## 1978

- Aerts, D. and Daubechies, I. (1978). About the structure-preserving maps of a quantum mechanical propositional system.
Helvetica Physica Acta,51,pp. 637-660. doi: 10.5169/seals-114964. download pdf.

Abstract:We study c-morphisms from one Hilbert space lattice (with dimension at least three) to another one; we show that for a c-morphism conserving modular pairs, there exists a linear structure underlying such a morphism, which enables us to construct explicitly a family of linear maps generating this morphism. As a special case we prove that a unitary c-morphism which preserves the atoms (i.e. maps one- dimensional subspaces into one-dimensional subspaces) is necessarily an isomorphism. Counterexamples are given when the Hilbert space has dimension 2.- Aerts, D. and Daubechies, I. (1978). Physical justification for using the tensor product to describe two quantum systems as one joint system.
Helvetica Physica Acta,51,pp. 661-675. doi:10.5169/seals-114965. download pdf.

Abstract:We require the following three conditions to hold on two systems being described as a joint system: (1) the structure of the two systems is preserved: (2) a measurement on one of the systems does not disturb the other one; (3) maximal information obtained on both systems separately gives maximal information on the joint system. With these conditions we show, within the framework of the propositional system formalism, that if the systems are classical the joint system is described by the cartesian product of the corresponding phase spaces, and if the systems are quantal the joint system is described by the tensor product of the corresponding Hilbert spaces.## 1979

- Aerts, D. and Daubechies, I. (1979). A connection between propositional systems in Hilbert spaces and Von Neumann algebra's.
Helvetica Physica Acta,52,pp. 184-199. doi: 10.5169/seals-115025. download pdf.

Abstract:A theorem of Bade proves that for a complete Boolean sublattice of a Hilbert space lattice the sublattice is equal to the set of projections in its Von Neumann algebra. We prove that this theorem does not hold for the physically interesting class of non-Boolean propositional systems embedded in a Hilbert space lattice; we derive however a necessary and sufficient condition under which the theorem does hold. This condition is automatically satisfied if the propositional system is Boolean.- Aerts, D. and Daubechies, I. (1979). Characterization of subsystems in physics.
Letters in Mathematical Physics,3,pp. 11-17. doi: 10.1007/BF00959533, download pdf.

Abstract:Working within the framework of the propositional system formalism, we use a previous study of the description of two independent physical systems as one big physical system to derive a characterization of a (non-interacting) physical subsystem. We discuss the classical case and the quantum case.- Aerts, D. and Daubechies, I. (1979). Mathematical condition for a sub-lattice of a propositional system to represent a physical subsystem with a physical interpretation.
Letters in Mathematical Physics,3,pp. 19-27. download pdf

Abstract:We display three equivalent conditions for a sublattice, isomorphic to a P(H), of the propositional system P(H) of a quantum system to be the representation of a physical subsystem. These conditions are valid for dim H > 2. We prove that one of them is still necessary and sufficient if dim H < 3. A physical interpretation of this condition is given.- Aerts, D. and Piron, C. (1979). The role of modular pairs in the category of complete orthomodular lattices.
Letters in Mathematical Physics,3,pp. 1-10. doi: 10.1007/BF00959532. download pdf.

Abstract:We study the modular pairs of a complete orthomodular lattice i.e. a CROC. We propose the concept of m-morphism as a mapping which preserves the lattice structure, the orthogonality and the property to be a modular pair. We give a characterization of the m-morphisms in the case of the complex Hilbert space to justify this concept.## 1980

- Aerts, D. (1980). Subsystems in physics described by bi-linear maps between the corresponding vector spaces.
Journal of Mathematical Physics,21,pp. 778-788. download pdf.

Abstract:We show that whenever a physical system is composed of two subsystems, there exists a (sigma1, sigma2)-linear map between their generalized Hilbert spaces which describes this composition. As a consequence, subsystems of a physical system described by a generalized Hilbert space over a division ring K are always described by a generalized Hilbert space over a subdivision ring of K.- Aerts, D. (1980). Why is it impossible in quantum mechanics to describe two or more separated entities.
Bulletin de l'Academie royale de Belgique, Classes des Sciences,66,pp. 705-714. download pdf.

Abstract:We prove that a theory that is different from a classical theory and has the property that orthogonal states of a physical system are compatible states (a property called orthomodularity in axiomatics) cannot describe two separated entities. Quantum mechanics has this property and as a consequence it is impossible to describe two separated entities in quantum mechanics. This is essentially due to the fact that quantum mechanics is too specific a theory. We indicate the direction for a generalization where the problem could be solved sharingdownavond ## 1981

- Aerts, D. (1981). Description of compound physical systems and logical interaction of physical systems. In E. G. Beltrametti and B. C. van Fraassen (Eds.),
Current Issues on Quantum Logic(pp. 381-405), Ettore Majorana, International Science Series, Physical Sciences, vol.8. Plenum Publishing Corporation. download pdf.- Aerts, D. (1981).
The One and the Many: Towards a Unification of the Quantum and Classical Description of One and Many Physical Entities.Doctoral dissertation, Brussels Free University.## 1982

- Aerts, D. (1982). Example of a macroscopical situation that violates Bell inequalities.
Lettere al Nuovo Cimento,34,pp. 107-111. doi: 10.1007/BF02817207. download pdf.

Abstract:We give an example of a classical macroscopical situation that violates Bell inequalities. The example shows a certain analogy with the system composed of two spin-1/2 particles in the singlet spin state.- Aerts, D. (1982). Description of many physical entities without the paradoxes encountered in quantum mechanics.
Foundations of Physics,12,pp. 1131-1170. doi: 10.1007/BF00729621. download pdf.

Abstract:We show that it is impossible in quantum mechanics to describe two separated systems. This is due to the mathematical structure of quantum mechanics. It is possible to give a description of two separated systems in a theory which is a generalization of quantum mechanics and of classical mechanics, in the sense that this theory contains both theories as special cases. We identify the axioms of quantum mechanics that make it impossible to describe separated systems. One of these axioms is equivalent to the superposition principle. We show how these findings throw a different light on the paradox of Einstein, Podolsky, and Rosen.## 1983

- Aerts, D. (1983). Classical-theories and non-classical theories as a special case of a more general theory.
Journal of Mathematical Physics,24,pp. 2441-2453. download pdf.- Aerts, D. (1983). The description of one and many physical systems. In C. Gruber (Ed.),
Foundations of Quantum Mechanics(pp. 63-148). Lausanne: AVCP. download pdf.- Aerts, D. and Daubechies, I. (1983). Simple proof that the structure-preserving maps between quantum-mechanical propositional systems conserve the angles.
Helvetica Physica Acta,56,pp. 1187-1190. download pdf.

