On some mathematical aspects of the Heisenberg relation

The Heisenberg commutation relation, QP P Q = ihI, is the most fundamental relation of quantum mechanics. Heisenberg’s encoding of the ad-hoc quantum rules in this simple relation embodies the character-istic 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 .