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