I have a long-percolating post that I hope to finish soon (when everything else is finished!) on “Why String Theory Must Be Right.” Not because it actually must be right, of course; it’s an hypothesis that will ultimately have to be tested against data. But there are very good reasons to think that something like string theory is going to be part of the ultimate understanding of quantum gravity, and it would be nice if more people knew what those reasons were.
Of course, it would be even nicer if those reasons were explained (to interested non-physicists as well as other physicists who are not specialists) by string theorists themselves. Unfortunately, they’re not. Most string theorists (not all, obviously; there are laudable exceptions) seem to not deem it worth their time to make much of an effort to explain why this theory with no empirical support whatsoever is nevertheless so promising. (Which it is.) Meanwhile, people who think that string theory has hit a dead end and should admit defeat — who are a tiny minority of those who are well-informed about the subject — are getting their message out with devastating effectiveness.
The latest manifestation of this trend is this video dialogue on Bloggingheads.tv, featuring science writers John Horgan and George Johnson. (Via Not Even Wrong.) Horgan is explicitly anti-string theory, while Johnson is more willing to admit that it might be worthwhile, and he’s not really qualified to pass judgment. But you’ll hear things like “string theory is just not a serious enterprise,” and see it compared to pseudoscience, postmodernism, and theology. (Pick the boogeyman of your choice!)
One of their pieces of evidence for the decline of string theory is a recent public debate between Brian Greene and Lawrence Krauss about the status of string theory. They seemed to take the very existence of such a debate as evidence that string theory isn’t really science any more — as if serious scientific subjects were never to be debated in public. Peter Woit agrees that “things are not looking good for a physical theory when there start being public debates on the subject”; indeed, I’m just about ready to give up on evolution for just that reason.
In their rush to find evidence for the conclusion they want to reach, everyone seems to be ignoring the fact that having public debates is actually a good thing, whatever the state of health of a particular field might be. The existence of a public debate isn’t evidence that a field is in trouble; it’s evidence that there is an unresolved scientific question about which many people are interested, which is wonderful. Science writers, of all people, should understand this. It’s not our job as researchers to hide away from the rest of the world until we’re absolutely sure that we’ve figured it all out, and only then share what we’ve learned; science is a process, and it needn’t be an especially esoteric one. There’s nothing illegitimate or unsavory about allowing the hoi-polloi the occasional glimpse at how the sausage is made.
What is illegitimate is when the view thereby provided is highly distorted. I’ve long supported the rights of stringy skeptics to get their arguments out to a wide audience, even if I don’t agree with them myself. The correct response on the part of those of us who appreciate the promise of string theory is to come back with our (vastly superior, of course) counter-arguments. The free market of ideas, I’m sure you’ve heard it all before.
Come on, string theorists! Make some effort to explain to everyone why this set of lofty speculations is as promising as you know it to be. It won’t hurt too much, really.
Update: Just to clarify the background of the above-mentioned debate. The original idea did not come from Brian or Lawrence; it was organized (they’ve told me) by the Smithsonian to generate interest and excitement for the adventure of particle physics, especially in the DC area, and they agreed to participate to help achieve this laudable purpose. The fact, as mentioned on Bloggingheads, that the participants were joking and enjoying themselves is evidence that they are friends who respect each other and understand that they are ultimately on the same side; not evidence that string theory itself is a joke.
It would be a shame if leading scientists were discouraged from participating in such events out of fear that discussing controversies in public gave people the wrong impression about the health of their field.
off-debate physics question, regarding gauge-invariance in QED. Isn’t it correct that Gauss’ law only says that physical states should be invariant under local gauge transformations that are trivial at spatial infinity?
Specifically, if I write the Gauss’ law constraint as C(x)=0 where C(x)=div E(x) -rho(x), then G=int C(x)theta(x) is a generator of a U(1) gauge transformation with parameter theta(x) only when theta(x) goes to zero at spatial infinity. For example, take theta(x) equal to a constant. In this case, the Gauss law constraint G=0 just gives (Electric flux = Q). So physical states do not have to be invariant under global U(1) transformations generated by Q (otherwise how do I get charged states in QED?).
Is this correct?
Diogenes,
I’m no expert on chiral gauge theories on the lattice, and would appreciate hearing from one about the exact state of this. My own experience doing lattice calculations involved several iterations of thinking I knew how to properly define something, only to find that there was a problem I hadn’t thought of once I coded the algorithm and started simulations. This is one reason I’d like to see an actual calculation of something. The literature I’ve tried to read contains various claims and the situation seems to me, as I wrote, unclear. Here’s Mike Creutz’s 2004 summary (in hep-lat/0406007) of the situation:
“While a lattice regularization of a full chiral gauge theory such as the standard model remains elusive, we may not be far off.”
