Momentum in Free Scalar Field

Question

I have a few issues with making the transition between these:

$\phi(x)=\int{\frac{d^3p}{2\pi^3}\frac{1}{\sqrt{2\omega_{\vec{p}}}}(a_{\vec{p}}e^{i \vec{p} \vec{x}}+ a^{\dagger}_{\vec{p}}e^{-i \vec{p} \vec{x}}})$

$\pi(x)=\int{\frac{d^3p}{2\pi^3}(-i)\sqrt{\frac{\omega_{\vec{p}}}{2}}(a_{\vec{p}}e^{i \vec{p} \vec{x}}- a^{\dagger}_{\vec{p}}e^{-i \vec{p} \vec{x}}})$.

So from the Lagrangian of the real scalar field, we get

$\pi(x)=\partial_0 \phi(x)$

which just means differentiating our $\phi(x)$ w.r.t time. But where does the $-i$ come from (as in the minus sign)? Also aren't the exponentials independent of time as $\vec{p} \vec{x}$ isn't a four vector? So now I'm wondering where the $\omega_{\vec{p}}$ has come from as well!

This post imported from StackExchange Physics at 2014-05-04 11:10 (UCT), posted by SE-user user13223423

Comments

Put in the time-dependence in the expansion for $\phi(x)$. It is supposed to solve the  Klein-Gordon equation.

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It's explained in Peskin and Schroeder in section 2.4.