Whats the deal with String Theory?

[Canum]

It is my understanding that String Theory is an attempt at combining the quantum theories with relativity, correct? My question is, why is this considerd so difficult? What about the two separate theories (quantum and relativity) makes them so hard to combine? Do they oppose each other or something?

 

[Mentat]

Originally posted by Canum
It is my understanding that String Theory is an attempt at combining the quantum theories with relativity, correct? My question is, why is this considerd so difficult? What about the two separate theories (quantum and relativity) makes them so hard to combine? Do they oppose each other or something?

Well, Canum, I suggest you read "The Elegant Universe", by Brain Greene, for an easy-to-understand explanation.

However, as to your question, you ask why is it so difficult to combine the two theories; the answer is that Quantum Theory is very good at describing the very small and Relativity is very good at describing the very large, but when one attempts to apply either theory to the opposite framework (apply Relativity to the very small or Quantum Theory to the very large) you get inconsistencies (which usually show up in the mathematics as infinities, and infinities are usually taken as an indication that something was wrong).

For an example, if I try to apply General Relativity to spacetime at sizes that approach the Planck's size, I would have to assume that it is flat, smooth, until perturbed by a gravitational field. Alas, this cannot be so, since Quantum Mechanics dictates that the exact state of something that small (and really, the exact state of anything) cannot be determined exactly, and thus odd bends and warpings and tunnels and cracks in the fabric of spacetime should be occurring all of the time at those very small levels.

Now, it'd be easy for one to just try to replace General Relativity (and I'm sure some have tried), but it's just too good at describing what it was made to describe, and so is rather invaluable to the Theoretical Physicist. So, instead of trying to eliminate one of these awesome theories, Theoretical Physicists are trying to find a theory that will unite the two theories (explaining away their apparent inconsistencies) along with uniting all of the four "forces" (electromagnetic, gravtational, strong and weak) into one mathematical equation. They call this theory the Theory of Everything (T.O.E. for short), and string theory is one of the possible (my favorite) candidates.

I hope this helps answer your question, though there is definitely more to it, and that's why I suggested "The Elegant Universe".

 

[Mentat]

Oh, btw, Welcome to the PFs, Canum.

 

[Canum]

However, as to your question, you ask why is it so difficult to combine the two theories; the answer is that Quantum Theory is very good at describing the very small and Relativity is very good at describing the very large, but when one attempts to apply either theory to the opposite framework (apply Relativity to the very small or Quantum Theory to the very large) you get inconsistencies (which usually show up in the mathematics as infinities, and infinities are usually taken as an indication that something was wrong).

But for either of these theories to be deemed correct descriptions of the physical universe, shouldn't they have to work on every scale? I'm having a difficult time understanding how a theory can be held up as true, when its results are only accurate when applied to certain dimensions.

Oh, btw, Welcome to the PFs, Canum.

Thanks

 

[Loren Booda]

General relativity represents gravitation by a continuous geometry of spacetime, whereas quantum mechanics assigns to physical properties mathematical probabilities at every measurable point.

G. R. limits our knowledge of reality by horizons beyond which object escape velocity would exceed the speed of light, but Q. M. limits our knowledge to within uncertainties of selective measurement and its complement.

G. R. is a theory "differentiated" from the spacetime cosmos downward, while Q. M. is a theory "integrated" from the phase space quantum upward.

 

[Canum]

G. R. is a theory "differentiated" from the spacetime cosmos downward, while Q. M. is a theory "integrated" from the phase space quantum upward.

Loren,
Not sure I understand that. Could you maybe rephrase and elaborate?

 

[marcus]

Originally posted by Canum
It is my understanding that String Theory is an attempt at combining the quantum theories with relativity, correct? My question is, why is this considerd so difficult? What about the two separate theories (quantum and relativity) makes them so hard to combine? Do they oppose each other or something?

at the most fundamental level what makes it difficult to quantize classical 1915 General Relativity is that the classical theory is "background independent"

ordinary quantum theories, quantum field theories, are constructed on some pre-established space+time geometry----which could be like normal 3D Euclidean space with a time line, or like the plain vanilla uncurved unexpanding 4D space of SPECIAL relativity, or
whatever----the flaw (from a GR standpoint) is pre-committment to any set geometry whatever.

to just begin defining the gear: waves, particles, strings, some fixed framework or geometric background has to be established---
but GR is different

in GR the shape of the spatial background is totally variable and dynamic, determined by the basic Einstein equation relating curvature to the distribution of matter and other energy---as the energy flows the curvature changes and as the curvature changes it guides the flow of matter and energy

the problem is not discrepancy of scale (as some posters suggest) although applicability at various scales is always an issue----the problem is that

GR lives on a completely free dynamic evolving geometry which emerges from the GR equations-----to precommit, even if you change it or perturb it later, trashes GR at its foundations.

String theory does not attempt to quantize GR. It is an attempt to arrive at an alternative explanatory model for gravity which will approximate the results of GR in certain situations at certain scales. There is a plethora of variant string theories and they are background dependent. I have not seen much evidence of numbers being predicted by this proliferating batch of stringy theories----numbers that could be checked against observation and experiment so as to help kill off some of the variants and select lines of development to pursue.

