It seems to me that there are fairly good reasons to assume that quantum theories need to rely in their formulation on infinite-dimensional spaces, cf. https://physics.stackexchange.com/q/149786 or https://physics.stackexchange.com/q/29740/ . @Arnold Neumaier wrote in a stackexchange thread:
With a finite-dimensional Hilbert space, the whole apparatus of practical QM is lost. Very little is left - no continuous spectra, no scattering theory, no S-matrix, no cross sections. No Dirac equation, no relativity theory, no relation between symmetry and conservation laws, no quantum fields.
It appears, however, that in some contexts, e.g. quantum gravity, people have been making claims to the effect that it is enough to consider finite-dimensional Hilbert spaces. Here's Ed Witten:
We discuss some general properties of quantum gravity in de Sitter space. It has been argued that the Hilbert space is of finite dimension. (http://cds.cern.ch/record/504347/files/0106109.pdf)
How is it possible to reconcile those two statements? Are the respective contexts vastly different in some sense?