Why I am REALLY disappointed about string theory

[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.

 

[suprised]

Oups, that turned out to be a Pandora's box. Let me put M-Theory aside, and comment on QCD.

Of course QCD is non-perturbatively defined, sorry for the imprecise way of writing. What I meant is that the degrees of freedom, in terms of which you write the QCD lagrangian, are ill-suited to describe IR physics, because they are strongly coupled there. They don't exist as asymptotic states! There is no scattering process where they would figure as incoming and outgoing states, at larges distances, so in this sense quarks and gluons are not meaningful observable quantities at low energies. Thus the usual QCD lagrangian is the "wrong" formulation to describe IR physics.(*)

One could say that the QCD lagrangrian encodes the observables of the UV theory in a direct, perturbative way, which is amenable to explicit computations, but the IR observables in such a complicated, non-perturbative way that it is practically useless for describing IR physics (I am talking about analytical, not numerical lattice computations). For describing IR physics, other variables, like meson fields, should be introduced. Similarly, for describing physics at strong coupling for a large number of colors, the good variables become type II strings on AdS5xS5 (putting Susy aside).

So I agree with Tom on most, perhaps not all points.

At any rate, the important issue we all agree on is that a QFT is more general than just perturbation theory, or Feynman diagrams, which is derived from some lagrangian containing weakly coupled degrees of freedom. While in the case of QCD a regime (namely the UV regime) with a weakly coupled lagrangian description exists, there are other theories, like certain M-branes, for which such a lagrangian formulation (apparently) does not exist, so that these are intrinsically quantum and not approximations to some classical theory. This is what one loosely refers to as "non-lagrangian theories".


(*) Edit: removed "of course, strictly speaking, it defines the theory everywhere, in the sense of spanning the complete Hilbert space".

 

[suprised]

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.)

Any good book on CFT should do. For example "Conformal Field Theory" by Di Francesco et al. Google for conformal bootstrap and you are led to page 186 of this book where this is explained (I have probs linking it).

 

[MTd2]

Tom, look at above again. I am not suprised. You wouldn't be surprised that I agree with you. I am just trying to understand him.

 

[tom.stoer]

@suprsised & MTd2: I perfectly understand, I see you understand - and I see that we agree in many points; so Pandora's box is now closed again (but do you know what Pandora did not set free when closing the box?)

I think there is one point where I do not agree, namely that "IR observables ... in such a complicated, non-perturbative way that it is practically useless for describing IR physics (analytically, not numerically)". But that's not our point here, b/c we discuss M-theory, not QCD.

But it seems that this is a much more severe problem in M-theory than in QCD; that weakly coupled versions do not span the whole theory space (whereas in QCD the degrees of freedom which are weakly coupled in the UV are complicated but do span the whole theory space).

One question: does that really mean that M-theory cannot be defined in the whole theory space in principle? has this been ruled out? or does it only mean that it's hard to solve M-theory but that there's still hope to find an appropriate definition which allows a global definition?

 

[MTd2]

Let me see if I understand what you mean.

Between the 5 fundamental superstring theories there are dual relations. But regarding 11d sugra compactified on a circle there might be nothing relevant on the sugra side of the duality, and that this is just a coincidence.

 

[tom.stoer]

I am not sure if I understand.

Let's try it differently:

1) are there results showing that a fundamental representation (globally valid in the full theory space) is not available in principle? or are we simply not clever enough to identify / construct it?

2) if this description does not exist: are the current approaches sufficient to cover the full theory space using the known "patches" (5 * ST, 1 * SUGRA, some M-theory corners)? or is still something missing?

 

[atyy]

Any good book on CFT should do. For example "Conformal Field Theory" by Di Francesco et al. Google for conformal bootstrap and you are led to page 186 of this book where this is explained (I have probs linking it).

Let's see if I got this straight. The view is nth order correlations are all that can be measured experimentally. So we specify the correlations directly by constraining their symmetry (instead of indirectly via a Hamiltonian or Lagrangian)?

 

[tom.stoer]

It's great that this thread is still kept alive ...

... but in order not to lose focus I would liketo come back to the basic questions:

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? Are there fundamental degrees spanning the whole theory space and allow for non-perturbative calculations?

Another bunch of questions: we have this web of dualities relating 5 * ST, 1 * 11-dim. SUGRA and 1* M-theory. As we still do not know what M-theory really IS: why can we be sure that M-theory is really the mother of all other theories just mentioned - and not "just another theory related by a certain duality"? And why do we expect that there are no other theories still to be identified? What makes M-theory special - not b/c of our expectations bjut b/c of the facts we know about it?

Do I still ask the right questions? If not, what are the appropriate ones?

 

[Haelfix]

The nonperturbative sector of string theory is something that they started to really probe in the 90s and 2ks and of course is where it starts to get difficult interpretation wise.

I believe its important to distinguish the word "M theory" in the sense of the 11 dimensional theory that when compactified on a circle yields type d=10 type IIA string theory and Mtheory in the sense of 'mother theory' which contains everything ever learned about string theory.

The former is just another limit of the stringy 'configuration' space and/or an equivalent description of the same physics of some other corner. The latter is something else entirely and the answer to the question 'what is string theory'?