I’m not aware of any major breakthroughs in this area since 2004.
onymous,
Again, I’m no expert, and haven’t thought much about this in a long time, but from what I remember, the global topology of the finite volume is not the problem, rather it is what is happening at the scale of the cut-off.
gs: right, gauge transformations that don’t vanish at infinity should be thought of as global transformations that act on the Hilbert space of the theory. The generator of those transformations is the (total) electric charge. Related issues are the definition of mass in GR, and that of the theta angle in QCD, where gauge transformations not vanishing at infinity are important.
Peter, it looks like the issue is more subtle than I had realized. Nielsen and Ninomiya first gave a proof using homotopy theory, relying on the topology of the Brillouin zone (Nucl.Phys.B185:20,1981). This is what I was (somewhat) familiar with. Later, they gave a more general argument (Phys.Lett.B105:219,1981) that claims to apply to any regulator satisfying some basic assumptions that one would not want to give up (gauge invariance, the right anomaly, an action bilinear in the fermions). The obvious perturbative regulators (dim reg, Pauli-Villars) each violate one of these assumptions.
Mark Srednicki, #347 and Moshe:
I can flesh out the mathematical side of the problem of the Gauss law constraint and its relation to indefinite metric representations (+ a different option).
First, one needs to know about the following two theorems:
1) Strocchi 1967:
If A_mu(x) is an operator valued distribution (on a Hilbert space) which transforms covariantly w.r.t. the Poincare group, then
Box A_mu =0 (field eq’n) and
partial^mu A_mu =0 (Gauss law)
implies that F_{munu} = 0 where F_{munu} = partial_nu A_mu – partial_mu A_nu
2) Barut + Raczka 1972:
A J-unitary representation of the Poincare group which is of zero mass and on tensor valued functions (e.g. Fock rep), must have J not equal to I, i.e. it must be on an indefinite metric space.
For the electromagnetic field, these two theorems force you (for the Fock representation) to go to an indefinite metric space, and hence the Gupta-Bleuler method of enforcing both the Gauss law and the field equation as state constraint conditions.
The justification is that after constraining, the final physical space is again a Hilbert space (i.e. the indefinite metric is eliminated), so the indefinite metric only appears for the nonphysical part of the theory.
However, as I have argued (with my co-author) in Rev. Math. Phys. 12, 1159 (2000), the indefinite metric can be avoided. The point is that there are implicit regularity assumptions for the representation of the two theorems above, and there is no reason to require regularity on nonphysical objects. So, if one takes the field algebra of observables, and allow (Hilbert space) representations to have nonregular behaviour on the nonphysical objects, then the theorems above do not hold anymore, and one can enforce the Gauss law constraint and field equation without any problems. In particular, we have done this for Gupta–Bleuler electromagnetism, and we obtained the same results than what one obtains via indefinite metric Fock rep. Moreover, the final physical algebra has a perfectly sensible Fock rep with all the right properties, but importantly, it did not come from constraining a Fock rep on the original field alg.
So the moral of the story is that for gauge theories one should not be married to indefinite metric representations in constraint problems, they can be avoided.
Chris,
Looking at Shaw’s slides, I’m not surprised the talk didn’t go well. There don’t seem to be any clear conclusions. Is he claiming to have constructed previously unknown string theories? I can’t tell …
And I don’t agree at all with your statement that while “it would be very nice to explain SM parameters, but establishing a mathematical framework free of inconsistencies has to be done first.” Historically, progress in physics has almost never been made this way. It almost always came by somebody futzing around with some not-fully-baked theory, until something interesting popped out. QED, for example, almost certainly does not exist as a stand-alone theory. Waiting around for some UV completion of it before calculating a few orders of g-2 would have been a very bad idea.
Peter,
I’m now completely mystified as to what that comment on gauge-group representations in your Orlando talk was supposed to mean if you agree that only the trivial rep has physical significance. But never mind.
For the benefit of those who might not know, I would like to point out that there is a huge literature on the possibility that the Higgs is dynamical, stretching back to the seminal late-70s papers of Weinberg and Susskind on technicolor (Susskind’s term). Technicolor models were extensively studied and developed, then slowly abandoned as they got pushed into tighter and tighter corners by observations. Susskind jumped off the ship pretty early (presumably sensing that it wasn’t going to float much longer), despite the fact that if technicolor had panned out, he would almost certainly have gotten a Nobel Prize for it.
