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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?
ziv
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    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? – Charlie Mar 29 '21 at 22:42
  • I think that I am confused because the dimensions. What does he mean when he say 1 dimension - (1+1) manifold? – ziv Mar 30 '21 at 11:55
  • @ziv (1+1)-dimensions means one temporal dimension and one spatial dimension. – J. Murray Apr 04 '21 at 14:41
  • @J.Murray Yes I know. I'm asking if Witten refer to (1+1) manifold or 1 manifold – ziv Apr 04 '21 at 14:58
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    Oh, I see. Sorry, I misunderstood your question. Witten is referring to a 1+0-dimensional spacetime. – J. Murray Apr 04 '21 at 15:09
  • @J.Murray So this dimension is purely spatial or is it 1 dimensional manifold that embedded in some way in (1+3) manifold? – ziv Apr 04 '21 at 15:50

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