But probably it does not matter because people are proceeding outside of the stringy context---there is a clear established way to quantize classical theories, called "canonical" quantization. We don't need an alternative explanation of gravity if we are satisfied with GR and can succeed in quantizing GR. All along since before 1950 there has been an ongoing effort to quantize GR while conserving background independence! People have been gradually working out a way to quantize GR in a way that preserves the essence of the theory
(which is the most precisely predictive model of gravity we have so far, and not lightly to be discarded).

This is finally getting done and the theory is beginning to make predictions, which as they are checked by observation or experiment will help refine and guide further development. This quantizing of classical 1915 GR, only just now happening, has nothing to do with stringy business but is a different kind of quantum gravity often called LQG (something of a misnomer since the loops attribute is not the essential element, what is essential is a straightforward quantization of the 1986 new variables version of GR with minimum additional structure, and this is done in various but interrelated ways, not always using loops).

So your question "what makes it hard" really applies most appropriately to direct quantizing GR (by LQG) and is an interesting, even historical, question. Why has it taken so long?
GR was born in 1915 and ordinary quantum mechanics in 1929 and people have known for 70 years what they had to do----quantize background independent GR---and people have struggled with it for that long and only in the 1990s began to make real progress.

 

[Loren Booda]

Canum,

The way I see it, General Relativity is most characteristically described at its largest spacetime scale, the universal horizon. There, quantum mechanics holds least sway (e. g., vanishingly apparent Hawking radiation).

Q. M. describes fundamentally the smallest unit of phase space, the quantum. Brane theory may diminish the accustomed inverse-square power of gravity approaching the quantum Planck length, ~10^-33cm.

Differentiating cosmological horizon curvature helps determine the geometrodynamics within. Quanta actions, on the other hand, are integrated using the Schroedinger equation to calculate probabilities from Planck's constant upward.

 

[Mentat]

Originally posted by Canum
But for either of these theories to be deemed correct descriptions of the physical universe, shouldn't they have to work on every scale? I'm having a difficult time understanding how a theory can be held up as true, when its results are only accurate when applied to certain dimensions.

Well, like I said, General Relativity is very good at describing macroscopic events. It is, as far as I know, flawless in this (if there were a real flaw, it would not be a theory). However, it breaks down at the quantum level.

The exact inverse is true of Quantum Theory...it is very good at describing microscopic phenomena; pretty much flawless (though not completely understood, and not at all able to be conceptualized), but its predictions become almost completely irrelevant at the macroscopic scales.

Since no one wants to get rid of either of them (they are so very good at what they do), they are trying to unify them.

 

[marcus]

Originally posted by Mentat
Well, like I said, General Relativity is very good at describing macroscopic events. It is, as far as I know, flawless in this (if there were a real flaw, it would not be a theory). However, it breaks down at the quantum level.

The exact inverse is true of Quantum Theory...it is very good at describing microscopic phenomena; pretty much flawless (though not completely understood, and not at all able to be conceptualized), but its predictions become almost completely irrelevant at the macroscopic scales.

Since no one wants to get rid of either of them (they are so very good at what they do), they are trying to unify them.

there is a difference of opinion here---relativists will say one thing and particle theorists say another

a review by a relativist will generally explain why the HEP community does want to get rid of GR----replace it with a background-based field theory similar to conventional or stringy theories of other forces.

the replacement for GR should predict the same results in the limit at lower energies or larger scales, but should (in the opinion of the HEP folk as seen by relativists) be fundamentally different because background-based rather than background-independent

in the view of relativists stringy theories do not "contain" GR because they live on fixed background space while GR models gravity by dynamic geometry

if you want to understand the viewpoint of GR side, which really is different from what string-folk say, there are some online reviews for which I can supply links---if you really want to know about it: its a different perspective and takes getting used to

anyway, what you say would not sound right to some people who see stringy theories as, in fact, a way of getting rid of General Relativity (chucking out the geometrical description of gravity, getting rid of GR fundamentally while retaining superficial resemblance of predictions at low energy limit)

 

[marcus]

Mentat
it is too bad that the idea of background-independence (also diffeomorphism invariance) is so hard to discuss at non-tech level but it is possible that you could get interested in it and find out about it.

what it means is very deep: GR lives on a 4D manifold (spacetime "continuum") where the points do not have physical meaning!
Einstein struggled with this from 1912 to 1915 and finally accepted it and published background-independent GR.
In describing how he wrestled with this he called it the problem of "the meaning of the coordinates".

We still have not fully assimilated this meaninglessness of spacetime points which is basic to GR----and makes it challenging to quantize---in scientific culture. Physicists sometimes talk as if this mental hurdle did not exist, as if one could "unify" GR with other fields on a fixed background basis and still have an intact functional version of GR! This is to ignore what is fundamental.

You might get a lot out of the non-technical parts of Rovelli's draft book "Quantum Gravity". It goes into the history and the philosophy of this and describes why relativist and particle theorist see the business differently and explains the fundamental difficulty of linking background-dependent and background-independent theories.

Large parts of "Quantum Gravity" don't have formulas or have only a few simple ones on any given page, and are very readable. Rovelli brings history, philosophy, and technical discussion together in a remarkable way----it is a graduate textbook, not a popularization, so the philosophical discussion is serious in a way that one rarely sees. The book is due to be
published by Cambridge U Press and the draft is online at
Carlo Rovelli's homepage. I don't know as I can make the issue
any clearer than Rovelli does in, say, chapter 2

Anyway it is not obvious that string-etc is an attempt to unify GR and quantum theories

and the fundamental difference is not strictly speaking one of scale

 

[jeff]

mentat,

Be aware that LQG is taken seriously only by a handful of people in the physics community. In fact, a fair idea of just how marginal the LQG program is and has always been can be gotten by searching the los alamos national archives under both high energy theory, and general relativity and quantum cosmology over the last 12 years under each of "ashtekar", "loop quantum gravity". "spin network", and "spin foam". You get approximately 200 papers in total, an average of only 17 papers per year!.

