Why I am REALLY disappointed about string theory

[suprised]

it is exactly this, namely that up to now nobody is able to explain what string theory fundamentally IS.

No one knows, but there are always speculative answers and this is a rich field for bullgarbageting.

Tongue-in-the-cheek answer: this big blob of theories that one sees to emerge ist just the set of consistent quantum theories that include gravity. And "string theory" is just the proper way to parametrize it in special regions, analogous to "gauge theory" (if one decouples gravity). The question about a fundamental theory of strings would be on a similar footing as the question about what underlies gauge theory - this may be just an ill-posed question.

Of course the hope of most people is that there is a) some underlying theory whose vacuum states are given by the big blob, and b) on top of that there would be some dynamical mechanism that would weigh differently or select certain vacua. But there is no reason for a) and b).

I personaly like the idea as explained above, in that the big blob is like an abstract topological manifold M and any local quantum theory, one writes down corresponds to choosing local coordinates on M. The choice of origin of these coordinates corresponds to the choice of background around which one expands perturbatively. This does not at all mean that the fundamental object, M, would be meaningless and arbitrary, it just cannot be globally be described by local QFT. The fundamental theory, if it exists, would be some coordinate-free and thus some kind of topological theory without any local degrees of freedom.

This ties together with what I said before: there are no more elementary local degrees of freedom than we already know. Going up in energy does not reveal any new stuff. Still there exists a fundamental theory, and choosing a vacuum state produces an infinite amount of local degrees of freedom by expanding around it. This abstract underlying theory may or may not have any non-trivial dynamics.


This abstract pre-geometric theory is, I guess, similar in spirit to what the LQG people aim for. So I don't see here a fundamental disagreement.

 

[tom.stoer]

The differenc between string and gauge theory seems to be that strings allow one to solve for (SUSY/SUGRA) gauge theories as low-energy theories, so string theory somehow unifies gauge theories. Gauge theory w/o string theory is a collection of different theories only, the relation between them is construction not solution. Another difference seems to be that different (sectors of) string theories seem to be related by dualities i.e. some dynamic principles, whereas different gauge theories are related by copying the construction principles only. Going from SU(M) to SU(N) does not involve any physical principle.

Aiming for a fundamental topological or algebraic theory seems to be reasonable. But that means there are two tasks: identify the pre-geometric setting and find a principle which breaks it up into several patches and which induces or generates local degrees of freedom.

Again regarding the most promising research direction: what is your advice?

 

[suprised]

..., whereas different gauge theories are related by copying the construction principles only. Going from SU(M) to SU(N) does not involve any physical principle.

Well there are non-perturbative dualities that relate different gauge groups to each other. For example in N=2 Susy gauge theory, at large VEVS the theory looks like a pure SU(2) gauge theory, say, but at small VEVs it looks like a U(1) gauge theory with extra matter fields. Another example are Seiberg dualities for N=1 Susy Gauge theories, which relate theories with SU(n) and SU(m) and different matter content to each other. These theories, though looking "completely different" perturbatively, become equivalent in the low energy limit. That's very analogous to what happens to the higher-dimensional string theories.

Aiming for a fundamental topological or algebraic theory seems to be reasonable. But that means there are two tasks: identify the pre-geometric setting and find a principle which breaks it up into several patches and which induces or generates local degrees of freedom.

Indeed, but this is the hard part... bullgarbageting how what things should be like is easy. But actually doing it is infinitely much harder. And often one finds, by doing actual computations, that things turn out quite differently than expected. So there is not much content in bullgarbageting, unfortunately!

So that's why I don't know what to comment eg on Fra's remarks... much of it sounds quite reasonable, but putting flesh to it and make it concretely work, is 99.999999% of the problem and that's why 99.999999% of such generic ideas don't lead to anywhere, unfortunately.

Most promising for what precisely - uncovering the "underlying structure" of string theory? I don't have any concrete idea, nor do I know anybody who would. This seems a bit to hard a question to attack directly. So most ppl look for simpler toy problems where they hope to learn something about the inner workings of the theory. Most of this is quite technical work which doesn't tell much to non-insiders.

 

[tom.stoer]

Yes, that exactly my question: What is the most promising research direction in order to identify the unique, underlying, pre-geomeric structure of string theory and to identify the fundamental principle that explains how string- / M-geometry and their degrees of freedom emerge from it?

 

[Fra]

Indeed, but this is the hard part... bullgarbageting how what things should be like is easy. But actually doing it is infinitely much harder. And often one finds, by doing actual computations, that things turn out quite differently than expected. So there is not much content in bullgarbageting, unfortunately!

So that's why I don't know what to comment eg on Fra's remarks... much of it sounds quite reasonable, but putting flesh to it and make it concretely work, is 99.999999% of the problem and that's why 99.999999% of such generic ideas don't lead to anywhere, unfortunately.

