Amazing bid by Thiemann to absorb string theory into LQG

[eigenguy]

Originally posted by selfAdjoint
Algebraic quantum theory was invented in the first place because standard quantum theory was mathematically defective. It still is.

From the view that QFT is only an approximation to a more fundamental way to describe nature (by strings for example) it's defects are not only irrelevant, they are to be expected. Thus the raison d'etre of AQFT collapses, along with your argument.

 

[selfAdjoint]

Originally posted by eigenguy
From the view that QFT is only an approximation to a more fundamental way to describe nature (by strings for example) it's defects are not only irrelevant, they are to be expected. Thus the raison d'etre of AQFT collapses, along with your argument.

And that of course would be why there is a million dollar prize for putting a rigorous underpinning under Yang_Mills theory - a prize that no string theorist I know of has called foolish.

 

[eigenguy]

Originally posted by selfAdjoint
And that of course would be why there is a million dollar prize for putting a rigorous underpinning under Yang_Mills theory - a prize that no string theorist I know of has called foolish.

I guess you are referring to the prize being offered by the clay institute to anyone who explains the theoretical underpinnings of the observed mass gap in the strong interactions described by yang-mills. Since yang-mills does not automatically mean QFT, and since it is unknown whether some reformulation of QFT or something more general (like string theory) will be required, my point stands.

 

[lethe]

Originally posted by eigenguy
I guess you are referring to the prize being offered by the clay institute to anyone who explains the theoretical underpinnings of the observed mass gap in the strong interactions described by yang-mills. Since yang-mills does not automatically mean QFT, and since it is unknown whether some reformulation of QFT or something more general (like string theory) will be required, my point stands.

um, actually, it does. the claymath problem is specifically about QFT.

 

[eigenguy]

Maybe LQG wasn't given a fair shake

Something just occurred to me. Suppose it turns out that LQG is wrong for the reasons that urs discovered. Wouldn't it stand to reason that if other physicists had given LQG a serious look, they would have seen this a long time ago? I believe that feynman said the physicists job is to prove themselves wrong as quickly as possible (Of course, from this point of view, the LQG camp would deserve most of the blame).

 

[eigenguy]

Originally posted by lethe
um, actually, it does. the claymath problem is specifically about QFT.

You will find the following description http://www.claymath.org/millennium/Yang-Mills_Theory/ :

Yang-Mills and Mass Gap

The laws of quantum physics stand to the world of elementary particles in the way that Newton's laws of classical mechanics stand to the macroscopic world. Almost half a century ago, Yang and Mills introduced a remarkable new framework to describe elementary particles using structures that also occur in geometry. Quantum Yang-Mills theory [note the word "field" is not used here or anywhere else in this paragraph] is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the "mass gap:" the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang-Mills theory and a mass gap and will require the introduction of fundamental new ideas both in physics and in mathematics.

Clearly, no assumption has been, nor should be, made about what the solution will look like.

 

[lethe]

Originally posted by eigenguy

Clearly, no assumption has been, nor should be, made about what the solution will look like.

ummm... what are you talking about?? this is a question about quantum Yang-Mills, which is a quantum field theory!

 

[eigenguy]

Originally posted by lethe
ummm... what are you talking about?? this is a question about quantum Yang-Mills, which is a quantum field theory!

Yang-mills refers to symmetry, in this case non-abelian gauge symmetry. Such symmetries can be incorporated into string theory.

 

[lethe]

Originally posted by eigenguy
Yang-mills refers to symmetry, in this case non-abelian gauge symmetry. Such symmetries can be incorporated into string theory.

OK, fine, string theory allows nonabelian gauge theories. but Yang-Mills theory is not string theory, it is a quantum field theory. The positive mass gap conjecture is not about string theory or some other as-yet-undetermined theory, it is about Yang-Mills theory.

 

[selfAdjoint]

Originally posted by eigenguy
Yang-mills refers to symmetry, in this case non-abelian gauge symmetry. Such symmetries can be incorporated into string theory.

Eigen, I am afraid you've got your foot in by your tonsils. The words Yang-Mills, followed by the word theory, refer to a class of Quantum Field Theories. If you want to refer to Yang-Mills symmetry, you say Yang-Mills symmetry. See for example

Peskin & schoeder section 15.2, the field theory associated with a non-commuting local symmetry is termed a non-Abelian gauge theory.

Ryder, section 3.5 The Yang-Mills field.

Both P & S and Ryder have in their indices, Yang-Mills theory, see non-Abelian gauge theory.

Yang-Mills theory was quantized by Veltzmann & 'tHooft, becoming thereby a Quantum Field Theory. It is this theory that is usually referred to as Y-M theory.

 

[eigenguy]

Originally posted by selfAdjoint
Eigen, I am afraid you've got your foot in by your tonsils. The words Yang-Mills, followed by the word theory, refer to a class of Quantum Field Theories. If you want to refer to Yang-Mills symmetry, you say Yang-Mills symmetry. See for example

Peskin & schoeder section 15.2, the field theory associated with a non-commuting local symmetry is termed a non-Abelian gauge theory.

