In QM we deal with the Schrödinger equation:1
$$i\frac{\partial}{\partial t}\psi = H \psi$$
the wave function $\psi(x)$ is the main object of interest: it can be interpreted as a scalar field, in the sense that it links every point in space with a scalar value, and the really neat thing is that the scalar value in question is nothing but the probability amplitude of the particle described by $\psi$ being in that position $x$. This links very neatly with the bra-ket notation: in QM we know that the probability amplitude of a particle in state $|\psi\rangle$ to be measured in state $|\phi\rangle$ is:
$$\langle \phi| \psi \rangle$$
this is nothing but the scalar product of the two kets, as defined in Hilbert space. It's simply an expression of Born's rule. This fits well with the concept of wave function $\psi (x)$ since we can interpret it as simply an infinite dimensional vector of scalar products:
$$\psi (x)=\begin{pmatrix}\langle x'|\psi\rangle\\\langle x'' | \psi \rangle\\.\\.\\.\end{pmatrix}\tag{1}$$
this last visual really helped me to understand the concept of wave function in relation with bra-ket representation.
Given all this I feel like I know what I am talking about when I speak of quantum states and wave functions, the problem is that this confidence completely breaks switching in the QFT framework. To expose my doubts let's stick to the Klein-Gordon field, where the problem is more evident.
Trying to marry QM with Special Relativity we can arrive to the Klein-Gordon equation:2
$$(\partial ^2 +m^2)\phi(x) =0 \ \ \ , \ \ \ \phi \in \mathbb{R}$$
this equation describes a real scalar field (we could deal with a complex, charged, field, but lets keep this simple).
Here the problems begin:
We could try interpreting $\phi$ as a wave function like we did in QM, but the problem is that the scalar product defined on the space of the solutions of the K-G equation is not positive definite! So we cannot interpret the scalar product as a probability amplitude, and so we cannot link the field $\phi$ with the bra-ket representation like we did in $(1)$.3
Furthermore when we dig a little deeper in the matter we find out that the state of the system now has to be represented in Fock's space, since we are stuck with a formalism that predicts a variable number of particles. So Hilbert space flyes out the window: the kets still represent the quantum state of the system, but now they are vectors in Fock's space, and I can't find a clear link of any kind with the field $\phi$ like we had before in $(1)$.
On top of this when we perform second quantisation, to impose the necessary commutation relations, we start interpreting $\phi (x)$ not as a true scalar field, but as a field of operators (hermitian operators to be precise, since we said $\phi \in \mathbb{R}$). This creates even more confusion since now the field is a field of operators, and linking it with the concept of quantum state, represented by a ket, or to the concept of particle, or probability amplitude, becomes more and more tricky.
Given this context my question is:
Is there a neat interpretation of the concept of field in QFT (pre or post second quantization), that conceptually links it to quantum ket states in Fock's space, like we had in QM?
[1]: Allow me to use natural units: $\hbar,c = 1$.
[2]: $\partial ^2$ is the d'Alembert operator.
[3]: I know that this was a problem also for the physicist working on this stuff one hundred years ago, including Dirac himself; he developed his equation trying to takle exactly this problem, but apparently turns out that the problem of positive definitiveness is not a problem at all, and the K-G equation is perfectly usable to describe spinless bosons.