For the record I fully agree with this and I certainly have no illusions that any of the things I suggests is easy. If it was easy, I would have done it already but I haven't. I certainly have not solve the problems! The way of reasoning towards solutions and problem formulations I suggests indeed also holds it's own hard problems, some of which are similar to those in ST. That is the one reason why I partly defended and sympathise with some of it's issues.

But I just think that just because it's so extremely complex, and that the journey from idea to concrete model is long, it's of even more importance so make sure we are working the the right direction, and it's true that we can not even for sure know that, but at minimum we should keep questioning our direction; it COULD save us lenghty detours. I've felt that this has sometimes been missing. In particular when experimental feedback is sparse, the science of model building becomes more important. When experimental feedback is ample model generation is not so important as it's quick to shoot down the wrong ones.

It becomes harder and harder to falsify theories, and that's why that framework whereby the candidates for falsification are important. We can't afford to spend time on carelessly built theories, because the lost investments if we are wrong are higher. This is another reason why I'm focusing more on a model building that is constructed as a rational learning or inference system. Here the "evolution" of the theory itself becomes a key point! A theory is no longer just a candidate that's either wrong and shot dead, or corroborated. It's something more, becase interesting things happen in particular when the theory is what we used to call "wrong". This however borderlines to the foundations of the scientific method, and also adaptive learning models.

There are similarities here also with string theory, as I see it. But probably more due to coincidence since the type of reasoning I advocate was not ever part of ST constructing principles, it's mostly similar to reflections that comes also from speculations towards solutions to some of the ST problems.

My original point in this thread (way back) was that I think it's superficial to dismiss ST just on the argument that it has no simple clean static timeless theory we can shoot down. Somehow it's noy what I seek either to be honest. I rather seek to understand the evolution of theories in the sense I've already mentioned, which by the way is one-2-on with the evolution of observers and particle properties.

Here it seems Tom has a slighly different critique.

But of course this is hard stuff. This is why trying to think in new terms may not be so bad after all.

/Fredrik

 

[marcus]

...This abstract pre-geometric theory is, I guess, similar in spirit to what the LQG people aim for. So I don't see here a fundamental disagreement.

Yes, that exactly my question: What is the most promising research direction in order to identify the unique, underlying, pre-geometric structure of string theory and to identify the fundamental principle that explains how string- / M-geometry and their degrees of freedom emerge from it?

Thanks, both. These observations go to the heart of the matter.

 

[tom.stoer]

your welcome :-)

it took us 255 posts to ask such a simple question, so I guess another 255 posts will suffice to derive 42 (or perhaps 10, 11,24 or 26)

 

[marcus]

your welcome :-)

it took us 255 posts to ask such a simple question,..

but many of those were interesting and enlightening posts, sometimes exceptionally so.

If as Surprised just said, he sees string/M and LQG having similar aims---they would not need to be precisely the same, for trading ideas to be productive---then we could try to learn something by comparison.

Maybe spin-networks (which are graphs labeled by group-reps) have something to suggest about the formulation of M-theory. A possibility even if seemingly remote.

Both your comments mentioned "abstract pre-geometry" as an important goal.

What would be a "post-geometry"?

Presumably to get a "post-geometry" one would throw away the continuum (the smooth manifold representing space or spacetime) and just consider the finite information which one can have.

[Information about what? ... the Umwelt? ...the space and matter relationships?...the experimenter's Experiment?...I'm sorry for the vagueness. The "what" is not mathematically represented, only the information about it.]

This is what I see happening in the two current papers that epitomize LQG and it's application to cosmology LQC: 1004.1780 and 1003.3483

Perhaps the idea is that at a very microscopic level we cannot tell if the world is smooth or not smooth. Does it even makes sense to represent it mathmatically as a set with some axiomatic structure? All we have, if we are lucky, is information from some measurements. The networks of LQG---the labeled graphs---represent that batch of information. So the approach as I see it could be called "post-geometry".

But I guess you could also think of it as an "atomic" pre-geometry. The nodes of the network are "chunks of volume" and the links of the network represent adjacency and the "glue of area" joining the chunks. Then if matter is to be added, fermions become labels on the chunks and Y-M fields are flux-labels through the glue-joints. Please don't take this concrete picture seriously . Maybe it helps sometimes to have two contradictory ways to view something, so I offer you the tension between seeing LQG as "pre-geometry" and as "post-geometry". Also since I can't claim expertise I urge anyone interested to read the March and April papers 1003.3483 and 1004.1780.

Conceivably glancing over at what the LQG are doing could help think of how the big M-gap could be filled.

 

[atyy]

your welcome :-)

it took us 255 posts to ask such a simple question, so I guess another 255 posts will suffice to derive 42 (or perhaps 10, 11,24 or 26)

Yeah, it's obvious string theory gets the wrong answer!

