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In quantum mechanical systems which have classical counterparts, we can typically recover classical mechanics by letting $\hbar \rightarrow 0$. Is recovering Einstein's field equations (conceptually) that simple in string theory?


marked as duplicate by Peter Shor , Dan, Qmechanic Jul 31 '13 at 12:49

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To recover Einstein's equations (sourceless) in string theory, start with the following world sheet theory (Polchinski vol 1 eq 3.7.2): $$ S = \frac{1}{4\pi \alpha'} \int_M d^2\sigma\, g^{1/2} g^{ab}G_{\mu\nu}(X) \partial_aX^\mu \partial_bX^\nu $$ where $g$ is the worldsheet metric, $G$ is the spacetime metric, and $X$ are the string embedding coordinates. This is an action for strings moving in a curved spacetime. This theory is classically scale-invariant, but after quantization there is a Weyl anomaly measured by the non-vanishing of the beta functional. In fact, one can show that to order $\alpha'$, one has $$ \beta^G_{\mu\nu} = \alpha' R^G_{\mu\nu} $$ where $R^G$ is the spacetime Ricci tensor. Notice that now, if we enforce scale-invariance at the qauntum level, then the beta function must vanish, and we reproduce the vacuum Einstein equations; $$ R_{\mu\nu} = 0 $$ So in summary, the Einstein equations can be recovered in string theory by enforcing scale-invariance of a worldsheet theory at the quantum level!

  • $\begingroup$ It seems that this says that in order to have string theory on curved background without that anomaly, the manifold has to be Ricci flat. But the question is whether Einstein's equations can be obtained from string theory in a classical limit. $\endgroup$ – MBN Feb 19 '13 at 11:25
  • $\begingroup$ @MBN I basically agree with the first statement (although the non-linear sigma model action I wrote down is just one example of a string theory). The question asked "Is recovering Einstein's field equations (conceptually) that simple in string theory?" The answer, as far as I am aware, is no, and I attempted to include the most relevant version of an answer that I could despite this fact. $\endgroup$ – joshphysics Feb 19 '13 at 15:22

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