Representations of Baer *-Semigroups and Quantum Logics in Hilbert Space

James C. T. Pool

Héctor Freytes, Graciela Domenech, Christian DE RONDE

Sylvia Pulmannová

David W. Cohen

G. T. Riittimalm

Croc lattices are of central importance in the mathematical foundation of quantum meehanies. They reflect the structure of the logical calculus of propositions concerning microphenomena. In this context, the mathematical consequences resulting from the special structure of the lattice are of particular interest. Beginning with the notion of hemimorphism of a eroc lattice, we get, in a natural way, an extension of the structure of the lattice by the Baer *-semigroup. We embed the Baer *-semigroup in an algebra with generalized *- and -operations and prove some theorems as consequences of the particular properties of the underlying croc lattice.

Peter Mittelstaedt

Self-referential consistency and inconsistency Universality implies self-referentiality In the preceding chapters, we have mentioned on several occasions that there are good reasons to consider quantum mechanics as universally valid. Indeed, during the last 70 years quantum mechanics has not been disproved by a single experiment. In spite of numerous attempts to discover the limits of applicability and validity of this theory, there is no indication that the theory should be improved, extended, or reformulated. Moreover, the formal structure of quantum mechanics is based on very few assumptions, and these do not leave much room for alternative formulations. The most radical attempt to justify quantum mechanics, operational quantum logic, begins with the most general preconditions of a scientific language of physical objects, and derives from these preconditions the logico-algebraic structure of quantum mechanical propositions [Mit 78,86], [Sta 80]. There are strong indications that from these structures (orthomodular lattices, Baer*-semigroups, orthomodular posets, etc.) the full quantum mechanics in Hilbert space can be obtained. Although simple application of Piron's representation theorem [Pir 76] does not lead to the desired result, [Kel 80], [Gro 90], there are new and very promising results [Sol 95] which indicate that the intended goal may well be achieved within the next few years. Together with the experimental confirmation and verification of quantum mechanics, these quantum logical results strongly support the hypothesis that quantum mechanics is indeed universally valid.