Amazing bid by Thiemann to absorb string theory into LQG

[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.

 

[lethe]

Originally posted by eigenguy
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.

well, i have seen you launch character assaults on people on this forum based on your impression of their knowledge. for this reason, i prefer it when your impression of my knowledge is very vague.

But I haven't heard of a target space "theory". I guess you probably meant what I just said anyway.

by that i just meant the low energy effective field theory in the target space. the details depend on which string theory you are looking at.

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.

X is a bosonic field.

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.

certainly it is not the center of mass momentum squared, but that is silly. the mass of a stringy excitation is indeed its momentum squared.

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?

perhaps...

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.

perhaps.

 

[eigenguy]

Originally posted by lethe
X is a bosonic field.

Well, X is bosonic anyway: It is both a spacetime and world-sheet boson. The kind of fields I'm talking about are defined as such because of their "point-likeness":, i.e., they have no internal degrees of freedom.

Originally posted by lethe
certainly it is not the center of mass momentum squared, but that is silly. the mass of a stringy excitation is indeed its momentum squared.

The mass squared of an open bosonic string state is a sum of a zero mode term which is the as you say the "centre of mass" momentum squared, and terms involving higher modes. But I don't recall coming across attributions of spacetime momentum to internal excitations. I'll have a careful look at this, since what you are saying seems intuitively true, but from the standpoint of my above comment, this would not help you.

What about my question about D-branes?

Originally posted by lethe
well, i have seen you launch character assaults on people on this forum based on your impression of their knowledge. for this reason, i prefer it when your impression of my knowledge is very vague.

It's not their knowledge, it's their motives and tactics. Pointing out that despite the authoritive air that always accompanies their comments (especially when criticizing mainstream views which they admit they don't pay attention to) touching on the subject of their "religion" (which they also never really understood, something that can be easily seen by looking at their rather curt exchanges with urs in this very thread.) they really don't know what the hell they are talking about, is just one way of preventing members from being suckered into turning away from reality and joining their irrational fanaticism. Make no mistake, when it comes to these guys, it's all about egos, their interest in physics is really just incidental and would have played out the same way whatever the subject was.

 

[lethe]

Originally posted by eigenguy
What about my question about D-branes?

it is not at all clear to me what your question is, but i think we have been off-topic on this thread for far too long. i was enjoying this thread a lot, and i only stepped into defend selfAdjoint from your false impressions about Yang-Mills theory.

 

[eigenguy]

Originally posted by lethe
it is not at all clear to me what your question is, but i think we have been off-topic on this thread for far too long.

Fine, but for what its worth, my question about D-branes was how do you reconcile the fact that they are described by the born-infeld action with your statement that string theory is really just a 2D QFT.

 

[marcus]

Reconnecting with the topic

We need to scroll back to page 20 of this thread to reconnect with the main substance of the discussion. Urs took exception to the fact that in LQG a hilbert space of (spin-labeled) knots serves to embody the states of the gravitational field.

Since the states start out embodied as spin networks, i.e. embedded spin-labeled graphs, to get abstract knots one has to take diffeomorphism equivalence classes. It is a common algebraic proceedure---factoring down by an equivalence relation----sometimes used in constructing, for example, the complex numbers.

Two spin-networks are to be considered equivalent if one can be smoothly deformed into the other.

That is, one mapped into the other by a diffeomorphism, or (if you like special effect movies) one network "morphed" into the other.

Only abstract knot-type info remains when the networks are grouped into diffeo-equivalence classes.

Urs argued that this algebraic way of realizing spatial
diffeo-invariance was not kosher quantum theory. Perhaps it invalidated the whole of LQG? SelfAdjoint mentioned that the proceedure was used in Algebraic Quantum Field Theory (AQFT).
Around this point, on page 20 of the thread, Urs said he had
contacted two authorities, Abhay Ashtekar and Hermann Nicolai,
about this.

This interesting issue arose because Thomas Thiemann did something analogous (implementing a certain relation algebraically) in his paper. An objection to the special case (in TT's paper) implied a general-case fundamental objection to the construction of the state space in LQG.

 

[marcus]

quoting from page 20, for continuity

I happened to be online around noon Germany time when Urs checked into this forum. But he just looked and went away. I think it is too bad the last 3 pages have been so off topic. So, in hopes of restoring a connection to the main thread, I will quote from page 20.

The first post here is from selfAdjoint.
=================================================

quote by selfAdjoint of something by Urs:
---------------------------------------------
Originally posted by Urs

selfAdjoint,

yes, thanks for pointing out that the first appearance of this idea is in equation (4.2), right.

Yes, these operators U exist and there is nothing wrong with the GNS construction as such. That's what I am trying so say all along: We can construct these operators U and demand that states be invariant under them - but that is not what we are told to do by standard quantum theory. Standard quantum theory says nothing about finding operator representations of the classical symmetry group. Instead it says that the first class constraints must vanish weakly.

The latter, in our case, implies nothing but the very familiar fact that the Klein-Gordon equation should hold!
-------------------------------------------------------------

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.


