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.
amused,
Regarding (ii), is there any reason to believe that a continuum limit can be taken, given that perturbation theory gives you Landau poles? In QCD asymptotic freedom comes to the rescue.
anon,
I’ve read The Trouble with Physics. I disagree with most of it, especially the accusations of blatant racism and sexism.
As for the failure of string theory, allow me to stipulate that. Yes, string theory has so far failed to make a definite prediction. But so has every other appraoch to the problem of quantum gravity. Lee has pushed the idea that violations of Lorentz invariance might turn up in one particular flavor of LQG. But there are versions of LQG that are Lorentz invariant from the start, so this cannot be said to be a prediction of LQG. It seems to me that this is exactly analogous to statements that certain classes of compactifications in string theory make certain predictions, but when all possible classes are considered, no prediction is common to all. Similarly, LQC makes no predictions that are common to all versions of it that are currently under investigation.
There is then a subjective issue of which framework is closer to making a prediction. Peter evaluates this in terms of the number of bong hits needed to believe any particular scenario, which is as good a measure as any.
For me, string theory currently needs about 2 bong hits, the Lorentz-invariant flavors of LQG maybe 4 (though if one of them works I would expect it to also have a string interpretation), and I’m passed out before we get to anything else.
Your opinion may differ. Lee’s certainly does.
So, how are we going to fund things? I believe it is an intellectually defensible position to say that none of this stuff should be funded. It’s all useless fluff that is not going to feed the hungry or house the homeless.
Assuming you don’t agree with that, I think the current system, while far from perfect, is pretty good. In his book, Lee says that there are about 200 people working on LQG. There are maybe 2000 (I’m guessing) working on string theory. I think this ratio is about right, when translated to the bong-hit scale (which I see as logarithmic). This sounds like an awful lot of people overall, and it is, but the difficulty is that it’s awfully hard to figure out which people to keep, if you’re going to cut some. The current system has many flaws, but it does have the virtue that many different people are making hiring decisions at many different institutions.
Dear Mark,
Many many thanks for the explanation (and for the optimism) on the Gauss law and non trivial representation. Let me try to think about it. (The “off debate” fragments are very nice, in my opinion. In this thread also some of the debate part seemed interesting.)
Gina
Mark,
Thanks. The standard paper for the overlap formulation is hep-th/9411108; for the GW formulation see hep-lat/9811032 and hep-lat/9904009.
For reviews, see, e.g, hep-th/0102028 and hep-lat/0009033. The stuff about anomalies and index theory that I mentioned was subsequent to these reviews but it should be easy enough to track down by a literature search if you are interested.
A small comment on the general debate here: The discussion often gives the impression that there is nothing else interesting and worth working on besides quantum gravity and Theories of Everything. In fact there are plenty of interesting “bottom up” things to do. For example, the QCD dynamics responsible for confinement and spontaneous breaking of chiral symmetry still seems far from understood. E.g. there is so far no definitive explanation for why the phase transitions in finite temperature QCD where deconfinement and chiral symm restoration set in occur at the same critical temperature (which lattice simulations have revealed to be the case). Isn’t it an interesting problem to try to explain that? I guess it’s pretty mundane compared to the glamour of quantum gravity, but on the other hand there’s surely a much better chance of making some real progress in understanding physics.
gs, re. #376:
To tell the truth I have no idea how things will work out regarding (ii). My guess, for what it’s worth, is that it will be possible to take a continuum limit but that the theory might not be in the phase that we would like it to be in, e.g the electroweak part in a confined phase. In that case I guess we would have to move on to lattice formulation of GUTs or maybe supersymmetric extensions of the SM…
“Mark
….A small comment on the general debate here: The discussion often
gives the impression that there is nothing else interesting and worth
working on besides quantum gravity and Theories of Everything. In
fact there are plenty of interesting “bottom up” things to do….”
