In quantum mechanics (both non-relativistic and relativistic), it is possible to study physical systems by looking for solutions of PDEs, whose solutions belong to suitable Hilbert spaces:
Schrödinger equation for spinless particles* provides solutions which are $C^2$-wavefunctions, which form a dense subset of the Hilbert space $\mathcal{H} = L^2(\mathbb{R}^3)$ of functions square-integrable over $\mathbb{R}^3$.
The Schrödinger-Pauli equation is a generalization for 1/2-spin particles, whose natural Hilbert space seems to be $\mathcal{H} = L^2(\mathbb{R}^3)\otimes \mathbb{C}^2$.
Dirac equation, include spin and antiparticles, and its solutions can be considered a dense subset of $\mathcal{H} = L^2(\mathbb{R}^3)\otimes \mathbb{C}^4$.
For "collision states" which are non-square-integrable wavefunctions, rigged Hilbert spaces can be used, for a mathematically rigorous formulation.
However, in the ordinary QFT approach we do not use evolution equations as above, nor do we make explicit to which type of Hilbert space the physical states belong. My question is:
Why is it not easy or useful to generalize the above approach to QFT?
