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!

Lorentz transforming differential operators on scalar fields

  1. Jan 19, 2015 #1
    1. The problem statement, all variables and given/known data

    I'm reading Peskin and Schroeder to the best of my ability. Other than a few integration tricks that escaped me I made it through chapter 2 with no trouble, but the beginning of chapter three, "Lorentz Invariance in Wave Equations", has me stumped. They are going through a proof that the Klein-Gordon equation [itex](\partial_\mu\partial^\mu+m^2)\phi=0[/itex] is Lorentz invariant, and I can't understand for the life of me how they came up with the transformations of differential operators.

    2. Relevant equations
    [tex]\partial_\mu\phi(x)\to\partial_\mu(\phi(\Lambda^{-1}x))=(\Lambda^{-1})^\nu_{\,\,\mu}(\partial_\nu\phi)(\Lambda^{-1}x)[/tex]
    3. The attempt at a solution
    I understand (at least intuitively) how a scalar field should transform on its own, which makes it easy to transform the mass term in the KG Lagrangian (we're working with no potential or interaction terms) which is proportional to the square of the field. The second equality in the equation above is the problem. At first I thought it had something to do with conjugation by the inverse of the transformation in question, but this doesn't seem right.
     
  2. jcsd
  3. Jan 19, 2015 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Try the chain rule ##\partial_\mu = (\partial x'{}^\nu/\partial x^\mu) \partial_{\nu'}## and the relation between ##\partial^\mu## and ##\partial_\mu##.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Lorentz transforming differential operators on scalar fields
Loading...