 

[tom.stoer]

No, it's correct.

42 = 2*24 - (10-4)

So you take twice the Leech lattice and subtract the number of compactified dimensions; it fits perfectly.

 

[tom.stoer]

It is interesting to compare LQG and ST from that perspective. Afaik Penrose already guessed spin networks w/o any indication from QG decades ago. And I don't know if the LQG guys had spin networks in mind when they started to identify the discrete structure underlying the loops / cylinder functions.

It is interesting how we construct our theories. First we try to write down an action with a huge symmetry - the larger the better. Then we work for decades to reduce these symmetries and identify the physical degrees of freedom. Crazy! There should be a shortcut from physical phenomena directly to physical degrees of freedom (I am not talking about global symmetries like flavour which are "physical"; I am talking about gauge symmetries, conformal and diffeomorphism invariance etc.).

Therefore a topological theory or an algebraic structure like spin foams is desirable for string theory. If local symmetries are an unphysical intermediate step we should try to find a different approach.

My propositions are elucidatory in this way: he who understands me finally recognizes them as senseless, when he has climbed out through them, on them, over them. (He must so to speak throw away the ladder, after he has climbed up on it.)
Wittgenstein​

 

[atyy]

No, it's correct.

42 = 2*24 - (10-4)

So you take twice the Leech lattice and subtract the number of compactified dimensions; it fits perfectly.

OK so LQG is dead

 

[Fra]

There should be a shortcut from physical phenomena directly to physical degrees of freedom

Not sure if this was part of your joke but this sounds like you are putting up a quest for a general understanding of the process wherby an observer, goes from observations to establishing a symmetry of his observations? And how the inferred symmetries kept by the observer evolve as new observations are made?

If so, I share that quest with you.

/Fredrik

 

[tom.stoer]

No, the idea to avoid unphysical symmetries was not part of the joke.

My observation is that gauge symmetries are unphysical symmetries; they introduce unphysical degrees of freedom which have to be reduced to the physical one by gauge fixing. Gauge symmetries are used to guess Larangians respecting certain physical symmetries as it is easier to guess one Lagrangian from which everything can be derived instead of guessing the Hamiltonian plus all other symmetry generators (e.g. the full Poincare algebra) with the correct interaction terms. If you have ever seen the gauge fixed QCD Hamiltonian you will understand why we cannot guess something like that.

The question is if this process "Lagrangian with huge unphysical symmetry => gauge fixing => physical degrees of freedom" which in LQG reduces the theory to a discrete structure hides some physical aspects (which we do not know) or if the whole process is somehow "physical" (even if we do not understand why).

 

[Fra]

The question is if this process "Lagrangian with huge unphysical symmetry => gauge fixing => physical degrees of freedom" which in LQG reduces the theory to a discrete structure hides some physical aspects (which we do not know) or if the whole process is somehow "physical" (even if we do not understand why).

I think this is a good question and I for one thinks there is more to understand here.

I expect that due to our different views on structural realism I have a different view of symmetry than you. The nature of symmetries in this sense, and what is physical and what's not, is even closely related to what elementso the theory that should correspond to observables in a measurement theory.

About your statement that gauge symmetry is non-physical, I understand what you mean but there is also another way of seeing it, where it's not so easy to tell if it's physical or not. I more think of gauge symmetries and transient symmetries in that sense that a completely unbroken symmetry is trivial(non-physical), and if represented will decay as it's a waste of complexity due to it's redundancy. But I'm not sure I would say that this means that they are unphysical because there is a physical reason (historical reason) why the originally broken symmetry, was restored, and then become trivial, and dissappeared from the representation.

I think these are also hard problems but these are things I've been thinking about and I'm confident that there are a lot of things in this "process" that is of physical relevance.

If you consider a totally unboken symmetry, no observer would ever be able to distinguish it, since inferring and establishing the symmetry to some degree of confidence can by understood from collecting data from cases where the symmetry is broken. Somehow the broken symmetry cases is what justifies the notion of the symmetry, but only transiently. This is what may first seem like a paradox; as much as the discussion of what's physical in GR. This is also similary to the discussion of observer invariance; and how observer invariance is observed. I see the symmetry discussion as closely parallell.

/Fredrik

 

[Fra]

Addition: Where I FULLY agree is that in some mathematical attempts to construct the theories, THEN we introduce what I would call "mathematical redundancies" that are clearly unphysical, and here I agree with you.