Abstract:We show that for any c-morphism phi from the lattice P(H) of closed subspaces of a complex Hilbert space H (dim H > 2) to another such P(H'), a conservation property for the angles holds: For every x, y in H, x different from zero and y different from zero, we have that the angle between x and y equals the angle between phi(x) and phi(y). This implies that every c-morphism is a m-morphism. Our proof uses Gleason's theorem; this result was suggested to us by the work of R. Wright.## 1984

- Aerts, D. (1984). Construction of a structure which enables to describe the joint system of a classical and a quantum system.
Reports in Mathematical Physics,20,pp. 421-428. doi: 10.1016/0034-4877(84)90077-6. download pdf.

Abstract:We consider the joint system of a classical system and a quantum system. We require the same conditions on the two systems as those that give the description of two classical systems by means of the cartesian product of the phase spaces and the description of two quantum systems by means of the tensor product of the Hilbert spaces. We set up a structure that enables us to describe the joint system of a classical system and a quantum system.- Aerts, D. (1984). Construction of the tensor product for lattices of properties of physical entities.
Journal of Mathematical Physics,25,pp. 1434-1441. download pdf.

Abstract:We construct the tensor product for the property lattices of two entities. We give a physical interpretation for this tensor product. We show that the tensor product is never an orthocomplemented lattice if both entities are nontrivial. It also never satisfies the covering law. This is the reason why this tensor product does not exist in quantum mechanics and does not exist in quantum logic. We analyze the relation of the tensor product with the usual description of two entities in quantum mechanics. At the end, we give a mathematical way of constructing this tensor product.- Aerts, D. (1984). The missing elements of reality in the description of quantum mechanics of the EPR paradox situation.
Helvetica Physica Acta,57,pp. 421-428. download pdf.

Abstract:We show that quantum mechanics is not a complete theory. We do not as in the case of Einstein Podolsky and Rosen derive this incompleteness by a logical reasoning ex absurdum, but indicate explicitly which are the missing elements of reality in the description by quantum mechanics of separated physical systems.- Aerts, D. (1984). How do we have to change quantum mechanics in order to describe separated systems. In S. Diner, D. Fargue, G. Lochak and F. Selerri (Eds.),
The Wave-Particle Dualism(pp. 419-431). Dordrecht: Springer. download pdf.

Abstract:Since we were able to show recently that quantum mechanics can not describe separated physical systems we analyse again the reasoning of Einstein-Podolsky-Rosen, and find that the most straight forward conclusion of this paradox is not correct. We indicate the missing elements of reality in the quantum mechanical description of separated physical systems. We show that Bell inequalities are satisfied iff the two physical systems are separated, whether the systems are quantum systems or classical systems is of no matter. We give an example of a classical macroscopical situation where Bell inequalities are violated.- Aerts, D. (1984). The missing elements of reality in the description of quantum mechanics of the EPR paradox situation.
Annales de la Fondation Louis de Broglie,9,pp. 163-175.## 1985

- Aerts, D. (1985). The physical origin of the Einstein Podolsky Rosen paradox. In G. Tarozzi and A. van der Merwe (Eds.),
Open Questions in Quantum Physics: Invited Papers on the Foundations of Microphysics(pp. 33-50). Dordrecht: Springer.- Aerts, D. (1985). The physical origin of the EPR paradox and how to violate Bell inequalities by macroscopical systems. In P. Lathi and P. Mittelstaedt (Eds.),
Symposium on the Foundations of Modern Physics: 50 years of the Einstein-Podolsky-Rosen Gedankenexperiment(pp. 305-320). Singapore: World Scientific.- Aerts, D. (1985). A possible explanation for the probabilities of quantum mechanics and a macroscopical situation that violates Bell inequalities. In P. Mittelstaedt and E. W. Stachow (Eds.),
Recent Developments in Quantum Logic, Grundlagen der Exacten Naturwissenschaften, vol.6, Wissenschaftverlag(pp. 235-251). Mannheim: Bibliographisches Institut.## 1986

- Aerts, D. (1986). A possible explanation for the probabilities of quantum mechanics.
Journal of Mathematical Physics,27,pp. 202-210. downlaod pdf.

Abstract:It is shown that a lack of knowledge about the measurements of a physical system gives rise to a nonclassical probability calculus for this physical system. It is also shown that the nonclassical probability calculus of quantum mechanics can be interpreted as being the result of a lack of knowledge about the measurements. Examples are given of macroscopic real systems that have a nonclassical probability calculus. A macroscopic real system that has a quantum probability calculus is also given, and more specifically a model for the spin of a spin-1/2 particle is contructed. These results are analysed in the light of the old hidden variable problem.- Duch, W. and Aerts, D. (1986). Microphysical reality.
Physics Today,39,pp. 13-14. download pdf.## 1987

- Aerts, D. (1987). The origin of the non-classical character of the quantum probability model. In A. Blanquiere, S. Diner and G. Lochak (Eds.),
Information, Complexity, and Control in Quantum Physics(pp. 77-100). Wien-New York: Springer-Verlag.## 1988

- Aerts, D. (1988). The description of separated systems and quantum mechanics and a possible explanation for the probabilities of quantum mechanics. In A. van der Merwe, G. Tarozzi and F. Selleri (Eds.),
Micro-physical Reality and Quantum Formalism: Volumes 1 and 2(pp. 97-115). Dordrecht: Springer. download pdf.## 1990

- Aerts, D. (1990). An attempt to imagine parts of the reality of the micro-world. In J. Mizerski, A. Posiewnik, J. Pykacz and M. Zukowski (Eds.),
Problems in Quantum Physics(pp. 3-25). Singapore: World Scientific. download pdf.

Abstract:Quantum mechanics is the theory used to 'describe' the processes that take place in the micro-world. From the start quantum mechanics has been a 'strange' theory, in the sense that it seemed to contradict in various ways the image of a micro-world consisting of 'objects' moving around in a three dimensional space, and interacting with each other in this three dimensional space. So from the advent of the theory a lot of disagreement existed as to the 'physical meaning' of this quantum theory, and a lot of discussions of a philosophical nature have taken place among the founding fathers. Only however during the last years experiments have been performed that, independently of the strangeness of the quantum theory, confront us directly with the strangeness of the reality of the micro-world. We have in mind the experiments on the EPR problem. In our opinion to be able to 'understand' the reality of this micro-world, it will be necessary to introduce new concepts, and become aware of old 'classical' prejudices. Certainly in not such a radical way as proposed by what is sometimes called the 'California interpretation' of quantum mechanics, but also in not such a vague way as is proposed by what is called the 'Copenhagen interpretation' of quantum mechanics. Since we nowadays have very 'specific' results, on very refined experiments, we should start 'imagining' how this 'micro-reality' is. The aim of this paper is to try something in this direction, and to propose what could be called a new discipline in theoretical physics. This discipline should investigate whether different kinds of realities (world-models) can correspond with the results of the experiments that we now have, and with the theoretical descriptions given by the quantum theory. And so although we agree that the quantum-world is a very strange one, our aim will be to show that it is not so strange as it looks at the first place. Just because 'a reality' can be much more complicated than one would imagine.## 1991

- Aerts, D. (1991). A mechanistic classical laboratory situation violating the Bell inequalities with 2sqrt(2), exactly 'in the same way' as its violations by the EPR experiments.
Helvetica Physica Acta,64,pp. 1-23. download pdf.

