Why I am REALLY disappointed about string theory

[suprised]

How is time evolution described when there is neither Hamiltonian nor Lagrangian?

There is of course an abstract Hamiltionian, one just cannot write it down explicitly.

 

[suprised]

Afaik the S-matrix approach has failed and I do not see how it could be raised from the dead. And I do not see why it should be easier to extract bound states physics from scattering states - even if there may be no scattering states in a certain regime at all.

Example: how would you extract the well-known QCD form factors or structure functions from the QCD S-matrix? analytically, not experimentally?

In fact what happens these days IS a resurrection of scattering matrix/amplitudes techniques, and this expressly goes against lagrangian formalism. Pages over pages of complicated feynman diagram calculations can be replaced by a few lines when employing the new twistor-based techniques. Just listen to recent talks of NAH, where he very strongly (perhaps a bit too strong) spells out how the old traditional QFT methods based on Feynman diagrams should be superseded by the new techniques.

 

[tom.stoer]

In fact what happens these days IS a resurrection of scattering matrix/amplitudes techniques, and this expressly goes against lagrangian formalism. Pages over pages of complicated feynman diagram calculations can be replaced by a few lines when employing the new twistor-based techniques. Just listen to recent talks of NAH, where he very strongly (perhaps a bit too strong) spells out how the old traditional QFT methods based on Feynman diagrams should be superseded by the new techniques.

Again it seems that you confuse QFT and the Lagrangian formalism with perturbation theory and Feynman diagrams. Neither Feynman diagrams nor perturbation theory are fundamental. Old traditional approaches based on Feynman diagrams are partially outdated, but not due to twistor strings or something like that, but due to non-perturbative methods developed (again) for QCD - based on Lagrangian or Hamiltonian techniques.

Where does this impression come from that writing down a Lagrangian automatically implies that it has to be treated perturbatively? Or that perturbation theory itself IS QFT?

Comparing QFT with Feynman diagrams is like comparing calculus with Tayor series.

I think we should stop this discussion.

So let me ask again: What does it mean that "a theory lacks a Lagrangian description"? Or that it "has no classical or weak coupling limit?" I doubt that you are able to proof that there is no PI available that fully describes the theory. What does it mean that "there is an abstract Hamiltonan which cannot be written down explicitly?"

 

[Finbar]

True, but irrelevant for strong coupling.


So to say that a Lagrangian is a classical concept is missleading. It is used in classical physics, it is used for quantization. But if you are able to guess a Lagrangian plus a PI measure plus observables this completely defines a quantum theory.

What if I hand you the Hilbert space of a theory and the observables. You can have a well defined quantum field theory without ever having to write a Lagrangian or a path integral.

As a matter of definition a Lagrangian is a classical concept. Yes, when you quantise a theory you use a quantity which has the same structure as the Lagrangian in the PI. But in a theory which has not been obtained by quantising a classical theory there is no Lagrangian any where in the theory.

 

[suprised]

Again it seems that you confuse QFT and the Lagrangian formalism with perturbation theory and Feynman diagrams.

Emphatically not!

Neither Feynman diagrams nor perturbation theory are fundamental.

This is exactly what I wrote over and over again.

Old traditional approaches based on Feynman diagrams are partially outdated, but not due to twistor strings or something like that, but due to non-perturbative methods developed (again) for QCD - based on Lagrangian or Hamiltonian techniques.

_Also_ due to twistor strings, and precisely this is my point. By now scattering processes are being computed in completely different way as before, which goes in the direction of analytical S-matrix.

Or that perturbation theory itself IS QFT?

Who ever wanted to claim this?

I think we should stop this discussion.

You bet.

 

[Haelfix]

I sense some confusion here.

The modern scattering amplitude research program is logically distinct from this business about certain theories without any classical lagrangian.

The latter are very much unique to a small subclass of conformal field theories (they don't necessarily have anything to do with string theory, although sometimes they do) that have no obvious or known classical starting point. That isn't to say that such a formulation isn't possible, its just that it is not necessary in order to define the theory. In order to see this properly, you really do have to know a lot about the mechanics of conformal field theory and study the models by themselves (eg specific rational conformal field theory etc). It is not at all obvious what one means by any of this, but then suffice is to say that such objects have been well studied now for twenty years, so the phenomenon is by no means new or controversial.