I just went to the next step to suggest that even beyond these "mathematical" and physically empty complexions of the system description, I still see physical process behind evolving symmetries. Ie where a symmetry is emergent, and later decayed. They plane I understand this on is on the pure inference and datacompression level where you consider economy of representation. Here one can imagine a stream of observations, that once enough data is acquired emerges with a symmetry. But if this symmetry remains unchallanged, then the value of wasting memory on this becomes less, so the symmetry can be recompressed into a more efficient structure (because it is not challanged). At some point the symmetry just becomes part of the background of the observer and takes up no memory. So the physical representation of the explicit symmetry is ur survival value only when it's not abundant in the environment. That's how I see it conceptually but it's still apparentely extremely difficult ti implement this. It's part of what I hope to do, but it's not easy.

/Fredrik

 

[tom.stoer]

Just a clarificaton: I do not talk about broken symmetries.

Another remark: gauge symmetries are a powerful principle as they allow one to construct renormalizable field theories and to derive conserved quantities. So as a tool they useful, but they are not "directly realized in nature".

SU(3)_Flavor is more or less observed directly.

SU(3)_Color is identified rather indirectly, via the fact that there are three quarks in a proton. So it's not totally unphysical. But the SU(3) Lagrangian contains unphysical gluons which have to be reduced to the physical subspace. But in the physical subspace there is no SU(3)_Color symmetry left.

Think about a two-particle system where the interaction depends only on V(x-y). That means that the system is invariant w.r.t. translations as x-y is an invariant. The "physical subspace" in QM is the sector of the Hilbert space with vanishing total momentum P, i.e. P|phys> = 0 (this is not enforced by the theory by used in practical calculations). No in this subspace there is no translation invariance anymore; it has been eliminated by setting P|phys> = 0. Something like that happens in gauge theories as well; the fully gauge fixed theory with physical degrees of freedom does not contain any gauge symmetry.

In LQG its rather similar (even so not every body agrees on the implementation of the constraints as they are on-shell instead of off-shell). Both the gauge symmetry (local Lorentz invariance) and diffeomorphism invariance vanish in the physical subspace, the space of spin networks.

But unfortunately it is by no means clear how to derive the theory w/o going through all the gauge fixing issues. Simply writing down an SF model and claiming that it is QG doesn not help. One has to show how it followes from a quantization + constraint procedure.

In LQG there is a second approach how to derive the SF models. One starts with a topological theory (w/o any physical degrees of freedom) and introduces a term which breaks the huge invariance and in parallel generates physical degrees of freedom = which increases the number of physical polarizations of the gravitational field from 0 to 2.

Now let's come back to strings

What would that mean in the context of string theory? One would have to identify a topological theory = a theory w/o physical degrees of freedom; in a second step one would have to introduce terms which break these symmetries and at the same time generate physical excitations. Because one starts with a topological theory one has a good chance to define it globally (even so one would not succeed in identifying discrete structures immediately as a detaour via smooth manifolds is still used; but this manifold need not be our spacetime, it could be some "dual" entity as well; so discrete structures could emerge from a dual structure; see for example the Fourier expansion on a smooth circle).

Has anybody thought about that possibility? Are topological strings a good starting point?

 

[Fra]

Ok, I guess you ask much more specific questions defined the context of some of the current programs. I have no good comments on that level, as I find some of these questions to be somewhat open wires or patches partially floating in the current effective frameworks. My own strategy is to restructure from much more basic levels and build my understanding from there; the connection to the floating patches are yet remote for me.

/Fredrik

 

[tom.stoer]

Of course this question goes to the string theory experts.

 

[suprised]

Just a clarificaton: I do not talk about broken symmetries.

But that's what I had in mind... ANY kind of background would correspond to a (partial) breaking of whatever symmetry the underlying theory has, and the physical excitations would be like Goldstone modes for that background. These ideas are of course not new. Already in the early days ppl were considering eg scattering amplitudes at infinite energy, where the vertex operator algebra simplifies, and tried to uncover "fundamental" symmetries this way. Certainly these attempts were naive but it the spirit was OK.

So I think the relevant concept would be symmetry breaking; gauge fixing, BRST etc is not a physical principle, it has to do with the formulation of the theory.

Has anybody thought about that possibility? Are topological strings a good starting point?

There are several notions of topological strings. What one usually means by that is a toy model for superstrings and this is not what is meant here. There are other topological theories like Chern Simons, BF etc, and this is what I had in mind. Many things have been tried, also in the context of string field theory. But AFAIK one never could make the big step between writing down some simple topological action, and then deriving something non-trivial from it. Many ppl feel that such attempts are too naive.

 

[tom.stoer]

My remark regarding "not talking about broken symmetries" was in the context of "gauge symmetries are unphysical". I did not consider symmetry breaking there and I just wanted to clarify this in my discussion with Fra.

Of course you are right in the other context of deriving a dynamical theory from a topological setting like BF theory. I know gravity as constraint BF theory from Plebanski and LQG/SF. Yes, this is somehow a breaking of the underlying symmetry, but rather different from standard symmetry breaking a la Goldstone and Higgs as it generates local degrees of freedom, something which the Higgs does not! The Higgs simply transforms an already existing scalar degree of freedom into a new polarization state = a vector degree of freedom.