02-09-2004 03:34 PM


=================================================
marcus:

quote by me, of something by ranyart
--------------------------------------------------------------
Originally posted by ranyart
Marcus a paper by Marolf and Rovelli from sometime ago may have a bearing on this thread:http://uk.arxiv.org/abs/gr-qc?0203056

Eight pages long and it has some far reaching aspects, even by Rovelli standards, take a good look and make some interesting insights
----------------------------------------------------------------


you know ranyart though I don't have the right to judge I have to say I think Rovelli's thoughts about quantum theory are among the most perceptive and sophisticated--especially in connection with relativity. he thinks about situations and measurments in an extremely concrete fashion.

I keep seeing Marolf's name, maybe he is another one who really thinks instead of just operating at a symbolic level.

Rovelli has a section, pages 62-68 in the book, where he talks about
"Physical coordinates and GPS observables"
it uses the Global Positioning Satellite system to illustrate something about general relativity. I haven't grasped it. have you looked at it?

Anyway thanks for the link.

what it means to me relative to this thread is the article you give is further evidence that Rovelli does not just quantize by rote, or by accepted procedures. He is one of the more philosophically astute people in knowing what is going on when he quantizes something. (IMHO of course


02-09-2004 04:08 PM

===========================================

arivero:


invariance of diathige and trope

quote:
--------------------------------------------------------------------------------
Originally posted by Urs

The standard theory of quantum physics instead tells us that we must impose the first class constraint of the theory weakly as an operator equation .

-----------------------------

Let me to notice the historical remark in Rovelli Living Review:


quote by arivero of something by Rovelli:
----------------------------------------------
The discovery of the Jacobson-Smolin Wilson loop solutions prompted Carlo Rovelli and Lee Smolin [182, 163, 183, 184] to ``change basis in the Hilbert space of the theory''
...The immediate results of the loop representation were two: The diffeomorphism constraint was completely solved by knot states (loop functionals that depend only on the knotting of the loops), making earlier suggestions by Smolin on the role of knot theory in quantum gravity [195] concrete; and (suitable [184, 196] extensions of) the knot states with support on non-selfintersecting loops were proven to be solutions of all quantum constraints, namely exact physical states of quantum gravity.

-------------------------------



It seems there are so sure of his technique that the review articles already forget to relate it to the constrains.

On other hand, Thiemann Hamiltonian constrain is a later development, dated 1996.



02-10-2004 03:18 AM

====================================

Urs:

Yes, it's kind of strange. The quantum constraints are not even mentioned anymore when it comes to 'solving' diffeo-invariance in LQG reviews. I believe this is a trap. At least people should be well aware that at this point standard canonical quantization is abandoned. Luckiliy, this has become clear now in the simpler example of quantization of the Nambu-Goto action by Thomas Thiemann.
====================================

It was soon after this that Urs reported he had written to both
Abhay Ashtekar and Hermann Nicolai about this perceived "non-standardness" of LQG.

I hope very much their replies can be forwarded to PF and are not
relegated solely to Jacques Distler's message board!

 

[selfAdjoint]

I guess I should put in something here. Of all the things in Thiemann's paper that are called arbitrary, I can find only one that I think really is arbitrary, and that is his representation of his Weyl algebra, display 6.20:

$$\omega_{\pm}(W_{\pm}(s)) := \delta_{s,0}$$

He adopts this wild and crazy representation (=1 on all his momentum networks, 0 on "the empty network") in default of being able to develop a real representation theory. The only authority for it he cites is his prior experience with LQG developments.

Now I am not able to show this myself, but it seems plausible that if you went with a representation that couldn't distinguish one momentum distribution from another, you might get a theory that couldn't recognized anomalies.

 

[marcus]

selfAdjoint, yes I remember that eqn. 6.20
It is in section "6.3 A specific example"

He warns us early on, as I recall, that he is opening up
a broad problem of finding all the representations and, in this
paper, only taking an initial "baby step" so to speak of
offering one representation, which IIRC he notes is not very interesting.

"In this paper we will content ourselves with giving just one
non-trivial example. Here it is:"

 

[eigenguy]

I thought that in quantum theory poincare picks up no anomaly. So maybe momentum distributions aren't relevant (unless you are talking about a different kind of momentum).

 

[marcus]

The conclusions in TT's paper are phrased in a cautious fashion, as if to say "if we extend this and it checks out then so-and-so"
so in the abstract:

"While we do not solve the...representation theory completely...we present [one solution]...

The existence of this stable solution is...exciting because raises the hope [that by looking further for more complicated solutions]...
we find stable, phenomenonologically acceptable ones in lower dimensional target spaces..."

So this first solution, which you point to in equation 6.20
is a drop in the bucket----he cautions us up front, reasonably enough.
I guess the point is that, as he says, even that one rather artificial unphysical case is indeed exciting. Because we did seem to get excited whether over at Distler's board or here at PF.
But realistically it has to be followed by substantially more research to mean anything, or?