Why are there gravitons in string theory

Sean Carroll has written a text for Nude Socialist. It has an optimistic name

• String theory: not dead yet. ;-)

After I read the full text, it looks fair even though I am flabbergasted by the very observation that some people apparently think that physicists can suddenly change their opinions about theoretical physics because of a campaign organized mainly by two crackpots.

If an activist such as Al Gore organizes such a campaign in climate science, he can scare all sane people and everyone starts to twist the numbers and publish higher, catastrophic estimates of the future warming. It is hard to figure out that what the scientists produce is a biased pile of nonsense because every number a priori seems as good as every other number.




But in theoretical physics, this is simply not possible. If someone scares you into studying a theory of quantum gravity that differs from string theory, it simply won't work. Ingredients won't fit together. When you're ordered to work on such an inconsistent theory, you will feel like an idiot after you write down the second equation. Rightfully so. The whole machinery will collapse just like if you replace gasoline in your Chrysler by used toilet paper.

Gravity in string theory

But back to the main topic. Some people ask why string theory inevitably predicts spin 2 massless particles that moreover interact as gravitons.



Let me explain. Consider a closed string - a loop of energy - that oscillates in a spacetime. There exist functions

• X^m (sigma,tau)

that describe the embedding of the two-dimensional worldsheet, a history of propagating string, in this spacetime. The laws controlling the oscillations may be described by a two-dimensional field theory defined on this worldsheet - a two-dimensional manifold with a spatial coordinate "sigma" along the string (X^m have periodic conditions in sigma, to make the string closed) and a temporal coordinate "tau".

Its equations of motion are essentially wave equations for the scalars X^m.

For every background geometry (and possibly other fields such as dilaton and gauge fields), there exists a two-dimensional action on the worldsheet. But only some geometries lead to a consistent string theory. In fact, the worldsheet theory must be invariant under conformal transformations - transformations of the worldsheet coordinates that preserve angles - because the internal geometry on the worldsheet and the choice of coordinates on the worldsheet must be unphysical: if they were not, we would introduce new, unwanted degrees of freedom (essentially new spacetime coordinates). Scaling is the most important conformal transformation.

How does the theory on the worldsheet change under scalings? A quantum field theory - and the theory on the worldsheet is an example - has various coupling constants. The change of a "running" coupling constant under scaling transformations is generally encoded in the so-called beta-function. For each coupling constant, you have one beta-function.



At the beginning, we mentioned that for every background geometry, we have one theory on the worldsheet. All numbers describing such a geometry, namely all components of the metric tensor

• g_{mn} (X^k)

as functions of spacetime coordinates, are thus coupling constants of the theory on the worldsheet. How much do they run? What are the beta-functions? It is not hard to see that for every value of a component of the metric tensor, there will be one beta-function. Consequently, the beta-functions will depend on X^k and they will carry two vector indices. Moreover, they can be seen to include second spacetime derivatives of the metric tensor. When you think about the "manifest" spacetime diffeomorphism symmetry, it is not hard to see that the full answer for the beta-functions must actually be proportional to

• R_{mn} (X^k).

If the worldsheet theory is consistent as a string theory, it must be scale-invariant, and the spacetime geometry must thus be Ricci-flat! We have just derived Einstein's equations from scale invariance of a two-dimensional theory. Or at least we have sketched the derivation. If we considered backgrounds with other fields (matter fields), they would also contribute to the beta-function. We would obtain Einstein's equations with the correct right-hand side.

State-operator correspondence

Strings can only propagate consistently on backgrounds that respect the laws of general relativity or its generalizations. Does it mean that there are gravitons? Yes, it does.

Take the worldsheet action for a particular spacetime geometry, and make an infinitesimal (epsilon) change of the spacetime geometry so that the geometry remains Ricci-flat (for example, add a gravitational wave). Look at the difference of these two actions (and divide by epsilon). In other words, differentiate the worldsheet action with respect to the spacetime metric. You will inevitably get another integral over the worldsheet coordinates:

• integral d sigma d tau V(sigma,tau)
  

Any infinitesimal variation of the spacetime metric is thus associated with an operator V(sigma,tau) on the worldsheet. We can make something even more interesting. Cut a very small disk from the worldsheet. The new, short boundary of the worldsheet will look like a closed string. The actual length of this boundary is actually unphysical, because of the scaling symmetry of the worldsheet theory. For every wave functional on this closed string - a possible state in the Hilbert space of states of a single closed string - there will exist a local operator, and vice versa.

This one-to-one map is known as the state-operator correspondence.



