Relativistic Quantum Mechanics suffers from structural problems which are traced back to the lack of a position operator $\hat{ x}$, satisfying $[\hat{ x},\hat{ p}]=i\hbar\hat{ 1}$ with the ordinary momentum operator $\hat{ p}$, in the basic symmetry group -- the Poincar\'e group. In this paper we provide a finite-dimensional extension of the Poincar\'e group containing only one more (in 1+1D) generator $\hat{ \pi}$, satisfying the commutation relation $[\hat{ k},\hat{ \pi}]=i\hbar\hat{ 1}$ with the ordinary boost generator $\hat{ k}$. The unitary irreducible representations are calculated and the carrier space proves to be the set of Shapiro's wave functions. The generalized equations of motion constitute a simple example of exactly solvable finite-difference set of equations associated with infinite-order polarization equations.
Categories: hep-th