I also don’t see how further knowledge of nonperturbative behavior of chiral gauge theories is going to help with this program, since the model-builders of old were willing to assume just about any minimally plausible behavior that would get them a workable model.
And I’m glad to learn that you think that “If string theorists want to believe … that … a compactification that describes the world will be found, will be predictive, get tested and verified, they’re welcome to do so.” I don’t know that I believe this will happen, but I do think it’s an open possibilty, and one with much greater promise than that offered by any other framework that is known today.
Moshe, one should be careful about which version of the algebra of gauge transformations one considers. In the polynomial or compact support version, one can combine nonzero charge with local triviality, because the local transformations generate an ideal. However, if one considers the Fourier or Laurent polynomial versions, as one does in string theory, no part of the gauge algebra can act trivially in the presence of nonzero charge.
This is easy to see for the centerless Virasoro algebra. Since
[L_m, L_-m] = 2m L_0,
a nonzero charge L_0 != 0 implies that all L_m != 0, and unitarity the requires an anomaly. In the polynomial version, where m >= 0, we can have L_0 != 0 and all other L_m = 0, but in that version there is no such thing as a central charge. Other gauge algebras work in exactly the same way. Thus, there seems to be only three alternatives:
1. Laurent polynomials are always allowed. Then there are new gauge anomalies, and string theory is wrong.
2. Laurent polynomials are never allowed. Then there is no such thing as a conformal anomaly, and string theory is wrong.
3. Laurent polynomials are allowed in string theory but not in Yang-Mills. This is clearly a case of double standards, aimed at hiding the fact that string theory is wrong.
Mark: Nonzero charge is an experimental fact, and string theory teaches us that Laurent polynomials are ok.
If you’re quantizing on the circle (as in string theory), then there is no “spatial infinity” where bona fide gauge transformations are supposed to go to the identity.
Sheesh!
Mark,
“I don’t know how to think up a better one. I’m very much open to hearing of a better one from someone else, but I don’t expect to;”
if you are searching for new ideas, start speaking with people outside the ST community. Organize a conference with people from LQG, twistors, whatever. Take the proposals of Peter Woit in a constructive way.
“I think quantum gravity is a very constrained problem, and that string theory is very likely to be the only solution to this problem. If it is the only solution, abandoning it now would be a foolish thing to do.”
The key is the if.
So quantize in a Fourier basis on the n-dimensional torus. The algebras are isomorphic.
Sheesh!
Mark,
I did not word this very well. Try this: “particle physicists are used to using a calculational framework that is as full of holes as a Swiss cheese. So-called “effective” field theory is not a theory, it is just recipe whose success in a limited sphere (QED) has led physicists to believe that it will also work for other types of interaction. Success here, though, has been limited, quantum gravity being a notable failure. The response of the vast majority has been to explore higher levels of abstraction, hoping that, whilst leaving it largely intact – they will be able to “explain” effective field theory and – hopefully – also solve some of the mathematical difficulties.
“I never felt optimistic about this program, but more than twenty years on, I feel even less optimistic. I believe that it is high time that “effective” field theory was abandoned altogether. Accurate values of g-2 and the Lamb shift may, possibly, elude us temporarily, but for mathematical consistency it is a price worth paying.”
Alright?
Mark,
The significance of the comment was:
1. We don’t understand non-trivial representations of the gauge group for any gauge theory in 3+1 d.
2. There’s no indication this matters for QED, little for QCD, but maybe it is important for completely understanding the SM, since it is a chiral gauge theory, and we don’t fully understand it non-perturbatively.
One way of seeing the problem is to think about it in Hamiltonian form. BRST then becomes essentially Lie algebra cohomology, where you construct the invariant, trivial piece of a representation by cancelling a sequence of non-trivial representations against each other. To do this, maybe you actually need to understand what those representations are.
Wow, this thread must have smashed all length records!
Since it came up above, here is a bit about the current situation re. chiral gauge theories on the lattice. To actually get to the stage of being able to calculate anything there are two main things that need to be done: (i) An acceptable lattice formulation of chiral gauge theories (and in particular the SM) needs to be developed, and then (ii) an appropriate phase transition in the lattice model needs to be found where correlators diverge and a continuum limit can be defined. Both of these things have been done in lattice QCD; there is a phase transition for bare coupling g -> 0 where a continuum limit can be defined, and the hadronic masses etc which have been extracted so far are in very good agreement with experiment.
