1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Conformal group, infinitesimal transformation

  1. Jun 1, 2012 #1
    1. The problem statement, all variables and given/known data
    In order to determine the infinitesimal generators of the conformal group we consider an infinitesimal coordinate transformation:
    [itex]x^{\mu} \to x^\mu+\epsilon^\mu[/itex]
    We obtain [itex]\partial_\mu\epsilon_\nu+\partial_\nu\epsilon_\mu=\frac{2}{d}(\partial\cdot\epsilon)\eta_{\mu\nu}[/itex] where d is the dimension of spacetime.
    Derive [itex](\eta_{\mu\nu}\Box+(d-2)\partial_\mu \partial_\nu)\partial\cdot\epsilon=0[/itex]

    2. Relevant equations

    3. The attempt at a solution
    I think I am really close to the solution, but somehow I don't arrive there.
    [itex]\Rightarrow \partial^\nu\partial_\mu\epsilon_\nu+\partial^\nu{\partial}_\nu\epsilon_\mu=\frac{2}{d}\partial^\nu({\partial}\cdot\epsilon)\eta_{\mu\nu}[/itex]
    [itex]\Rightarrow \partial_\mu(\partial\cdot\epsilon)+\Box{\epsilon}_\mu=\frac{2}{d}\partial_\mu(\partial\cdot\epsilon)[/itex]
    [itex]\Rightarrow (d-2)\partial_\mu(\partial\cdot\epsilon)+d\cdot{\Box}{\epsilon}_\mu=0[/itex]
    [itex]\Rightarrow (d-2)\partial_\mu\partial_\nu(\partial\cdot\epsilon)+d\partial_\nu\Box{\epsilon}_\mu=0[/itex]
    I think I need to use [itex]d=\eta_{\mu\nu}\eta^{\mu\nu}[/itex] now, but I don't get the right result.
    Can somebody help me?

    Best regards, physicus
  2. jcsd
  3. Jun 1, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The first term in (**) is symmetric in ##\mu\nu##, so we can write (**) as

    $$(d-2)\partial_\mu\partial_\nu(\partial\cdot\epsilon)+\frac{d}{2} \Box ( \partial_\nu{\epsilon}_\mu + \partial_\mu{\epsilon}_\nu)=0.$$

    Alternatively, you can take the divergence of (*) with ##\partial^\mu## and add it to (**) to get this equation. It should be clear what to do next.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook