FINITE-DIFFERENCE EQUATIONS IN RELATIVISTIC QUANTUM-MECHANICS

Relativistic quantum mechanics suffers from structural problems which are traced back to the lack of a position operator ($) over cap x, satisfying [($) over cap x, ($) over cap p] = i ($) over bar h1 with the ordinary momentum operator ($) over cap p, in the basic symmetry group-the Poincare group. In this letter we provide a finite-dimensional extension of the Poincare group containing only one more generator ($) over cap pi (in 1 + 1 dimensions), satisfying the commutation relation [($) over cap kappa, ($) over cap pi] = i ($) over bar h1 with the ordinary boost generator ($) over cap kappa. The unitary irreducible representations are calculated and the carrier space proves to be the set of Shapiro's wavefunctions. The generalized equations of motion constitute a simple example of exactly solvable finite-difference set of equations associated with infinite-order polarization equations.