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The action for the relativistic point particle with mass $m \geq 0$ in a curved background is given by:

\begin{equation} S[X] = \int_{\tau_0}^{\tau_1} d\tau \left[ e(\tau)^{-1} g_{\mu \nu}(X(\tau)) \dot{X}^\mu \dot{X}^\nu - e(\tau) m^2 \right].\tag{1} \end{equation}

Is there a difference in the canonical/constraint/gauge structure between the cases $m=0$ and $m>0$?

I am asking, since I have the feeling, that in the massive case the gauge choice corresponding to an affine parametrization is natural, while in the massless case there exists no natural gauge choice.

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1 Answer 1

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  1. The action (1) has a world-line (WL) reparametrization gauge symmetry in both the massive and massless case.

  2. An important difference is that in the massless case one cannot eliminate/integrate out the einbein field $e$, cf. e.g. this Phys.SE post.

    In contrast such elimination in the massive case leads to the well-known square-root action, cf. e.g. my Phys.SE answer here.

  3. If we gauge-fix $e={\rm const}>0$, then the Euler-Lagrange (EL) equation for the gauge-fixed action (1) is in both cases a geodesics equation (without an inhomogeneous term, cf. e.g. this Phys.SE post.) This means that two different parametrizations are affinely related.

    Whether the geodesic is time-like or light-like then comes down to boundary conditions (BCs).

    Affine parametrization (in the sense that spacetime arclength is affinely related to the WL parameter) holds in the gauge-fixed case for time-like geodesics, but does not make sense for light-like geodesics.

    See also e.g. this related Phys.SE post.

  4. For the corresponding Hamiltonian formulation, see e.g. this Phys.SE post.

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  • $\begingroup$ Thanks for your answer. Is the impossibility of the elimination of $e$ in the massless case somehow encoded in "abstract" properties of the constraints/the constraint algebra? $\endgroup$
    – warpfel
    May 22, 2023 at 14:00
  • $\begingroup$ I updated the answer. $\endgroup$
    – Qmechanic
    May 22, 2023 at 14:06
  • $\begingroup$ Thanks. I still don't understand, if the last statement of my question is true (or at least meaningful), i.e. if in the massive case the gauge choice corresponding to an affine parametrization is natural, while in the massless case there exists no natural gauge choice. Could you maybe comment on that? $\endgroup$
    – warpfel
    May 22, 2023 at 14:24
  • $\begingroup$ Thanks for the further specification of the answer. Sorry, but I still don't understand it completely. You wrote: "Affine parametrization (in the sense that spacetime arclength is affinely related to the WL parameter) holds in the gauge-fixed case for time-like geodesics, but does not make sense for light-like geodesics.". It is clear to me, that it does not make sense, since there is no non-zero arclength for null geodesics. But can one see this also intrinsically from the gauge/constraint structure without referring to the spacetime in which the worldline is embedded? $\endgroup$
    – warpfel
    May 22, 2023 at 19:18
  • $\begingroup$ Ok, I think I got it. The difference is really quite subtile. I will write it up today or tomorrow. Thanks for your help! $\endgroup$
    – warpfel
    May 23, 2023 at 10:49

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