Inserting the operator V(sigma,tau) at the point (sigma,tau) of the worldsheet is therefore equivalent to cutting a small disk around (sigma,tau) and integrating over all possible initial conditions on this circle weighted by an appropriate wave functional. For every local operator V(sigma,tau), there exists a state.

But previously, we have found an operator V(sigma,tau) for every possible infinitesimal variation of the background, e.g. for every gravitational wave. When we combine this old result with the most recent one, we see that for every gravitational wave, we discover one state of the closed string. In other words, closed strings will always have a state in their Hilbert space that is canonically associated with a change of the spacetime geometry.

Because closed strings in the Minkowski space - the simplest example (that approximates very well any background whose radius of curvature is much longer than the string scale, a typical distance scale associated with string theory) - are really described by a pile of ordinary harmonic oscillators, it is not hard to see that the states associated with the operators V(sigma,tau) corresponding to an infinitesimal perturbation of the spacetime geometry are spin 2 particles.

The operator V(sigma,tau) for a gravitational wave looks like

• exp(i k.X(sigma,tau)) partial_+ X^m(sigma,tau) partial_- X^n(sigma,tau)
  

It depends on a spacetime momentum vector "k". The complex exponential is multiplied by a holomorphic derivative of one "X" and the anti-holomorphic derivative of another "X". The two free indices "m,n" give it a spin equal to two.

Do the corresponding particles - closed strings with a particular vibration on them - interact as gravitons? You bet. When you deduce the interaction rates among these spin 2 particles composed of a vibrating closed string, the state-operator correspondence guarantees that they will respect the overall equations of motion given by Ricci-flatness - a condition that we have derived from the conformal symmetry.

There are other ways to see that the interactions of the string-theoretical gravitons must exactly respect the rules of general relativity. Every consistent gauge-invariant theory of spin 2 massless particles must inevitably have the diffeomorphism symmetry built in it which essentially guarantees that the theory is a version of general relativity.

You can indeed check that the spin 2 particles obtained from the vibrating closed strings are massless, gauge-invariant, and their interactions are consistent. That assures that the scattering amplitudes will coincide with those obtained from general relativity (in the long distance limit). You may also verify this conclusion by explicit calculations.

AdS/CFT

There are other approaches to string theory that have been shown to describe the same laws of physics. Holography in anti de Sitter spaces is a popular example. In this picture, there is a non-gravitational theory defined on the boundary of the anti de Sitter space at infinity and the key statement is that this theory is equivalent to a theory in the bulk.

The bulk theory is inevitably a gravitational theory. In other words, it is a consistent theory of quantum gravity, also known as string/M-theory. How can you see that there must always be gravity in the AdS bulk?

Well, the reasons are somewhat similar to the worldsheet arguments above. A particle that can go to the boundary of the AdS space corresponds to a local operator on the boundary. In this case, we are interested in the stress-energy tensor on the boundary, a rather special kind of a local operator. For every component of the stress-energy tensor of the boundary theory, there must exist a physical particle in the bulk. Once again, one can prove that the interactions of these particles must be consistent with the diffeomorphism symmetry: they must be gravitons.

There are other reasons why the bulk theory is always a gravitational theory. In non-gravitational theories, the entropy stored in a volume V is proportional to the volume. In gravitational theories, however, the maximum entropy you can squeeze into this volume is carried by a black hole and the black hole entropy is only proportional to the surface area. Gravitational theories secretly carry a small number of degrees of freedom than what you would naively think. This is a key fact that makes holography possible.

Matrix theory

In the BFSS Matrix theory, only highly supersymmetric backgrounds are well-understood. The origin of gravitons in any Matrix theory is always analogous to the case of the maximally supersymmetric, 11-dimensional background of M-theory.

If you have 32 supercharges, you can prove that there are states that preserve one half of the supercharges, namely 16. The broken generators can be combined into 8 complex pairs - 8 creation plus 8 annihilation operators in a fermionic harmonic oscillator. Each such an operator raises or lowers the spin by 1/2. They're exactly enough to climb from

• j_z = -2 ... to ... j_z = +2

because there are (+2 - (-2)) / (1/2) = 8 steps in between. 32 supercharges thus guarantee, once again, that the spin j=2 is the highest spin included in the simplest supermultiplet. Supersymmetry - one of the symmetries that can also be proven in Matrix theory - also implies that general relativity must be included in the effective action, by spacetime arguments.

There are other approaches to string/M-theory that can be proven to describe the same local physics in spacetime. In all cases, one can also explicitly show that the graviton is a part of the story. The methods to find the gravitons that we sketched above look very diverse but in overlapping situations, they are related.

