0
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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...

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.