Ryder, section 3.5 The Yang-Mills field.

Both P & S and Ryder have in their indices, Yang-Mills theory, see non-Abelian gauge theory.

Yang-Mills theory was quantized by Veltzmann & 'tHooft, becoming thereby a Quantum Field Theory. It is this theory that is usually referred to as Y-M theory.

Obviously if you look in a QFT book the definitions you find will be made in terms of QFT. But ST came after QFT, and in ST you will find discussions of non-abelian gauge theory in which the term yang-mills is used without reference to QFT.

However none of this matters. My point was that the status of the problem of finding a rigorous formulation of QFT is tied to the question of whether QFT is just an approximation to some more fundamental theory. If it is, then we shouldn't be upset or surprised if in fact QFT can never be formulated in a completely rigorous way. Note that the basic legitimacy of my point really does not depend on the definition of yang-mills.

Btw, I got this from weinberg, so go argue with him.

 

[eigenguy]

By the way selfadjoint,

I pm'ed you again asking what you thought of haag's book. I'd really like to know, especially about what it says about the GNS construction.

 

[lethe]

Originally posted by eigenguy
Obviously if you look in a QFT book the definitions you find will be made in terms of QFT.

actually, if you look in any book where the author knows what he is talking about, you will find yang-mills defined as a field theory. any properly trained physicist knows this, and would not say otherwise. of course this includes the authors of many popular quantum field theory books, but also many other books, including string theory books.

But ST came after QFT, and in ST you will find discussions of non-abelian gauge theory in which the term yang-mills is used without reference to QFT.

can you please provide a reference to a string theory book which refers to Yang-Mills theory without meaning it as a field theory?

However none of this matters. My point was that the status of the problem of finding a rigorous formulation of QFT is tied to the question of whether QFT is just an approximation to some more fundamental theory. If it is, then we shouldn't be upset or surprised if in fact QFT can never be formulated in a completely rigorous way. Note that the basic legitimacy of my point really does not depend on the definition of yang-mills.

even if Yang-Mills theory is only a low energy effective theory, it still makes sense to ask questions about its properties. whether it is a consistent theory. the is a derivation in Peskin and Schroeder that shows that no matter what the high energy theory, there should be a renormalizable low energy effective quantum field theory describing it at some energy scale.

the Claymath problem asks for some properties of this theory to be put on a firm mathematical basis.

Btw, I got this from weinberg, so go argue with him.

if your point is that quantum field theory is irrelevant, then i would say that you have misinterpreted Weinberg, he would almost certainly say no such thing. so i will argue with you.

of course, you can prove me wrong quite easily: provide references. papers? page numbers? just show me where Weinberg thinks that solving nonperturbative Yang-Mills would be irrelevant.

 

[eigenguy]

Originally posted by lethe
if your point is that quantum field theory is irrelevant

Of course that's not my point and of course that's not what weinberg says.

However, I believe I am wrong and you and selfadjoint are right about the expression "yang-mills theory" meaning QFT. But I also think one can refer to a non-abelian gauge symmetry as a yang-mills type symmetry in theories that aren't field theories.

But I was initially responding to the following post

Originally posted by selfAdjoint
Urs, I'm going to quit this discussion because we are talking past each other. Thiemann has two things, after the dust settles: he has a very persuasive model of the string, laid out in his section 6.2, and he has the classic results of "local quantum physics" as Haag puts it. His achievement is to apply the latter to the former. Now you say this is not what you are told to do by standard quantum theory. So much the worse for standard quantum theory. Algebraic quantum theory was invented in the first place because standard quantum theory was mathematically defective. It still is.

So I can't convince you and I'm afraid you can't convince me.

My point was that it may be that if QFT is an approximation to a more fundamental theory, we should not be upset or surprised if it turns out there is no rigorous way to formulate QFT so that the algebraic or any other approach to doing so may be destined to fail. But of course QFT is valid in it's domain of applicability.

 

[lethe]

Originally posted by eigenguy
My point was that it may be that if QFT is an approximation to a more fundamental theory, we should not be upset or surprised if it turns out there is no rigorous way to formulate QFT so that the algebraic or any other approach to doing so may be destined to fail. But of course QFT is valid in it's domain of applicability.

you do know that string theory is a 2D quantum field theory, right?

 

[eigenguy]

Originally posted by lethe
string theory is a 2D quantum field theory, right?

I didn't say the new theory must be ST, and ST is not a theory of fields, but reduces to one in the low energy limit.

 

[lethe]

Originally posted by eigenguy
A theory of strings on the world-sheet is not the same as an ordinary 2D theory of fields, the difference being due to the extended nature of strings. ST is not a theory of fields, but reduces to one in the low energy limit.

you are not correct.

 

[eigenguy]

Originally posted by lethe
you are not correct.

So you are saying that ST can be understood completely in ordinary field theoretic terms?

 

[lethe]

Originally posted by eigenguy
So you are saying that ST can be understood completely in ordinary field theoretic terms?

i am only saying what you see me saying. do not put words in my mouth that you did not see me say.