By comparison, over the same period of time there have been tens of thousands of string theory papers. String theory dominates quantum gravity research since it's our only theory of quantum gravity. LQG is meant to be a quantum gravity theory, but it's not. It's nothing more than a very simple though interesting but failed attempt to achieve what strings have, to become another genuine quantum theory of gravity. Given the amount of LQG propaganda that is spread throughout this forum - which is one of the few places where LQG is popular, even though no one here really understands it - I must encourage you and others in the strongest terms to verify this by either phoning a university physics department with a string theory group or by submitting a question to sci.phys.research.

I will be addressing this issue more fully in the future.

By the way, Greg Bernhardt, the owner of this site pm'ed me to the effect that he agrees with me (he also asked me to be broach the subject gently).

 

[selfAdjoint]

Jeff, I can't help but find this post irrelevant. I am sure that in other contexts you would be eager to disavow the idea that correctness in science is to be determined by headcount.

The response of physicists to the various initiatives of stringy physics has been described (perhaps unfaily caricatured) as a feeding frenzy. Certainly an awful lot of the papers on hep-th will not be cited say, five years from now. This could be true of the LQG papers too, of course, but which is which cannot be determined until the physical consequences are worked out.

 

[marcus]

I am reminded of a Usenet SPR post from Jeffery Winkler that I quoted at the head of a nearby thread called "Interesting Argument Among String Thinkers". It was 27 september post, if I remember, and here's part of what it said:

"...I don't think there's anything wrong with the anthropic principle. If you take M-theory + inflationary cosmology + anthropic principle, you get the majority view among physicists.

Jeffery Winkler

http://www.geocities.com/jefferywinkler"

there is an argument floating around based on "the majority view among physicists" but I am not sure as to how well-founded or even useful---in actual science that is.

To me, the most interesting critique of string thinking is from within (e.g. Tom Banks) and from those (e.g. Peter Woit) completely outside quantum gravity. Of course there is always the question of what is propaganda and in whose interest and what is it being used to enforce. LQG researchers rarely bother to criticize string-think but simply describe what is distinctive and different about their approach and why they think it is interesting.
this is not a propaganda attack and if there is "Emperor's Clothes"-type criticism it is more apt to be internal or come from a different quarter

 

[marcus]

just for my own pleasure, in case the original suffers substantial editing later, I am saving the post to which selfAdjoint just replied:

----------------
mentat,

Be aware that LQG is taken seriously only by a handful of people in the physics community. In fact, a fair idea of just how marginal the LQG program is and has always been can be gotten by searching the los alamos national archives under both high energy theory, and general relativity and quantum cosmology over the last 12 years under each of "ashtekar", "loop quantum gravity". "spin network", and "spin foam". You get approximately 200 papers in total, an average of only 17 papers per year!.

By comparison, over the same period of time there have been tens of thousands of string theory papers. String theory dominates quantum gravity research since it's our only theory of quantum gravity. LQG is meant to be a quantum gravity theory, but it's not. It's nothing more than a very simple though interesting but failed attempt to achieve what strings have, to become another genuine quantum theory of gravity. Given the amount of LQG propaganda that is spread throughout this forum - which is one of the few places where LQG is popular, even though no one here really understands it - I must encourage you and others in the strongest terms to verify this by either phoning a university physics department with a string theory group or by submitting a question to sci.phys.research.

I will be addressing this issue more fully in the future.

By the way, Greg Bernhardt, the owner of this site pm'ed me to the effect that he agrees with me (he also asked me to be broach the subject gently).

__________________
Keep it about the physics.
------------------------------------------------

Edit: by Jeff obviously

 

[Avron]

I am absolutely taken aback in awe of how good this forum has become. You guys are great...to say the least.
I stopped looking at the old forum on yahoo because of all the hair-braned, psuedo-intellectual bunk that was the bulk of it. Had the forum been this good then, I would never have left.
I commend you all for bringing this forum to what it should always have been.
Is it now moderated or something? LOL...whatever the thing is, I'm glad for it.

Avron Prosini

 

[jeff]

Originally posted by Avron
I am absolutely taken aback in awe of how good this forum has become. You guys are great...to say the least.
I stopped looking at the old forum on yahoo because of all the hair-braned, psuedo-intellectual bunk that was the bulk of it. Had the forum been this good then, I would never have left.
I commend you all for bringing this forum to what it should always have been.
Is it now moderated or something? LOL...whatever the thing is, I'm glad for it.

Avron Prosini

Dude, I have a feeling there is plenty more of this to come, so tell all your friends to join. Things around here could get real interesting in the next little while.

 

[jeff]

Originally posted by selfAdjoint
Jeff, I can't help but find this post irrelevant. I am sure that in other contexts you would be eager to disavow the idea that correctness in science is to be determined by headcount.

