Commutator of complex Klein-Gordon solution with total momentum

[Dixanadu]

Homework Statement

Hey guys,

So I have to show the following:

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

where

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

and

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

Homework Equations

$\mathcal{N}_{a}=a^{\dagger}(\vec{k})a(\vec{k})$
$\mathcal{N}_{b}=b^{\dagger}(\vec{k})b(\vec{k})$

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!

 

[stevendaryl]

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

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

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

where $E = k^0$