Explaning gravitation via string theory

[clerk]

This may be a silly question since i am relatively new to the string theory. While going through a quantization of the bosonic string , I see a tower of states appearing corresponding to different modes of excitations of the bosonic string. Now I was trying to figure out how this whole machine is working to explain ordinary gravitational attraction between say -the Earth and a ball. Say the graviton mode is excited (considering closed string excitations)..so where are these closed strings situated?In particular how does this string know what music to play to..( I mean how does it know that it has to excite the precise mode that generates gravitons and thus lead to gravitational attraction somehow??)

 

[tom.stoer]

I am not an expert in string theory but I guess that the gravitational field between two massive bodies is a kind of 'coherent excitations of strings'.

Look at ordinary QED; you can chose the A°=0 gauge which eliminates the unphysical A° field (which does not have a conjugate momentum); in addition you can fix the residual gauge symmetry of time-indep. gauge transformations respecting A°=0 which is generated by the Gauss law constraint; solving the Gauss law eliminates a second unphysical photon and defines a physical Hilbert space with two physical = transversal d.o.f. Within this physical Hilbert space the j°A° term in the Lagrangian is converted into an interaction term

j° Δ^-1 j° → ∫ d³x d³y ρ(x) |x-y|^-1 ρ(y)

As you can see in this gauge the Coulomb term is not due to virtual photons but seems to be a 'static interaction term'; I guess that within string theory you do have similar mechanisms such that gravitation is not only due to gravitons = string excitations but that there is such a non-perturbative description as well.

Would be interesting to know whether anybody has calculated the 'gravitational potential' plus 'stringy corrections'

 

[LBloom]

Clerk,

I think a strong background in QFT would certainly help in understanding if you don't have one already. The point is this, when quantizing the closed string a massless spin 2 boson naturally appears. This is naturally identified with the graviton, another spin 2 boson. Of course this doesn't give us Einstein's equations (and when taking the classical limit, Newton's Equations of Gravitation).

That's because we're working in the perturbative framework. Gravitons naturally appear when we take the linearized form of Einstein's equations, that is when we are doing perturbation theory against some fixed background.

The interesting thing is where we go from here. Supposing there exists a spin 2 boson, can we derive its physics in the nonperturbative realm? I think Feynman showed (as referenced in his Lectures on Gravitation) that a consistent QFT for a spin 2 boson will naturally lead to Einstein's equations. By summing up the infinite number of perturbation terms, a nonperturbative result comes out. Such a particle will give rise to a QFT that is nonrenormalizable.

So to sum it up:

-String Theory predicts the existence of a massless spin two boson.
-QFT says particles interact (and transmit forces) by exchanging virtual particles (photons for QED, gravitons in a quantum theory of GR).
-A consistent theory of QFT for spin two bosons leads to Einsteins Equations.
-Most importantly, a spin two massless boson in QFT leads to a nonrenormalizable theory. In String Theory, we don't have such a problem.

 

[tom.stoer]

Everything is fine except for

...
-QFT says particles interact (and transmit forces) by exchanging virtual particles (photons for QED, gravitons in a quantum theory of GR).

As I have tried to explain in my last post exchange of virtual particles does not cover all physics in QFT; the situation becomes worse in non-abelian models where the differential operator to be inverted is field-dependend and creates a gauge-field depended, non-local current-current interactions; that means that you can never expect to get all physically meaningful results in the IR from perturbation theory.

I expect that similar effects will be present in string theory as well.

 

[LBloom]

Tom,

You're probably right and I certainly don't think that we can gain full knowledge from ST from its perturbative framework. I was emphasizing the perturbative because I was trying to answer Clerk's question on how "does this string know what music to play to..( I mean how does it know that it has to excite the precise mode that generates gravitons and thus lead to gravitational attraction somehow??)" and that in it's most basic framework we explain interactions via the exchange of virtual particles and the train of logic I've seen that justifies ST as theory of QG.

 

[tom.stoer]

The interesting find is that ST in its perturbative formulation reproduces the Einstein equations (more precisely: Ricci-flatness which is equivalent to Einstein equation in vacuum) as a consistency condition (vanishing of beta-function).

Unfortunately this does *not* explain gravity of massive bodies b/c this is only a condition for a background introduced by hand but not determined dynamically. In order to "explain" gravity one must look at non-perturbative and manifest background-independent formulations of ST. As long as these are not developed, ST stays a theory of perturbative quantum gravity compatible with certain (static?) backgrounds.

I do not know whether such an "explanation" is already known, but it would be very interesting in order to answer the original question "how ... to explain ordinary gravitational attraction between say - the Earth and a ball".

 

[clerk]

Thanks for all the help Tom Stoer,L.bloom.
I think I have roughly got what you wanted to say.. would like to see how a spin two boson of QFT naturally leads to einstein's equations..Actually I had no idea that Feynman has a book on it, will surely look for it. Thanks.
And can you please explain me how we get something like non-perturbative string theory? How does it differ from the string theory that I am reading? Is it something different from the polyakov action where we couple 26 scalar fields to 2d worldsheet metric? Is this coupling a perturbation?

 

[tom.stoer]

The Polyakov action reads

$$S = \frac{T}{2}\int d^2\sigma \, \sqrt{-h}h^{ab}\,G_{\mu\nu}(X)\,\partial_a X^\mu(\sigma) \,\partial_b X^\nu(\sigma)$$

Latin indices refer to the 2-dim. world sheet of the string, Greek indices refer to the 26-dim. target spacetime. h is the world-sheet metric, G is the metric of the target space.

Usually one quantizes the theory on 4-dim. Minkowski spacetime * compactified space. Using superstrings the 26 is reduced to 10 and there are numerous quantizations known for 4-dim. Minkowski spacetime * 6-dim. Calabi-Yau space (there different construction schemes; Calabi-Yau is not the only one).

The problem is that this G is not a dynamical entity but is introduced by hand!

So string theory in this formulation does not determine the metric of spacetime. It is rather the other way round: one fixes a background spacetime G and constructs a quantization. There are allowed spacetimes like Minkowski * Calabi-Yau, there are forbidden spacetimes where consistent quantizations do not exist (the anomaly cancelation introduces the restriction of Ricci-flatness), and there are spacetimes where no quantization is known (whatever that means).

As far as I understand M-theory, branes, dualities, fluxes etc. there are numerous other ideas what a background means and how to introduce it, but the general problem remains the same: the background is not a dynamical entity and for each background the quantum theory looks different; there is not one single string theory, but an large (finite or infinite?) number of different quantum theories on different backgrounds .

b/c gravity is determined by the spacetime curvature it seems to me that string theory does not explain gravity but 'only' perturbative quantum gravity on top of fixed backgrounds; one could say that "strings don't curve spacetime". So the picture of full quantum gravity in terms of strings is by no means complete (it is not complete for other approaches, either).

 

[clerk]

Thanks a lot .
Interestingly,I found a section in Zwiebach where he discusses gravitational lensing via cosmic strings in chapter 7. Here he says " Although a string does not exert gravitational attraction it affects the geometry of planes orthogonal to the string". Makes it sound as if they do curve space time.. however , as we know of today,no cosmic string has been detected..

 

[tom.stoer]

I haven't Zwiebach at hand, but all what I know is that spacetime curvature in ST is always put in by hand i.e. classically and that there is no dynamical backreaction due to propagating strings on a certain background