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I found the following expression for the action of a (free, I think) relativistic particle in my notes but I can't remember from what it came from: $$ S = \int_{0}^{N} \left [ \frac{1}{4}\eta_{\mu\nu}{\mathrm{d} X^\mu \over \mathrm{d} s}{\mathrm{d} X^\nu \over \mathrm{d} s} -m^2c^2 \right]\mathrm{d}s$$

I'm not even sure what some of the symbols mean; no idea what $N$ is for example. Big note-taking fail of my part!

Any help is appreciated.

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What do you mean with ''what it came from"? [0:N] are just the intial and final eigentime. (Any Lagrangian needs to be integrated over a finite time interval (and in the field case any Lagrangian density between two Cauchy surfaces). In the relativistic single particle case the eigentime takes the role of time. – Arnold Neumaier Nov 30 '12 at 16:34
Related: physics.stackexchange.com/q/44947/2451 – Qmechanic♦ Nov 30 '12 at 16:40
It looks like the "alternative" action that is usually discussed in introductory string theory treatments, like eq 1.8 here. But I don't see the einbein field...? – twistor59 Nov 30 '12 at 20:26

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