Finding ##\partial^\mu\phi## for a squeezed state in QFT

[Sciencemaster]

TL;DR Summary

I'm trying to apply an operator to a massless and minimally coupled squeezed state, I'm having trouble calculating ##\partial^\mu\phi## but due to a sum over k and the ladder operators.

I'm trying to apply an operator to a massless and minimally coupled squeezed state. I have defined my state as $$\phi=\sum_k\left(a_kf_k+a^\dagger_kf^*_k\right)$$, where the a_k operators are ladder operators and f_k is the mode function $$f_k=\frac{1}{\sqrt{2L^3\omega}}e^{ik_\mu x^\mu}$$ (assuming periodic boundary condition in a three-dimensional box of side L where k is the wave number).
However, I'm having trouble calculating ##\partial^\mu\phi## due to the sum over k and the ladder operators. I would very much appreciate it if someone could help me through the math of this step!

 

[strangerep]

Well, ##\partial_\mu## is a linear operator, so you can apply it term by term in the sum. As for the ladder operators, if they're independent of ##x## you can just treat them like constants during the partial differentiation.

Btw, you'll need to use a different dummy summation index in the exponent so as not to conflict with the free index ##\mu## on ##\partial_\mu##. E.g., change ##k_\mu x^\mu## to ##k_\alpha x^\alpha##.

 

[vanhees71]

One should also note that this doesn't describe a state but a field operator in terms of free-field energy eigenmodes or a neutral scalar field. The ##\hat{a}_k## are annihilation and ##\hat{a}_k^{\dagger}## in Fock space.