Is there a reason(both physical and mathematical) why the Klein-Gordon field is represented as a fourier expansion in the second quantization as opposed to other mathematical expansions? Be gentle with the answers!
Fourier expansion is used as a change of basis method that makes our calculations simpler and more in context. Usually we are working with particles with precise momentum. Momentum is a far more useful quantity than position when doing experiments.
Also the Feynman rules can be obtained in terms of momentum as well. Furthermore as ACuriousMind mentioned in the comments above, we obtain nice Fourier modes which we interpret as creation and annihilation of particles in some momentum eigenstate.