The Calculus of Effective Quantum Logic

Peter Mittelstaedt

Peter Mittelstaedt

It has been shown by Birkhoff and v. Neumann (1936) and by Jauch and Piron (1963,1964,1968) that the subspaces of Hilbert space constitute an orthocomplemented quasi-modular lattice Lq, if one considers between two subspaces (elements) a, b the relation a⊆b and the operations a∩b, a∪b, a⊥. Furthermore, since the subspaces can be interpreted as quantum mechanical propositions, and since the operations ∩,∪ ,⊥ have some similarity with the logical operations ⋀ (and), ⋁ (or) and ⌝ (not), the question has been raised already by Birkhoff and v. Neumann, whether the lattice of subspaces of Hilbert space can be interpreted as a propositional calculus, sometimes called quantum logic. There are many kinds of lattices which can be interpreted as a propositional or logical calculus. A Boolean lattice LB of propositions corresponds to the calculus of classical logic and an implicative (Birkhoff, 1961) lattice Li has as a model the calculus of effective (intuitionistic) logic.

J. M. Jauch, C. Piron

In this paper we present a short introduction and survey of the calculus often called "Quantum Logic", and we point out the difficulties which prevent us from considering this calculus as a logic. It is furthermore shown that the probability calculus for a quantal system is a generalization of the classical probability calculus, which gives a more general meaning to the notion of random variable.

Peter Mittelstaedt

Othman Qasim Malhas

Abstract In much the same way that it is possible to construct a model of hyperbolic geometry in the Euclidean plane, it is possible to model quantum logic within the classical propositional calculus.