Abstract:We present a macroscopical mechanistic classical laboratory situation, and a classical macroscopical entity, and coincidence measurements on this entity, that lead to a violation of the Bell inequalities corresponding to these coincidence measurements. The violation that we obtain with these coincidence measurements is exactly the same as the violation of the Bell inequalities by the well known coincidence measurements of the quantum entity of two spin 1/2 particles in a singlet spin state. With this we mean that it gives rise to the same numerical values for the expectation values and the same numerical value 2sqrt(2) for the expression used in the Bell inequality. We analyze the origin of the violation, and can formulate the main difference between the violation of Bell inequalities by means of classical entities and the violation of Bell inequalities by means of quantum entities. The making clear of this difference can help us to understand better what the quantum-violation could mean for the nature of reality. We think that some classical concepts will have to be changed, and new concepts will have to be introduced, to be able to understand the reality of the quantum world.- Aerts, D. (1991). A macroscopical classical laboratory situation with only macroscopical classical entities giving rise to a quantum mechanical probability model. In L. Accardi (Ed.),
Quantum Probability and Related Topics, Volume VI(pp. 75-85). Singapore: World Scientific. download pdf.

Abstract:We propose a macroscopical classical physical entity, giving a detailed description of the preparation apparatuses and the preparations (states) of this entity. We consider experiments that can be performed on the entity, and give a detailed description of the measurement apparatuses, and the measurements used in these experiments. We investigate the collection of probabilities for the outcomes of the measurements the entity being prepared in a given state. Therefore we use the ordinary meaning of probability as approximate relative frequency of repeated experiments, hence experiments consisting of equivalent measurements on equivalently prepared entities. We show that the collection of probabilities that results in this way for our macroscopical entity is the same as the collection of probabilities for the outcomes of the Stern- Gerlach spin measurements on a spin 1/2 quantum entity prepared in a given spin state. By analyzing in which way this purely classical physical situation gives rise to a quantum probability model, we propose an explanation for the non classical probability structure of the quantum probability model. We conclude by showing that this explanation is plausible from a physical point of view, and if accepted makes disappear a lot of the paradoxical nature of the quantum formalism, in the sense that the quantum probabilities do not have to be interpreted any more as 'ontological' or 'intrinsically' present in nature itself.- Aerts, D., Apostel, L., De Moor, B., Hellemans, S., Lesthaege, R., Maex, E., Van Belle, H., Van der Veken, J., Van Geen, R. and Van Landschoot, J. (1991).
Wereldbeelden, Van Fragmentering naar Integratie,. Kapellen: Pelckmans. download pdf.- Aerts, D. and Reignier, J. (1991). The spin of a quantum entity and problems of non-locality. In P. Lahti and P. Mittelstaedt (Eds.),
Symposium on the Foundations of Modern Physics 1990: Quantum Theory of Measurement and Related Philosophical Problems(pp. 9-19). Singapore: World Scientific. download pdf.

Abstract:We introduce a possible definition for the concept of non-locality in the quantum world, which seems to us a minimal operational definition, taking into account the results of actually performed experiments and reasonings about possible 'gedanken' experiments. The definition is the following: An entity is "non local" if it is possible to prepare it in a state such that it can be influenced from macroscopically separated regions of space by (macroscopically) local apparatus acting only in one (or several) of these separated regions at one time. We discuss two examples of spin superposition experiments which clearly show that quantum entities are non-local. In particular, we show that the familiar Stern- Gerlach experiment allows a nice illustration of this non-locality.- Aerts, D. and Reignier, J. (1991). On the problem of non-locality in quantum mechanics.
Helvetica Physica Acta,64,pp. 527-547. doi: 10.5169/seals-116311. download pdf.

Abstract:Many experimental results and theoretical reasonings on "gedanken" experiments suggest that quantum entities behave in a "non-local" way. In order to make the discussion more precise, we propose an operational definition of "non-locality". We discuss two examples of spin superposition experiments which illustrate the use of this definition. They show that quantum entities like a neutron or an atom are non-local. We make an analysis of the difficulty to imagine such non-local entities and to conciliate this non-locality with the locality of classical entities.## 1992

- Aerts, D. (1992). The construction of reality and its influence on the understanding of quantum structures.
International Journal of Theoretical Physics,31,pp. 1815-1837. doi: 10.1007/BF00678294. download pdf.

Abstract:We present a formalism to analyze 'the construction of reality'. We want to use this formalism to understand some aspects of the reality of the classical and the quantum world on a deeper level. One of the aims is to detect 'hidden' prejudices' that are 'unconsciously' used to interpret and criticize certain parts of reality, that do perhaps not obey these prejudices. It will follow that some of the difficulties that we have to understand the quantum reality are due to 'pre-scientific' prejudices about the way in which we think that reality has to be, prejudiced which we have completely forgotten about. We will try to put in evidence some of these 'pre-scientific' prejudices. To do this we will have to introduce new concepts. We will see however that the concepts needed to understand the aspect of the quantum reality that we want to investigate in this paper are not mysterious and ununderstandable. They exist and are used frequently in situations of everyday life. Therefore we will introduce these concepts by means of examples of everyday life, such that they are intuitively clear. The formalism is not yet completed, a lot of steps are still unfinished, and have to be investigated further. But the methodology is clearly stated. In this sense it can be a starting point for further research on the understanding of other parts of 'human' reality.- Aerts, D. and Van Bogaert, B. (1992). Mechanistic classical laboratory situation with a quantum logic structure.
International Journal of Theoretical Physics,31,pp. 1839-1848. doi: 10.1007/BF00678295. download pdf.

Abstract:The difference between quantum entities and classical entities can be noticed in many different ways. Quantum logic has been profoundly interested in analyzing this difference and trying to understand it. Our aim is to represent a macroscopical classical mechanistic laboratory situation, and to show that this situation entails a non-classical logical structure. The example has been presented already some time ago by one of us showing it to have a quantum probability model, and analyzing the effect of this on a possible understanding of the origin of quantum probabilities. In this paper we will make a similar attempt but now concentrating on the logical aspects of the example.## 1993

- Aerts, D. (1993). Quantum structures due to fluctuations of the measurement situations.
International Journal of Theoretical Physics,32,pp. 2207-2220. doi: 10.1007/BF00672993. download pdf.

