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Foundations of Quantum Physics: a General Realistic and
 Operational Approach, Int. J. Theor. Phys
, 1999
"... We present a general formalism with the aim of describing the situation of an entity, how it is, how it reacts to experiments, how we can make statistics with it, and how it ‘changes ’ under the influence of the rest of the universe. Therefore we base our formalism on the following basic notions: (1 ..."
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Cited by 35 (25 self)
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We present a general formalism with the aim of describing the situation of an entity, how it is, how it reacts to experiments, how we can make statistics with it, and how it ‘changes ’ under the influence of the rest of the universe. Therefore we base our formalism on the following basic notions: (1) the states of the entity; they describe the modes of being of the entity, (2) the experiments that can be performed on the entity; they describe how we act upon and collect knowledge about the entity, (3) the probabilities; they describe our repeated experiments and the statistics of these repeated experiments, (4) the symmetries; they describe the interactions of the entity with the external world without being experimented upon. Starting from these basic notions we formulate the necessary derived notions: mixed states, mixed experiments and events, an eigen closure structure describing the properties of the entity, an ortho closure structure introducing an orthocomplementation, outcome determination, experiment determination, state determination and atomicity giving rise to some of the topological separation axioms for the closures. We define the notion of sub entity in a general way and identify the morphisms of our structure. We study specific examples in detail in the light of this formalism: a classical deterministic entity and a quantum entity described by the standard quantum mechanical formalism. We present a possible solution to the problem of the description of sub entities within the standard quantum mechanical procedure using the tensor product of the Hilbert spaces, by introducing a new completed quantum mechanics in Hilbert space, were new ‘pure ’ states are introduced, not represented by rays of the Hilbert space.
Contextualizing concepts using a mathematical generalization of the quantum formalism
 Trends in Cognitive Science
, 2000
"... We outline the rationale and preliminary results of using the State Context Property (SCOP) formalism, originally developed as a generalization of quantum mechanics, to describe the contextual manner in which concepts are evoked, used, and combined to generate meaning. The quantum formalism was deve ..."
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Cited by 28 (18 self)
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We outline the rationale and preliminary results of using the State Context Property (SCOP) formalism, originally developed as a generalization of quantum mechanics, to describe the contextual manner in which concepts are evoked, used, and combined to generate meaning. The quantum formalism was developed to cope with problems arising in the description of (1) the measurement process, and (2) the generation of new states with new properties when particles become entangled. Similar problems arising with concepts motivated the formal treatment introduced here. Concepts are viewed not as fixed representations, but entities existing in states of potentiality that require interaction with a context—a stimulus or another concept—to ‘collapse ’ to an instantiated form (e.g. exemplar, prototype, or other possibly imaginary instance). The stimulus situation plays the role of the measurement in physics, acting as context that induces a change of the cognitive state from superposition state to collapsed state. The collapsed state is more likely to consist of a conjunction of concepts for associative than analytic thought because more stimulus or concept properties take part in the collapse. We provide two contextual measures of conceptual distance—one using collapse probabilities and the other weighted properties—and show how they can be applied to conjunctions using the pet fish problem.
Quantum mechanics: structures, axioms and paradoxes
 in Quantum Mechanics and the Nature of Reality
, 1999
"... We present an analysis of quantum mechanics and its problems and paradoxes taking into account the results that have been obtained during the last two decades by investigations in the field of ‘quantum structures research’. We concentrate mostly on the results of our group FUND at Brussels Free Univ ..."
