I've been trying to understand some constructions in QFT in a more precise manner and this has been extremely difficult.
The first one is the quantization of the real massless scalar field which satisfies the massless Klein-Gordon equation $\Box \phi = 0$. In that case, one usually "loosely" says that this is entirely connected to the simple harmonic oscillator quantization and to the Fock space construction.
More precisely, one says that the quantum field solution to this equation is:
$$\phi(x)=\int \dfrac{d^3 p}{(2\pi)^3}(a_pe^{-i p_\mu x^\mu}+a_p^\dagger e^{i p_\mu x^\mu})$$
where $a_p$ are operators. One then shows that the commutation relations $[\phi(x),\pi(y)]=i\delta(x-y)$ and $[\phi(x),\phi(y)]=[\pi(x),\pi(y)]=0$ are equivalent to $[a_p,a_q^\dagger]=(2\pi)^3\delta(p-q)$ and $[a_p,a_q]=[a_p^\dagger,a_q^\dagger]=0$.
Thus one merely shifts the problem from finding $\phi(x)$ to finding $a_p$.
However then one just says that those $a_p$ acts as ladder operators on a Fock space, such that there is one state $|0\rangle$ such that $a_p |0\rangle = 0$ for all $p$ and $a_p^\dagger |0\rangle = |p\rangle$ are one-particle states of a Fock space, which allows one to solve the problem.
Trying to summarize: what is being implied, I believe is: if $a_p$ are operators in a Hilbert space $ \mathcal{E}$ satisfying $[a_p,a_q^\dagger]=(2\pi)^3\delta(p-q)$ and $[a_p,a_q]=[a_p^\dagger,a_q^\dagger]=0$, then $\mathcal{E}$ is indeed a Fock space
$$\mathcal{F}_+(\mathcal{H}) = \bigoplus_{n=0}^\infty S_{+} \mathcal{H}^{\otimes n} ,$$
being $\mathcal{H}$ the state space spanned by $|p\rangle = a_p^\dagger|0\rangle$.
Here $S_+$ means to take the symmetric tensor product. I just want to know how to make this a little more precise and actually prove this result.
Now, I don't want anything extremely rigorous and fancy with $^\ast$-algebras and so forth. So I don't mind considering the continuous kets $|p\rangle$, indeed I'm realy fine with Dirac's formalism.
I just want to make this derivation a little bit clearer to take out the impression that this came out of thin air.
So how should this be done? I believe one first has to show that there is $|0\rangle$, then define $\mathcal{H}$ and prove $\mathcal{E}$ is isomorphic to $\mathcal{F}_+(\mathcal{H})$, but I mithg be wrong. How do I make this construction a little bit better and show this result?
EDIT: Fine, by the discussion in comments I already understand that unlike in Quantum Mechanics, in QFT there is no unique construction (up to unitary equivalence) that leads to $\phi(x)$ obeying the canonical commutation relations. What I want to know is how to derive the construction made by Physicsits which relies on the Fock space.