Abstract:We want to analyze in this paper the meaning of the non-classical aspects of quantum structures. We proceed by introducing a simple mechanistic macroscopic experimental situation that gives rise to quantum-like structures. We use this situation as a guiding example for our attempts to explain the origin of the non-classical aspects of quantum structures. We see that the quantum probabilities can be introduced as a consequence of the presence of ˇuctuations on the experimental apparatuses, and show that the full quantum structure can be obtained in this way. We define the classical limit as the physical situation that arises when the fluctuations on the experimental apparatuses disappear. In the limit case we come to a classical structure but in between we find structures that are neither quantum nor classical. In this sense, our approach not only gives an explanation for the non-classical structure of quantum theory, but also makes it possible to define, and study the structure describing the intermediate new situations. By investigating in which way the non-local quantum behavior disappears during the limiting process we can explain the 'apparent' locality of the classical macroscopical world. We come to the conclusion that quantum structures are the ordinary structures of reality, and that our difculties of becoming aware of this fact are due to pre-scientific prejudices, of which some of them we shall point out.- Aerts, D. 1993,
De Muze van het Leven, Quantummechanica en de Aard van de Werkelijkheid,Pelckmans, Kapellen, Agora Kok, Kampen.- Aerts, D., Durt, T., Grib, A., Van Bogaert, B. and Zapatrin, A. (1993). Quantum structures in macroscopical reality.
International Journal of Theoretical Physics,32, pp. 489-498. doi: 10.1007/BF00673355. download pdf.

Abstract:We want to show in this paper that it is possible to construct macroscopical entities that entail a quantum logical structure. We do this by means of the introduction of a simple macroscopical entity and study its structure in terms of lattices and graphs, and show that the lattice is non-Boolean.- Aerts, D., Durt, T. and Van Bogaert, B. (1993). A physical example of quantum fuzzy sets and the classical limit.
Tatra Mountains Mathematical Publications,1,pp. 5-15. download pdf.

Abstract:We present an explicit physical example of an experimental situation on a physical entity that gives rise to a fuzzy set. The fuzziness in the example is due to fluctuations of the experimental apparatus, and not to an indeterminacy about the states of the physical entity, and is described by a varying parameter epsilon. For zero value of the parameter (no fluctuations), the example reduces to a classical mechanics situation, and the corresponding fuzzy set is a quasi-crisp set. For maximal value (maximal fluctuations), the example gives rise to a quantum fuzzy set, more precisely a spin- 1/2 model. In between, we have a continuum of fuzzy situations, neither classical, nor quantum. We believe that the example can make us understand the nature of the quantum mechanical fuzziness and probability, and how these are related to the classical situation.- Aerts, D., Durt, T. and Van Bogaert, B. (1993). Quantum probability, the classical limit and nonlocality. In K. V. Laurikainen and C. Montonen (Eds.),
Symposium on the Foundations of Modern Physics 1992: The Copenhagen Interpretation and Wolfgang Pauli(pp. 35-56). Singapore: World Scientific. download pdf.

Abstract:We investigate quantum mechanics using an approach where the quantum probabilities arise as a consequence of the presence of fluc tuations on the experimental apparatuses. We show that the full quantum structure can be obtained in this way and define the classical limit as the physical situation that arises when the fluctuations on the experimental apparatuses disappear. In the limit case we come to a classical structure but in between we find structures that are neither quantum nor classical. In this sense, our approach not only gives an explanation for the non-classical structure of quantum theory, but also makes it possible to define, and study the structure describing the intermediate new situations. By investigating in which way the non-local quantum behaviour disappears during the limiting process we can explain the 'apparent' locality of the classical macroscopical world.## 1994

- Aerts, D. (1994). Quantum structures, separated physical entities and probability.
Foundations of Physics,24,pp. 1227-1259. doi: 10.1007/BF02148566. download pdf.

Abstract:We prove that if the physical entity S consisting of two separated physical entitie S1 and S2 satisfies the axioms of orthodox quantum mechanics, then at least one of the two subentities is a classical physical entity. This theorem implies that separated quantum entities cannot be described by quantum mechanics. We formulate this theorem in an approach where physical entities are described by the set of their states, and the set of their relevant experiments. We also show that the collection of eigenstate sets forms a closure structure on the set of states, that we call the eigen-closure structure. We derive another closure structure on the set of states by means of the orthogonality relation, and call it the ortho-closure structure, and show that the main axioms of quantum mechanics can be introduced in a very general way by means of these two closure structures. We prove that for a general physical entity, and hence also for a quantum entity, the probabilities can always be explained as being due to the presence of a lack of knowledge about the interaction between the experimental apparatus and the entity.- Aerts, D. (1994). Continuing a quest for the understanding of fundamental physical theories and the pursuit of their elaboration.
Foundations of Physics,24,pp. 1107-1111. doi: 10.1007/BF0205785. download pdf.- Aerts, D. (1994). Quantummechanica. In L. Apostel and F. Verbeure (Eds.),
Verwijdering of Ontmoeting?(pp. 123-142). Kapellen: Pelckmans. download pdf.- Aerts, D. (1994). Het spel van de biomousa: een beeld van ontdekking en creatie. In D. Aerts, L. Apostel, B. De Moor, S. Hellemans, E. Maex, H. Van Belle and J. Van der Veken (Eds.),
Cirkelen om de Wereld, Concrete Invulling van het Wereldbeeldenproject(pp. 19-56). Kapellen: Pelckmans.- Aerts, D., Apostel, L., De Moor, B., Hellemans, S., Maex, E., Van Belle, H. and Van der Veken, J. (1994).
Worldviews, from Fragmentation towards Integration.Brussels: VUBPress. download pdf.- Aerts, D., Apostel, L., De Moor, B., Hellemans, S., Maex, E., Van Belle, H. and Van der Veken, J. (1994).
Cirkelen om de Wereld, Concrete Invulling van het Wereldbeeldenproject.Kapellen: Pelckmans.- Aerts, D. and Durt, T. (1994). Quantum, classical and intermediate: a measurement model. In K. V. Laurikainen, C. Montonen and K. Sunnaborg (Eds.),
Symposium on the Foundations of Modern Physics.Gives Sur Yvettes, France: Editions Frontieres. download pdf.

Abstract:We present a measurement model where the origin of the quantum probabilities lies in the presence of fluctuations between the measurement apparatus and the physical system. First we make a reasoning where we show that the measurement process cannot be described by the unitary Schrodinger evolution only. Afterwards we present our model of measurement and show the necessity of developping a more general structure than orthodox Hilbert space quantum mechanics to resolve the measurement problem.- Aerts, D. and Durt, T. (1994). Quantum, classical and intermediate, an illustrative example.
Foundations of Physics,24,pp. 1353-1369. doi: 10.1007/BF02283037. download pdf.

Abstract:We present a model that allows to build structures that evolve continuously from classical to quantum, and we study the intermediate situations, giving rise to structures that are neither classical nor quantum. We construct the closure structure corresponding to the collection of eigenstate sets of these intermediate situations, and demonstrate how the superposition principle disappears during the transition from quantum to classical. We investigate the validity of the axioms of quantum mechanics for the intermediate situations.## 1995

- Aerts, D. (1995). Quantum structures: an attempt to explain their appearance in nature.
International Journal of Theoretical Physics,34,pp. 1165-1186. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0111071. doi: 10.1007/BF00676227. dowload pdf.