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Cited by 14 (9 self)
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We present an analysis of quantum mechanics and its problems and paradoxes taking into account the results that have been obtained during the last two decades by investigations in the field of ‘quantum structures research’. We concentrate mostly on the results of our group FUND at Brussels Free University. By means of a spin 1 2 model where the quantum probability is generated by the presence of fluctuations on the interactions between measuring apparatus and physical system, we show that the quantum structure can find its origin in the presence of these fluctuations. This appraoch, that we have called the ‘hidden measurement approach’, makes it possible to construct systems that are in between quantum and classical. We show that two of the traditional axioms of quantum axiomatics are not satisfied for these ‘in between quantum and classical’ situations, and how this structural shortcoming of standard quantum mechanics is at the origin of most of the quantum paradoxes. We show that in this approach the EPR paradox identifies a genuine incompleteness of standard quantum mechanics, however not an incompleteness that means the lacking of hidden variables, but an incompleteness pointing at the impossibility for standard quantum mechanics to describe separated quantum systems. We indicate in which way, by redefining the meaning of density states, standard quantum mechanics can be completed. We put forward in which way the axiomatic approach to quantum mechanics can be compared to the traditional axiomatic approach to relativity theory, and how this might lead to studying another road to unification of both theories.
Being and change: foundations of a realistic operational formalism
 in Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics
, 2002
"... The aim of this article is to represent the general description of an entity by means of its states, contexts and properties. The entity that we want to describe does not necessarily have to be a physical entity, but can also be an entity of a more abstract nature, for example a concept, or a cultur ..."
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Cited by 14 (12 self)
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The aim of this article is to represent the general description of an entity by means of its states, contexts and properties. The entity that we want to describe does not necessarily have to be a physical entity, but can also be an entity of a more abstract nature, for example a concept, or a cultural artifact, or the mind of a person, etc..., which means that we aim at very general description. The effect that a context has on the state of the entity plays a fundamental role, which means that our approach is intrinsically contextual. The approach is inspired by the mathematical formalisms that have been developed in axiomatic quantum mechanics, where a specific type of quantum contextuality is modelled. However, because in general states also influence context – which is not the case in quantum mechanics – we need a more general setting than the one used there. Our focus on context as a fundamental concept makes it possible to unify ‘dynamical change ’ and ‘change under influence of measurement’, which makes our approach also more general and more powerful than the traditional quantum axiomatic approaches. For this reason an experiment (or measurement) is introduced as a specific kind of context. Mathematically we introduce a state context property system as the structure to describe an entity by means of its states, contexts and properties. We also strive from the start to a categorical setting and derive the morphisms between
A theory of concepts and their combinations I: the structure of the sets of contexts and properties
, 2005
"... ..."
in press), A case for applying an abstracted quantum formalism to cognition
 in Mind in Interaction
"... This chapter outlines some of the highlights of efforts undertaken by our group to describe the role of contextuality in the conceptualization of conscious experience using generalized formalisms from quantum mechanics. Conscious experience is filtered not just through innate categories to give rise ..."
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Cited by 10 (4 self)
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This chapter outlines some of the highlights of efforts undertaken by our group to describe the role of contextuality in the conceptualization of conscious experience using generalized formalisms from quantum mechanics. Conscious experience is filtered not just through innate categories to give rise to stimulusresponse reflexes, but also learned categories, including concepts such as ‘container’, ‘democracy’, ‘truth’ and ‘falsehood’. The meanings of these concepts are not rigid or static but shift fluidly depending on context, increasing dramatically our potential to both inform and be informed by the world. As Edelman and Tonini (2000, p. 101) put it: “Every act of perception is, to some extent, an act of creation, and every act of memory is, to some degree, an act of imagination.” Elements of conscious experience, such as perceived stimuli, retrieved memories, and concepts are considered entities of the personal cognitive sphere, referred to collectively as ‘conceptual entities’. These can be modelled by considering them as configurations of properties, which are consistently testable through personal and interpersonal cognitive processes such as perception, reflection, and social interactions. Specifically, we are concerned with the interface between the conceptual entity and its extraneous surroundings: the context. The interrelation and concatenation of concepts consciously experienced as a stream of thought is particularly affected by context, being influenced not only by the everfluctuating
Towards a General Operational and Realistic Framework for Quantum Mechanics and Relativity Theory
, 2004
"... We propose a general operational and realistic framework that aims at a generalization of quantum mechanics and relativity theory, such that both appear as special cases of this new theory. Our framework is operational, in the sense that all aspects are introduced with specific reference to events ..."
