Finite-Difference Equations in Relativistic Quantum Mechanics

Abstract

Relativistic Quantum Mechanics suffers from structural problems which are traced back to the lack of a position operator $\hat{x}$, satisfying $[\hat{x},\hat{p}]=i\hslash \hat{1}$ with the ordinary momentum operator $\hat{p}$, in the basic symmetry group -- the Poincaré group. In this paper we provide a finite-dimensional extension of the Poincaré group containing only one more (in 1+1D) generator $\hat{\pi }$, satisfying the commutation relation $[\hat{k},\hat{\pi }]=i\hslash \hat{1}$ 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.

10 LaTeX pages, final version, enlarged (2 more pages)

Cited by in corpus (1)

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