BF theory seemed to me rather artificial. One starts with a topological action - which is nice - and then constrains it in order to generate gravity. How can this step be motivated? I mean, why should one consider this to be physical if one did not knew that gravity should emerge? Is there a deeper principle behind it?

Regarding gauge fixing, BRST etc. we agree.

 

[MTd2]

It's work in progress by Witten.

He anounced that he will publish the ideas of his thought on citation 14 of his new paper:

http://arxiv.org/abs/1009.6032

 

[tom.stoer]

http://arxiv.org/abs/1009.6032
A New Look At The Path Integral Of Quantum Mechanics
Edward Witten
(Submitted on 30 Sep 2010)
Abstract: The Feynman path integral of ordinary quantum mechanics is complexified and it is shown that possible integration cycles for this complexified integral are associated with branes in a two-dimensional A-model. This provides a fairly direct explanation of the relationship of the A-model to quantum mechanics; such a relationship has been explored from several points of view in the last few years. These phenomena have an analog for Chern-Simons gauge theory in three dimensions: integration cycles in the path integral of this theory can be derived from N=4 super Yang-Mills theory in four dimensions. Hence, under certain conditions, a Chern-Simons path integral in three dimensions is equivalent to an N=4 path integral in four dimensions.

therein

[14] E. Witten, “Fivebranes and Knots, I,” to appear.

... and show exactly how a quantum path integral in N = 4 super Yang-Mills theory on a four-manifold with boundary can reproduce the Chern-Simons path integral on the boundary, with a certain integration cycle. This has an application which will be described elsewhere [14]. The application involves a new way to understand the link between BPS states of branes and Khovanov homology of knots

I am sorry, but can anybody explain to me how this could guide us towards a more fundamental understanding of what string theory really is? Isn't this "yet another reformulataion"?

 

[MTd2]

I don't think so. This paper belongs to the same area and line of research of his Fields Prize.

 

[qsa]

Perhaps the idea is that at a very microscopic level we cannot tell if the world is smooth or not smooth. Does it even makes sense to represent it mathmatically as a set with some axiomatic structure? All we have, if we are lucky, is information from some measurements. .

This is the best statement I have read on PF. while the status of virtual particle as something between mathematical and real I can really understand. But GR statement that space-time is curved bugles my mind. Although, it is easy to see how it is a good modeling scheme just like virtual particles, but it is much less satisfying. I think statistical mechanics is the way to go.

 

[mitchell porter]

I don't know what else Witten's paper will lead to, but I believe it is indirectly relevant to quantizing M-theory. In fact, the philosophy is that M-theory is somehow "inherently quantum" - it has a classical limit, but the theory itself is not to be obtained by starting with that limit and "quantizing" it according to known procedures.

I have become aware of two specific technical issues. One is that the worldvolume theory of the M5-brane is "non-Lagrangian". The "geometric Langlands program" is somehow relevant here. The other is that there is no fundamental dilaton field in M-theory, so you can't construct a perturbative expansion as one does in string theory, where the dilaton field strength is the expansion parameter. arXiv:hep-th/0601141 talks about how this looks from the M-brane perspective.

I think these investigations by Witten into new perspectives on quantization pertain to these problems. Note that in the first part of this paper, he identifies an ordinary quantum-mechanical system with an "A-model" construction from topological string theory. If you turn that around, he's starting from within string theory and getting a quantum theory. Also, Chern-Simons fields show up in M-brane worldvolumes, so the second part may be relevant too.

 

[MTd2]

What is the meaning of something being non-lagrangian?

 

[suprised]

Not all QFT's have a lagrangrian description, in particular, strongly coupled ones, which cannot be represented in this way.

 

[MTd2]

Oh, that's quite a new thing for me! Well, but there's a hamiltonian description, right?

 

[suprised]

Not really. One needs to make sense of what one writes down, at the quantum level. Usually one needs to have a theory with some small parameter, like a coupling constant, and writes the theory as a perturbative series around the free theory, with this parameter as expansion variable. In this way one can compute the quantum corrections to the operators in the lagrangian or hamiltonian in a systematic manner; this is the content of the renormalization procedure.

But as has been pointed out above, not all theories are of this kind, like the M5 brane or non-critical strings in 6d or interacting conformal theories. There is no small parameter to expand into, so there exists no perturbative description of such theories and thus, no Hamiltonian or Lagrangian one would know how to write down starting from the classical one; since there is no classical one to start with.

Sometimes this is not even necessary, for example 2d conformal field theories like the minimal models. The correlators of those theories can be determined purely from consistency conditions, and one never needs to (nor even could) write down a lagrangian for them.

