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: 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],


    [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]


    [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


    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:

    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


    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]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted