In quantum mechanical systems which have classical counterparts, we can typically recover classical mechanics by letting ℏ→0. Is recovering Einstein's field equations (conceptually) that simple in string theory?
Answer
To recover Einstein's equations (sourceless) in string theory, start with the following world sheet theory (Polchinski vol 1 eq 3.7.2): S=14πα′∫Md2σg1/2gabGμν(X)∂aXμ∂bXν
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 α′, one has βGμν=α′RGμν
where RG 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; Rμν=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!
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