For chiral gauge theories we have not yet progressed beyond (i). For a long time there was a major obstacle to even getting started on (i), namely the Nielsen-Ninomiya no-go theorem. The naive lattice discretization of the Dirac operator gives rise to “doublers” (spurious fermion species) and N-N basically says that there is no way to fix this without ruining the chiral nature of the theory, i.e. ruining the possibility to decompose the massless lattice fermion action into decoupled left- and right-handed pieces. In the ’90’s there was a major breakthrough though, and this problem got solved. The developments came from two directions, going by the names “overlap formulation” and “Ginsparg-Wilson (GW) formulation” respectively, which converged into two mathematically equivalent ways of formulating chiral gauge theories on the lattice. From the GW perspective, the solution to the original problem is that the notion of chirality gets modified on the lattice: Lattice Dirac operators satisfying the so-called GW relation (with the overlap Dirac operator being the main explicit example) break usual chiral symmetry but have instead an exact lattice-deformed version of chiral symm which can be used to decompose the action into left- and right-handed pieces as in the continuum. Thus a not-obviously-inviable lattice formulation of chiral gauge theories is obtained.
To be acceptable though, the formulation should reproduce the things we already know about continuum chiral gauge theories; in particular the chiral gauge anomalies and their cancellation in certain fermion representations. This is actually a big challenge. In the continuum there is a beautiful connection between chiral gauge anomalies and families index theory for the Dirac operator coupled to gauge fields. In particular, there are obstructions to the vanishing of anomalies corresponding to non-trivial topological structure of the index determinant line bundle over the orbit space of gauge fields. It would seem at first sight that a lattice formulation could never reproduce this, since in finite volume (e.g. on the 4-torus) the space of lattice spinor fields is finite-dimensional but to have a non-trivial index theory the operators generally must be acting on an infinite-dimensional vectorspace. Amazingly though, the the overlap/GW lattice formulation is able to handle this; an index bundle can be defined in a natural way over the orbit space of lattice gauge fields, connected to anomalies in the lattice model in the same way as in the continuum setting, and the obstructions to vanishing of anomalies are found to reproduce the continuum ones. For me this is a very powerful indication that the overlap/GW formulation must be the right approach to chiral gauge theories on the lattice and is deserving of interest and attention.
The resolution of (i) in the overlap/GW formulation is not yet compete though. We need to explicitly construct a “chiral fermion measure” as a function of the lattice gauge field such that the resulting chiral fermion determinant is gauge invariant in fermion representations satisfying the usual anomaly cancellation conditions. This has been done so far only for gauge group U(1); the non-abelian cases remain. It’s a hard technical problem. Let’s imagine that it eventually gets solved though, and consider the challenge of (ii). Unlike lattice QCD, the lattice chiral gauge theories are likely to have a complicated phase structure and it might not be obvious which phase transition to use to define the continuum limit. In fact this is already known to be the case for lattice QED. Another big problem will be that the chiral fermion determinants are complex-valued – numerical lattice simulation methods require real (positive) fermion determinants. My own hope (no doubt naive and perhaps completely wrong) is that both of these obstacles may be overcome as follows. First, I would expect that there should be a phase transition in the limits where the bare couplings g’ and g” of the U(1) and SU(2) gauge fields of the lattice electroweak theory both go to infinity, and that this would be the appropriate phase transition for defining the continuum limit. (In fact I would expect the analogous thing to be true for lattice QED, but I don’t know the current state of knowledge on that.) Then, it should be possible to analytically make strong coupling expansion in g’ and g” to get expressions for the quantities of interest which don’t directly contain the chiral fermion determinants and can therefore (hopefully) be calculated via numerical lattice simulations (and maybe even analytically in some cases for electroweak theory without QCD).
At any rate, this is one example of a non-string formal particle theory topic which seems pretty interesting but is career suicide to work on in the present string era.
Larsson wrote:
There’s no spatial infinity on an n-torus either.
So that case is entirely irrelevant to the situation that had been under discussion (until you tried to hijack it into a discussion of your pet theory), namely the status of “gauge” transformations that don’t go to the identity at spatial infinity.
Normally, I don’t bother responding to your ceaseless self-promotions. But there was a serious physics discussion going on, and your remarks promised to hopelessly confuse it.