Of course we now can say a lot about them. For instance that such an object must be a conformal field theory if it is to always stay strongly coupled (at any scale), follows from simple renormalization group arguments.

Now as to the renaissance of the scattering amplitudes business. Well it does share a lot in common with the old Smatrix program, but it is also distinct. Most of the theories considered for instance N = 4 SYM or even plain old QCD, are theories that do have a lagrangian description. Also it is not so much about defining a new theory (like in the 60s where the SMatrix program was trying to replace quantum field theory), as it is to reexpress existing theories in a particular way such that calculations become more tractable.

 

[negru]

Well the ultimate goal is to replace QFT. Nobody cares about the calculations. The whole point is finding why some calculations are so simple, and it has already been found that it's not a QFT by any means behind everything.

 

[Haelfix]

Yes and no.

The original SMatrix program wanted to zero in on the unique theory of the strong interactions, by inputting certain requirements (like unitarity, analyticity, certain crossing symmetries etc).

Of course nowdays that seems crazy to us. There are a million different lagrangians possible with those rules, so in hindsight it was silly for them to expect to do such a thing.

By contrast the modern program is looking specifically at the theories that have already had their lagrangians worked out and where we know they are relevant to the real world (or at least the almost real world). They aren't working with something that doesn't exist yet.

Of course the *hope* is that the end result *generalizes* into something new (perhaps some crazy mathematical generalization of a Grassmanian), and so in that sense you are correct.

 

[negru]

Oh yeah, I agree it's not exactly the same as the Smatrix program. The direction is opposite (do calculations -> find principles) but I was just saying that the ultimate goal is the same (get rid of qft and lagrangians).

 

[atyy]

There is of course an abstract Hamiltionian, one just cannot write it down explicitly.

The latter are very much unique to a small subclass of conformal field theories (they don't necessarily have anything to do with string theory, although sometimes they do) that have no obvious or known classical starting point.

Is this eg. what's mentioned in http://arxiv.org/abs/hep-th/9108028 for 2D CFTs?

The CFTs in http://www.ictp.it/media/101047/schwarzictp.pdf that were thought not to have Lagrangian descriptions aren't 2D - presumably the technique can be extended to some higher D CFTs?

Also, are the CFTs without explicit Lagrangian description fixed points of any renormalization flow?

 

[tom.stoer]

OK, sorry for this detour.

Is there an idea how a simple picture of a theory "w/o Lagrangian" would look like? What about it's fundamental objects or d.o.f.? What essentially "defines" such a theory? (a QFT can be defined via Lagrangian + quantization or via a Hamiltonian + a Hilbert space with inner product) What about its relation to (SUSY) gauge theory / SUGRA (which are still the low-energy limits for string theories).

 

[S.Daedalus]

I may misunderstand the depth of the discussion, but isn't it just generally true that there are equations of motions that don't follow from any Lagrangian? So if you just had those, you'd have a 'theory without a Lagrangian description' -- or is this something conceptually fundamentally different?

 

[suprised]

Is there an idea how a simple picture of a theory "w/o Lagrangian" would look like? What about it's fundamental objects or d.o.f.? What essentially "defines" such a theory? (a QFT can be defined via Lagrangian + quantization or via a Hamiltonian + a Hilbert space with inner product)

Its easier for me to eludicitate specific examples.

Again, as prime example consider 2d CFTs. The point is that some of them ("minimal models") can be solved purely by consistency, and this does not require any lagrangian as input. The input is the existence of a stress tensor (hamiltonian if you like) with a central charge in its OPE, plus basic consistency requirements like crossing symmetry, modular invariance, perhaps unitarity. That's enough to determine the spectrum (partition function), and all correlations functions.

As said, this is independent of whether a lagrangian exists or not. Often lagrangians do exist in the sense that they define the theory in the UV, and then there is a RG flow to the IR where the theory becomes conformal. Under the RG flow many objects become strongly quantum corrected/renormalized, so the UV desription is not be useful for doing actual computations for the CFT. Right at the conformal RG fixed point, there may not exist a useful lagrangian formulation at all! So despite there may be a lagrangian definition of the theory in the UV, it may be of almost no help for solving the theory in the IR.