I'm not worried about people who like me can evaluate theories on their own. I'm worried about the 99.9 % of the people here who've decided to invest time (and as you know this sort of physics is difficult and demands much energy) in learning about what drives current research in QQ - which of course is SMT (String/M-theory, or as I like to call it these days, S&MT) and not LQG or any of the other small scale research programs - but are easily mislead, intentionally or not, into believing that because it's not "background-independent", they should forget about SMT, and thus virtually everything that's going on in theoretical physics these days. For example, I don't think you appreciate how monumentally important SMT has and continues to be in moving forward ideas in connection with QFT, cosmology, and mathematics. As well, the people in LQG are constantly on the lookout for a way to connect LQG to SMT. Do you think they'd be doing that if they felt the same way about SMT as some of the people here do?

Also, we're not talking about politics in which opinion and fact are closely mixed. We're talking about an exact science in which there is a an extremely strong relation between what is currently the most popular idea and what is in fact currently our best idea. I don't think the right place for neophytes to start is with polemical papers or the jaundiced and ill-informed personal opinions of others. Consider the thread begun by marcus entitled "String irrelevant to quantizing General Relativity (quantum spacetime geometry)". Now, technically, he was talking about quantizing GR directly which is not what SMT does (this is in fact why LQG was so unlikely to ever work), but I'm pretty sure he knew that the people here would not pick up on that subtlety.

 

[marcus]

Originally posted by Avron
I am absolutely taken aback in awe of how good this forum has become. You guys are great...to say the least.
I stopped looking at the old forum on yahoo because of all the hair-braned, psuedo-intellectual bunk that was the bulk of it. Had the forum been this good then, I would never have left.
I commend you all for bringing this forum to what it should always have been.
Is it now moderated or something? LOL...whatever the thing is, I'm glad for it.

Avron Prosini

Avron this is a cheering comment although you could be more specific---or you might illustrate what you mean and what you'd like to see more of.

I didnt see the "old forum on yahoo" you mention. Was that a Doctor Kaku board or a Greg board?

I was delighted by the term "hair-braned" to describe a category of bunk. It has definite possibilities.

 

[Mentat]

Originally posted by marcus
there is a difference of opinion here---relativists will say one thing and particle theorists say another

a review by a relativist will generally explain why the HEP community does want to get rid of GR----replace it with a background-based field theory similar to conventional or stringy theories of other forces.

the replacement for GR should predict the same results in the limit at lower energies or larger scales, but should (in the opinion of the HEP folk as seen by relativists) be fundamentally different because background-based rather than background-independent

in the view of relativists stringy theories do not "contain" GR because they live on fixed background space while GR models gravity by dynamic geometry

if you want to understand the viewpoint of GR side, which really is different from what string-folk say, there are some online reviews for which I can supply links---if you really want to know about it: its a different perspective and takes getting used to

anyway, what you say would not sound right to some people who see stringy theories as, in fact, a way of getting rid of General Relativity (chucking out the geometrical description of gravity, getting rid of GR fundamentally while retaining superficial resemblance of predictions at low energy limit)

I find this odd (though I must confess that you're making sense), since Michio Kaku (one of the strongest supporters of String Theory) has always described it as the next step that Einstein had been trying to take, but couldn't. He (Kaku) always says that String Theory is just like Relativity, except that it requires more dimensions. Now, of course, I know that this is an over-simplification, but I didn't know that string theory was trying to negate GR, in it's attempt at unification.

 

[marcus]

Originally posted by Mentat
I find this odd (though I must confess that you're making sense), since Michio Kaku (one of the strongest supporters of String Theory) has always described it as the next step that Einstein had been trying to take, but couldn't. He (Kaku) always says that String Theory is just like Relativity, except that it requires more dimensions. Now, of course, I know that this is an over-simplification, but I didn't know that string theory was trying to negate GR, in it's attempt at unification.

Mentat, I know you can handle technical exposition (with equations and such) but I'm going to suggest a few pages of
plain words with little or no math, in case you want a different person's perspective.

Pages 6 through 9 of Rovelli's "Living Reviews in Relativity" article
the topic headings (he follows an outline) are

2. Quantum Gravity: Where are we?
2.1 What is the problem? The view of a high energy physicist.
2.2 What is the problem? The view of a relativist.

It is a stock presentation of the cultural split between General Relativity and string thought, and the two approaches to quantizing gravity. The other reviews and survey papers I can think of go over the same situation in much the same way.

I hope this link works

http://www.livingreviews.org/Articles/Volume1/1998-1rovelli

I will get some other links but have to go now. If the link doesn't work I will try to correct it as soon as I have a moment

Living Reviews in Relativity is put on line by the Max Planck Institute of Gravitation Physics (Albert Einstein Institute) at Berlin
they invited Rovelli to do one of the first articles, I gather

 

[jeff]

Originally posted by Mentat
I find this odd (though I must confess that you're making sense), since Michio Kaku (one of the strongest supporters of String Theory) has always described it as the next step that Einstein had been trying to take, but couldn't. He (Kaku) always says that String Theory is just like Relativity, except that it requires more dimensions. Now, of course, I know that this is an over-simplification, but I didn't know that string theory was trying to negate GR, in it's attempt at unification.

Odd? You mean you find it odd that michio kaku, string theorist and full professor at CCNY who graduated at the top of his class at harvard knows less then marcus? The best people from whom to learn string theory is from string theorists, not marcus or LQG people. Now, I recently addressed this issue in detail when selfAdjoint specifically asked me too. I repost it here for your convenience. If you have any questions, don't hesitate to ask.