A reply to Gina

Unsuccessfully linked elsewhere: Falsifiability in physics (direct link here)

Gina is one of the more sensible participants of the laymen's discussions about string theory. While the majority of others who think that Smolin's and Woit's comments make any sense are deeply confused people incapable of a rational manipulation with ideas related to theoretical physics, Gina at least sometimes tries to use his or her brain a little bit.

As we can see in this new text, this positive appraisal certainly doesn't hold universally.

Weakest link

Joe writes that a chain of reasoning is only as strong as its weakest link. You would think: what an obvious fact. Gina conjectures that there are cases when it's not true because the weakest links are strengthened by the stronger links. Well, I happen to think that if this occurs, we can be pretty sure that the person is doing something very different from a rational analysis.

Many links can make a lot of sense but if a whole argument depends on a link that cannot be replaced by another link and if this link is weak, the whole argument is weak, too. I simply can't imagine how a rational person could ever disagree with this fact. Weak links can only be ignored if there are other links that can be substituted instead of them into the logical chain and that can play their role, either completely or at least partially.

Gina also writes:

• Certainly Joe’s line is not so positive when it comes to string theory. Some of the links (or steps) are not only weak but yet non-existing.

Well, I think that this pair of sentences is thoroughly incompatible with a scientific mode of thinking. There are two basic problems with these sentences:

• It is not explained what statements are actually being discussed in these sentences: "string theory" is not a statement
  
• The statements are really untrue as long as the actual subject is any important and generally accepted well-known insight of string theory: how can someone be so "certain" about a statement that is demonstrably incorrect?
  

Let me explain a little bit. If science is done properly, we don't reduce all questions to the question "string theory Yes/No" that would be discussed with the help of oversimplified sociological pseudo-arguments: physicists leave this job to the crackpots and journalists.

Instead, physicists are talking about much more concrete, less ambitious questions that can actually be grasped and studied by the scientific method. It is nice if someone is ambitious but if he can't find any chain with strong links that makes any sense, the ambitions are useless or worse.

In junk sciences, things are very different. For example, the "climate change science" tries to transform every observation to an argument showing that "global warming is true", and the statement "global warming is true" is then used to justify any favorite assertion or policy of the people who like this particular junk science.

The similarity with God - something that always sits at the center of all arguments - is obvious. This new kind of God always plays a role in any system of ideas that the climate change "scientists" and their fans produce. Moreover, the links between this new kind of God and the actual observations or predictions are usually weak and sometimes they are upside-down.

But high-energy physics is not another junk science. High-energy physicists actually talk about concrete chains of arguments and facts and their relationships - relationships that usually circumvent the "big questions" and God - and they must very carefully realize that the chain is as weak as its weakest link. It is much more reasonable to connect two things by a chain if these two things are close to one another in which case the chain can often be extremely strong: we don't need to connect A and B by a long chain from A to God to B. When these things are done properly, we can demonstrate things like

• local quantum field theories of the known types can't lead to a predictive theory of quantum gravity at the Planck scale
  
• black holes inevitably have thermodynamic properties such as entropy and temperature
  
• string theory is finite to all orders of perturbation theory and probably beyond
• huge classes of theories at a comparable level of complexity as string theory are much less consistent than string theory
  
• a conformal field theory is equivalent to stringy quantum gravity in the anti de Sitter bulk
• experimentally established physics of gauge theories is connected with - in fact, equivalent to - gravitational physics in higher-dimensional spacetime that includes stringy phenomena
• vacua of string theory are connected
  

and thousands of others in such a way that all links are strong. Whoever thinks that the physicists are so stupid that they are consciously using chains where links are weak has been manipulated by factories producing fog and lies such as Peter Woit and Lee Smolin. Their statements are simply not true and every good theoretical physics grad student knows why.

The fact that many people repeat Woit's and Smolin's stupidities doesn't make these stupidities less stupid. The fact that it has become fashionable in certain communities to say these things doesn't make them intelligent either. Whoever repeats all these silly comments about divergences of string theory, doubts about Maldacena's duality, comments about the heresy of believing a large number of solutions to the fundamental equations of quantum gravity, or speculations about a hypothetical weakness of the string-theoretical picture in general is just an overly simple-minded person whom I assertively encourage to realize his or her severe intellectual limitations.

The weakest link in the string-theoretical image of the world is a direct experimental evidence. Everyone realizes that this is a weak link. It is not easy to get experimental data about quantum gravity. This unsurprising fact is a real difficulty for everyone who actually tries to think about these things and find answers, as opposed to a "critic" who only wants to spread hatred and lies.

No string theorist would dispute that the absence of a direct experimental proof is one of the weakest links in the chain of reasoning that leads us to think that string theory describes the real world at the most fundamental level. It is probably the weakest one. One can try to replace this weakest link by other, more theoretically flavored links, but everyone knows that the replacement is not perfect.