-amused-
Dear amused
I think that you’re preaching to the choir here. If you take a look at Mark’s publications (you can search for him on the Stanford Public Information Retreival System (SPIRES); go to: http://www.slac.stanford.edu/spires/ and type in your search), I think you’ll find that while he has done some (important: see his paper “IIB or not IIB”) work on string theory, this work represents only about 6 of his 100+ papers. His thesis was on lattice gauge theory, and since then he has done important work on axions, technicolour models, Grand Unified Theories, supersymmetric model building, early universe cosmology and inflation, baryogenesis, black-hole thermodynamics (see his paper “Entropy and Area”), temperature anisotropies in the cosmic microwave background radiation, and cosmic reheating after inflation.
He is an acknowledged world expert on the particle astrophysics of dark matter, having coauthored the first paper computing neutralino relic densities in the full susy standard model, and he has continued to be a leading authority in the field over the succeeding quarter century, with over thirty papers on the subject to date. He is also the author of an exceptionally clear and pedagogical textbook on Quantum Field Theory, which just appeared this last February.
You should not assume that just because someone defends the value of continuing to work on understanding string theory, that they work primarily (or even at all) on string theory. There is considerable interest in string theory among cosmologists, astrophysicists, particle phenomenologists, and heavy-ion nuclear theorists, because it acts as an “imagination stretcher” (to use a phrase first coined by Bjorken) for the fields with which it interacts.
amused,
Thanks for the references. I agree completely that there’s plenty of interesting “bottom up” stuff to do. So do all the string theorists that I talk to. I don’t know anyone who wouldn’t be very interested in results on properties of strongly-coupled chiral gauge theories from lattice simulations. (And let’s not forget that we learned a tremendous amount about strongly coupled supersymmetric gauge theories from the work of Seiberg and other string theorists, all of whom would be thrilled if they could extend their results to nonsupersymmetric cases.)
As for Landau poles, I wouldn’t think that they would be much of a problem; they mean you won’t be able to go all the way to the continuum limit, but you never do anyway (in an actual simulation).
Dear Peter, concerning your comment #363:
“…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.”
Do you mean this comment:
1) For classical BRST (which is rigorous via Stasheff e.a., so I won’t comment), or
2) For quantum BRST, but within the physics understanding of it (for which my understanding falls short, so I won’t comment), or
3) For quantum BRST, but from a mathematical point of view?
If it case (3) could you please give me a reference? The maths of quantum BRST (one of my pet projects) may be a point of argument.
My apologies for adding noise to the string debate (I’m also an opponent – see my post #204), but the BRST-issue is something which matters to me independently.
Diogenes,
I’m blushing. Yes, I like to work on lots of different things; apparently I have a poor attention span. For the past few years I’ve mostly been working on quantum chaos theory, work that doesn’t show up on spires.
I haven’t done much work on string theory because I’ve haven’t had many good ideas about it. (The IIB or not IIB stuff was a lucky exception.) But the people I know who do work on it are all incredibly smart, in all the best ways. I’m offended by the ridiculous caricature of an insular community of blinkered blowhards that some people seem to hold to.
String theory feels right to me. I didn’t always think that way. I first heard about string theory in 1982. I was a postdoc at Princeton, and one day Ed Witten and I were the only two people who showed up. (I think it was a Monday holiday and we were the only ones who didn’t notice.) At lunch, Ed asked me if I’d ever read any of the papers of Green and Schwarz. I replied that I’d never heard of Green and Schwarz. He then told me about string theory, that there were three different versions, one that gave phi-cubed theory in 26 dimensions, one that gave N=4 U(N) supersymmetric gauge theory, and one that gave N=8 supergravity. “I admit that it’s hard to see the charm of phi-cubed theory in 26 dimensions,” he said, “but you’ve got to admit that the other two cases are interesting!” So, after lunch, I went to the preprint library and found the latest paper of Green and Schwarz. It was about computing the mobius amplitude in Type I theory. It was horribly technical and completely incomprehensible to a neophyte. “No way can this stuff be relevant for real physics,” I thought. I made no more effort to understand string theory.
Some time after the November Revolution of 1984, I had a job in Santa Barbara, and Ed came to town. There had been recent papers with a solution of the doublet-triplet splitting problem in stirng-based models. (If you don’t know what that is, it doesn’t matter.) I told Ed that I found that impressive. He laughed. “That it’s a finite theory of quantum gravity doesn’t impress you, but a solution to the doublet-triplet splitting problem does? Well, I expect that everyone will soon be working on it, for one reason or another!” He went on to say that he thought the next major advance would be a solution to the cosmological constant problem. “We’ve been trying to solve it without having the right theory of quantum gravity,” he said. “Now that we do have the right theory, it should be much easier.” (All quotes from ancient memory and almost certainly not perfectly accurate.)