Firstly, any correct theory of quantum gravity must in the appropriate limits give rise to helicity-2 excitations of the gravitational field because GR reduces in it's weak-field approximation to the theory of a helicity-2 excitation that couples to itself and everything else in a way that respects the general covariance of GR. In the other direction, such a self-interacting theory implies GR. Now for the hard part:

Transition amplitudes in ST are defined in a 1st quantized formalism based on the world-sheet action

S_G = - (1/4πα′) ∫ dμ_γγ^abG_μν(X)∂_aX^μ∂_bX^ν

in which the basic fields X^μ of the theory embed the world-sheet with metric γ^ab and measure dμ_γ in a background spacetime with metric G_μν. Recall that in QFT the tree level feynman diagram for an interaction consists of a vertex where legs representing incoming and outgoing particles meet. Analogously, for closed strings we have a sphere with punctures to which are glued the ends of "world-tubes" representing incoming or outgoing strings. The invariance, known as weyl-invariance, of S_G under rescalings γ^ab → e^φγ^ab of the world-sheet metric allows the projection (continuous deformation) of world-tubes onto the punctures, effectively sealing each one by insertion of a point sitting at which is a vertex operator defined in terms of X^μ and it's world-sheet derivatives and carrying the quantum numbers of the original incoming/outgoing string state vector: This is known as the state-operator correspondence, an example of which is given at the end of this post. Higher order interactions are obtained as compact oriented boundaryless surfaces of genus g with a vertex operator insertion V_i(k_i) for each incoming/outgoing closed string of momentum k_i. Hence, amplitudes for n external string states have the form of a sum of path-integrals with insertions

<V₁(k₁)&sdot;&sdot;&sdot;V_n(k_n)> ~ &sum;_g=0,1,2,... &int;_g D&gamma;_abDX^&mu; V₁(k₁)&sdot;&sdot;&sdot;V_n(k_n)e^-S_G.

Now, take

G_&mu;&nu;(X) = &eta;_&mu;&nu; + &epsilon;_&mu;&nu;(X)

with

&epsilon;_&mu;&nu;(X) = &int; d²⁶k &epsilon;_&mu;&nu;(k)e^ik&sdot;X

everywhere small compared to &eta;_&mu;&nu;. Then

e^-S_G = e^{-(S_&eta; + S_&epsilon;)} = e^-S_&eta; &sum;_n=0,1,...(- 4&pi;&alpha;&prime;)^-n(1/n!) &int; d²⁶k₁&sdot;&sdot;&sdot;d²⁶k_n V(k₁)&sdot;&sdot;&sdot;V(k_n)

in which

V(k) &equiv; &epsilon;_&mu;&nu;(k)V^&mu;&nu;(k) &equiv; &epsilon;_&mu;&nu;(k) &int; d&mu;_&gamma; &gamma;^ab&part;_aX^&mu;&part;_bX^&nu;e^ik&sdot;X

is a vertex operator coupling strings to fluctuations in the background metric G_&mu;&nu;. Note that like all vertex operators, V is an integral over the world-sheet since it can be inserted at any point. Next, observe that &epsilon;_&mu;&nu; picks out the symmetric part of V^&mu;&nu;, so V is the vertex operator of a spin-2 state. Also, since the state-operator correspondence (see the example at the end of this post) requires that vertex operators transform like the string state vectors they represent, they must include the factor e^ik&sdot;X to transform properly under spacetime translations X^&mu; &rarr; X^&mu; + a^&mu;. Now, any insertion must respect the local weyl symmetry of the theory. In particular, demanding that V be weyl-invariant requires (see polchinski I Chap 3.6)

k² = k²&epsilon;_&mu;&nu;(k) = 0 &harr; &uArr;&epsilon;_&mu;&nu;(X) = &uArr;G_&mu;&nu;(X) = 0

k^&mu;&epsilon;_&mu;&nu;(k) = 0 &harr; &part;^&mu;&epsilon;_&mu;&nu;(X) = &part;^&mu;G_&mu;&nu;(X) = 0,

&epsilon;^&mu;_&mu;(k) = 0 &harr; &epsilon;^&mu;_&mu;(X) = 0.

In addition to showing that the spin-2 excitations are massless, because the ricci tensor R_&mu;&nu; satisfies

R_&mu;&nu; &prop; &part;_&mu;&part;_&nu;&epsilon;_&lambda;^&lambda; - 2&part;_&lambda;&part;_(&mu;&epsilon;_&mu;)^&lambda; + &uArr;&epsilon;_&mu;&nu; + O(&epsilon;²),

this also shows that to leading order in metric fluctuations, weyl-invariance in the pure helicity-2 theory requires that the background G_&mu;&nu; satisfy the vacuum einstein equations R_&mu;&nu; = 0.

Because massless states are transversally polarized, V must be invariant under the shift

&epsilon;_&mu;&nu;(k) &rarr; &epsilon;_&mu;&nu;(k) + k_&mu;&xi;_&nu; + k_&nu;&xi;_&mu;

by longitudinal polarizations. In terms of the metric, this gauge-invariance

&epsilon;_&mu;&nu;(X) &rarr; &epsilon;_&mu;&nu;(X) + k_&mu;&xi;_&nu;(X) + k_&nu;&xi;_&mu;(X)

is an infinitesimal diffeomorphism generated by the vector field &xi;^&mu;(X) in the approximation where O(&epsilon;²) terms are neglected and under which R_&mu;&nu; = 0 is invariant. In fact R_&mu;&nu; = 0 is the only spacetime diffeo-invariant equation that reduces to &uArr;G_&mu;&nu;(X) = 0 in the linearized limit.