Other, related example: massive soliton scattering theories in 2d. Some of them can be obtained by perturbing CFTs. Some of them are integrable and have a factorized S-matrix, and can be solved just by solving the consistency condition of the S-matrix (crossing relations, Yang-Baxter eqs, etc). In fact there exists a huge variety of quantum integrable systems whose scattering matrices can be exactly determined without ever needing a lagrangian formulation. What is common to them is an underlying algebraic structure, typically Lie algebras, which is strong enough as to fix the theories solely on the basis of consistency.

Third example: N=2 gauge theory in d=4. Certainly there exists a definition of the theory at high energies where it is weakly coupled, namely in term of an asymptotically free SU(2) gauge theory, say. In this region one can write down a well-defined lagrangian, in terms of weakly coupled local, "fundamental" degrees of freedom.

However at low energies, this description breaks down because the theory becomes strongly coupled. In fact one knows that at some point, the gauge fields will decay into monopole-dyon pairs, and should not be considered as "fundamental" any more; rather they can be viewed as bound states of monopoles and dyons (at high energies it is the other way around; so what about the notion of "fundamental"). Certainly the original lagrangian in the UV is not a good way to describe the physics in this regime, involving gauge fields which do not even exist as good quantum operators in the IR. Non-perturbative phenomena like the decay of a gauge field into a monopole-dyon pair cannot be easily captured in terms of the UV lagrangian.

The whole point of Seiberg and Witten was to de-emphasize the role of an underlying lagrangian, and rather to focus on global consistency conditions in order to determine the effective action in a direct manner.
This works so miraculously well because of the properties of extended SUSY, which restricts quantum corrections and also allows to make non-perturbatively exact statements via the BPS property.

One can say that most of the progress in non-perturbative physics in the last 15 years was precisely because one did not think in terms of lagrangians. Typically the trick is to use some minimal input (algebraic symmetries, SUSY and BPS property, integrability) and then solving the theory (or part of a theory) via consistency conditions.

This is also the current way to think about certain theories in 6d (eg non-abelian non-criticial strings), for which a lagrangian formulation is either not known or even does not exist. The art is to make use of BPS properties etc in order to make some rough statements about properties of the theory, without having a complete definition at hand. Probably this applies to what is called M-theory as a whole, which many criticize because there is no known fundamental, first principle definition. Again, the art is to obtain non-trivial results even in the absence of a complete definition.

 

[tom.stoer]

I agree to most of what you said, except for the last statement, that "probably this applies to what is called M-theory as a whole, which many criticize because there is no known fundamental, first principle definition. Again, the art is to obtain non-trivial results even in the absence of a complete definition".

Perhaps I should de-emphasize the Lagrangian as well; it is useful for three reasons:
a) starting point for quantization
b) identifying symmetries
c) pertubation theory / semiclassical limit

In principle I don't want to quantize a theory (always handwaving), but I want to have a well-defined quantum theory. So a) goes away.
If there is a different way to identify a symmetry it's fine. Think about the n-dim. harmonic oscillator with an SU(n) symmetry which is directly visible looking at the Hamiltonian and the generators w/o ever referring to a Lagrangian. So b) goes away, too.
c) is relevant only because 99% of all QFT courses and textbooks talk about it and b/c it worked for the SM; lucky guys. So c) goes away as well.

My main point is different.

In order to define a theory I have to define the theory.

Wow! In order to understand M-theory I have to know what it really is. I do not care about a specific formulation (Lagrangian, Hamiltonian, full OPE, ...) but I have to be able to say on a few sheets of paper (or in a few talks) what the theory really is. w/o being able to condense my knowledge regarding the theory in such a way, there is always the danger that the "theory" is nothing else but a collection of loosely related facts and discoveries. It's like looking at atoms and molecules before QM, like "QM" in 1920, like particle physics before the discovery of the SM ...

The problem with M-theory is that there are two different objectives
1) a theory should work fapp (in the sense of "shut up and calculate")
2) a theory should provide a sound / well-defined and small but powerful basis
and that M-theory still fails in both cases; it is not useful practically and it has no well-defined basis theoretically.

So either we transform it into a toolbox from which I can calculate a huge number of (new) predictions rather easily, or we identify its fundamental description (or we do both :-)

I do not care about the specific formulation and of course there is no reason why it must be "Lagrangian" in the sense as discussed before. But it has to be more than just a collection of facts, relations, formulas, dualities etc.