 

[atyy]

Not all QFT's have a lagrangrian description, in particular, strongly coupled ones, which cannot be represented in this way.

Is that proved? I read on Motl's site that ABJM is a Lagrangian for something people used to think didn't have one.

 

[suprised]

Is that proved? I read on Motl's site that ABJM is a Lagrangian for something people used to think didn't have one.

Look, this depends on the case. I was talking generically. Sometimes miracles happen ;-)

 

[atyy]

Look, this depends on the case. I was talking generically. Sometimes miracles happen ;-)

OK, thanks - I hope the next miracle is that string theory can be tied up with scotch tape

Edit: I see that has already been tried! http://arxiv.org/abs/0810.3005

 

[atyy]

Look, this depends on the case. I was talking generically. Sometimes miracles happen ;-)

BTW, in the classical case, I think all equations can be derived from Lagrangians by just adding Lagrange multiplers, but that's not always useful because of the extra variables that are really constants. Is this formal trick truly absent in the string theory cases with no Lagrangian, or is it just not useful?

 

[MTd2]

Suprised,

I am really confused. How do you know that there is a theory without eigenvalues or equation of motion?

What about the hamiltionian of these minimal models? There is the virasoro algebra, which does have a hamiltonian.

 

[suprised]

BTW, in the classical case, I think all equations can be derived from Lagrangians by just adding Lagrange multiplers, but that's not always useful because of the extra variables that are really constants. Is this formal trick truly absent in the string theory cases with no Lagrangian, or is it just not useful?

The problem is what one means by the operators resp. fields one adds - they need to be defined quantum mechanically. In the absence of a perturbative renormalization scheme, where you would start from a classcial operator or field, what do you write down for it explicitly?

As for L_0, the Hamiltonian for a mininal model CFT, you never need to write it in terms of classical fields, the only thing you need to know is the commutation relations and this suffices to solve for the correlation functions. In some cases one can do it, eg for a free theory (let's better not get into free field realizations of minimal models etc), or in supersymmetric theories where some objects can be protected from quantum corrections. But in general one doesn't know how to write down a quantum operator of a strongly interacting theory, nor determine its correlation functions.

 

[MTd2]

So, you have something that gives eigenvalues from minimal models, even though there is no classical counterpart. For a theory, one needs values to measure, so I don't see a problem in this. I mean, this is science, you have a black box, shake it, and see the outcome.

What I want to know is, how do you know that there is a theory without anything to measure? I don't understand how your 1st paragraph answer this.

 

[suprised]

What I want to know is, how do you know that there is a theory without anything to measure? I don't understand how your 1st paragraph answer this:

Well there are correlation functions that are in general non-trivial and that can be measured. This is independent from whether a perturbative Lagrangian exists or not. If not, it is hard to compute them. Even defining what your quantum operators, or observables are, is already non-trivial.

 

[atyy]

The problem is what one means by the operators resp. fields one adds - they need to be defined quantum mechanically. In the absence of a perturbative renormalization scheme, where you would start from a classcial operator or field, what do you write down for it explicitly?

As for L_0, the Hamiltonian for a mininal model CFT, you never need to write it in terms of classical fields, the only thing you need to know is the commutation relations and this suffices to solve for the correlation functions. In some cases one can do it, eg for a free theory (let's better not get into free field realizations of minimal models etc), or in supersymmetric theories where some objects can be protected from quantum corrections. But in general one doesn't know how to write down a quantum operator of a strongly interacting theory, nor determine its correlation functions.

Could you give examples of some papers that use this approach? (I'm a biologist, so it'll be all over my head anyway, so even very abstruse ones are fine.)

 

[tom.stoer]

One needs to make sense of what one writes down, at the quantum level. Usually one needs to have a theory with some small parameter, like a coupling constant, and writes the theory as a perturbative series around the free theory, with this parameter as expansion variable. In this way one can compute the quantum corrections to the operators in the lagrangian or hamiltonian in a systematic manner; this is the content of the renormalization procedure.

But as has been pointed out above, not all theories are of this kind, like the M5 brane or non-critical strings in 6d or interacting conformal theories. There is no small parameter to expand into, so there exists no perturbative description of such theories and thus, no Hamiltonian or Lagrangian one would know how to write down starting from the classical one; since there is no classical one to start with.

This is a misconception

There is absolutely no reason why "quantization" must always mean "perturbative quantization".

You can start with the QCD Lagrangian, derive a Hamiltonian, gauge fix this Hamiltonian using non-perturbative techniques (like unitary transformations) which avoids perturbative gauge fixing a la Fadeev-Popov, BRST etc.

You end up with a fully quantized theory w/o any need for perturbation expansion.

 

[suprised]

This is a misconception
You can start with the QCD Lagrangian, derive a Hamiltonian, gauge fix this Hamiltonian using non-perturbative techniques (like unitary transformations) which avoids perturbative gauge fixing a la Fadeev-Popov, BRST etc.