In sum, weyl-invariance requires spin-2 excitations be massless and couple in a gauge-invariant way, that is, it requires the general covariance of GR, justifying the interpretation of helicity-2 excitations as gravitons.

State-operator correspondence for the graviton vertex operator:

Define world-sheet coordinates

z = e^{-i&sigma; + &tau;} , z* = e^{i&sigma; + &tau;}

with &sigma; = &sigma; + 2&pi; the periodic coordinate along the string and &tau; the time coordinate on the world-sheet. We then have

V &prop; &epsilon;_&mu;&nu;&int;d²z &part;_zX^&mu;(z)&part;_z*X^&nu;(z*)e^{ik&sdot;X(z,z*)}

in which we've taken the world-sheet metric in "conformal gauge" so that it effectively drops out. Then (up to proportionality) the state-operator correspondence is

&part;_zX^&mu;(0) &harr; &alpha;_-1^&mu; , &part;_z*X^&mu;(0) &harr; (&alpha;_-1^&mu;)* , e^{ik&sdot;X(0,0)} &harr; |0;0;k>

where &alpha;_-1^&mu; and (&alpha;_-1^&mu;)* excite left- and right-moving n = 1 modes.

Putting these together gives

V &harr; &epsilon;_&mu;&nu;&alpha;_-1^&mu;(&alpha;_-1^&nu;)*|0;0;k>.

 

[marcus]

Originally posted by Mentat
I find this odd (though I must confess that you're making sense), since Michio Kaku (one of the strongest supporters of String Theory) has always described it as the next step that Einstein had been trying to take, but couldn't. He (Kaku) always says that String Theory is just like Relativity, except that it requires more dimensions. Now, of course, I know that this is an over-simplification, but I didn't know that string theory was trying to negate GR, in it's attempt at unification.

Mentat, I am not responding to the immediately preceding post but still, when I have a moment, trying to see how best you might (or someone in your situation might) get some perspective on the quantum gravity problem and the deep division between the two main approaches to it.
I have already suggested 3 or 4 general overview pages in Rovelli (there are similar more recent discussions by people on the General Relativity side but Rovelli's will do for starters)

For balance, maybe you would like to look at a very recent comparison made by a STRING THEORIST. It is always good to get balanced views. There is a July 2003 paper called
"Loops versus Strings" by E. Alvarez that was presented to a general audience of particle physicists at a conference this summer called "What comes after the Standard Model"

I don't have time to get the link now. But I will fetch it later and edit it in.

 

[Ramanujan12]

LOOOOOOOL!.....LOOOOOOOL!.....
LOOOOOOOL!.....LOOOOOOOL!.....
LOOOOOOOL!.....LOOOOOOOL!.....
LOOOOOOOL!.....LOOOOOOOL!.....
LOOOOOOOL!.....LOOOOOOOL!.....



http://groups.msn.com/shamanism/mathandphysics.msnw?action=ShowPhoto&PhotoID=9

 

[marcus]

Originally posted by Mentat
I find this odd (though I must confess that you're making sense), since Michio Kaku (one of the strongest supporters of String Theory) has always described it as the next step that Einstein had been trying to take, but couldn't. He (Kaku) always says that String Theory is just like Relativity, except that it requires more dimensions. Now, of course, I know that this is an over-simplification, but I didn't know that string theory was trying to negate GR, in it's attempt at unification.

despite rather many interruptions I want to continue pursuing this thread

here is the link I promised to Enrique Alvarez "Loops versus Strings"

http://arxiv.org/gr-qc/0307090

the article is interesting and credible for several reasons

he is actively engaged in string research currently
(around 16 string/brane articles since mid-1999 show in arxiv, you can check in arxiv to see how much they have been cited by other authors---compare e.g. Kaku zero articles since the one in mid-1999 and that one has been cited only once, according to the arxiv citation-bot)

the article is recent----mid 2003

the article was an invited talk at the mid-2003 HEP
conference at Portoroz called "What comes beyond the standard model?"----that is, other HEP theorists want to hear what Alvarez has to say

the guy is senior and broadly knowledgeable----so capable of doing an intelligent overview and making a useful string vs loop comparison

CAVEAT: his viewpoint (and that of his HEP audience) is very
UN-general relativity, his perspective is HEP/QFT/string. So you don't get a relativist perspective. I cannot vouch for his article
beyond mentioning the circumstantial evidence that he seems to be senior, currently active in research, and respected.

Mentat, the situation in quantum gravity is currently exciting, the picture is changing, the split between General Relativity and the relativists on one hand and the (far mor numerous, almost mob) of string/braners on the other is fascinating, or so I find. It points to a deep division in the foundations of physics concerning the nature of space and time (this is what "background-independence" is about---and the problem that string-thinking doesn't have it). So these things are IMHO very worth trying to follow, even enlightening, maybe historical in some sense.

I don't know of any other 2003 article of the "Loops versus Strings" overview/comparison type from the string side! This
Alvarez one is all I could find! So I offer it to you or anyone who wants to follow the action, to look over.

