As I understand it, a relativistic *quantum field* is an operator-valued function of spacetime that transforms under some *finite dimensional irreducible representation* of the Lorentz* group:
\begin{equation}
U(\Lambda)\Psi^{\alpha}U(\Lambda^{-1}) = D(\Lambda^{-1})^\alpha{}_{\beta}\Psi^{\beta}(\Lambda x)
\end{equation}
Particle states (which I will write as $|\mathbf{p},\lambda \rangle = a^\dagger (\mathbf{p},\lambda) |0\rangle$, with $\mathbf{p}$ being the 3-momentum and $\lambda$ being the helicity), on the other hand, must preserve unitarity due to Quantum Mechanics and therefore should transform under unitary representations of the Poincaré* group

\begin{equation}
\Lambda|\mathbf{p},\lambda\rangle = |\mathbf{p}',\lambda'\rangle D^{s}(R(\Lambda,p))^{\lambda'}{}_{\lambda} 
\end{equation}

where $p'^{\mu} = \Lambda^{\mu}{}_{\nu}p^\nu$ and $\lambda $s are the helicities. Note that $\alpha, \beta$ are the Lorentz indices. The reason fields are introduced is because we can *embed* internal particle degrees of freedom transforming under an irreducible unitary representation (and hence infinite-dimensional) in a finite dimensional representation as

\begin{equation}
\Psi^{\alpha}(x) = \sum_{\lambda}\int \tilde{d}p[a(\mathbf{p},\lambda)u^{\alpha}(\mathbf{p},\lambda)e^{ipx} + \text{ negative energy term}]
\end{equation}
where $\tilde{d}p$ is the Lorentz invariant integration measure.
My questions are
1. How does one make contact with quantum mechanics? The book I am reading (Wu Ki Tung *Group Theory in Physics*, Chapter 10) says that the *wavefunction* for a state $|\phi\rangle$ is given by
\begin{equation}
\phi^\alpha(x) = \langle 0 | \Psi^{\alpha}(x) |\phi\rangle.
\end{equation}
So I assume this object describes the particle in the position basis. According the Peskin and Schroeder (chapter 2, eq. 2.42), the field operator *creates* a particle at position $x$ as such. But then what is the *wavefunctional* of the system, i.e., the object that describes the probability density corresponding to a particular configuration of the fields all throughout spacetime.

2. This is probably very naive but the book talks about relating the *Lorentz* symmetry of the fields to the *Poincaré* symmetry of the one-particle states. But I thought that fields are required to transform under the full Poincaré group, since Matthew D. Schwartz in his *QFT and the Standard Model* says that (chapter 8 titled *Spin 1 and Gauge Invariance*, page 110) scalar fields at all points in spacetime form a representation of translations since $\phi(x) \rightarrow \phi(x+a)$ under the action of translations. So I would expect that a natural demand to make is that the fields transform under finite dimensional representations of the Poincaré group.