Well, Ed was right about some things and wrong about others. Time will tell what’s right and wrong about our current ideas.
Hendrik,
I do mean quantum BRST, but as far as I know in the case of 4d gauge theory all that exists are physicist’s arguments, no well-formulated mathematical theory. There’s a long story about different versions of BRST, and their precise formulation in QM and 1+1 d QFT, and it would take a long posting to say sensible things about this topic. My only point was that, no matter what your version of quantum BRST, you are using homological techniques to isolate an invariant piece of some non-trivial representation, and understanding how this works out requires working with non-trivial representations.
Diogenes,
Don’t worry, I know who Prof. Srednicki is and that he has made many important contributions across a wide range of topics. My comment was mostly sparked by his amusing description of the relative number of bong hits needed to believe in string theory and LQG. After reading it I thought: hey, there are still topics that are interesting to work on without any bong hits, right? I’m sure Mark and many others knew this, but it still seems worth mentioning now and again since otherwise an impression gets created that it has to be either strings or LQG.
Mark,
Thanks for your remark about Landau poles and the continuum limit – that’s good to bear in mind in this business.
As for string theorists also being interested in non-string bottom-up work, that’s definitely my impression as well, at least for the more senior ones. On the other hand, the reality is that if you want a string/brane theorist to hire you then you had better be working on their topic. (That is of course true across other topics as well.) Since most people in formal particle theory are doing strings/branes, that leaves very few possibilities for someone working on something else. I would have liked to continue working on lattice chiral gauge theories, but felt obliged to drop it some time ago and focus on other things to have a chance of remaining employable. This isn’t a moan about my own situation (which is actually ok for the time being); I just think it is a shame that there are interesting non-string topics in formal particle theory but young people would be putting themselves at a disadvantage careerwise by working on them.
Dear Peter,
Thanks, I appreciate your reply.
I know there are many BRST’s, some ill-defined & inequivalent. I asked the question because Hamiltonian BRST is defined independently from gauge theories (e.g. by Henneaux) purely as a constraint reduction algorithm, and in this form can be analyzed mathematically.
“I’ve read The Trouble with Physics. I disagree with most of it, especially the accusations of blatant racism and sexism.
As for the failure of string theory, allow me to stipulate that. Yes, string theory has so far failed to make a definite prediction. But so has every other appraoch to the problem of quantum gravity…” – Mark Srednicki.
As Not Even Wrong points out, even the few things string should predict in an ad hoc way are totally wrong: using the measured weak SU(2) and electromagnetic U(1) forces, supersymmetry predicts the SU(3) force incorrectly high by 10-15%, when the experimental data is accurate to a standard deviation of about 3%.
It’s also not true that no other theories make definite predictions in gravity; what you should right is that the only other approaches string theorists take seriously and actually read are failures. It’s a big difference, mainly focussed on the idea that gravity is definitely due to spin-2 gravitons which interact with one another, creating massive problems at very high energy. There’s no evidence for that. So the framework of ideas you are interested in is entirely speculative, and not necessarily scientific.
I’ve had a paper published in a peer-reviewed journal (not a gravity-related journal!) which in 1996 predicted the 1998 observational discovery that there is no cosmological slowing down on the expansion, because any quantum gravitational mechanism should suffer from gauge boson redshift (energy degradation) when exchange occurs between rapidly receding masses, over large distances in this universe. It also post-dicts the gravitational coupling constant acurately. Even Nobel Laureate Phil Anderson points out that the simplest resolution of the cc is that it is zero:
“the flat universe is just not decelerating, it isn’t really accelerating” – http://blogs.discovermagazine.com/cosmicvariance/2006/01/03/danger-phil-anderson/#comment-10901
Supporting a cc is zero, so exchange radiation redshift effects weaken the gravitational coupling constant and cause the lack of gravitational deceleration, there is Lunsford’s unification of electromagnetism and general relativity http://cdsweb.cern.ch/search?f=author&p=Lunsford%2C+D+R which was censored off arxiv without explanation despite being published in a peer-reviewed journal, Int. J. Theor. Phys.: 43 (2004) no. 1, pp.161-177. This shows that unification implies that the cc is exactly zero.
