real problems of

QFT (confinement, for example). I repeat (for the fourth or fifth

time) the path-integral measure on Euclidean lattices is fine

(and mathematical physicists, by which I mean people who

know measure theory agree). Trying to convince yourself it isn’t

won’t lead to progress. Proving that there is a continuum limit

is not the main issue – there is no doubt that

an asymptotically free theory has a continuum limit as the

bare coupling vanishes. The problem is to understand the properties

of that limit. Then, once that is done, one can try to clean

things up mathematically.

My philosophy on such problems is that we need to study them

any way we can that yields a bit of progress. For myself, this usually (though not always) means inventing techniques rather than theorems.

I think we’re talking past each other. I agree that the questions you discuss are all interesting. I agree that lattice gauge theories are beautiful, well-defined things. I’m not convinced, however, that the existence of the continuum limit is necessarily a boring technical detail. I was trained as a physicist; you won’t find me arguing that all progress must necessarily involve rigorous theorems. Nothing could be further from the truth. However, I think it is important that things do get cleaned up in the end, and it is somewhat disturbing to me that QFT has not yet been cleaned up. If it’s just ugly technical analysis, I’d like to know that. If it’s a sign of some deeper obstruction, I’d like to know that, too.

This all goes back to the old question of what is a QFT. I think Wilson has certainly led us in the direction of a possible answer in the case of Lagrangian field theories. I want to know what the general structure this leads to. As a physicist studying the standard model, perhaps this isn’t so interesting, but as someone who wants to study thing beyond the standard model, understanding fully what a QFT is strikes me as very interesting.

Not to belabor the obvious too much, but this isn’t an either/or proposition; there are lots of interesting questions out there. If I had more cajones, maybe I’d even work on it, but I don’t at the moment, and I think I’ll stick with my derived categories for the time being. Technical analysis was never my strong point anyways :).

Aaron,

The reference by Borcherds is just to the well-known fact that the things physicists write as “Dx”, a putative infinite-dimensional version of the translation-invariant Lebesgue measure, don’t make sense, even for a free particle moving in R^n. But there is a measure in this case that does make sense, Wiener measure. Even in quantum mechanics, to make sense of the path integral as a measure, you need to do two things:

1. Euclideanize

2. Include some version of [tex] e^{-\frac{1}{2}\int |{\dot x}|^2 [/tex] in the measure to get Wiener measure, not the non-existent Lebesgue measure

Aaron,

I never said the existence of the limit is a boring detail. I said

it is a secondary detail.

Hmmm — I read Borcherds’s statement as more general than that (ie, the nonexistence of measures on field spaces — besides the Gaussian case, of course), but looking around Wikipedia, it looks like he probably does mean the more basic statement you refer to, however.

I thought there were some more general no-go theorems, but I can’t seem to think of where I heard about them at the moment.

(And, after googling a bit, it doesn’t look like the Gaussian measure exists in higher dimensions, but it’s not too hard to get around that in a well-defined way)

Not to belabor the obvious too much, but this isn’t an either/or proposition; there are lots of interesting questions out there.Is there a list of well-formulated questions about QFT out there somewhere, kind of like a Hilbert’s problems?