Abstract:We explain the quantum structure as due to the presence of two effects, (a) a real change of state of the entity under influence of the measurement and, (b) a lack of knowledge about a deeper deterministic reality of the measurement process. We present a quantum machine, where we can illustrate in a simple way how the quantum structure arises as a consequence of the two mentioned effects. We introduce a parameter epsilon that measures the size of the lack of knowledge on the measurement process, and by varying this parameter, we describe a continuous evolution from a quantum structure (maximal lack of knowledge) to a classical structure (zero lack of knowledge). We show that for intermediate values of epsilon we find a new type of structure, that is neither quantum nor classical. We apply the model that we have introduced to situations of lack of knowledge about the measurement process appearing in other regions of reality. More specifically we investigate the quantum-like structures that appear in the situation of psychological decision processes, where the subject is influenced during the testing, and forms some of his opinions during the testing process. Our conclusion is that in the light of this explanation, the quantum probabilities are epistemic and not ontological, which means that quantum mechanics is compatible with a determinism of the whole.- Aerts, D. (1995). The game of the biomousa: A view of discovery and creation. In D. Aerts, L. Apostel, B. De Moor, S. Hellemans, E. Maex, H. Van Belle and J. Van der Veken (Eds.),
Perspectives on the World, an Interdisciplinary Reflection(pp. 17-48). Brussels: VUBPress. download pdf.- Aerts, D., Apostel, L., De Moor, B., Hellemans, S., Maex, E., Van Belle, H. and Van der Veken, J. (1995).
Perspectives on the World, an Interdisciplinary Reflection.Brussels: VUBPress.- Aerts, D. and Aerts, S. (1995). Applications of quantum statistics in psychological studies of decision processes.
Foundations of Science,1,pp. 85-97. doi: 10.1007/BF00208726. download pdf.

Abstract:We present a new approach to the old problem of how to incorporate the role of the observer in statistics. We show classical probability theory to be inadequate for this task and take refuge in the epsilon-model, which is the only model known to us capable of handling situations between quantum and classical statistics. An example is worked out and some problems are discussed as to the new viewpoint that emanates from our approach.## 1996

- Aerts, D. (1996). Framework for possible unification of quantum and relativity theories.
International Journal of Theoretical Physics,35,pp. 2399-2416. doi: 10.1007/BF02302456. download pdf.

Abstract:We put forward a framework, inspired by recent axiomatic and operational approaches to generalized quantum theories, wherein we investigate the possibility of unifying quantum theories and relativity theories. The framework concentrates on a detailed analysis of a general construction of reality, that can be used in both, quantum and relativity theories. By means of this construction of reality we clarify some well known conceptual problems that stand in the way for a conceptual unifcation of quantum and relativity theories on a more profound physical level than the purely mathematical algebraic level on which now unification attempts are investigated. More specifically we concentrate on the problem of 'what is physical reality' in quantum and relativity theories.- Aerts, D. (1996). Relativity theory: what is reality?.
Foundations of Physics,26,pp. 1627-1644. doi: 10.1007/BF02282126. download pdf.

Abstract:In classical Newtonian physics there was a clear understanding of 'what reality is'. Indeed in this classical view, reality at a certain time is the collection of all what is actual at this time, and this is contained in 'the present'. Often it is stated that three dimensional space and one dimensional time have been substituted by four dimensional space-time in relativity theory, and as a consequence the classical concept of reality, as that what is 'present', cannot be retained. Is reality then the four dimensional manifold of relativity theory? And if so, what is then the meaning of 'change in time' ? This problematic confronts a geometric view (as the Einsteinian interpretation of relativity theory) with a process view (where reality changes constantly in time). In this paper we investigate this problem, taking into account our insight in the nature of reality as it came by analyzing the problems of quantum mechanics. We show that with an Einsteinian interpretation of relativity theory, reality is indeed four dimensional, but there is no contradiction with the process view, where this reality changes in time.- Aerts, D. (1996). Quantum Physics at the Einstein meets Magritte conference.
International Journal of Theoretical Physics,35,pp. 2213-2214. doi: 1007/BF02302441. download pdf.- Aerts, D. and D'Hooghe, B. (1996). Operator structure of a nonquantum and nonclassical system.
International Journal of Theoretical Physics,35,pp. 2285-2298. doi: 10.1007/BF02302447. download pdf.

Abstract:There exists a connection between the vectors of the Poincare-sphere and the elements of the two dimensional complex Hilbert space. This latter space is used to describe spin 1/2 measurements. We will now use this connection to study the intermediate cases of a more general spin 1/2 measurement model which has no representation in a Hilbert space. We will construct the set of operators of this general model, and investigate under which circumstances it is possible to define linear operators. Because no Hilbert space structure is possible for these intermediate cases, it can be expected that no linear operators are possible and it is shown that under very plausible assumptions this is indeed the case.## 1997

- Aerts, D. and Aerts, S. (1997). Application of quantum statistics in psychological studies of decision processes. In B. C. van Fraassen (Ed.),
Topics in the Foundation of Statistics.Dordrecht: Springer. download pdf.

Abstract:We present a new approach to the old problem of how to incorporate the role of the observer in statistics. We show classical probability theory to be inadequate for this task and take refuge in the epsilon-model, which is the only model known to us capable of handling situations between quantum and classical statistics. An example is worked out and some problems are discussed as to the new viewpoint that emanates from our approach.- Aerts, D. and Aerts, S. (1997). The hidden measurement formalism: quantum mechanics as a consequence of fluctuations on the measurement. In M. Ferrero and A. van der Merwe (Eds.),
New Developments on Fundamental Problems in Quantum Physics(pp. 1-6). Dordrecht: Springer. download pdf.

Abstract:We expose a formalism, that we have called the `hidden measurement formalism', where the quantum structure is due to the presence of `fluctuations' on the interaction between the system and the measurement apparatus. In this formalism the quantum probabilities are not ontological but arrise as a consequence of `lack of knowledge' about this interaction. We study the quantum classical limit and the EPR problem in the light of this explanation.- Aerts, D., Aerts, S., Coecke, B., D'Hooghe, B., Durt, T. and Valckenborgh, F. (1997). A model with varying fluctuations in the measurement context. In M. Ferrero and A. van der Merwe (Eds.),
New Developments on Fundamental Problems in Quantum Physics(pp. 7-9). Dordrecht: Springer. download pdf.

Abstract:In the `hidden measurement formalism', we try to explain the emergence of the quantum probability from the presence of fluctuations in the measurement context. We use a model that was constructed by Aerts et al. as a metaphorical model to extend these ideas.- Aerts, D., Coecke, B., D'Hooghe, B. and Valckenborgh, F. (1997). A mechanistic macroscopical physical entity with a three dimensional Hilbert space quantum description.
Helvetica Physica Acta,70,pp. 793-802. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0111074. doi: 10.5169/seals-117052. download pdf.

Abstract:It is sometimes stated that Gleason's theorem prevents the construction of hidden-variable models for quantum entities described in a more than two-dimensional Hilbert space. In this paper however we explicitly construct a classical (macroscopical) system that can be represented in a three-dimensional real Hilbert space, the probability structure appearing as the result of a lack of knowledge about the measurement context. We briefly discuss Gleason's theorem from this point of view.- Aerts, D., Coecke, B., Durt, T. and Valckenborgh, F. (1997). Quantum, classical and intermediate I: a model on the poincare sphere.
Tatra Mountains Mathematical Publications,10,p. 225. download pdf.