This is why the whole stringy framework is harmful. I can’t imagine Smolin censoring out alternatives to the degree that string theory does. The accusations of racism and sexism are not blatant: where they are made in every case a reference is given. They’re not invented accusations. Face the facts, please.
anon. said
“The accusations of racism and sexism are not blatant: where they are made in every case a reference is given. They’re not invented accusations. Face the facts, please.”
This is absolutely and demonstrably false. I quote from TWP:
“There are rules and ethics of confidentiality that prevent me from giving examples, but there are several detailed studies that tell the story (2).”
(2) references the important and useful MIT study on women in science, but it certainly does not document the “blatant prejudice” claimed by Lee.
Lee is using a cheap rhetorical device by trying to associate LQG with women and minorities and critics of LQG with sexists and racists. I find it one of the more deplorable parts of his book.
anon,
You wrote “As Not Even Wrong points out, even the few things string should predict in an ad hoc way are totally wrong …”
I have no idea what “should predict” and “in an ad hoc way” are supposed to mean. Either string theory makes a prediction, or it doesn’t. Currently, string theory, considered as a general framework, does not make any definite predictions for the parameters of the Standard Model (or any plausible extension thereof). There may (or may not) be one (or more, maybe many more) compactifications of string theory that are consistent with the Standard Model. Each of these (if one or more exists) will make definite predictions (in principle) for all observable physics. I say “in principle”, because we will only be able to do the relevant calculations in a regime where there is a suitable expansion parameter, and it is logically possible that there is no such regime, other than the low-energy limit where weakly coupled field theory works. (On the other hand, even strongly coupled string theory might eventually be tackled with numerical methods, just as we can now tackle certain strongly coupled gauge theories.)
You write, “[Smolin’s] accusations of racism and sexism are not blatant: where they are made in every case a reference is given.”
Lee does not give any details of any of case he has personally witnessed, citing confidentiality rules. His references (that, he says, “tell the story”) are http://web.mit.edu/fnl/women/women.html and http://www.aps.org/educ/cswp . The latter page no longer exists; CSWP is the Committee on the Status of Women in Physics, and their reports can now be found at http://www.aps.org/programs/women/reports/index.cfm .
I invite everyone to compare these reports with what Lee says in his book.
Dear all,
I found the off-debate discussion on non trivial representations, QED and Gauss laws very interesting and also the related more advanced comments. (Also there are quite yummy off debate comments like Mark’s memories and comments about Witten, Susskind and others.)
I feel very fortunate to have received three different explanations #339 #347 and #355 on why non-trivial representations cannot occur in QED without violating Gauss’ rule. (It looks from what Moshe and Mark said that their argument applies in much greater generality, but I am happy to think just on the “simple” case of QED.)
Mark commented that his explanation is not just an elaboration of Moshe’s but rather a different one and that the connections between these two explanations is actually quite deep. This type of different explanations for the same phenomenon and then deeper relations between the explanations themselves that sometimes lead to further understanding always stroke me as a very nice feature of science.
(Actually, I did not understand the bottom line of Hendrik’s #355 comment, beyond some technical statement from mathematics that of course I did not understand. Is it a different explanation why non trivial representation would violate the Gauss law or perhaps a different view? I do not mind not understanding the mathematics, of course, but please please Hendrik do explain what is the bottom line of your comment.)
Everybody (including Peter) seem to agree that invoking non-trivial representations is not a good idea to proceed in the context of QDE (and Moshe and Mark point out that this seem to apply also to cases where Peter original suggestions may refer to.)
So let me ask the following: Is it actually impossible (or perhaps, why is it impossible) to offer a mathematical model for QED which will involve non trivial representations and will give precisely the same empirical predictions as the current model (including Gauss law, of course)?
