Why I am REALLY disappointed about string theory

[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.