But on the other hand there are quite a few recent overviews of developments in quantum gravity from the loopers or, to put it more clearly, the GR people involved in quantizing GR.
Smolin has a 2003 paper called "How far are we from a theory of quantum gravity?" which has been extensively cited, to mention just one, and Rovelli even has this book "Quantum Gravity" in current draft form at his site.

I will get some links for these things, and also check the arxiv bot to get citation numbers for some current Ashtekar, Smolin or Rovelli work---it is an interesting quantitative measure of how important or useful a research paper has been---I've mostly been judging by the citations I see in other papers I happen to read, a partially subjective assessment which I believe is also important to make. Bear in mind that QGR is a small field and the numbers are small, but there seems to be a shift underway (in science a small active minority can sometimes bring about change)

Oh BTW you mentioned the fact that, while GR appeared in 1915, Einstein spent much of his time in later life working on a unified theory of other stuff besides gravity. The essence of the 1915 GR is its background-independence---you don't specify a static spacetime geometry ahead of time but let the geometry be dynamic and variable. I DON'T KNOW if Einstein's later efforts at unification retained this essential, and new, view of space and time! He may have given up and gone back to some fixed background approach like "special" relativity. Today's particle/field theories (QFT, Standard Model, whatever) are fixed-background.
My GUESS would be that Einstein would have kept to the dynamic GR spacetime and tried to bring other forces besides gravity into the picture which as we know is very hard and would help explain why he met with frustration. In effect that line of effort is only just beginning to get results and make testable predictions! So what you decide, on a verbal level, to call a continuation of Einstein's project is kind of subjective and depends on how detailed a picture you have of how he was trying to construct a unified model.

 

[Mentat]

Originally posted by marcus
Mentat, I know you can handle technical exposition (with equations and such) but I'm going to suggest a few pages of
plain words with little or no math, in case you want a different person's perspective.

Thank you. I may normally love technical exposition as much as layman explanation, but I've got a horrible headache, and it shows no sign of subsiding (only getting worse [b(]), and so a little easy explanation is most welcome.

Pages 6 through 9 of Rovelli's "Living Reviews in Relativity" article
the topic headings (he follows an outline) are

2. Quantum Gravity: Where are we?
2.1 What is the problem? The view of a high energy physicist.
2.2 What is the problem? The view of a relativist.

It is a stock presentation of the cultural split between General Relativity and string thought, and the two approaches to quantizing gravity. The other reviews and survey papers I can think of go over the same situation in much the same way.

I hope this link works

http://www.livingreviews.org/Articles/Volume1/1998-1rovelli

I will get some other links but have to go now. If the link doesn't work I will try to correct it as soon as I have a moment

Living Reviews in Relativity is put on line by the Max Planck Institute of Gravitation Physics (Albert Einstein Institute) at Berlin
they invited Rovelli to do one of the first articles, I gather

It works. I will begin reading it ASAP. Thank you.

 

[jeff]

Originally posted by Mentat
Thank you. I may normally love technical exposition as much as layman explanation, but I've got a horrible headache, and it shows no sign of subsiding (only getting worse [b(]), and so a little easy explanation is most welcome.



It works. I will begin reading it ASAP. Thank you.

Mentat,

Learning about string theory from papers specifically written to deride it is like learning about judiasm by studying the qu'ran. You understand what I mean? LQG really is quite dead in the physics community. In fact it never really was totally alive. Marcus harps on background independence, but what's much more difficult and important is producing a genuine QGT, which string theory is and LQG isn't and never has been. Why are you so resistent to simply verifying what I and marcus are saying? Marcus, doesn't really understand the problem with LQG, which is why he continues with this ridiculous program of steering everyone away from what's actually going on in QG. Remember, the Q in LQG means quantum field theory, and marcus doesn't understand QFT, as he indicated when he told me he'd be unable to contribute to any group study QFT thread. In any event, if you go through his posts you'll find that he never responds to any posts on the subject of quantum theory. I don't think it's fair for marcus to cause people to waste time on learning something that's both difficult and irrelevant.

Let me give you an example of how twisted this whole thing is by telling you an interesting story about the development of LQG over the past year or so. One key indicator of the viability of any QGT is it's ability to reproduce hawking's famous relation between the entropy of any black hole and the area of it's event horizon. Unlike string theory - which by the way has gone considerably further beyond the original QFT result - LQG was unable to produce hawking's famous formula. But last year, something happened that for the first time generated genuine excitement outside of the 20 or so people that work full time on LQG. Consider a schwarzschild black hole spacetime M. If we perturb M, it will "ring" like a bell. Someone noticed something about this phenomenon that seemed to suggest how LQG might produce the hawking black hole entropy relation. Unfortunately though, there was a problem. It appeared that if this argument was taken seriously, LQG would be inconsistent with the existence of matter, clearly not a good thing. Now, marcus picked up on the good part, but didn't really understand the bad part. After I made three or four attempts to convince him that he wasn't seeing the big picture, it finally began to sink in. There were a number of suggestions made about how to keep matter in LQG without sacrificing the hawking relation, which by the way I very politely brought to marcus's attention. However, it was recently shown that the whole thing was only a coincidence since it only works in the case of a schwarzschild black hole. If things had panned out, it would have been the first and only thing LQG ever predicted correctly. I private messaged marcus to say if he liked, he could explain what happened since he'd been following it. But instead he decided to say nothing about it. Again, I encourage you to verify all of this for yourself, if not for you, at least for other members.

