WIGNER APPROACH TO NONCANONICAL QUANTIZATIONS

Abstract

The Wigner problem,i.e. the search for quantum mechanical commutation relations consistent with the Heisenberg evolution equations of a given form is studied. In the framework of recently proposed generalization of the Wigner approach the classical analogy is postulated for the form of the time evolution equations only and the forms of the time evolution and symmetry generators are nota priori assumed. Instead of that the set of basic, physically justified algebraic relations is required to have a Lie algebra structure. Here the problem is formulated for the system of two particles interacting harmonically and a noncanonical Lie algebra of fundamental quantum mechanical quantities, alternative to the standard canonical one, is found. The solution of the nonrelativistic problem suggests what kind of algebras could be investigated as its relativistic analogues.