State Property Systems and Orthogonality

Abstract

The structure of a state property system was introduced to formalize in a complete way the operational content of the Geneva-Brussels approach to the foundations of quantum mechanics (Aerts, D. the category of state property systems was proven to be equivalent to the category of closure spaces (Aerts, D., Colebunders, E., van der Voorde, A., and van Steirteghem, B., International Journal of Theoretical Physics, 38, 359-385, 1999; Aerts, D., Colebunders, E., van der Voorde, A., and van Steirteghem, B., The construct of closure spaces as the amnestic modification of the physical theory of state property systems, Applied Categorical Structures, in press). The first axioms of standard quantum axiomatics (state determination and atomisticity) have been shown to be equivalent to the T 0 and T 1 axioms of closure spaces (van Steirteghem, B.van der Voorde, A., Separation Axioms in Extension Theory for Closure Spaces and Their Relevance to State Property Systems, Doctoral Thesis, Brussels Free University, 2001), and classical properties to correspond to clopen sets, leading to a decomposition theorem into classical and purely nonclassical components for a general state property system (Aerts, D., van der Voorde, A., and Deses, D., Journal of Electrical Engineering, 52, 18-21, 2001; Aerts, D., van der Voorde, A., and Deses, D. International Journal of Theoretical Physics; Aerts, D. and Deses, D., Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Computation, and Axiomatics, World Scientific, Singapore, 2002). The concept of orthogonality, very important for quantum axiomatics, had however not yet been introduced within the formal scheme of the state property system. In this paper we introduce orthogonality in an operational way, and define ortho state property systems. Birkhoff's well known biorthogonal construction gives rise to an orthoclosure and we study the relation between this orthoclosure and the operational orthogonality that we introduced. KEY WORDS: orthogonality; state property systems; ortho state property systems; ortho axioms.

Failure of Standard Quantum Mechanics for the Description of Compound Quantum Entities

“…We cannot go into the details of the attempts that have been made to interpret the orthocomplementation in a physical way, and refer to [20,21,1,2,4] for those that are interested in this problem. Also in [22], [23], [17], [13] and [15] this problem is considered in depth.…”

A theory of concepts and their combinations I

“…Applying analogous techniques to those in Aerts et al (1999, 2002), it can be proven that λ (M) is a closure space. Further structural results about SCOP can be obtained along the lines of Aerts et al (2001 2005), Aerts and Deses (2002, 2005). Having introduced this closure space the supremum can be given a topological meaning, namely for e , f ∈M we have λ ( e ∨ f )= λ ( e )∪ λ ( f ) where λ ( e )∪ λ ( f ) is the closure of λ ( e )∪λ( f ), which is obtained exactly, in the case of the linear Hilbert space introduced in Aerts and Gabora (2005), by adding the superposition states to the set λ ( e )∪λ( f ).…”

“…Most of the results obtained in quantum axiomatics can be applied readily to concepts being considered as entities with properties. Borrowing from the study of State Property Systems (Aerts, 1999a, b, Aerts et al , 1999, 2002, 2001; Aerts and Deses, 2002, 2005) we introduce the function Equation 26, Equation 27 and Equation 28 that has been called the “Cartan Map” in the study of State Property Systems. Clearly we have for a ∈L Equation 29 and for a , b ∈L Equation 30 The Cartan Map introduces a closure space, namely κ (L).…”

Undecidable Classical Properties of Observers

“…The problem of the 'inverse property' is then treated by the introduction of ortho-axioms on LS [2], and the role of the inversion is played by the orthocomplementation. For a Boolean sub-lattice of LS, or for an entity that is classical, (i.e.…”

“…So far we have described a system can have properties, which partitions its set of states, and the observer is the one who indicates the outcome, which in turn partitions his set of states (2). These are two very basic desiderata of the process of observing a property.…”