Abstract:Following an approach, that we have called the hidden-measurement approach, where the probability structure of quantum mechanics is explained as being due to the presence of fluctuations on the measurement situations, we introduce explicitly a variation of these fluctuations, with the aim of defining a procedure for the classical limit. We study a concrete physical entity and show that for maximal fluctuations the entity is described by a quantum model, isomorphic to the model of the spin of a spin 1/2 quantum entity. For zero fluctuations we find a classical structure, and for intermediate fluctuations we find a structure that is neither quantum nor classical, to which we shall refer as the 'intermediate' situation.- Aerts, D., Coecke, B., Durt, T. and Valckenborgh, F. (1997). Quantum, classical and intermediate II: the vanishing vector space structure.
Tatra Mountains Mathematical Publications,10,p. 241. download pdf.

Abstract:We put forward an approach where physical entities are described by the set of their states, and the set of their relevant experiments. In this framework we will study a general entity that is neither quantum nor classical. We show that the collection of eigenstate sets forms a closure structure on the set of states. We also illustrate this framework on a concrete physical example, the epsilon-example. this leads us to a model for a continuous evolution from the linear closure in vector space to the standard topological closure.- Aerts, D. and Veretennicoff, I. (1997). Niet-ruimtelijkheid als werktuig. In J. Van Pelt,
Grenzeloze Wetenschap: Dertig Gesprekken met Vlamingen over Onderzoek. Leuven-Apeldoorn: Garant. download pdf.## 1998

- Aerts, D. (1998). The hidden measurement formalism: what can be explained and where paradoxes remain.
International Journal of Theoretical Physics,37,pp. 291-304. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0105126. doi: 10.1023/A:1026670802579. download pdf.

Abstract:In the hidden measurement formalism that we develop in Brussels we explain the quantum structure as due to the presence of two effects, (a) a real change of state of the system under influence of the measurement and, (b) a lack of knowledge about a deeper deterministic reality of the measurement process. We show that the presence of these two effects leads to the major part of the quantum mechanical structure of a theory describing a physical system where the measurements to test the properties of this physical system contain the two mentioned effects. We present a quantum machine, where we can illustrate in a simple way how the quantum structure arises as a consequence of the two effects. We introduce a parameter epsilon that measures the amount of the lack of knowledge on the measurement process, and by varying this parameter, we describe a continuous evolution from a quantum structure (maximal lack of knowledge) to a classical structure (zero lack of knowledge). We show that for intermediate values of epsilon we find a new type of structure that is neither quantum nor classical. We analyze the quantum paradoxes in the light of these findings and show that they can be divided into two groups: (1) The group (measurement problem and Schrodingers cat paradox) where the paradoxical aspects arise mainly from the application of standard quantum theory as a general theory (e.g. also describing the measurement apparatus). This type of paradox disappears in the hidden measurement formalism. (2) A second group collecting the paradoxes connected to the effect of non-locality (the Einstein-Podolsky-Rosen paradox and the violation of Bell inequalities). We show that these paradoxes are internally resolved because the effect of non-locality turns out to be a fundamental property of the hidden measurement formalism itself.- Aerts, D. (1998). The entity and modern physics: the creation-discovery view of reality. In E. Castellani (Ed.),
Interpreting Bodies: Classical and Quantum Objects in Modern Physics(pp. 223-257). Princeton: Princeton University Press. download pdf.

Abstract:The classical concept of 'physical entity', be it particle, wave, field or system, has become a problematic concept since the advent of relativity theory and quantum mechanics. The recent developments in modern quantum mechanics, with the performance of delicate and precise experiments involving single quantum entities, manifesting explicit non-local behavior for these entities, brings essential new information about the nature of the concept of entity. Such fundamental categories as space and time are put into question, and only a recourse to more axiomatic descriptions seems possible. In this contribution we want to put forward a 'picture' of what an 'entity' might be, taking into account these recent experimental and theoretical results, and using fundamental results of the axiomatic physical theories (describing classical as well as quantum entities) such as they have been developed during the last decade. We call our approach the 'creation-discovery view' because it considers measurements as physical interactions that in general entail two aspects: (1) a discovery of an already existing reality and (2) a creation of new aspects of reality during the act of measurement. We analyze the paradoxes of orthodox quantum mechanics in this creation-discovery view and point out the pre-scientifc preconceptions that are contained in the well-known orthodox interpretations of quantum mechanics. Finally we identify orthodox quantum mechanics as a first order non classical theory, and explain in this way why it is so successful in its numerical predictions.- Aerts, D. (1998). Kwantumtheater.
Etcetera,15,64, p. 7. download pdf.- Aerts, D. (1998). Synthesis and analysis, interdisciplinarity and foundations.
Foundations of Science,3,pp. 203-206. doi: 10.1023/A:1009663005061. download pdf.- Aerts, D., Broekaert, J. and Smets, S. (1998). Inconsistencies in constituent theories of world views: quantum mechanical examples.
Foundations of Science,3,pp. 313-340. doi: 10.1023/A:1009686326442. download pdf.

Abstract:We put forward the hypothesis that there exist three basic attitudes towards inconsistencies within world views: (1) The inconsistency is tolerated temporarily and is viewed as an expression of a temporary lack of knowledge due to an incomplete or wrong theory. The resolution of the inconsistency is believed to be inherent to the improvement of the theory. This improvement ultimately resolves the contradiction and therefore we call this attitude the 'regularising' attitude; (2) The inconsistency is tolerated and both contradicting elements in the theory are retained. This attitude integrates the inconsistency and leads to a paraconsistent calculus; therefore we will call it the paraconsistent attitude. (3) In the third attitude, both elements of inconsistency are considered to be false and the 'real situation' is considered something different that can not be described by the theory constructively. This indicates the incompleteness of the theory, and leads us to a paracomplete calculus; therefore we call it the paracomplete attitude. We illustrate these three attitudes by means of two 'paradoxical' situations in quantum mechanics, the wave-particle duality and the situation of non locality.- Aerts, D. and Rohrlich, F. (1998). Reduction. Foundations of Science,
3,pp. 27-35. doi: 10.1023/A:1009626018626. download pdf.## 1999

- Aerts, D. (Ed.) (1999).
Einstein meets Magritte: An Interdisciplinary Reflection about Science, Nature, Art, Human Action and Society. Dordrecht: Springer.- Aerts, D. (1999). Foundations of quantum physics: a general realistic and operational approach.
International Journal of Theoretical Physics,38,pp. 289-358. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0105109. doi: 10.1023/A:1026605829007. download pdf.

