Do QFTs with a physical cut-off not respect the postulates of Quantum Mechanics? Wilsonian renormalization says that it's fine to have a physical cut-off. But I am thinking that such theories do not respect the postulates of Quantum Mechanics. Is this true?
Theories with a physical cut-off $\Lambda$ describe a state $|\psi\rangle$ in the Hilbert space of wavefunctionals. So far, so good.
But the time evolution of this state is supposed to be a path integral over all intermediate states. Each basis state in the Hilbert space, i.e. each 3D field configuration must count as an intermediate state. However, theories with a physical cut off do not respect this postulate, because these theories ban the high energy configurations in their path integral definition.
This makes it look like we're modifying a QM postulate in an ad-hoc way. This shouldn't have worked because the path integral evolution postulate is intimately tied to other alternative postulates like the Schrodinger equation (which is synonymous with the unitarity of the theory), or the Moyal bracket from phase space QM.
Nevertheless, effective theories produce the correct predictions, so there must be some justification of this. One answer can be that there may exist an underlying non-field theory which obeys the QM postulates, from which the field theories may be derived. But this answer would be speculation. This would also imply that QFTs are not consistent Quantum Theories by themselves. Are the field theories with a physical cut-off on a solid mathematical foundation or not?
 A: Depending on the cutoff-scheme, it's easier/harder to convince yourself that there are no problems. Easiest cutoff to consider is a lattice-cutoff, where you discretise space onto a $L^3$ grid (but leave time continuous). This is sometimes called the 'Hamiltonian formalism'. In this case, you have:

*

*A UV cutoff given by $|p| < \frac{\pi}{a}$, set by the physical lattice spacing $a$.

*A IR cutoff given by $|p| > \frac{\pi}{aL}$, set by the physical width of the box your theory lives in. (You can remove this IR cutoff if you want by taking the box length $L \to \infty$. Correlation functions $\langle \phi(x) \phi(y) \rangle$ for fixed $x,y$ will remain constant in this limit)

*A completely formally well-defined, finite-dimensional hilbert space $\mathcal{H}$, as well as a hamiltonian operator $H$. The dimension of the hilbert space will generically scale like $d^{L^3}$ where $d$ is how many degrees of freedom you have per-site.

An example that is talked about a lot is the Kogut-Susskind Hamiltonian formulation of lattice-QCD.
