Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
2) The (3) is just a shorthand for the 3-dimensional Dirac delta, ## δ(p_x - q_x) δ(p_y - q_y) δ(p_z - q_z) ##.
1) My knowledge here is rather shaky, but since no one has answered you, I'll give it a shot. Think of a superposition of 2 sine functions with frequencies p and q. For p≠q, they interfere constructively and destructively, so that the integral over all the reals vanishes. For p=q, they interfere constructively everywhere, giving a nonzero contribution. The time dependent ladder operators here do the same: their product vanishes for p≠q, leaving only the contribution for p=q, which is just what δ means.
So I suppose you'd have to replace the operators by the δ before you take out the time dependence, but at that point I'm lost as well.
Hope that helps.
You still need to do the last step, right? You need to calculate [tex] \langle 0 | a_p a^\dagger_q |0 \rangle [/tex] To calculate this, you need to express it in terms of a commutator and a product [itex] \langle 0 | a^\dagger_q a_p |0 \rangle [/itex] which gives zero. The commutator piece will give you the three-dimensional delta function.
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