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When we study a relativistic point particle (say, at the beginning of a string theory course), we looked at einbeins, and that's because they were equivalent to the action $\int d\tau\sqrt{\eta_{\mu\nu}\dot{x}^\nu\dot{x}^\nu}$ then the einbein action worked. However why doesn't the action $\int d\tau ~\eta_{\mu\nu}\dot{x}^\nu\dot{x}^\nu$ work?

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

Qmechanic
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Toby Peterken
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1 Answers1

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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.

Qmechanic
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