If I understood Mark’s explanation such a possibility will require a large amount of “cancellations” because individual terms will go contrary to Gauss’ law; perhaps it also follows from what Mark said (but I am not sure about it) that such a model is not possible under the roof of QFT. From Moshe’s comment I find it hard to understand if this is impossible or just “unlikely” or “unknown”. Moshe uses the words “The only known way to do it is to impose local gauge invariants…” so it not clear to me if not “known” means “mathematically impossible” or really just unknown.
Why do I think this question may be interesting?
First of all, there is a rare consensus from Peter to Mark that it is not interesting, so by some sort of Marphy’s rule maybe this may be a place to look again 🙂 .
Second, while such a possibility will make no difference for QED, it may make some difference for cases where similar arguments are not only more complicated but also are more questionable.
Third, of course, QED seems the simplest, for this case (as I remember from Peter’s book) the frightening name “representation theory” is just plain old “trigonometry”.
(Also a fourth reason may be that I vaguely remember reading somewhere that there is no fully accurate mathematics model for QED.)
Two little less off-debate remarks:
I would like to support what Mark said about string theorists. I doubt if anybody really knows if string theory is the theory, but there are very good reasons to say that string theorists are the theorists; in terms of their wide horizons, intellectual and technical power, and creativity. Of course, there are very bright people in other fields of science and outside science as well. Perhaps even in wall street (but I do not think the smartest people go, as Horgan and G. Johnson would like us to believe, to wall street). There is something special about string theorists.
Another small comment is that I looked at the slides of Peter’s talk. It looks like a very good talk which refer mainly to the first half of his book which is not about string theory. (I do not like Peter’s anti string polemic and repeatedly criticized various aspects of it but the the first half of his book is, in my opinion, very good.) There is no harm if among the 40-60 slides there is one slide with some of Peter’s own ideas (or even wild speculations) on what direction he feels it is fruitful to proceed.
Mark´#371,
I don’t understand this comment. By also probing our system with divergent transformations, we can learn more about it than if we only use local and global transformations. Why is it illegal to learn more about the system?
Consider CFT as an example. If we wish, we can restrict attention to positive conformal transformations – L_m with m > 0. There is nothing inconsistent with this restriction, and the positive L_m in isolation generate a gauge symmetry; there is no anomaly in the positive sector and thus a nilpotent BRST operator. If you insist on preserving this gauge symmetry, introducing the negative L_m is illegal; it leads to an anomaly. The reason is that we must now consider a tower of systems, where your original anomaly-free system is the ground state. Your original gauge generators still annihilate this vacuum, but not the tower above it.
At least in CFT, the introduction of this gauge-violating tower leads to much non-trivial information. One should also observe that this line of reasoning has led to dramatic discoveries: the multi-dimensional Virasoro algebra, lowest-energy representations thereof, and a formulation of physics which takes the quantum nature of the observer into account. One feels that with such a spectacular track record, this idea must be an important part of The TRUTH.
Re: Thomas Larsson Comment 393
Dear Dr. Larsson
In the case of two dimensional critical phenomena (statistical field theory) the conformal symmetry is a global symmetry of a conventional field theory, and the scale anomaly can give anomalous scaling at the critical point (ie. a representation of the Virasoro algebra with c not equal to zero).
This symmetry is not gauged so there is no problem with an anomaly in it.
In the case of the string world sheet the conformal symmetry is a (conformal) gauge fixed remnant of local general coordinate and Weyl symmetries. As such it is a gauge symmetry and we do not allow it to be anomalous. That is the origin of the critical dimension in string theory, which arises from requiring that we have enough world-sheet “matter’ fields contributing positive central charge, to cancel the contribution of the reparametrization ghosts which contribute negative central charge, to give total central charge zero and cancel the anomaly.
Ad hoc is Latin “for the purpose required”, in other words, predicting something that we already know. This is no use unless it also predicts something else that can be tested.
The problem with the standard model and string theory is not that string theory can’t predict any of the parameters, but that it increases the number of parameters from 19 to at least 125 (for the minimally supersymmetric standard model).
So your statement that “string theory, considered as a general framework, does not make any definite predictions for the parameters of the Standard Model (or any plausible extension thereof” is plain misleading; you should be writing something more along the lines that string theory makes the empirically developed standard model uncheckable by creating a landscape of about 10^500 models; supersymmetry increases the number of fine tuned parameters from 19 to at least 125 without explaining any of them; far from even attempting to explain any physical reality, string theory makes a complete mess of physcs and due to the landscape has no possibility of ever achieving anything. This isn’t a new problem, Feynman was complaining about it just before he died nearly twenty years ago!