I am still with David Gross who asked exactly these questions:

• WHAT IS STRING THEORY?
  This is a strange question since we clearly know what string theory is to the extent that we can construct the theory and calculate some of its properties. However our construction of the theory has proceeded in an ad hoc fashion, often producing, for apparently mysterious reasons, structures that appear miraculous. It is evident that we are far from fully understanding the deep symmetries and physical principles that must underlie these theories. It is hoped that the recent efforts to construct covariant second quantized string field theories will shed light on this crucial question.
  
• We still do not understand what string theory is.
  We do not have a formulation of the dynamical principle behind ST. All we have is a vast array of dual formulations, most of which are defined by methods for constructing consistent semiclassical (perturbative) expansions about a given background (classical solution).
  
• What is the fundamental formulation of string theory?

What could be such a fundamental definition of string- / M-theory?

 

[mitchell porter]

We may be closing on a fundamental definition of string theory in anti de Sitter space, by analysis of which CFTs have string duals - see http://arxiv.org/abs/0907.0151" . If that is achieved, then it will be a matter of somehow extending this perspective to string theory in flat space and in de Sitter space.

 

[tom.stoer]

Does that mean that string theories will then be constructed for certain "superselection sectors" or topologies and can then live within one such sector with full dynamical spacetime? Or are there further restrictions like CY or other compactification such that these theories again will only cover a very small portion of full theory space?

 

[mitchell porter]

Each AdS/CFT dual pair is a superselection sector which, on the string side of the duality, has the form "M-theory on some product AdS_n x (compact space)_(11-n)", but that is a statement about asymptotic geometry. It is presumed that for each such "sector", topological fluctuations are allowed in the interior. But to change the boundary conditions at infinity - e.g. by adding a non-compact finite-tension brane that extended to infinity - would require an infinite amount of energy. That's the picture I get.

 

[tom.stoer]

That's a nice picture. Superselection sectors which require an infinite amount of energy for "tunneling processes" are to be expected.

 

[qsa]

But to change the boundary conditions at infinity - e.g. by adding a non-compact finite-tension brane that extended to infinity - would require an infinite amount of energy. That's the picture I get.

is there any specific reference

 

[mitchell porter]

is there any specific reference

http://arxiv.org/abs/hep-th/0204196" : "Since most stable D-branes in AdS are infinite in size they are also infinitely massive, and so represent superselection sectors of the Yang-Mills." (The YM theory here being the boundary dual to the string theory.) The reason is that if a string or brane has finite tension, it has a finite energy density, which means infinite total energy when integrated over infinite volume.

In the next sentence (page 2) they also mention that finite-size (and thus finite-energy) branes can be created quantum mechanically.

Also see http://physics.stackexchange.com/qu...1-form-potentials-in-string-theory/3481#3481".

 

[mitchell porter]

Combining the recent themes of "what is M-theory" and nonlagrangian theories:

I have maintained for a while that exploration of the ABJM theory might be the best avenue for the understanding of M-theory, since ABJM (a worldvolume theory for M2-branes) should be completely equivalent to M-theory on AdS4 x S7/Z_k. And I've just found some papers on obtaining the classical M5-brane within ABJM (http://arxiv.org/abs/0909.3101" ).

I can explain a little of this. M-theory on a circle, S1, is equivalent to the Type IIA string at strong coupling. And the manifold S7 is equivalent to CP3 x S1. Notice in the previous paragraph the quotient by Z_k. k is a parameter in ABJM, an integer coefficient in the Chern-Simons action called the "level". You think of Z_k as acting on this S1 factor in the S7, so the k->infinity limit of ABJM corresponds to Type IIA on AdS4 x CP3.

But when you go from M-theory to Type IIA, (unwrapped) M2-branes become D2-branes and (wrapped) M5-branes become D4-branes. (Wrapped M2-branes become IIA strings, and unwrapped M5-branes become "NS5-branes"; by wrapped and unwrapped, I mean with respect to the M-theoretic 11th dimension, the S1 which we are quotienting out of existence.) Furthermore, a D4-brane with an internal magnetic flux can be constructed as a bound state of infinitely many D2-branes. So this relationship, known from Type IIA string theory and lifted to M-theory, will become the basis of making M5-branes out of M2-branes. What we need to do is to understand the emergence of the M5-brane in terms of the M2-brane worldvolume theory. And apparently it turns out that some of the degrees of freedom in ABJM become a fuzzy 3-sphere (made out of the eigenvalues of matrices which correspond to noncommutative degrees of freedom, but I haven't read that part yet), which provides the extra 3 transverse directions needed for the M5-brane.

