When defining quantum fields as a sum of creation and annihilation operators for each momenta, we do it in analogy with the simple example of the harmonic oscillator in quantum mechanics. But why do we assume that the coefficients in the expansion can be interpreted in the same way as in the case of the harmonic oscillator? Is this just an assumption that seems to work, or is it a good explanation for it? For in the case of the harmonic oscillator we had an Hamiltonian and so could easily prove that a and a* could be interpreted as creation and annihilation operators respectively. But when defining the quantum field, we are not given an Hamiltonian, but still assumes that we could expand or solution in terms of those operators.
Sorry if my explanation is poorly written, by I think you get the point.
Sorry if my explanation is poorly written, by I think you get the point.