Finite-Difference Equations in Relativistic Quantum Mechanics

TL;DR

This paper introduces a finite-dimensional extension of the Poincaré group to address structural issues in relativistic quantum mechanics, leading to exactly solvable finite-difference equations with potential implications for wave function representations.

Contribution

It proposes a novel finite-dimensional extension of the Poincaré group with an additional generator, providing new representations and exactly solvable finite-difference equations in relativistic quantum mechanics.

Findings

Derived unitary irreducible representations involving Shapiro's wave functions.

Developed a set of exactly solvable finite-difference equations.

Connected finite-difference equations to polarization equations.

Abstract

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.