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Srednicki 2.8 / Inverse Lorentz Transformation

  1. Sep 1, 2010 #1
    1. The problem statement, all variables and given/known data

    I need to demonstrate the relation [tex][\varphi(x),M^{\mu\nu}]=\matchal{L}^{\mu\nu}\varphi(x)[/tex]

    where
    [tex]\mathcal{L}^{\mu\nu}\equiv \frac{\hbar}{i}(x^\mu\partial^\nu-x^\nu\partial^\mu)[/tex].

    2. Relevant equations

    [tex]U(\Lambda)^{-1}\varphi(x)U(\Lambda) = \varphi(\Lambda^{-1}x)[/tex]

    [tex]\Lambda = 1+\delta\omega[/tex]

    [tex]\Lambda^\mu_\nu = \delta^\mu_\nu + \delta\omega^\mu_\nu[/tex]

    [tex]U(1+\delta\omega) = 1 + \frac{i}{2\hbar}\delta\omega_{\mu\nu}M^{\mu\nu}[/tex]


    3. The attempt at a solution

    I can pretty easily evaluate the LHS of the expression to give

    [tex]\varphi(x) + \frac{i}{2\hbar}\delta\omega_{\mu\nu} [\varphi(x),M^{\mu\nu}][/tex],

    but I am having a ton of trouble with deciphering what is meant by [tex]\varphi(\Lambda^{-1}x)[/tex], particularly how I am supposed to get derivatives of the type [tex]\partial^\mu[/tex] when I had thought [tex]x\equiv x^\mu[/tex] here, implying derivatives of the kind [tex]\partial_\mu[/tex].

    I tried, for instance, defining an inverse transformation so

    [tex]\bar{x}^\mu = {\Lambda_\nu}^\mu x^\nu = (\delta^\mu_\nu - \delta\omega^\mu_\nu)x^\nu[/tex] and finding a Taylor series about x, but I don't really know how to do this in terms of indices. I get

    [tex]\varphi(\Lambda^{-1}x) = \varphi(x^\mu-\delta\omega^\mu_\nu x^\nu)[/tex] so that

    [tex]\sim \varphi(x) - \delta\omega^\mu_\nu x^\nu\bar{\partial_\mu}\varphi(\Lambda^{-1}x)[/tex], which is sort of close, but not right.

    If I magically change the inverse transformation to [tex]\Lambda_{\mu\nu}x^\nu = x_\mu - \delta\omega_{\mu\nu}x^\nu[/tex] it works, but I can't justify this.

    I think the problem is with my inverse transformation and taking the derivative. Help would be greatly appreciated.
     
    Last edited: Sep 1, 2010
  2. jcsd
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