Algebraic structures and observations

quantales for a noncommutative logic - theoretic approach to quantum mechanics

Mario Piazza

pp. 381-393

in: Claudio Garola, Arcangelo Rossi (eds), The foundations of quantum mechanics, Berlin, Springer, 1995

Abstract

This paper is a series of intertwining observations about the connection between the logic-theoretic noncommutativity and a logical foundation of quantum mechanics. We will analyze noncommutativity, both from an algebraic and prooftheoretic point of view, w.r.t. the quantum mechanics notion that the order of observation making is central to their description. To this end, we will present the sequential conjunction ⊗ : A ⊗ B) means "A at time t ₁ and then B at time t ₂".The thread running through our discourse is given by quantales, i.e. algebraic structures introduced by Mulvey as models for the logic of quantum mechanics, which offer an appropriate algebraic (and topological) tool for describing noncommutativity.