You end up with a fully quantized theory w/o any need for perturbation expansion.

No. The theory is not well defined at low energies. How do you compute scattering processes between nucleons with it? How do the nucleon operators look in terms of the fields you have in this lagrangian, to start with?

 

[tom.stoer]

?

Of course it is difficult to solve for the eigenstates, but that does not mean that the theory isn't well defined (in the physical sense).

Look at a very simple example, the harmonic oscillator. Nobody would solve it based on a plane wave expansion once knowing the Hermite functions. But the problem IS well-defined in terms of plane waves.

 

[suprised]

Let's say it better: you must give the theory a non-perturbative meaning - the fields you are writing down in the QCD lagrangian are strongly coupled in the IR, and don't represent the relevant degrees of freedom. So one needs to "choose better coordinates" and use a different (if you like dual) formulation of the theory, which might eg be an effective meson theory, and only in this new formulation you can meaningfully talk about long-distance correlation functions etc. The transition this new formulation is extremely complicted and thus never had been done analytically, only numerically. In the new formulation, the lagrangian of the UV theory, the associated Feyman rules, etc, don't play any direct role any more.

 

[MTd2]

So, what would be the "mathematical formula" of a theory without a lagrangian and without a hamiltonian? Certainly at the least minimum a Hamiltonian is necessary otherwise an experiment cannot be done! It doesn't seem that the non availability of a practical calculation is the same thing as not having a hamiltonian or lagrangian definition...

 

[tom.stoer]

you must give the theory a non-perturbative meaning - the fields you are writing down in the QCD lagrangian are strongly coupled in the IR, and don't represent the relevant degrees of freedom.

They do not represent the observable degrees of freedom in the IR, but they "span" the entire Hilbert space.

So one needs to "choose better coordinates" and use a different (if you like dual) formulation of the theory, which might eg be an effective meson theory, ...

that means you try to solve the theory; OK

... and only in this new formulation you can meaningfully talk about long-distance correlation functions etc.

no, that's not true; think about coherent states: they are totally different from plane waves; plane waves are not suitable for many problems in quantum optices; nevertheless formulating the problem in terms of plane waves is correct - it's complicated but mathematically well-defined.

The transition this new formulation is extremely complicted and thus never had been done analytically, only numerically. In the new formulation, the lagrangian of the UV theory, the associated Feyman rules, etc, don't play any direct role any more.

Yes it's complicated. But there is no reason to focus on Feynman rules. They are a mathematical tool only. The misconception is that in ordinary QFT textbooks there is no clear distinction between the definition of a theory in terms of a perturbation expansion and the solution of a certain class of problems in terms of a perturbation expansion. Looking at many QFT textbooks one could come tothe conclusion that Feynman rules are required to define the theory; as we have learned in the meantime this is misleading or even wrong.

 

[suprised]

I am not saying the theory has no Hamiltonian, rather one cannot write it down. In general you don't even know the relevant degrees of freedom to use in terms of which you may want to write it.

See above the discussion with Tom: the QCD lagrangian has quarks psi and gluons as perturbative degrees of freedom. Now how do you write the theory at low energies, where these fields are strongly coupled? How do you now which other operators are the relevant ones? In the QCD case you have a crude idea of what happens, as there is the naive picture of mesons and nucleons being composed out of quarks.

But now imagine a different theory, which is strongly coupled as well, but has no underlying QCD lagrangian, and no weakly coupled degrees of freedom; like eg the M5 brane theory; so what variables would you choose, if all possible ones are strongly coupled?

PS: ok now I need to go to bed, more tomorrow.

 

[tom.stoer]

... the QCD lagrangian has quarks psi and gluons as perturbative degrees of freedom.

This is not correct! They are not "perturbative" degrees of freedom only!

Do you know how the gauge-fixed QCD Hamiltonian looks like? Do you know that it is essentially non-perturbative?

But now imagine a different theory, which is strongly coupled as well, but has no underlying QCD lagrangian, and no weakly coupled degrees of freedom; like eg the M5 brane theory; so what variables would you choose, if all possible ones are strongly coupled?

You do not need quarks and gluons as weakly coupled degrees of freedom. Everything is fine even in a regime where they are strongly coupled.

 

[atyy]

I was under the impression that most physicists think QCD is completely non-perturbatively defined (but maybe the Clay Institute differs?). I had assumed suprised was talking about neither QCD nor AdS/CFT?

