I am learning about adv quantum and field theory and i have run across something unfamiliar mathematically. In several instances the author simpy expands the field or a wave function as a fourier transform. that is they assume the field or wave function is simply the transform of two other functions. I took pde and fourier analysis and this type of assumption never came up. I know it is a fact that non pathological functions can always be represented as a transform however It sounds like when ever you model any sort of wave trains in field theory the SOP is to just assume the field or wave function is the fourier transform of two other functions. one function traveling in one direction and the other tranveling in the other direction. Can anyone point me to a quick overview of this type ot thinking. it makes sense to me i just have never seen it done. Also is there a deeper reason for doing this it seems like just an added complication to me.