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Commutator of complex Klein-Gordon solution with total momentum

  1. Oct 29, 2014 #1
    1. The problem statement, all variables and given/known data
    Hey guys,

    So I have to show the following:

    [itex][P^{\mu} , \phi(x)]=-i\partial^{\mu}\phi(x)[/itex],

    where

    [itex]\phi(x)=\int \frac{d^{3}k}{(2\pi)^{3}2\omega(\vec{k})} \left[ a(\vec{k})e^{-ik\cdot x}+b^{\dagger}(\vec{k})e^{ik\cdot x} \right][/itex]

    and

    [itex]P^{\mu}=\int \frac{d^{3}k}{(2\pi)^{3}2\omega(\vec{k})} k^{\mu} \left[ \mathcal{N}_{a}(\vec{k})+\mathcal{N}_{b}(\vec{k}) \right][/itex]

    2. Relevant equations

    [itex]\mathcal{N}_{a}=a^{\dagger}(\vec{k})a(\vec{k})[/itex]
    [itex]\mathcal{N}_{b}=b^{\dagger}(\vec{k})b(\vec{k})[/itex]

    3. The attempt at a solution

    So all I've done is calculated the commutator as normal and collected all the terms, and I've got:

    JGYzoM.png
    The only difference is that I've dropped the functional dependence on k to make it shorter.

    I'm stuck at this point - not quite sure how to proceed! please help!!
     
  2. jcsd
  3. Oct 29, 2014 #2

    stevendaryl

    User Avatar
    Staff Emeritus
    Science Advisor

    Hmm. Since [itex]P^\mu[/itex] and [itex]\phi(x)[/itex] both involve integrals, the product should be a double integral. And you need to distinguish between the integration for [itex]P^\mu[/itex] and the integration for [itex]\phi(x)[/itex]. So use [itex]k[/itex] for the first integral and [itex]k'[/itex] for the second.

    You need the commutation relation on the operators [itex]a^\dagger, a, b^\dagger, b[/itex]. There are different conventions, so you have to look up which one your textbook is using, but it's something like this:

    [itex][ a^\dagger(k), a(k') ] = -(2 \pi)^3 2E \delta^3(k - k')[/itex]

    where [itex]E = k^0[/itex]
     
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