Algebraic computations in quantum logics

Abstract

Mathematical description of quantum phenomena requires an event structure more general than a Boolean algebra. For this purpose, Birkhoff and von Neumann proposed the notion of an orthomodular lattice. Its typical feature is the existence of so–called non–commuting events, which are not simultaneously observable (like position and momentum, according to Heisenberg’s uncertainty principle). There is an old open problem whether the word problem for orthomodular lattices is solvable. Is it possible to decide whether two formulas are equivalent? In Boolean algebras, an easy positive answer is given by a transformation of the formula to a unique normal form. This requires the commutativity, associativity, and distributivity of the Boolean operations (disjunction and conjunction). In orthomodular lattices, the corresponding lattice operations (join and meet) violate distributivity. This disables the use of normal forms. We looked for alternative approaches. E.g., the join is only one of six orthomodular lattice operations generalizing the disjunction. In the thesis, we study the question whether some of the 96 binary operations in orthomodular lattices admit normal forms similar to the classical logic. The first question was which operations satisfy the associative identity, eventually under the assumption that some variables commute. Then we studied monotonicity because it is related to distributivity over the meet and join. The conclusion is that there is no pair of operations in orthomodular lattices admitting “Boolean–like” normal forms. In the last chapter we study “Moufang–like” identities, which were inspired by the algebras of quaternions and octonions. These identities generalize associativity and may enable further progress. As a by–product, we proved interesting, yet unknown, properties of some orthomodular lattice operations (e.g., the Sasaki projection). These new tools simplify algebraic computations and give a chance to develop algorithms more general than the current specialized software.