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I'm following Witten's essay: What Every Physicist Should Know About String Theory . When applying path integral in 1D Witten mention:

Part of the process of evaluating the path integral in our quantum gravity model is to integrate over the metric on the one-manifold, modulo diffeomorphisms. But up to diffeomorphism, the one-manifold has only one invariant, its total length τ, which we will interpret as the elapsed proper time.

So my question is why this is true:

  1. Why is the proper time $\tau$ invariant?
  2. Why is it the the only invariant?
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    $\begingroup$ In the point particle theory, you would expect your physical quantities to not depend on how you parameterise the points on the worldline of the particle, hence $\tau$ (the worldline's length) is diffeomorphism invariant. As for why it is the only invariant, 1D is not that interesting, is it that surprising that it has minimal symmetries? $\endgroup$
    – Charlie
    Commented Mar 29, 2021 at 22:42
  • $\begingroup$ I think that I am confused because the dimensions. What does he mean when he say 1 dimension - (1+1) manifold? $\endgroup$
    – ziv
    Commented Mar 30, 2021 at 11:55
  • $\begingroup$ @ziv (1+1)-dimensions means one temporal dimension and one spatial dimension. $\endgroup$
    – J. Murray
    Commented Apr 4, 2021 at 14:41
  • $\begingroup$ @J.Murray Yes I know. I'm asking if Witten refer to (1+1) manifold or 1 manifold $\endgroup$
    – ziv
    Commented Apr 4, 2021 at 14:58
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    $\begingroup$ Oh, I see. Sorry, I misunderstood your question. Witten is referring to a 1+0-dimensional spacetime. $\endgroup$
    – J. Murray
    Commented Apr 4, 2021 at 15:09

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