A Klein-Gordon field on a Minkowski background can be written in the following expansion
$$ \hat{\phi}(x) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}} (\hat{a}_p e^{-ip x} + \hat{a}^\dagger_p e^{ip x}). $$
Here, $\hat{a_p}$ , $\hat{a}_p^\dagger$ are the usual creation/ annihilation operators satisfying CCR and the positive frequency modes (including the pre factors) satisfy the KG equation and form a complete set of orthonormal vectors wrt the KG inner product, spanning a Hilbert space (?).
Now, physically one expects the Hilbert space of this system to be formed by a set of field configurations. And similar to the wave function in QM, one can write a wave functional $\Psi[\phi]$ that gives an amplitude for each field configuration. Do these wave functionals also form a Hilbert space and how do they relate to the Hilbert space of the modes I described above? I am a bit confused.
Also, in more axiomatic treatments of QFT it is said that the fields are operator-valued distributions that output an operator on a Hilbert space after being smeared with a test-function. How does this point of view relate to the ones I mentioned above? The physics notation makes this obscure to me. I’d guess this is related to the wave functional. If someone can concretely point out which object corresponds to which (e.g. what is being smeared), that would be great.
