Thursday, May 24, 2007

Why are there gravitons in string theory

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


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.


  1. (A part of) My paper “GR-friendly description of quantum systems”
    (IJTP, DOI 10.1007/s10773-007-9474-3, )
    may be of interest to members of this community.
    It is also available at (PDF).
    Best regards,

  2. Fantastically lucid explanation, Lubos. I hope this is in the book!