In the context of the Polyakov action, the action for a relativistic point particle $$ S_P = \frac{1}{2} \int \mathrm{d}\tau \, e(\tau) \left(\frac{1}{e^2(\tau)}\left(\frac{\mathrm{d} X^\mu(\tau)}{\mathrm{d} \tau}\right)^2 - m^2\right), $$ where $\tau$ is an arbitrary parameter, and $X^\mu(\tau)$ and $e(\tau)$ are independent degrees of freedom, I recently read that invariance under local diffeomorphisms $\tau \to \tilde{\tau} - \epsilon(\tau)$ requires $X^\mu(\tau)$ and $e(\tau)$ to transform as $$ X^\mu(\tau) \to X^\mu(\tau) + \frac{\mathrm{d} X^\mu(\tau)}{\mathrm{d} \tau} \epsilon(\tau),\\ e(\tau) \to e(\tau) + \frac{\mathrm{d}}{\mathrm{d} \tau}\bigl(e(\tau) \epsilon(\tau)\bigr). $$ My question is, why do we require invariance under local diffeomorphisms in the first place. What does it represent physically?
2 Answers
The short answer is that the dynamics of a relativistic point particle should not depend on our parametrization of its world-line.
Perhaps the above conclusion becomes clearer if we integrate out the the einbein$^1$ field $e(\tau)$. Let us here for simplicity assume that the point particle is massive $m>0$, and leave the massless case $m=0$ as an exercise for the reader.
If we integrate out the einbein field $e(\tau)$, we get the square root action for a relativistic point particle, cf. e.g. this Phys.SE post.
This square root action is (up to a multiplicative constant) the change in proper time, cf. my Phys.SE answer here.
In other words, the action has a nice geometric interpretation manifestly independent of parametrization.
Similarly, the Polyakov action for a string can be reduced to the Nambu-Goto action, which has a geometric intepretation as the area of the worldsheet, independent of parametrization, cf. e.g this Phys.SE post.
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$^1$ The einbein action for a relativistic point particle is also discussed in this Phys.SE post and links therein.
In general field theories diffeomorphisms represent mere change of coordinates: in fact, given $U, V$ as two charts on a manifold $\mathcal{M}$ such that $U \cap V\neq \emptyset$, if $f\colon U\to V$ is a change of coordinates (i. e. change of observer) we want the two observers to measure the same equations of motion in $U\cap V$, namely the action must be invariant under $f$.
For $f$ to be a good change of coordinates, it is usually required that it is a diffeomorphism, since one wants it to be invertible (so that you can go both ways) and differentiable, the derivatives being invertible as well (so that you can calculate velocities both ways).
