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Transformation question; first order shift of a scalar field

  1. Jun 27, 2013 #1
    Hi to all! I have the following transformation

    [itex] \tau \to \tau' = f(\tau) = t - \xi(\tau). [/itex]

    Also I have the action

    [itex] S = \frac{1}{2} \int d\tau ( e^{-1} \dot{X}^2 - m^2e) [/itex]

    where [itex] e = e(\tau) [/itex]. Then in the BBS String book it says that

    $$ {X^{\mu}}' ({\tau}') = X^{\mu}(\tau)$$

    and that the first order shift is

    $$ \delta X^{\mu} = {X^{\mu}}'(\tau) - X^{\mu}(\tau) = \xi(\tau)\dot{X}^{\mu}$$

    Can someone explain why this is true? How can I realize it?

    Then it say similarly for the ##e(\tau)## that

    $$ \delta e = e'(\tau) - e(\tau) = \frac{d}{d\tau} (\xi e) $$

    again, I cannot see how this comes!

    Thank you in advance.
  2. jcsd
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