Are there any quantum mechanical systems where the position vector lives in a finite dimensional Hilbert Space? I don't have much to add to the title. Are there any quantum mechanical systems where the position vector lives in a finite dimensional Hilbert Space? If so, please provide example(s).
 A: We know that for any system, the position and momentum satisfy the commutation  relation
$$[\hat{x},\hat{p}]=i\hbar $$
Take the trace of the both hand side
$$\text{Tr}(\hat{x}\hat{p}-\hat{p}\hat{x})=i\hbar\text{Tr} (I)$$
$$0=i\hbar \text{Tr}\ (I)\ \ !!! $$
It's clear that such a commutation relation can't be satisfied for finite-dimensional cases. Thus the operator these operators $\hat{x}$ and $\hat{p}$ live in infinite dimensional space.
A: It depends a bit what you mean but there are meaningful definitions of discrete phase space, where operators are labelled by discrete (and cyclic) indices that can be interpreted as position and momentum.
The simplest example is spin.  You can label
\begin{align}
\sigma_z\to \sigma_{10}\, ,\qquad \sigma_x\to \sigma_{01}, 
\qquad \sigma_{0}:=\mathbb{1}\to \sigma_{00}\, ,\qquad 
\sigma_y\to \sigma_{11}
\end{align}
so that, up to constant factors
\begin{align}
\sigma_{a_1b_1}\sigma_{a_2b_2}\propto \sigma_{a_1+a_2,b_1+b_2} \, ,\tag{1}
\end{align}
where addition is taken modulo $2$.  Moreover, it turns out that the eigenstates of $\sigma_{10}$ and $\sigma_{01}$ are "complementary" in the sense that the modulus square of the overlap $\vert\langle \psi^{a_1,b_2}_{k}\vert \psi^{a_2,b_2}_j\rangle\vert^2=\frac{1}{2}$ is constant and independent of the choice of eigenstates.  Thus, if you prepare a system in the state $\vert\psi^{0,1}_j\rangle$, you know nothing about the outcomes of the "complementary" operator $\sigma_{1,0}$ since every measurement outcome of $\sigma_{1,0}$ is equally probable.  This is the discrete version of "knowing the position exactly means you know nothing about the momentum".  The eigenstate of $\sigma_{01}$ are related to those of $\sigma_{10}$ by a Fourier-transform.  Here, position and momentum can be taken to be the eigenvalues of $\sigma_{10}$ and $\sigma_{01}$, or taken to be in the set $\{0,1\}$ (by simple shift).  Either way, the "position" and "momentum" are discrete and the eigenstates live in a finite-dimensional Hilbert space.
You can search GoogleScholar for "discrete phase space" and you'll come up with additional examples.  This is closely related to the notion of mutually unbiased bases, on which there is considerable literature.
The construction used for the Pauli matrices can be generalized whenever the dimension $d$ is a prime number.  The property (1) generalizes (modulo $d$) for the generalized Pauli group as discussed in

Patera J, Zassenhaus H. The Pauli matrices in n dimensions and finest gradings of simple Lie algebras of type $A_{n− 1}$. Journal of Mathematical Physics. 1988 Mar;29(3):665-73.

If $d$ is a power of a prime, such as $d=4=2^2$ or $d=9=3^2$, then one must use finite fields (not the fields of quantum field theory or electric field, but algebraic finite field theory).  This discrete phase space picture was initially proposed in

Gibbons KS, Hoffman MJ, Wootters WK. Discrete phase space based on finite fields. Physical Review A. 2004 Dec 3;70(6):062101.

If $d$ is a composite, such as $d=6=2\times 3$, the problem of constructing a discrete phase space and associated complete set of mutually unbiased bases is unsolved although heroic attempts have been made, as in

Brierley S, Weigert S. Constructing mutually unbiased bases in dimension six. Physical Review A. 2009 May 14;79(5):052316.

The grading structure of (1) has also been exploited in the $d\to\infty$ limit, in

Zeitlin V. Finite-mode analogs of 2D ideal hydrodynamics: Coadjoint orbits and local canonical structure. Physica D: Nonlinear Phenomena. 1991 Apr 2;49(3):353-62.

