What's the rationale of replacing the Fourier coefficients in a field expansion by operators? Let's take a look on the particular case of the Fourier expansion of the Klein-Gordon field:
$$\psi (x,t) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_0(p)}[a(p)e^{i(E_0(p)t-px)}+a^\star (p)e^ {-i(E_0(p)t-px)} ]$$
Other fields have similar structures with but due to their spins they have extra features like the two components of the Dirac field or a polarization vector in case of spin-1 fields.
After the expansion is introduced it is usually just stated that the coefficients are replaced by operators which basically is the recipe second quantization. The operators are the creation and annihilation operators, which promote particles to different values of the three-momentum or created from, or sent back to the vacuum.
I'm not sure I understand the reason why this is done. The wavefunction expansion is just a one-particle wavefunction expressed in normal modes. Why should you replace the coefficients by creation and annihilation operators? It's clear to me that the theory needs to handle more particles and particles created from the vacuum. But how is this achieved from the one-particle wavefunction? The operators operate on the normal modes in the expansion. Does this create more particles? Or is it the creation of a particle from the vacuum that counts?
 A: I remember asking the same question in my QFT class. I have two things to offer:

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*A general idea during the development of the theory was that the fields themselves had to be quantized, and thus might need an operator just as $x$ or $p$. Take for example the electric field of a charged particle. If that particle is in superposition, what should the resulting field be? It seems not sufficient to just have fields as a non-quantized external source, as it is in regular Quantum Mechanics, to describe such a situation. The field itself should also be capable of superposition. Looking back today, we don't generally treat fields as observable operators as we treated e.g. $x,p$ in QM, but this was an original motivation for field operators. This is mentioned in Weinberg's chapter 1.


*If we accept that fields should be operators, we can ask what the most general operator solution is to this equation. Here I can't offer you a rationale off the top of my head, but I asked this question to a professor of mathematical QFT at a well-known university, who let me know that this is the most general self-adjoint operator solution to the Klein Gordon equation. So it seems that this is forced on us once we accept (1).