I think it was thought for awhile that the 11 dimensional theory really was the master nonperturbative description of all of string theory, but for one reason or another that point of view is passe.

As to what objects 'span' the full stringy Hilbert space. Well, its highly ambigous. The naive answer would be to say the fundamental string. But then that object doesn't exist in the 11 dimensional Mtheory (only M2 and M5 branes) which is the nonperturbative limit of maximal supergravity. However Mtheory is dual to type IIA/type IIB which are derived starting from the fundamental string.

How are you supposed to interpret that? In some sense you want to mod out this 'theory space' by all the dualities and look for whatever mathematical objects or structures that remain and are irreducible. Obviously, this is technically a formidable task and so remains only speculation.

 

[tom.stoer]

Haelfix, thanks for this clarification. So we will distinguish explicitly between the M2/M5-theory and the mother-theory. Very important point!

 

[suprised]

Indeed the current way of thinking by many is that the M2/M5-theory (whose low energy limit is 11-dim sugra) is one of many limits, or "coordinate patches", of some "mother" theory, and unfortunately the word "M-Theory" has been used for both.

One of the questions is whether there are other, more "fundamental" degrees of freedom, out of which the various theories we know, emerge. Somewhat similar to QCD, where the non-abelian gauge theory with quarks plays the role of a necessary UV completion of the meson theory; at high energies, "new" degrees of freedom need to be liberated, and render the theory consistent at high energies.

As for strings, they are supposedly UV complete; there is no reason (known to me at least) why new, "more elementary" degrees of freedom would be needed when going up in energy. As we know eg. from state counting in black holes, the string spectrum seems just right to yield a consistent theory. So what we call string theory is morally close to QCD and not close to the meson theory.

This supports the picture that all there is, is the "big blob" master theory, which can be approximated at different "coordinate patches" by different perturbative theories, and there are no further, "more elementary" microscopic degrees of freedom that would be revealed by going to high energies.

All what happens, say when moving from "theory patch A" to "theory patch B", that certain degrees of freedrom, that were non-perturbative in theory A, start becoming relevant and turn into the weakly coupled degrees of freedom of theory B when the approproate duality frame is chosen to represent them.

On of the current questions of the present thread, is, I think, whether the whole theory blob can be reconstructed out of one patch, say, the M2/M5-patch, when taking the full non-perturbative quantum theory into account.

I believe the answer depends on what one means by patch - does one throw away information, or not, when restricting to a patch? Let's come back to the QCD example, please forgive me, but I think it is helpful.

The analog of the big mother blob of theories is "abstract QCD", the fully non-perturbatively defined theory, with a certain definite Hilbert space. Analogs of coordinate patches are the quark-gluon theory, which is the patch relevant in the UV; the meson theory, which is the relevant patch in the IR (and if you like, type II strings on AdS5xS5 which is the relevant patch at strong coupling at large N; and there might be more).

Let's focus in the meson theory patch. Can quarks and gluons be reconstructed from it? When taking at face value, not, I believe; the theory has less degrees of freedom than the quark-gluon model, because by definition the meson theory results from integrating out, or throwing away many degrees of freedom, and a lot of information is lost in this way. In other words, the meson model is an incomplete theory that does not faithfully represent the full QCD Hilbert space.

Conversely, going to the UV, does the quark-gluon patch represent faithfully the abstract QCD Hilbert space including all the meson fields, etc? I don't think that this is a trivial question. Somehow one believes "yes, when all non-perturbative information is taken into account", which boils down to the question how the patch is defined. But can one actually meaningfully define it such that it faithfully represents QCD at all energies? We do know that all the quantities one uses to write the UV theory become ill-defined in the IR, and naive extrapolation fails because one hits a singularity, or phase transition, at approx 1 GeV. A priori it is not clear whether one can extrapolate the quark-gluon model past this singularity or not (note again that this is not a question about the abstract QCD that is defined everywhere, but rather about whether the quark-gluon model faithfully represents it everywhere).
I would be tempted to say, not, but on the other hand, the numerical computations of lattice QCD seem to say the opposite.

Sorry again for the QCD detour, but I think this is morally close to this remark, when we replace "M2/M5-theory" by the "quark gluon theory" and "mother-theory" by "abstract QCD":

Haelfix, thanks for this clarification. So we will distinguish explicitly between the M2/M5-theory and the mother-theory. Very important point!

 

[MTd2]

What is the failure of the proposed non perturbative matrix theories?

 

[suprised]

What is the failure of the proposed non perturbative matrix theories?

It's not a failure. They are just useful in their domain of validity, as are the other approximations. In particular they mess up in lower dimensions.

 

[suprised]

Let's see if I got this straight. The view is nth order correlations are all that can be measured experimentally. So we specify the correlations directly by constraining their symmetry (instead of indirectly via a Hamiltonian or Lagrangian)?

Right. Sometimes one can solve a theory despite a lagrangian or hamiltonian is not explicitly known.

 

[MTd2]

It's not a failure. They are just useful in their domain of validity, as are the other approximations. In particular they mess up in lower dimensions.

What I meant is why do the fail to be THE m theory?