 

[MTd2]

I guess I know what suprised wants to find. Among the fundamental string theories, you have S-symmetry. One can go from one to another, come/go, by taking the strong/weak of each one. I will show an exert from wikipedia:

"S-duality relates type IIB string theory with the coupling constant g to the same type IIB string theory with the coupling constant 1 / g. Similarly, type I string theory with the coupling g is equivalent to the SO(32) heterotic string theory with the coupling constant 1 / g. Perhaps most amazing are the S-dualities of type IIA string theory and E8 heterotic string theory with coupling constant g to the higher dimensional M-theory with a compact dimension of size g."

http://en.wikipedia.org/wiki/S-duality

Regarding the last one, there are 2 main objects in 11d sugra, M2 and M5 branes. This theory is supposedly the low energy of m-theory, but given that it is a non renormalizable one, finding its true quantized versions in m-theory is not trivial.

Due the s-duality, they are related to D3 and D5 branes on E8 heterotic strings. These branes are somehow related by topological relations in their connectijons, called gerbes (D3) and twisted gerbes (D5). Since these live in a renormalizable theory, string theory, there is hope that using the relations found for them using M2 and M5 branes forms it is somehow possible to find their quantum version and thus the m-theory itself. Notice that the dimensionality of D3 branes is the same of chern simons topological theory, so maybe what witten is doing now it is to find a new symmetries between M2 and M5 branes ( due the S-duality).

So, you have a theory that supposedly exists due to these considerations, m-theory, as well its probable fundamental objects, M2 and M5. But you cannot find them so fast because they are related by a dual relation of coupling constants. Finding corresponding objects among string theory is straightforward, relatively speaking, because you have both theories from the beginning. This is not the case though with m theory.

 

[tom.stoer]

I know S duality and of course I agree with the description below

... Among the fundamental string theories, you have S-symmetry.

"S-duality relates type IIB string theory with the coupling constant g to the same type IIB string theory with the coupling constant 1 / g. Similarly, type I string theory with the coupling g is equivalent to the SO(32) heterotic string theory with the coupling constant 1 / g. Perhaps most amazing are the S-dualities of type IIA string theory and E8 heterotic string theory with coupling constant g to the higher dimensional M-theory with a compact dimension of size g."

I know that one conjectures the existence of M-theory b/c due to these dualities.

Let me first comment on a few statements before coming back to my conclusion:

Regarding the last one, there are 2 main objects in 11d sugra, M2 and M5 branes. This theory is supposedly the low energy of m-theory, but ... finding its true quantized versions in m-theory is not trivial.

Agreed.

Since these live in a renormalizable theory, string theory, there is hope that using the relations found for them using M2 and M5 branes forms it is somehow possible to find their quantum version and thus the m-theory itself.

Of course any attempt to identify the underlying M-theory is welcome.
(Perturbative) renormalizability of string theory is a bold statement
- afaik the superspace measure beyond two loops has not yet been constructed
- finiteness up to all orders has not been derived rigorously
- convergence of the summed perturbation series is not to be expected
So perturbative renormalizability does not really help. It was helpful in QCD b/c of asymptotoc freedom only.

... you have a theory that supposedly exists due to these considerations, m-theory, as well its probable fundamental objects, M2 and M5.

I have seen different conjectures regarding its fndamental objects (branes, matrices, ...) but let's assume for the moment that M2 and M5 branes are inded what we are looking for.

But you cannot find them so fast because they are related by a dual relation of coupling constants. Finding corresponding objects among string theory is straightforward, relatively speaking, because you have both theories from the beginning. This is not the case though with m theory.

First you say that M2 and M5 branes are the fundamental objects; then you say that you can't identify them b/c you do not know M-theory. That's somehow contradictory.

Please have a look at QCD again:
1) one had a web of relations (not dualities) like chiral symmetry considerations, current algebra, (chiral) bags and non-rel. quarks model (which somehow already used the fundametal degrees of freedom, but in a "dressed" version)
2) the fundamental degrees of freedom where not known; later they where conjectured from deep inelastic scattering, but still the dynamics (Lagrangian, Hamiltonian) was not known.
3) due to asymptotic freedom it was possible to define the theory perturbatively - in a certain regime!
4) again later it was possible to define the theory by different methods and in different regimes using the same fundamental degrees of freedom.
Please note that all the effective theories mentioned above did not help mathematically in defining the theory! There were indications regarding what the underlying theory must reproduce, but w/o experiments or w/o an educated guess SU(3) would never have been identified!

Assume for a moment that the same applies to M theory. As we cannot be sure what its fundamental degrees of freedom are and as S duality cannot be proven rigorously (but only in certain limits) it is not clear if the above mentioned results really allow us to identify the fundamental degrees of freedom. Why do we assume that just this rather special M2 / M5 based theory is the true fundamental theory - and not "just another effective theory"?

In QCD the major break through was to identify fundamental degrees of freedom that were valid in the whole theory space, nut just in a specific regime! Restricting M-theory via M2 / M5 branes to a certain regime might be a step into the wrong direction as we are moving away from our main target to construct a theory valid in full theory space.