The logic of quantum mechanics incorporating time dimension

TL;DR

This paper extends the algebraic logic of quantum mechanics by incorporating time-dependent tense operators, forming a dynamic algebra that models propositions with temporal dependence.

Contribution

It introduces a purely algebraic method to include tense operators in quantum logic, connecting them with time preference relations and dynamic algebra structures.

Findings

Tense operators form dynamic pairs and a dynamic algebra.

Connections established between tense operators and logical connectives.

Methods to derive time preference relations from given tense operators.

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.