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Homework Help: Polyakov action, reparameterisation q, string theory

  1. Dec 26, 2017 #1
    1. The problem statement, all variables and given/known data

    qt.png

    i am stuck on part d , see below
    2. Relevant equations

    parts a to c are fine

    polyakov action:
    ## \frac{1}{2} \int \frac{1}{e(t)} \frac{dX^u}{dt}\frac{dX_u}{dt}-m^2 e(t) dt ##

    EoM of ##e(t)##:
    ##\frac{-1}{(e(t))^2} \frac{dX^u}{dt}\frac{dX_u}{dt}-m^2=0## [1]


    you plug the EoM of ##e(t)## (which is equivalent to the mass-shell constraint) into the polykov action to recover the Nambu-Goto action.

    3. The attempt at a solution

    More than anything, I am confused as to why we are given the mass, since the dimension of the space-time is not given, ##u=0,1...d##, ##d## is not specified. If it were ##d=1## I guess we would use it as something like computing the other space-time coordinate.

    Whilst ##e(t)## has a transformation rule, ##X^u(t)## just acts as a scalar on the world-sheet in string theory? or in this case on the world-line, and as such has no transformation rule and so it is just a case of plugging in.

    I interpret the trajectory in terms of ##t## as (apologies i have done ##\tau=t## and will also do ##\tau'=t'##) ##X^0(t)=t## and ##X^i(t)=0 ## for ##i=0,1,..,d##

    So that the mass-shell constraint reads ##\frac{\partial X^0}{\partial t}\frac{\partial X_0}{\partial t}=1/2 ## and via the chain rule of t' and t it is easy to check that this is consistent.

    So simply plugging in I get ## X^0(t'(t))=t' ^2##

    However I'm guessing this is wrong since I have no idea why we are given the mass, any help much appreciated,ta.
     
  2. jcsd
  3. Dec 31, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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