The Heisenberg picture is the only one where relativistic quantum field theory can be rigorously **defined**, e.g., through the Wightman axioms. The latter imply the existence of a 4-dimensional group of translations generated by the 4-momentum vector $p$. Once one select a future-like direction to determine time, one gets from $p$ the Hamiltonian $H=p_0$, and can use it to define time-dependent states $\psi(t)$ in the usual way. Together with the spatial part of the momentum, this gives a conventional, frame-dependent Schroedinger representation of the states.

Thus **the Schroedinger picture exists but is frame dependent.** You can regard any operator $A$ on the Hilbert space as a Schroedinger observable and find its expectation as time changes in the usual Schroedinger way, and translate it to the Heisenberg picture in the usual way such that $A$ becomes explicitly time-dependent and the state is fixed.

However, in 4-dimensional relativistic interacting quantum field theories, fields must be smeared in space and in time in order to produce densely defined operators (rather than distributions). Thus, unlike in the nonrelativistic case, the fixed-time spatial fields $\phi(x)=\Phi(t,x)|_{t=0}$, where $x$ is 3-dimensional, are not well-defined objects.

Thus** although the Schroedinger picture exists it does not represent local fields at a fixed time.** The attempt to pretend it did leads to the problems mentioned in Dirac's paper. In this sense, the relativistic field theory cannot be made well-defined without transcending the Schroedinger picture.

The above is completely independent of scattering theory. In scattering theory one has to construct the asymptotic Hilbert space. Unlike the interacting Hilbert space, the asymptotic Hilbert space is a Fock space and must therefore be defined by a limiting procedure. This is what is done in the Haag-Ruelle theory. Note that because of the Lorentz invariance of the future cone, the resulting asymptotic space does not depend on the choice of the time direction, as long as it points into the future cone.

There are presentations of the Haag-Ruelle theory that are a little more in the spirit of the Heisenberg picture; see the comments at the end of p.379 of Volume 3 of Reed and Simon. One can probably work almost completely in the Heisenberg picture if one works out the algebra of asymptotic constants (in analogy with the nonrelativistic case treated in Section 3.4 of Volume 3 of Thirring), but to show that the corresponding asymptotic operators have a Fock representation one apparently needs to go through some Schroedinger-like computations.

The meaning of the S-matrix in the Heisenberg picture is the following: There are frame-dependent isometries $\Omega_\pm$ from the interacting Hilbert space to the asymptotic Hilbert space such that for any two states $\phi,\psi$ in the Hilbert space of the interacting theory, the asymptotic states $\phi_-=\Omega_-\phi$ and $\psi_+=\Omega_+\psi$ satisfy the limiting relation $\lim_{t\to\infty}\langle\phi|e^{-itH/\hbar}|\psi\rangle=\langle\phi_-|\psi_+\rangle=\langle\phi|S|\psi\rangle$, where $S=\Omega_-^*\Omega_+$ is the S-matrix. This makes no reference at all to Schroedinger states. The latter are used only to construct the unitary mappings. But of course one can rewrite the limiting relation also in the Schroedinger picture by splitting $t$ into a sum of two times that separately go to infinity, and then attach these two times to the state.