I am learning string theory, as I understand, gravitons exist as modes in string excitations, and also other particles. It gave me this picture: a lot of strings fulling in the spacetime, excitations on strings are particles. But QFT gave me another picture that quantum fields exist in spacetime, excitations of these fields are particles, particles can be created from vacuum and annhilated. So, if I consider a finite number of strings flying in spacetime, how come it could describe a picture as quantum fields describes since spacetime regions that strings didn't sweep over are completely vacuum. Do we need string field theory to cover QFT?

Another question is about classical gravity. Given the picture I had about string theory, how explicitly the classical limit of this picture could come out as a total geometry as gravitions pass the interaction of gravity while classical gravity theory is pure geometrical?

  • $\begingroup$ Possible duplicates for the ST$\to$ GR reduction: physics.stackexchange.com/q/1073/2451 and links therein. $\endgroup$ – Qmechanic Sep 10 '14 at 16:20
  • $\begingroup$ I checked the link. It roughly described the question I posted here, but only half of my question. My question mainly focuses on two aspects: String to geometrical description of GR, String to QFT. The most-recommended answer in the link is not answering my question, and I think it's better to bring this question to public again since that one hasn't been answer for years. I also got one answer in my question, for me it is better than the answer in the link. And also, the second part of my question are waiting for an answer too. $\endgroup$ – Peng Liu Sep 11 '14 at 12:16

Classical fields emerge when there is a large (but not definite) number of particles in a coherent state. For a simple example, for a scalar field $\phi(x)$ we can write a state that describes a classical configuration as something like $$ \exp\left(\int d^D p\; \tilde\phi(p) a^\dagger_p\right)|0\rangle. $$ Note that this isn't an eigenstate of particle number, but a superposition of states with all particle numbers. (This is what I meant by an indefinite number of particles.)

For string theory, this appears rather nicely when we write a path integral for a string moving on a background $G_{\mu\nu}$, with action $$ S=-\frac{1}{2l_s^2}\int d^2\sigma \sqrt{-\gamma}\gamma^{ab}\partial_a X^\mu \partial_b X^\nu G_{\mu\nu}. $$ If we write $G_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, then $e^{-S}$ in the path integral looks just like the flat space oath integral with an exponential of graviton vertex operators inserted. In other words, the curved spacetime background path integral is the same as doing the path integral in a coherent state of gravitons.

To emphasise, there need not be any definite number of gravitons, nor need the gravitons be localised in space. But in any curved region, there will be some nonzero probability of finding gravitons there.

From this point of view, the dynamics of the field come from a rather odd place: the field equations are requirements for the consistency of the world-sheet theory. (An alternative way is to compute scattering amplitudes of gravitons and match the results to a low-energy effective field theory).

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  • $\begingroup$ Thanks! To sum up, your opinion is that classical gravity theory should emerge from the low-energy effective field theory. Am I getting your point? Another question about QFT, in string theory the world-sheet theory shall give rise to certain gauge symmetries that also appeared in QFT, I cannot see why this property makes string theory more fundamental than QFT. I don't know why these symmetries in world-sheet could go to symmetries of quantum fields in spacetime. $\endgroup$ – Peng Liu Sep 11 '14 at 2:19

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