1
$\begingroup$

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?

Improve this question
$\endgroup$
1
$\begingroup$

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.

Improve this answer
$\endgroup$
2
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.