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Cited by 6 (1 self)
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We propose a general operational and realistic framework that aims at a generalization of quantum mechanics and relativity theory, such that both appear as special cases of this new theory. Our framework is operational, in the sense that all aspects are introduced with specific reference to events to be experienced, and realistic, in the sense that the hypothesis of an independent existing reality is taken seriously. To come to this framework we present a detailed study of standard quantum mechanics within the axiomatic approach to quantum mechanics, more specifically the GenevaBrussels approach, identifying two of the traditional 6 axioms as `failing axioms'. We prove that these two failing axioms are at the origin of the impossibility for standard quantum mechanics to describe a continuous change from quantum to classical and hence its inability to describe macroscopic physical reality. Moreover we show that the same two axioms are also at the origin of the impossibility for standard quantum mechanics to deliver a model for the compound entity of two `separated' quantum entities. We put forward that it is necessary to replace these two axioms in order to proceed to the more general theory.
Contextual logic for quantum systems
 Journal of Mathematical Physics
, 2005
"... In this work we build a quantum logic that allows us to refer to physical magnitudes pertaining to different contexts from a fixed one without the contradictions with quantum mechanics expressed in nogo theorems. This logic arises from considering a sheaf over a topological space associated to the ..."
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Cited by 6 (4 self)
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In this work we build a quantum logic that allows us to refer to physical magnitudes pertaining to different contexts from a fixed one without the contradictions with quantum mechanics expressed in nogo theorems. This logic arises from considering a sheaf over a topological space associated to the Boolean sublattices of the ortholattice of closed subspaces of the Hilbert space of the physical system. Differently to standard quantum logics, the contextual logic maintains a distributive lattice structure and a good definition of implication as a residue of the conjunction.
Reality and Probability: Introducing a New Type of Probability Calculus
 in Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Computation and Axiomatics
, 2002
"... We consider a conception of reality that is the following: An object is `real' if we know that if we would try to test whether this object is present, this test would give us the answer `yes' with certainty. The knowledge about this certainty we gather from our overall experience with the world. ..."
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Cited by 6 (4 self)
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We consider a conception of reality that is the following: An object is `real' if we know that if we would try to test whether this object is present, this test would give us the answer `yes' with certainty. The knowledge about this certainty we gather from our overall experience with the world. If we consider a conception of reality where probability plays a fundamental role, which we should do if we want to incorporate the microworld into our reality, it can be shown that standard probability theory is not well suited to substitute `certainty' by means of `probability equal to 1'. We analyze the di#erent problems that arise when one tries to push standard probability to deliver a conception of reality as the one we advocate. The analysis of these problems lead us to propose a new type of probability theory that is a generalization of standard probability theory. This new type of probability theory is a function to the set of all subsets of the interval [0, 1] instead of to the interval [0, 1] itself, and hence its evaluation happens by means of a subset instead of a number. This subset corresponds to the di#erent limits of sequences of relative frequency that can arise when an intrinsic lack of knowledge about the context and how it influences the state of the physical entity under study in the process of experimentation is taken into account. The new probability theory makes it possible to define probability on the whole set of experiments within the GenevaBrussels approach to quantum mechanics, which was not possible with # Published as: D. Aerts, "Reality and probability: introducing a new type of probability calculus", in Probing the Structure of Quantum Mechanics: Nonlocality, Computation and Axiomatics, eds. D. Aerts, M. Czachor and T. Durt...
State property systems and orthogonality
 International Journal of Theoretical Physics. See
"... The structure of a state property system was introduced to formalize in a complete way the operational content of the GenevaBrussels approach to the foundations of quantum mechanics [7, 8, 9], and the category of state property systems was proven to be equivalence to the category of closure spaces ..."
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Cited by 5 (4 self)
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The structure of a state property system was introduced to formalize in a complete way the operational content of the GenevaBrussels approach to the foundations of quantum mechanics [7, 8, 9], and the category of state property systems was proven to be equivalence to the category of closure spaces [9, 10]. The first axioms of standard quantum axiomatics (state determination and atomisticity) have been shown to be equivalent to the T0 and T1 axioms of closure spaces [11, 12, 13], 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 [14, 15, 16]. 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 a 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. 1