 

[eigenguy]

Originally posted by lethe
i am only saying what you see me saying. do not put words in my mouth that you did not see me say.

I'm certainly not trying to put words in your mouth. Could you help me understand your view of the relation between ST and QFT. Keep in mind, I'm no expert and do not claim to be and am quite happy to admit I'm wrong the moment I believe that I am. I should point out that weinberg explains that he wrote his QFT books to address the possibility that a final theory does not necessarily have to be a field theory, and uses the example of ST to make his point. So basically, I've just been giving my best understanding of weinberg's views. Also, I'm pretty sure the mathematics of ST goes well beyond that of QFT and this is what is most germaine to my argument about whether we should expect there be a way to rigorously formulate it.

 

[lethe]

Originally posted by eigenguy
I'm certainly not trying to put words in your mouth. Could you help me understand your view of the relation between ST and QFT.

physically, string theory is not a spacetime quantum field theory, since in spacetime, it has strings instead of points. however, mathematically, it is simply a 2D quantum field theory. it uses all the standard techniques of conformal field theory.

in short: if quantum field theory is broken for some reason, then so is string theory.

Keep in mind, I'm no expert and do not claim to be and am quite happy to admit I'm wrong the moment I believe that I am.

i will keep that in mind.

 

[arivero]

Hmm 2D Conformal Field Theory is surely simpler than 4D QFT. All these 1+1 theories enjoy a symmetry group a lot more restricted than 3+1 Lorentz.

 

[eigenguy]

Originally posted by lethe
physically, string theory is not a spacetime quantum field theory, since in spacetime, it has strings instead of points. however, mathematically, it is simply a 2D quantum field theory. it uses all the standard techniques of conformal field theory.

But ST includes D-branes, what about them?

 

[lethe]

Originally posted by eigenguy
But ST includes D-branes, what about them?

what about them?

 

[eigenguy]

Originally posted by lethe
what about them?

Can D-branes be understood purely in terms of the world-sheet theory? Btw, are you a ST expert? I'm sure urs can clear this up.

 

[selfAdjoint]

Originally posted by eigenguy
By the way selfadjoint,

I pm'ed you again asking what you thought of haag's book. I'd really like to know, especially about what it says about the GNS construction.

And I replied, didn't you get it?

 

[eigenguy]

Originally posted by selfAdjoint
And I replied, didn't you get it?

Oops! I just checked and yes I got it. Sorry about that. I think I will order a copy though since it sounds like it will be useful and I know it has many very worthwhile insights. I'll let you know how I'm doing with it after I've had a chance to peruse it a bit. Thanks a bunch!

 

[lethe]

Originally posted by eigenguy
Can D-branes be understood purely in terms of the world-sheet theory?

yes. of course, they can also be understood in target space theory, but that is a good thing.

Btw, are you a ST expert?

i prefer to stay anonymous.

 

[Haelfix]

We;re deviating too much from the original thread, but indeed ST is intimately linked to quantum field theory and the mathematical machinery behind it.

If QFT has a mathematical error at some point, (for instance Fadeev Popov quantization) then its nearly guarenteed to pop out in perturbative ST as well.

Presumably, the nonperturbative sector of ST is something different and new, but no one knows what that is either.

 

[eigenguy]

Originally posted by lethe
i prefer to stay anonymous.

As you should, and I would never ask you to compromise that. It's just that I'm studying polchinski volume I now and if you are ahead of me that would be good to know, assuming you like talking about it, which you seem to.

Originally posted by lethe
yes. of course, they can also be understood in target space theory, but that is a good thing.

Okay, so let me comment in on this specifically in terms of what I've read in polchinski. Then I want to take a closer look at the basic issue you raised about ST really being a 2D QFT because if you are right and I'm not getting this, then I really have to reexamine things. I'm going to state things in a matter of fact way, so don't assume I'm pointing something out because I think you don't already know it.

So first target space "theory". The polyakov action is an example of a so-called non-linear sigma model embedding the world-sheet in a target space which here is spacetime. On the other hand, the dynamics of D-branes is governed by the born-infeld action whose relation to 2D CFT is not entirely clear to me. It seems there must be some connection though because D-branes arise by T-duality from open strings on backgrounds involving wilson lines. Perhaps you can explain this further. But I haven't heard of a target space "theory". I guess you probably meant what I just said anyway.

Now on the ST-QFT connection. I guess what you are saying is that in some very real sense ST can be broken down to or understood in terms of what could be legitimately viewed in some sense as field theory. I don't think the basic configuration variables X^\mu are fields in the sense of QFT. For example, string rest mass is not equal to the square of their 4-momentum, but includes contributions from it's internal oscillations as well. In fact the mode oscillators give rise to infinite dimensional algebras that (I think) are missing from ordinary field theory. Maybe we are using different definitions of field?

Originally posted by Haelfix
ST is intimately linked to quantum field theory

Yes, in that it appears in it's low energy limit. But I don't think inconsistencies in QFT necessarily imply inconsistencies in whatever M-theory turns out to actually be.