Diogenes,
As I pointed out, the subalgebra generated by L_m with m > 0 contains no reference to the conformal anomaly. Hence this subalgebra admits a nilpotent BRST operator, and can be viewed as a gauge symmetry. The BRST operator for the full Virasoro algebra is not nilpotent, and hence a global symmetry.
What seems to be little known among a younger string theorists is that also the subcritical free string can be quantized consistently, despite the conformal anomaly. Look at the first sentence of section 2.4 of GSW, which clearly states that the free string can be quantized with a ghost-free spectrum for all D <= 26. That is D <= 26, not D = 26. The interacting string runs into trouble when D < 26, but the free subcritical string is a nice example of a consistent, anomalous gauge theory. More precisely, a classical gauge theory which becomes a non-gauge theory after quantization.
Correction to something I wrote in the discussion of lattice chiral gauge theories in #364: The part in the discussion of (ii) starting with “My own hope (…) is …” is complete nonsense! I guess I must have been thinking about the possibility to investigate strongly coupled chiral gauge theories via strong coupling expansion in the lattice model and forgotten that I was supposed to be talking about the SM. Instead of what I suggested, the sensible thing to do re. (ii) would be to explore the phase structure of the lattice model and hopefully find a phase that has the right properties for the real world (confined quarks and gluons, deconfined leptons photons and W,Z bosons…; it should certainly not be a phase where everything is confined, which is where you would end up if my mistaken suggestion in #364 was followed). Then look for points at the boundary of that phase region in the space of couplings at which correlation lengths diverge and a continuum limit can be defined.
I suppose it could happen that there is a whole family of points where a continuum limit can be taken. This would give a “landscape” of possible continuum theories, and we would have to find which (if any) describe the real world. Well, at least that landscape would be easier to explore than the string theory one!
As mentioned in #364, a practical difficulty is that chiral fermion determinants are complex-valued and therefore cannot be handled by numerical lattice simulation methods. However, it may be possible to still investigate the the theory qualitatively in a “quenched approximation”. In lattice QCD the quenched approx is setting the fermion determinant equal to 1 – that might sound drastic and ad hoc, but there are theoretical reasons for why it is quite reasonable, and quenched lattice QCD has shown itself to give quite a reasonable description of the real thing. However, in an anomaly-free chiral lattice gauge theory, setting chiral fermion determinants equal to 1 might not be as justifiable as in QCD – something more elaborate may be required. (I have an idea for what the `something’ should be, and I’m sure you’re all just dying to know, but this isn’t the time or the place…)
Dear Thomas:
The book by GSW was published in 1987. A lot of the non-critical strings have been studied since. It would seem by your comment that your information is somewhat out of date.
The modern resolution for those theories (subcritical) is that the Lioville mode does not decopule. When you take it into account you get a non-trivial CFT on the worldsheet, and the criticality of the string is restored by the Liouville mode.
Also, the so-called linear dilaton backgrounds are studied
as examples of “non-critical” string theories. This is an active topic of research and many young string theorists do study those setups. They might be using different language than the one you are used to.
The no-ghost theorem was, I believe, proven in the 1970s. That people have studied Liouville theory later does not invalidate it.
David B (erenstein?),
In fact, Liouville theory is the reason why I believe that spontaneous symmetry breaking may be traded for anomalous symmetry breaking. Classically, the worldsheet metric has no physical components; it has three components but there are three gauge symmetries to cancel them, two diffeos and one conformal. After quantization, the trace of the metric becomes physical. Depending on your formalism, this can manifest itself in different ways:
1. If you start by gauge-fixing diffeos, you get a conformal anomaly.
2. If you start by gauge-fixing conformal transformations, you get a diff anomaly, cf hep-th/9501016 by Roman Jackiw.
3. If you add a self-interacting scalar field with c = 26-D, the total anomaly cancels, but you have an extra Goldstone boson.
Either way, you get one extra degree of freedom.