Conditional probabilities on orthomodular lattices

A definition of generalized probability on an orthomodular lattice which includes as particular cases the classical probability space and non-commutative probability theory on a von Neumann algebra is proposed. In this generalized structure the problem of conditioning with respect to Boolean σ-subalgebras is examined.     

Sites and tours in orthoalgebras and orthomodular lattices

A block of an orthoalgebra (or of an orthomodular lattice) is a maximal Boolean subalgebra. A site is the intersection of two distinct blocks. L is block (site)-finite if there are only finitely many blocks (sites). We introduce a certain type of subalgebra of an orthoalgebra which is a subortholattice if the orthoalgebra is an ortholattice (and therefore an orthomodular lattice) and which is block finite if the orthoalgebra is site finite. The construction yields a cover of a site-finite orthoalgebra or orthomodular lattice L by block-finite substructures of the same type and having the same center as L. Every site-finite orthomodular lattice is commutator finite.In memory of Charles H. Randall.     

Automorphisms of orthomodular lattices and symmetries of quantum logics

Given a group G, there is a proper class of pairwise nonembeddable orthomodular lattices with the automorphism group isomorphic to G. While the validity of the above statement depends on the used set theory, the analogous statement for groups of symmetries of quantum logics is valid absolutely.     

Bell-type inequalities in orthomodular lattices. I. Inequalities of order 2

We study Bell-type inequalities of ordern with emphasis on the casen = 2 in the framework of the structure of an orthomodular lattice, which is a logicoalgebraic model of quantum mechanics. We give necessary and sufficient conditions for the validity of Bell-type inequalities of order 2. In particular, we study Bell-type inequalities in various structures connected with a Hilert space, and we give a characterization of Boolean algebras via the validity of certain Bell-type inequalities.     

Propositional systems and measurements. III. Quasitensorproducts of certain orthomodular lattices

Continuing our investigations on propositional systems without assumption of the covering law, we introduce a quasi-tensor-product of a complete atomic orthomodular lattice with a complete atomic Boolean lattice. This product has a universal property with respect to postulates on propositional systems of coupled physical systems. We use it to describe measurements on a purely quantal object by a purely classical apparatus and find no nontrivial proposition of the object to be commensurable with its quantal negation. If the object is not purely quantal, the central propositions are commensurable. By this, it is shown directly that useful apparatuses must have a quantal microstructure.Dedicated to the 60th birthday of Professor G. Ludwig     

Bell-type inequalities in orthomodular lattices. II. Inequalities of higher order

This paper is a continuation of the first part and it is devoted to the study of Bell-type inequalities of order at least 3 in orthomodular lattices. We give some necessary and sufficient conditions for the validity of Bell-type inequalities of order 3 and also, more generally, for those of ordern.     

Commutativity in orthomodular posets

An abstract characterization of the commutation relation in orthomodular posets is given. This characterization is a generalization of Guz's result. In particular, if an orthomodular poset P is Boolean, then aCb iff a ∧ b exists in P.A method of constructing nonregular Boolean orthomodular posets is presented.     

Measures on infinite-dimensional orthomodular spaces

We classify the measures on the lattice of all closed subspaces of infinite-dimensional orthomodular spaces (E, ) over fields of generalized power series with coefficients in . We prove that every -additive measure on can be obtained by lifting measures from the residual spaces of (E, ). The measures being lifted are known, for the residual spaces are Euclidean. From the classification we deduce, among other things, that the set of all measures on is not separating.Research supported by the Swiss National Science Foundation.     

A particular non-atomistic orthomodular poset

Every atomic orthomodular lattice is atomistic. We show that the corresponding statement for orthomodular posets fails. The result is of interest in the study of the Algebraic Structure of Quantum Mechanics, see [5].     

Algebras of bounded operators on nonclassical orthomodular spaces

A Hermitian space is called orthomodular if the Projection Theorem holds: every orthogonally closed subspace is an orthogonal summand. Besides the familiar real or complex Hilbert spaces there are non-classical infinite dimensional examples constructed over certain non-Archimedeanly valued, complete fields. We study bounded linear operators on such spaces. In particular we construct an operator algebraA of von Neumann type that contains no orthogonal projections at all. For operators inA we establish a representation theorem from which we deduce thatA is commutative. We then focus on a subalgebra which turns out to be an integral domain with unique maximal ideal. Both analytic and topological characterizations of are given.     

Spectral decompositions of operators on non-Archimedean orthomodular spaces

The most central property of an infinite-dimensional Hilbert space is expressed by the projection theorem: Every orthogonally closed linear subspace is an orthogonal summand. Besides the obvious Hilbert spaces, there exist other infinite-dimensional orthomodular spaces. Here we study bounded linear operators on an orthomodular spaceE constructed over a field of generalized power series with real coefficients. Our main result states that every bounded, self-adjoint operator gives rise to a representation ofE as the closure of an infinite orthogonal sum of invariant subspaces each of which is of dimension 1 or 2. The proof combines the technique of reduction modulo the residual spaces with theorems on orthogonal decompositions of finite matrices over fields of power series.     

Jauch-Piron orthomodular posets and propositional systems: A comparison

We compare the structures obtained via orthomodular posets and via propositional systems, discussing some examples of the links between them. Despite some analogies, the two structures are fundamentally different.     

Quantic lattices

The category of quantic lattices is defined. All the multiplicative lattices, such asresiduated lattices andorthomodular lattices, turn out to be objects of this new category.     

Minimal inflation

Using the universal X-superfield that measures in the UV the violation of conformal invariance we build up a model of multifield inflation. The underlying dynamics is the one controlling the natural flow of this field in the IR to the goldstino superfield once SUSY is broken. We show that flat directions satisfying the slow-roll conditions exist only if R-symmetry is broken. Naturalness of our model leads to scales of SUSY breaking of the order of 10^11–13 GeV, a nearly scale-invariant spectrum of the initial perturbations and negligible gravitational waves. We obtain that the inflaton field is lighter than the gravitino by an amount determined by the slow-roll parameter η. The existence of slow-roll conditions is directly linked to the values of supersymmetry and R-symmetry breaking scales. We make cosmological predictions of our model and compare them to current data.     

The role of the modular pairs in the category of complete orthomodular lattice

We study the modular pairs of a complete orthomodular lattice i.e. a CROC. We propose the concept ofm-morphism as a mapping which preserves the lattice structure, the orthogonality and the property to be a modular pair. We give a characterization of them-morphisms in the case of the complex Hilbert space to justify this concept.     

Gleason-type theorems for signed measures on orthomodular posets of projections on linear spaces

We consider some orthomodular posets which are not lattices and the Gleason-type theorems for signed measures on them.