Mathematically, embedding the algebra of local gauge transformation into the larger algebra also containing global and divergent transformations is the most natural thing in the world to do. If you do that, gauge anomalies are forced upon you. But you can of course insist that embedding local transformations into a larger structure is an evil and illegal thing to do. After all, why on earth should a string theorist be interested in new mathematics with obviuos connections to gauge symmetries?
As everyone has surely noticed, the crucial point is the existence of a two-sided grading with the global charge generators at degree zero. E.g. on the torus, the gauge generators are J^a(m) = exp(im.x)Q^a, where Q^a are the global charges. Since [J^a(m), J^b(-m)] = if^abc Q^c, nonzero charges imply that the whole shebang is nonzero; no ideal of local transformations which can be represented trivially. Divergent transformations have really nothing to do with it.
amused,
Thanks a lot, that’s exactly the sort of informative explanation of the situation I was hoping for!
Chris,
Presumably you understand that abandoning effective field theory means abandoning everything we know about particle physics.
I also don’t see why quantum gravity should be considered a failure of effective field theory. Effective field theory predicts that, below the Planck scale, general relativity should work just as it does.
The situation is analogous to that of the V-A theory of weak interactions; this works just fine below the scale of the W mass. Above that scale, you need an ultraviolet completion.
Ignoring gravity, the Standard Model provides you with one, up to a much higher scale (wherever the first Landau pole is).
So, we need an ultraviolet completion of gravity. That’s what all string theorists and LQG theorists are trying to construct.
Dear Thomas:
When one studies string theory one always finds L_0=0 for all physical states.
This equation is part of the Virasoro constraints for the total conformal algebra that includes the ghosts.
Your algebraic argument, although correct, is irrelevant for physical situations in string theory. In fact, in string theory the total central charge vanishhes.
Thomas,
You say that “Nonzero charge is an experimental fact.” Of course it is! And it’s always observed to be carried by dynamical particles (e.g. electrons), consistent with Gauss’ Law.
You say that “Mathematically, embedding the algebra of local gauge transformation into the larger algebra also containing global and divergent transformations is the most natural thing in the world to do.”
OK, as a mathematically challenged physicist, I’m happy to agree that this is mathematically natural.
“If you do that, gauge anomalies are forced upon you.”
Oops! This is bad! Maybe it’s natural, but now I sure don’t want to do it!
“But you can of course insist that embedding local transformations into a larger structure is an evil and illegal thing to do.”
Well, if it leads to anomalies in gauge symmetries that I’m trying to preserve, then, yes, it’s illegal.
“After all, why on earth should a string theorist be interested in new mathematics with obviuos connections to gauge symmetries?”
Because, according to you, it leads to bad physics, which I have no interest in constructing.
Paolo,
You wrote, “if you are searching for new ideas, start speaking with people outside the ST community. Organize a conference with people from LQG, twistors, whatever.”
Already done!
KITP Miniprogram: The Quantum Nature of Spacetime Singularities (January 8-26, 2007); Martin Bojowald, Robert H. Brandenberger, Gary T. Horowitz, Hong Liu. organizers. http://online.itp.ucsb.edu/online/singular_m07
I didn’t organize it, but I attended as much as I could. (For those who don’t know, Bojowald is a LQG theorist, Horowitz is a general relativist and string theorist, Liu is a string theorist, and Brandenberger is a cosmologist.)
Paolo,
In response to my statement that “”I think quantum gravity is a very constrained problem, and that string theory is very likely to be the only solution to this problem. If it is the only solution, abandoning it now would be a foolish thing to do,” you replied, “The key is the if.”
Of course! Those who think string theory is not worth pursuing, should not pursue it. Those who think it is, should.
Ever Peter now appears to agree with this.
“Those who think string theory is not worth pursuing, should not pursue it. Those who think it is, should.
Ever Peter now appears to agree with this.” – Mark Srednicki
If you’d read Not Even Wrong or The Trouble with Physics, maybe you’d see that the problem is somewhat different: a tyranny of failed ideas which just demands more research and more money every time it fails.
The problem is that it’s celebrated in advance of actually achieving anything. This was how aether went wrong. You need to check the theory in advance of acclaiming it (however, that’s easier said than done because the failures get renamed anomalies, i.e., the failure to predict the cc, and the failure of supersymmetry to predict the observed strong SU(3) force from the other forces at low energy).
amused, very interesting. Is there a review article available on this? Or a standard set of papers?