It is extremely important to realize and acknowledge what the weak links actually are. It is extremely important not to hide weak links. It is however equally important not to create the impression that some additional weak links exist if they don't. It is illegitimate to spread lies about weak links, even in the context when you make loads of dopes happy by spreading these lies.

Maldacena's duality

Gina tells us that he or she sees nothing wrong with a research program aimed to find weak points in or counter arguments to the strong Maldacena’s conjectures. Well, there are only three wrong things about this research program:

• this research program is demonstrably wrong because it contradicts things that have been established by careful analyses
  
• the very motivation of the research program is biased
• there is nothing such as strong and weak Maldacena's correspondence.
  

Whoever tries to think that there simply must exist weak links of the AdS/CFT correspondence and is not ready to change his or her opinion even when robust calculations show that the whole duality is unbreakable is a bigot. Moreover, the very approach trying to selectively look for weak links shows a prejudice of the researcher. The scientific community - and in most cases even individual scientists - should try to find out whether a statement is true instead of selectively trying to look for negative or positive arguments only. Gina's recommendation is just another example of a lack of scientific integrity. It is shocking if he or she doesn't realize that.

Needless to say, when things are done properly, we can see that there exist no weak links in the chain of reasoning that leads to the AdS/CFT correspondence, at least in the backgrounds that have been widely studied such as the N=4 gauge theory in four dimensions. Indeed, to falsify Maldacena's duality, it would be enough to find one calculation that would demonstrate that the AdS/CFT correspondence doesn't work. No such calculation exists.

Gina claims that Polchinski has argued that finding errors in a duality is a priori misguided. Polchinski has never claimed so. It is misguided to be looking for errors in "strong" Maldacena's correspondence and it is misguided to talk about "weak" Maldacena's correspondence but it is only misguided a posteriori, after the checks have been carefully made in thousands of papers, checks that imply that no such weak links exist and no available definition of a "weak" Maldacena's correspondence can survive a scrutiny.

Temptation of rigor

Gina shows us that he or she has completely misunderstood the facts about rigor being useless or worse when the assumptions are physically wrong. He or she asks:

• If there are scientific reasons to reject the argument based on “Rehren duality” that is fine, but what does it have to do with rigor in general?

Well, the reason why these two questions are closely related is that Lee Smolin, among many others, is trying to oversell physically meaningless constructions such as the Rehren duality by claiming that the Rehren duality is "rigorous". It may be rigorous but from a physics viewpoint, it is a naive tautology about non-holographic theories, a tautology that has nothing to do with the actual non-trivial content of holography in quantum gravity. Joe only gives his explanations because he has finally noticed that some people misunderstand that by formulating some ideas in a "rigorous" framework, they don't become physically valid.

As our understanding of the laws of physics expands, we are becoming more familiar with the importance of different characteristics of scientific hypotheses. We learn which characteristics eventually decide about the validity and consistency of a physical hypothesis or a framework, and which characteristics are superficial and physically irrelevant clothes.

The Rehren approach is an approach that tries to be careful about superficial characteristics that are physically irrelevant - the approach cares about the shape of the wooden earphones that you know from the cargo cult sciences - while it completely ignores the things that are essential for physical theories. The standard particle physics and string theory approach is the opposite one.

• But, in my opinion, it does not come across very well when (even justified) weaknesses are portrayed as advantages.

No one has ever claimed that a weakness is an advantage. Joe has just explained the obvious fact that having the Rehren style of rigor is not a physical advantage because it is not. And the lack of the Rehren style of rigor is not a physical disadvantage. The actual physical arguments ultimately boil down to an indirect analysis of observations and after the dust is settled, they imply that Maldacena's duality is profound and true while the Rehren duality is a vacuous and silly tautology. The validity of physical theories has nothing to do with the beauty of the fonts in which the equations are written or with the Rehren style of rigor.

Retractions

Gina is surprised that we hardly see any retractions - statements like "I was wrong on this point". Well, Lee Smolin has had more than 25 years to understand why his ideas from 100+ of his mutually incompatible papers are wrong and make these retractions because many leading figures whom I know as well as many other physicists such as your humble correspondent have spent quite a lot of time with explaining him his mistakes in detail.

I fear that if he hasn't listened to rational arguments for more than 25 years (or if he has never remembered the discussions after they ended), he will never do so. Physicists who understand their science probably won't think that Lee's comments and papers make many sense but millions of confused laymen and thousands of journalists will and that is apparently what Lee Smolin really cares about.

And that's the memo.