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When we studies the beginnings of string theory, we looked at einbeins, and that because they were equivalent to the action $\int \sqrt{\eta_{\mu\nu}\dot{x}^\nu\dot{x}^\nu}$ then the einbein action worked. However why doesnt the action $\int \eta_{\mu\nu}\dot{x}^\nu\dot{x}^\nu$

I mean my question isnt about this action per se, but more general, when do two classically equivalent actions give the same quantum results?

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There is likely not an exhaustive classification for when pairs of classically equivalent theories are equivalent quantum mechanically. The best one can do is probably to present a relatively short list of known examples. The most famous pairs are square root actions vs. non-square root actions, cf. e.g. Nambu-Goto vs. Polyakov action in string theory or the point-particle analogue.

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  • $\begingroup$ Are there any theorems for checking/ would help me check specific examples? Obviously the poisson bracket only defines the quantum commutators up to $\hbar ^2$, otherwise this would trivial $\endgroup$ – Toby Peterken Oct 9 '20 at 15:38
  • $\begingroup$ in string theory you can check whether two 2D CFT’s are equivalent by computing all tree-level 3pt amplitudes and all 1-pt torus (ie 1-loop) amplitudes. (i implicitly have closed strings in mind.) if these are the same the remaining correlators should also match due to modular invariance. i don’t know of any shortcuts .. by the way, these two CFT’s needn’t have the same “classical” equations. this is a vast subject ... in general you check equivalences of two theories by matching correlations functions (which is also true in AdS/CFT, etc). matching symmetries are a good first indicator. $\endgroup$ – Wakabaloola Oct 10 '20 at 11:44

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