On some mathematical aspects of the Heisenberg relation

Abstract

The Heisenberg commutation relation, QP−PQ = iħI, is the most fundamental relation of quantum mechanics. Heisenberg’s encoding of the ad-hoc quantum rules in this simple relation embodies the characteristic indeterminacy and uncertainty of quantum theory. Representations of the Heisenberg relation in various mathematical structures are discussed. In particular, after a discussion of unbounded operators affiliated with finite von Neumann algebras, especially, factors of Type II1, we answer the question of whether or not the Heisenberg relation can be realized with unbounded self-adjoint operators in the algebra of operators affiliated with a factor of type II1.