Although various schemes for anhomomorphic logics for quantum mechanics have been considered in the past we shall mainly concentrate on the quadratic or grade-2 scheme. In this scheme, the grade-2 truth functions are called coevents. We discuss properties of coevents, projections on the space of coevents and the master observable. We show that the set of projections forms an orthomodular poset. We introduce the concept of precluding coevents and show that this is stronger than the previously studied concept of preclusive coevents. Precluding coevents are defined naturally in terms of the master observable. A result that exhibits a duality between preclusive and precluding coevents is given. Some simple examples are presented.
Although various schemes for anhomomorphic logics for quantum mechanics have been considered in the past we shall mainly concentrate on the quadratic or grade-2 scheme. In this scheme, the grade-2 truth functions are called coevents. We discuss properties of coevents, projections on the space of coevents and the master observable. We show that the set of projections forms an orthomodular poset. We introduce the concept of precluding coevents and show that this is stronger than the previously studied concept of preclusive coevents. Precluding coevents are defined naturally in terms of the master observable. A result that exhibits a duality between preclusive and precluding coevents is given. Some simple examples are presented.
Anhomomorphic logic is a new interpretation of Quantum Theory (due to R. Sorkin). It is a histories formulation (c.f. consistent histories, quantum measure theory). In this approach, reality is a co-event, which is essentially an assignment of a truth value {True, False} to each question. The way this assignment is done mimics classical physics in as much as possible allowing however for sufficient flexibility to accommodate quantum 'paradoxes', as is shown by the analysis of Kochen-Specker theorem. In this contribution, after briefly reviewing the approach, we will examine how probabilistic predictions can arise. The Cournot principle and the use of approximate preclusions will play a crucial role. Facing similar problems in interpreting probability as in classical probability theory, we will resort to the weak form of Cournot principle, where possible realities will be preclusive co-events and the quantum measure is used to obtain predictions. Examples considered, includes the fair coin and the double slit pattern arguably one of the most important paradigms for quantum theory.
A classical logic exhibits a threefold inner structure comprising an algebra of propositions , a space of "truth values" V , and a distinguished family of mappings ϕ from propositions to truth values. Classically is a Boolean algebra, V = 2 , and the admissible maps ϕ: 2 are homomorphisms . If one admits a larger set of maps, one obtains an anhomomorphic logic that seems better suited to quantal reality (and the needs of quantum gravity). I explain these ideas and illustrate them with three simple examples.
An anhomomorphic logic $\ascript ^*$ is the set of all possible realities for a quantum system. Our main goal is to find the "actual reality" $ϕ_a\in\ascript ^*$ for the system. Reality filters are employed to eliminate unwanted potential realities until only ϕa remains. In this paper, we consider three reality filters that are constructed by means of quantum integrals. A quantum measure μ can generate or actualize a $ϕ\in\ascript ^*$ if μ(A) is a quantum integral with respect to ϕ for a density function f over events A. In this sense, μ is an "average" of the truth values of ϕ with weights given by f. We mainly discuss relations between these filters and their existence and uniqueness properties. For example, we show that a quadratic reality generated by a quantum measure is unique. In this case we obtain the unique actual quadratic reality.
An anhomomorphic logic A * is the set of all possible realities for a quantum system.Our main goal is to find the "actual reality" φ a ∈ A * for the system.Reality filters are employed to eliminate unwanted potential realities until only φ a remains.In this paper, we consider three reality filters that are constructed by means of quantum integrals.A quantum measure µ can generate or actualize a φ ∈ A * if µ(A) is a quantum integral with respect to φ for a density function f over events A. In this sense, µ is an "average" of the truth values of φ with weights given by f .We mainly discuss relations between these filters and their existence and uniqueness properties.For example, we show that a quadratic reality generated by a quantum measure is unique.In this case we obtain the unique actual quadratic reality."Reality is merely an illusion, albeit a very persistent one.
A set of statements about the properties of a quantum system is looked at as at a partially ordered set of subspaces of finite or infinite dimensional Hilbert space. The operation of ordering is performed on a set of propositions comparing the truth values of these propositions and on the set of subspaces as the opera- tion of inclusion. Based on the required properties these structures are translated into operations on the lattice. The correspondence with Heisenberg uncertainty principle is shown there. Furthermore, it is shown that the lattices correspond- ing to the subspaces of infinite dimensional Hilbert space are not modular. This property is replaced with weaker property of orthomodularity, when operation of the negation is added. Following the work of G. Birkhoff and J. von Neumann, the structure of quantum logic is looked for in projective spaces, which are in- troduced either arithmetically or axiomatically. The examples of quantum logic, their physical implementation and eventual implementation in projective spaces are analysed. 1
The paper is the contribution to quantum toposophy focusing on the abstract orthomodular structures (following Dunn-Moss-Wang terminology). Early quantum toposophical approach to "abstract quantum logic" was proposed based on the topos of functors [E,Sets] where E is a so-called orthomodular preorder category – a modification of categorically rewritten orthomodular lattice (taking into account that like any lattice it will be a finite co-complete preorder category). In the paper another kind of categorical semantics of quantum logic is discussed which is based on the modification of the topos construction itself – so called quantos – which would be evaluated as a non-classical modification of topos with some extra structure allowing to take into consideration the peculiarity of negation in orthomodular quantum logic. The algebra of subobjects of quantos is not the Heyting algebra but an orthomodular lattice. Quantoses might be apprehended as an abstract reflection of Landsman's proposal of "Bohrification", i.e., the mathematical interpretation of Bohr's classical concepts by commutative C∗-algebras, which in turn are studied in their quantum habitat of noncommutative C∗-algebras – more fundamental structures than commutative C∗-algebras. The Bohrification suggests that topos-theoretic approach also should be modified. Since topos by its nature is an intuitionistic construction then Bohrification in abstract case should be transformed in an application of categorical structure based on an orthomodular lattice which is more general construction than Heyting algebra – orthomodular lattices are non-distributive while Heyting algebras are distributive ones. Toposes thus should be studied in their quantum habitat of "orthomodular" categories i.e. of quntoses. Also an interpretation of some well-known systems of orthomodular quantum logic in quantos of functors [E,QSets] is constructed where QSets is a quantos (not a topos) of quantum sets. The completeness of those systems in respect to the semantics proposed is proved.