The logic of quantum mechanics incorporating time dimension

Abstract: Similarly as classical propositional calculus is based algebraically on Boolean algebras, the logic of quantum mechanics was based on orthomodular lattices by G. Birkhoff and J. von Neumann and K. Husimi. However, this logic does not incorporate time dimension although it is apparent that the propositions occurring in the logic of quantum mechanics are depending on time. The aim of the present paper is to show that so-called tense operators can be introduced also in such a logic for given time set and given time preference relation. In this case we can introduce these operators in a purely algebraic way. We derive several important properties of such operators, in particular we show that they form dynamic pairs and, altogether, a dynamic algebra. We investigate connections of these operators with logical connectives conjunction and implication derived from Sasaki projections. Then we solve the converse problem, namely to find for given time set and given tense operators a time preference relation in order that the resulting time frame induces the given operators. We show that the given operators can be obtained as restrictions of operators induced by a suitable extended time frame.