# Ladder operators in terms of Klein Gordon real field

an exercise asks me to explicit the functions $${a}(k)$$ and $$a^{*}(k)$$ in term of the real field $$\phi$$ and its temporal derivate $$\partial_0 \phi$$ if the general solution of the Klein-Gordon equation is $$\phi(x,t) = \frac{1}{(2\pi)^3} \int \frac{d^3k}{\sqrt{2w_k}}[a(k)e^{-i\, k\cdot x}+ a^{*}(k) e^{i \, k \cdot x}],$$ where $$k \cdot x = w_kt - \vec{k}\cdot\vec{x}$$.

I wrote down in explicit form the Klein Gordon equation, such as $$(\partial^{\mu}\partial_{\mu} + m^2)\phi=0 \Rightarrow \ddot{\phi} = (\nabla^2 - m^2) \phi$$ and inserted the general solution on it but I didn't find the solution.

Can someone help me?

Here is a hint: Your expression for $$\phi(x)$$ is in the Heisenberg picture ($$x = (\vec{x}, t)$$), so you can find $$\pi(x) = \partial_0 \phi(x)$$ simply by differentiating with respect to $$t = x^0$$. Then, you should be able to find $$a(\vec{k})$$ and $$a^\dagger(\vec{k})$$ from the values at $$t = 0$$ (the Schrödinger picture fields $$\phi(\vec{x})$$ and $$\pi(\vec{x})$$. Try inverting the Fourier transform...