I've seem in several books and lecture notes the quantization of the free KG field, and perhaps because I'm a kind of person that feels umconfortable with "hand waving" constructions, I still feel the need for a somwhat a little more rigorous approach which I found nowhere.
To summarize, the way this is done everywhere is:
First take the Fourier transform of the KG equation $\Box\phi = 0$ and realize that in Fourier space we get the time evolution of uncountably many harmonic oscillators, that is, the equation $\partial_t \hat{\phi}(t,\mathbf{p})+\omega_p^2 \hat{\phi}(t,\mathbf{p})=0$ with $\mathbf{p}$ fixed.
From this one directly states that the quantum field is $$\phi(x,t)=\int \dfrac{d^3 p}{(2\pi)^3}(a_p e^{-i p_\mu x^\mu}+a_p^\dagger e^{ip_\mu x^\mu})$$
importantly here no one has ever defined the operators $a_p$. At this point one claims that this is just a consequence of the field being equivalent to uncountably many decoupled oscillators and claims one is just using the results from QM of the harmonic oscillator, but not much of a precise construction is made.
After that derives that the commutation relations for $\phi,\pi$ are equivalent to the commutation relations $[a_p,a_q^\dagger]=i\delta(p-q)$ and $[a_p,a_q]=[a_p^\dagger,a_q^\dagger]=0$ working formaly. Remember that neither $\phi(x)$, nor $a_p$, nor the space where they act was ever defined.
One just claims that it is obvious that the space where these operators are acting is a Fock space (though nothing is said about the fact that the usual Fock space is built over countably many Hilbert spaces, while here we have uncountably mane oscillators in the Classical Picture as explained in (1)). Another issue that is never tackled is that the Fock space requires tacking the symmetric or antisymmetric tensor product, and it is not made clear which of them and how this was derived.
To make tings worse, with having never defined $a_p$ one just claims that there is one state $|0\rangle$ such that $a_p |0\rangle = 0$ and that for each $p$ fixed the results for the ladder operators from the Harmonic Oscilator can be carried over and adapted.
I mean, I understand that rigor is complicated in QFT. I've read a little bit about that. But this is another story: here things are coming out of thin air!
As an example, I don't have a problem with Dirac's formalism in QM, even though making it rigorous is a lot complicated, but I'm fine with it because the assumptions are made clear in most QM books and the derivations are almost always done without anything coming out of thin air. The derivation of the spectrum and eigenstates of the SHO for example is carried out in detail and in a prceise fashion in many books.
Now thiz quantization procedure has a lot of gaps that are not explained. The relation between the field, the harmonic oscilators and the Fock space is used all the time, but never made precise. One just claims things without much explanation.
What is really going on here? How can all this construction and these relations be made precise? What can be done to at least make the assumptions clear and the derivation also clear? How can we construct all this in a more comprehensive manner?
