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. Why do we need infinite-dimensional Hilbert spaces in physics?, Does the Hilbert space of the universe have to be infinite dimensional to make sense of quantum mechanics?). @Arnold Neumaier wrote in another 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 that `the Hilbert space of quantum gravity in asymptotically de Sitter space time has a finite dimension N'. 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?

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. Why do we need infinite-dimensional Hilbert spaces in physics?, Does the Hilbert space of the universe have to be infinite dimensional to make sense of quantum mechanics?). @Arnold Neumaier wrote in another 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 that `the Hilbert space of quantum gravity in asymptotically de Sitter space time has a finite dimension $N$'. 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?

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. Why do we need infinite-dimensional Hilbert spaces in physics?, Does the Hilbert space of the universe have to be infinite dimensional to make sense of quantum mechanics?). @Arnold Neumaier wrote in another 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 that `it is not enough to merely have a finite dimensional Hilbert space'. 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?

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. Why do we need infinite-dimensional Hilbert spaces in physics?, Does the Hilbert space of the universe have to be infinite dimensional to make sense of quantum mechanics?). @Arnold Neumaier wrote in another 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 that `the Hilbert space of quantum gravity in asymptotically de Sitter space time has a finite dimension N'. 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?