Abstract: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 orthoclosure 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 subentity 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.- Aerts, D. (1999). The general introduction of Einstein meets Magritte, an introduction to the series. In D. Aerts, D., J. Broekaert, and E. Mathijs (Eds.),
Einstein meets Magritte: An Interdisciplinary Reflection(pp. vii-xiii). Dordrecht: Springer. download pdf.- Aerts, D. (1999). The scholar, Terpsichore and the barfly. In D. Aerts, J. Broekaert and E. Mathijs (Eds.),
Einstein meets Magritte: An Interdisciplinary Reflection(pp. 1-11). Dordrecht: Springer. download pdf.- Aerts, D. (1999). The stuff the world is made of: physics and reality. In D. Aerts, J. Broekaert and E. Mathijs (Eds.),
Einstein meets Magritte: An Interdisciplinary Reflection(pp. 129-183). Dordrecht: Springer. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0107044. download pdf.

Abstract:Taking into account the results that we have been obtained during the last decade in the foundations of quantum mechanic we put forward a view on reality that we call the 'creation discovery view'. In this view it is made explicit that a measurement is an act of a macroscopic physical entity on a microphysical entity that entails the creation of new elements of reality as well as the detection of existing elements of reality. Within this view most of the quantum mechanical paradoxes are due to structural shortcomings of the standard quantum theory, which means that our analysis agrees with the claim made in the Einstein Podolsky Rosen paper, namely that standards quantum mechanics is an incomplete theory. This incompleteness is however not due to the absence of hidden variables but to the impossibility for standard quantum mechanics to describe separated quantum entities. Nonlocality appears as a genuine property of nature in our view and makes it necessary to reconsider the role of space in reality. Our proposal for a new interpretation for space makes it possible to put forward an new hypothesis for why it has not been possible to unify quantum mechanics and relativity theory.- Aerts, D. (1999). Participating in the world: research and education in a changing society. In D. Aerts, S. Gutwirth, S. Smets and L. Van Langenhove (Eds.),
Science, Technology and Social Change(pp. 1-34). Dordrecht: Springer. download pdf.- Aerts, D. (1999). The game of the biomousa: a view of discovery and creation. In D. Aerts, H. Van Belle and J. Van der Veken (Eds.),
Worldviews and the Problem of Synthesis(pp. 95-125). Dordecht: Springer. download pdf.- Aerts, D. (1999). From freedom to freedom of movement. In D. Aerts, J. Broekaert and W. Wijns (Eds.),
A World in Transition, Humankind and Nature(pp. 43-82). Dordecht: Springer. download pdf.- Aerts, D. (1999). Quantum mechanics: structures, axioms and paradoxes. In D. Aerts and J. Pykacz (Eds.),
Quantum Mechanics and the Nature of Reality(pp. 141-205). Dordrecht: Springer. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0106132. download pdf.

Abstract: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.- Aerts, D. (1999). Creativity and science.
Foundations of Science,4,pp. 111-112. doi: 10.1023/A:1009610911185. download pdf.- Aerts, D., Broekaert, J. and Gabora, L. (1999). Formal and informal representations of science.
Foundations of Science,4,1-2. doi: 10.1023/A:1009664701532. download pdf.- Aerts, D., Broekaert, J. and Gabora, L. (1999). Nonclassical contextuality in cognition: Borrowing from quantum mechanical approaches to indeterminism and observer dependence. In R. Campbell (Ed.), Dialogues in Psychology,
10, Proceedings of 'Mind IV', Dublin.- Aerts, D., Broekaert, J. and Mathijs, E. (Eds.) (1999).
Einstein meets Magritte: An Interdisciplinary Reflection. Dordrecht: Springer.- Aerts, D., Broekaert, J. and Mathijs, E. (1999). Somewhere over the rainbow, an introduction to the book. In D. Aerts, J. Broekaert and E. Mathijs (Eds.),
Einstein meets Magritte: An Interdisciplinary Reflection(pp. xv-xix). Dordrecht: Springer. download pdf.- Aerts, D., Broekaert, J. and Smets, S. (1999). The liar paradox in a quantum mechanical perspective.
Foundations of Science,4,pp. 115-132. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0007047. doi: 10.1023/A:1009610326206. download pdf.

Abstract:We propose a method to model the truth behaviour of cognitive entities taking into account the possible influence of the cognitive person on the truth behaviour of the entity. Hereby we specifically apply the mathematical formalism of quantum mechanics because of the fact that this formalism allows the description of real contextual influences, i.e. the influence of the measuring apparatus on the physical entity. We concentrated on the typical situation of the liar paradox and have shown that (1) the truth-false state of this liar paradox can be represented by a quantum vector in a finite dimensional complex Hilbert space and the different interpretative interactions by the actions of the corresponding quantum projections, (2) the typical oscillations between false and truth - the paradox - is now quantum dynamically described by a Schrodinger equation. We analyse possible philosophical implications of this result.- Aerts, D., Broekaert, J. and Smets, S. (1999). A quantum structure description of the liar paradox.
International Journal of Theoretical Physics,38,pp. 3231-3239. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0106131. doi: 10.1023/A:1026686316673. download pdf.

Abstract:In this article we propose an approach that models the truth behavior of cognitive entities (i.e. sets of connected propositions) by taking into account in a very explicit way the possible influence of the cognitive person (the one that interacts with the considered cognitive entity). Hereby we specifically apply the mathematical formalism of quantum mechanics because of the fact that this formalism allows the description of real contextual influences, i.e. the influence of the measuring apparatus on the physical entity. We concentrated on the typical situation of the liar paradox and have shown that (1) the truth-false state of this liar paradox can be represented by a quantum vector of the non-product type in a finite dimensional complex Hilbert space and the different cognitive interactions by the actions of the corresponding quantum projections, (2) the typical oscillations between false and truth - the paradox - is now quantum dynamically described by a Schrodinger equation. We analyse possible philosophical implications of this result.- Aerts, D., Broekaert, J. and Weyns, W. (Eds.) (1999).
A World in Transition, Humankind and Nature. Dordrecht: Springer.- Aerts, D., Castagnino, M., Durt, T., Gangui, A. and Gunzig, E. (1999). Cosmology and quantum mechanics in Peyresc.
International Journal of Theoretical Physics,38,pp. 3-8. doi: 10.1023/A:1026620804900. download pdf.- Aerts, D. and Coecke, B. (1999). The creation-discovery-view: towards a possible explanation of quantum reality. In M. L. Dalla Chiara, R. Giuntini and F. Laudisa (Eds.),
Language, Quantum, Music: Selected Contributed Papers of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence, August 1995. Dordrecht: Springer. download pdf.

