Fourier transform field solutions

[zwoodrow]

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.

 

[mathman]

When describing em radiation (light, etc.), it can be considered as a function of time. Taking the Fourier transform gives the frequency spectrum.

 

[Hepth]

When describing em radiation (light, etc.), it can be considered as a function of time. Taking the Fourier transform gives the frequency spectrum.

I'm not really sure that's what he's asking.

Pages 20-21 of Peskin and Schroeder go through this for the Klein-Gordon equation. The deeper reason is that the field, such as:
$$\phi \left(x\right) = \int \frac{d^3 p}{\left(2 \pi\right)^3} \frac{1}{\sqrt{2 \omega_p}} \left(a_p e^{i px} + a^{\dag}_p e^{-i px}] \right)$$

I think he's asking why there are two terms, one for the positive solutions, one for the negative. The answer I believe is that instead of each solution existing by itself, we write it as an "independent oscillator" which contains its own creation and annihilation operators for creating and destroying states, and is a linear combination of the two separate states.

Remember back in differential equations that if two solutions to the same differential eq. are sin(kx) and cos(kx), we write the FULL solution as the linear combination of those two.