Explicit expressions for the creation and annihilation operators What are the explicit expressions for the creation and annihilation operators $\hat{a_\vec p}$ and $\hat{a}^{\dagger}_\vec p$ for bosons? I can't find them anywhere, as every source seems to introduce them when quantizing the fields
$$\phi(\vec x) = \int \frac{d³p}{(2\pi)³} \frac{1}{\sqrt{2\omega_{\vec p}}}  \left( \hat{a_\vec p} e^{i\vec p \cdot \vec x} + \hat{a}^{\dagger}_\vec p  e^{-i\vec p \cdot \vec x}\right)  $$
$$\pi(\vec x) = \int \frac{d³p}{(2\pi)³} (-i) \sqrt{\frac{\omega_{\vec p}}{2}}  \left( \hat{a_\vec p} e^{i\vec p \cdot \vec x} - \hat{a}^{\dagger}_\vec p  e^{-i\vec p \cdot \vec x}\right)  $$
Without giving an explicit expression for them.
I would like to know, because e.g. calculating the Hamiltonian for the Klein-Gordon field requires me to know what effect switching the sign of the momentum has, i.e. what  $\hat{a}_{-\vec{p}}$ and $\hat{a}^\dagger_{-\vec{p}}$ are. 
 A: Note that you can also write $\phi$ and $\pi$ as
$$ \phi(\vec{x}) = \int \frac{d^3 p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_{\vec{p}}}}
\left(a_\vec{p} + a_{-\vec{p}}^\dagger \right) e^{i\vec{p} \cdot \vec{x}} $$
$$ \pi(\vec{x}) = \int \frac{d^3 p}{(2\pi)^3} (-i) \sqrt{\frac{\omega_{\vec{p}}}{2}}
\left(a_\vec{p} - a_{-\vec{p}}^\dagger \right) e^{i\vec{p} \cdot \vec{x}} $$
(equations (2.27) and (2.28) in Peskin and Schroeder). From here, you can take the Fourier transform to get
$$ a_\vec{p} + a_{-\vec{p}}^\dagger = \sqrt{2\omega_{\vec{p}}} \int d^3 x\, \phi(\vec{x}) e^{-i\vec{p} \cdot \vec{x}} $$
$$ a_\vec{p} - a_{-\vec{p}}^\dagger = i\sqrt{\frac{2}{\omega_{\vec{p}}}} \int d^3 x\, \pi(\vec{x}) e^{-i\vec{p} \cdot \vec{x}} $$
and adding these together gives
$$ a_\vec{p} = \int d^3 x \left(\sqrt{\frac{\omega_\vec{p}}{2}} \phi(\vec{x}) + \frac{i}{\sqrt{2\omega_\vec{p}}} \pi(\vec{x}) \right) e^{-i\vec{p} \cdot \vec{x}} $$
and then, by Hermitian conjugation,
$$ a_\vec{p}^\dagger = \int d^3 x \left(\sqrt{\frac{\omega_\vec{p}}{2}} \phi(\vec{x}) - \frac{i}{\sqrt{2\omega_\vec{p}}} \pi(\vec{x}) \right) e^{i\vec{p} \cdot \vec{x}}. $$
Using these equations, you can explicitly verify the commutation relation $[a_\vec{p}, a_\vec{q}^\dagger] = (2\pi)^3 \delta^{(3)}(\vec{p} - \vec{q})$. In practice, it seems like you rarely need the explicit expressions for the ladder operators; remembering the commutation relation is usually enough.