 

[Mentat]

jeff,

I hope others don't construe your post as a personal attack against marcus. I'm rather positive that that's not what you are doing, but instead just exposing the likelyhood that there is much more to string theory than marcus is aware of (we all have things we don't know...if we didn't we'd never learn anything ). However, you have left me torn. On the one hand, I have been interested in and awed by string theory since I first heard of it. On the other hand, I strive to remain open-minded, and so I want to learn as much as I can about LQG as well, since it seems to be the "competition" for TOE.

If you are right about LQG, then I won't waste my time on it, but if marcus is right, then I will regret not having taken a closer look at it. I just don't know which of you is right on this matter.

 

[Mentat]

Originally posted by Ramanujan12
LOOOOOOOL!.....LOOOOOOOL!.....
LOOOOOOOL!.....LOOOOOOOL!.....
LOOOOOOOL!.....LOOOOOOOL!.....
LOOOOOOOL!.....LOOOOOOOL!.....
LOOOOOOOL!.....LOOOOOOOL!.....



http://groups.msn.com/shamanism/mathandphysics.msnw?action=ShowPhoto&PhotoID=9

What's with the collage of math, Ramanujan12?

 

[marcus]

copy of jeff's post

--------------------------------------------------------------------------------
Originally posted by Mentat
...<snip>...It works. I will begin reading it ASAP. Thank you.
--------------------------------------------------------------------------------



Mentat,

Learning about string theory from papers specifically written to deride it is like learning about judiasm by studying the qu'ran. You understand what I mean? LQG really is quite dead in the physics community. In fact it never really was totally alive. Marcus harps on background independence, but what's much more difficult and important is producing a genuine QGT, which string theory is and LQG isn't and never has been. Why are you so resistent to simply verifying what I and marcus are saying? Marcus, doesn't really understand the problem with LQG, which is why he continues with this ridiculous program of steering everyone away from what's actually going on in QG. Remember, the Q in LQG means quantum field theory, and marcus doesn't understand QFT, as he indicated when he told me he'd be unable to contribute to any group study QFT thread. In any event, if you go through his posts you'll find that he never responds to any posts on the subject of quantum theory. I don't think it's fair for marcus to cause people to waste time on learning something that's both difficult and irrelevant.

Let me give you an example of how twisted this whole thing is by telling you an interesting story about the development of LQG over the past year or so. One key indicator of the viability of any QGT is it's ability to reproduce hawking's famous relation between the entropy of any black hole and the area of it's event horizon. Unlike string theory - which by the way has gone considerably further beyond the original QFT result - LQG was unable to produce hawking's famous formula. But last year, something happened that for the first time generated genuine excitement outside of the 20 or so people that work full time on LQG. Consider a schwarzschild black hole spacetime M. If we perturb M, it will "ring" like a bell. Someone noticed something about this phenomenon that seemed to suggest how LQG might produce the hawking black hole entropy relation. Unfortunately though, there was a problem. It appeared that if this argument was taken seriously, LQG would be inconsistent with the existence of matter, clearly not a good thing. Now, marcus picked up on the good part, but didn't really understand the bad part. After I made three or four attempts to convince him that he wasn't seeing the big picture, it finally began to sink in. There were a number of suggestions made about how to keep matter in LQG without sacrificing the hawking relation, which by the way I very politely brought to marcus's attention. However, it was recently shown that the whole thing was only a coincidence since it only works in the case of a schwarzschild black hole. If things had panned out, it would have been the first and only thing LQG ever predicted correctly. I private messaged marcus to say if he liked, he could explain what happened since he'd been following it. But instead he decided to say nothing about it. Again, I encourage you to verify all of this for yourself, if not for you, at least for other members.


__________________
Keep it about the physics.

 

[Mentat]

So...uh...why'd you copy his post? I was kind of expecting a response to it .

 

[Ramanujan12]

Mentat.

The collage there is an unarranged string of mathematical (QST [inter] holographic theory) crunched together. With lack of a uniform language and higher-order symmetry (hypersymmetry) configured yet. To many notes not enough spaces [null]. More forthcomming...

 

[Ramanujan12]

BRST and World-Sheets.

The world is not written on a sheet.

Actually no wait...it is!. It's the language that's written on the sheet we haven't translated it into mathematics yet.

the Word-Sheet and the World-Sheet; we have not a Worldperfect spread for this darn sheet.

 

[jeff]

Originally posted by Mentat
So...uh...why'd you copy his post? I was kind of expecting a response to it .

Congratulations, you've asked the $64,000 question.

 

[jeff]

Originally posted by Mentat
Well, according to M-Theory, the Universe is not expanding or contracting, it's doing both. This is a result of the t duality, which I've brought up on numerous threads before.

Let me explain the aspect of T-duality that causes this misconception - including in me (but only for a couple of hours ) when I'd first heard about it as an undergrad. Like particles, strings can carry momentum. But strings also carry winding number, and in particular, they can wind around compact dimensions. Consider just one compact dimension, a circle, say with radius R. T-duality says that there's a mathematically different but physically equivalent description of this system in which the circle has radius &prop; 1/R so that a small circle in the original description becomes a big circle in the new but physically identical description. Thus T-duality is a symmetry relating string theories compactified on small and large tori (tori are higher dimensional generalizations of the circle. For example, the circle is a 1-torus, and the 2-torus is the surface of a donut). From this it should be clear how easy it is to goof if T-duality isn't explained properly when you first hear about it.