So, progress continues...

 

[p-brane]

I agree to most of what you said, except for the last statement, that "probably this applies to what is called M-theory as a whole, which many criticize because there is no known fundamental, first principle definition. Again, the art is to obtain non-trivial results even in the absence of a complete definition".

Do you know what the relation between the 11-dimensional supergravity theory discovered by Witten and the theory he christened "M-theory" actually is?

 

[tom.stoer]

Yes.

It was assumed that M-theory is really the "mother" from which all other string theories incl. SUGRA can be derived in certain limits. In the meantime it became clear that this is not the case but that M-theory is just another new "coordinate patch" in the whole "theory space", so M-theory is - unfortuantely - by no means the "mother".

Some argued that it may not even be desirable or required to have one unique fundamental theory but that this "atlas of mutually dual patches" covering the "theory space" would be sufficient. Perhaps this is a philosophical question only, but as others (including Gross) seem to disagree, I would say that it's worth thinking about it (SU(3) and quarks have been identified exactly by searching for such a fundamental description below the hadrons).

 

[p-brane]

...It was assumed that M-theory is really the "mother" from which all other string theories incl. SUGRA can be derived in certain limits...

Let me restate my question in slightly modified form. "M-theory" as christened by Witten is a quantum theory. How is it related to the aforementioned 11-dimensional supergravity theory?

 

[qsa]

Let me restate my question in slightly modified form. "M-theory" as christened by Witten is a quantum theory. How is it related to the aforementioned 11-dimensional supergravity theory?

it is sort of a long short story. read this and then maybe you can ask more specific question:



http://arxiv.org/PS_cache/hep-th/pdf/0101/0101126v2.pdf

 

[suprised]

Let me restate my question in slightly modified form. "M-theory" as christened by Witten is a quantum theory. How is it related to the aforementioned 11-dimensional supergravity theory?

Well 11d sugra is the low energy limit of M-theory! That was the original definition of the latter.

 

[qsa]

Let me restate my question in slightly modified form. "M-theory" as christened by Witten is a quantum theory. How is it related to the aforementioned 11-dimensional supergravity theory?

it is sort of a long short story. read this and then maybe you can ask more specific question:



http://arxiv.org/PS_cache/hep-th/pdf/0101/0101126v2.pdf

 

[p-brane]

Well 11d sugra is the low energy limit of M-theory! That was the original definition of the latter.

Saying it slightly differently, M-theory in the original sense of Witten is the as yet unknown quantum theory having the 11d sugra as it's classical limit. By contrast, we already have quantum theories for the five standard string theories.

The paper qsa refers to is a review of the earliest effort at discovering what this theory might look like. It is called Matrix mechanics and has D0-branes as it's fundamental degress of freedom.

I know that qsa, surprised and lot's of other people posting in this thread already know this very well. These questions aren't really directed at them.

Anyways, M-theory may be viewed as the master quantum theory underlying all five string theories. Nowadays M-theory is typically used to denote the single master theory underlying the theories at every point in moduli space. I think this may be the only way M-theory has been used in this thread.

So my next question is what precisely is moduli space?

 

[Orbb]

So my next question is what precisely is moduli space?

It's the parameter space of vacuum (lowest-energy) configurations of a given theory. In that sense, the different String Theories correspond to picking specific points in the full parameter space of vacua of M-Theory (correct me if I'm wrong).

 

[tom.stoer]

... M-theory may be viewed as the master quantum theory underlying all five string theories. Nowadays M-theory is typically used to denote the single master theory ... I think this may be the only way M-theory has been used in this thread.

No, not really.

It may be the case that M-theory can be formulated such that it becomes this single master theory, but unfortunately this cannot be deduced from the existing formulation. So using M-theory in the sense of "single master theory" cannot be fully justified and we must distinguish between certain knowledge and future research.

I think we have made ​​this distinction in most cases.