Abstract:We present a realistic interpretation for quantum mechanics that we have called the 'creation discovery view' and that is being developed in our group in Brussels. In this view the change of state of a quantum entity during an experiment is taken to be a 'real change' under influence of the experiment, and the quantum probability that corresponds to the experiment is explained as due to a lack of knowledge of a deeper deterministic reality of the measurement process. The technical mathematical theory underlying the creation discovery view that we are elaborating we have called the 'hidden measurement formalism'. We present a simple physical example: the 'quantum machine', where we can illustrate easily how the quantum structure arises as a consequence of the two mentioned effects, a real change of the state, and a lack of knowledge about a deeper reality of the measurement process. We analyze non-locality in the light of the creation discovery view, and show that we can understand it if we accept that also the basic concept of 'space' is partly due to a creation: when a detection of a quantum entity in a non-local state occurs, the physical act of detection itself 'creates' partly the 'place' of the quantum entity. In this way the creation discovery view introduces a new ontology for space: space is not the all embracing theater, where all 'real' objects have their place, but it is the structure that governs a special type of relations (the space-like relations) between macroscopic physical entities. We bring forward a number of elements that show the plausibility of the approach and also analyze the way in which the presence of Bell-type correlated events can be incorporated.- Aerts, D., Coecke, B and Smets, S (1999). On the origin of probabilities in quantum mechanics: creative and contextual aspects. In G. Cornelis, S. Smets and J. P. Van Bendegem (Eds.),
Metadebates on Science(pp. 291-302). Dordrecht: Springer. download pdf.- Aerts, D., Colebunders, E., Van der Voorde, A. and Van Steirteghem, B. (1999). State property systems and closure spaces: a study of categorical equivalence.
International Journal of Theoretical Physics,38,pp. 359-385. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0105108. doi: 10.1023/A:1026657913077. download pdf.

Abstract:We show that the natural mathematical structure to describe a physical entity by means of its states and its properties within the Geneva-Brussels approach is that of a state property system. We prove that the category of state property systems (and morphisms), SP, is equivalent to the category of closure spaces (and continuous maps), Cls. We show the equivalence of the 'state determination axiom' for state property systems with the 'T0 separation axiom' for closure spaces. We also prove that the category SP0 of state determined state property systems is equivalent to the category L0 of based complete lattices. In this sense the equivalence of SP and Cls generalizes the equivalence of Cls0 (T0 closure spaces) and L0, proven in (Erne 1984).- Aerts, D., Aerts, S., Durt, T. and Leveque, O. (1999). Classical and quantum probability in the epsilon model.
International Journal of Theoretical Physics,38,pp. 407-429. doi: 10.1023/A:1026662013986. download pdf.

Abstract:We describe the probabilistic study of a hidden variable model in which the origin of the quantum probability is due to fluctuations of the internal state of the measuring apparatus. By varying the intensity of these fluctuations from zero to a maximal value, we describe in a heuristic manner the transition from classical behavior to quantum behavior. We characterize this transition in terms of the Accardi-Fedullo inequalities. This is a review article in which we gather our recent contributions to the subject, most of which have not been published in article form.- Aerts, D., Gutwirth, S., Smets, S. and Van Langehove, L. (Eds.) (1999).
Science, Technology and Social Change. Dordrecht: Springer.- Aerts, D., Mathijs, E. and Mosselmans, B. (Eds.) (1999).
Science and Art. Dordrecht: Springer.- Aerts, D. and Pykacz, J., (Eds.) (1999).
Quantum Structures and the Nature of Reality.Dordrecht: Springer.- Aerts, D. and Pykacz, J. (1999). Quantum structures and the nature of reality, an introduction to the book. In D. Aerts and J. Pykacz,
Quantum Structures and the Nature of Reality(pp. xv-xviii). Dordrecht: Springer. download pdf.- Aerts, D., Van Belle, H. and Van der Veken, J. (Eds.) (1999).
Worldviews and the Problem of Synthesis. Dordrecht: Springer.## 2000

- Aerts, D. (2000). The description of joint quantum entities and the formulation of a paradox.
International Journal of Theoretical Physics,39,pp. 485-496. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0105106. doi: 10.1023/A:1003609031040. download pdf.

Abstract:We formulate a paradox in relation to the description of a joint entity consisting of two subentities by standard quantum mechanics. We put forward a proposal for a possible solution, entailing the interpretation of 'density states' as 'pure states'. We explain where the inspiration for this proposal comes from and how its validity can be tested experimentally. We discuss the consequences on quantum axiomatics of the proposal.- Aerts, D. (2000). Quantum, mimesis and the social sciences.
Foundations of Science,5, 1-2. doi: 10.1023/A:1026511218440. download pdf.- Aerts, D., Aerts, S., Broekaert, J. and Gabora, L. (2000). The violation of Bell inequalities in the macroworld.
Foundations of Physics,30,pp. 1387-1414. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0007044. doi: 10.1023/A:1026449716544. download pdf.

Abstract:We show that Bell inequalities can be violated in the macroscopic world. The macroworld violation is illustrated using an example involving connected vessels of water. We show that whether the violation of inequalities occurs in the microworld or in the macroworld, it is the identification of nonidentical events that plays a crucial role. Specifically, we prove that if nonidentical events are consistently differentiated, Bell-type Pitowsky inequalities are no longer violated, even for Bohm's example of two entangled spin 1/2 quantum particles. We show how Bell inequalities can be violated in cognition, specifically in the relationship between abstract concepts and specific instances of these concepts. This supports the hypothesis that genuine quantum structure exists in the mind. We introduce a model where the amount of nonlocality and the degree of quantum uncertainty are parameterized, and demonstrate that increasing nonlocality increases the degree of violation, while increasing quantum uncertainty decreases the degree of violation.- Aerts, D., Cattaneo, G., Dvurecenski, A. and Giuntini, R. (2000). Interdisciplinary studies of quantum structures.
Foundations of Physics,30,pp. 1331-1335. doi: 10.1023/A:1026420315635. download pdf.- Aerts, D., D'Hondt, E. and Gabora, L. (2000). Why the disjunction in quantum logic is not classical.
Foundations of Physics,30,pp. 1473-1480. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0007041. doi: 10.1023/A:1026457918361. download pdf.

Abstract:In this paper, the quantum logical 'or' is analyzed from a physical perspective. We show that it is the existence of EPR-like correlation states for the quantum mechanical entity under consideration that make it nonequivalent to the classical situation. Specifically, the presence of potentiality in these correlation states gives rise to the quantum deviation from the classical logical 'or'. We show how this arises not only in the microworld, but also in macroscopic situations where EPR-like correlation states are present. We investigate how application of this analysis to concepts could alleviate some well known problems in cognitive science.- Aerts, D. and Van Steirteghem, B. (2000). Quantum axiomatics and a theorem of M.P. Soler.
International Journal of Theoretical Physics,39,pp. 497-502. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0105107. doi: 10.1023/A:1003661015110. download pdf.

Abstract:Three of the traditional quantum axioms (orthocomplementation, ortho modularity and the covering law) show incompatibilities with two products introduced by Aerts for the description of joint entities. Inspired by Soler's theorem and Holland's AUG axiom, we propose a property of 'plane transitivity', which also characterizes classical Hilbert spaces among infinite dimensional orthomodular spaces, as a possible partial substitute for the 'defective' axioms.## 2001

- Aerts, D. (2001). Transdisciplinary and integrative sciences in sustainable development. In M. K. Tolba (Ed.),
Our Fragile World, A Forerunner of the Encyclopedia of Life Support Systems(pp. 1203-1214). Aldates, Oxford: Baldwin House. download pdf.

Abstract:We put forward two fundamental problems with which humanity is confronted in relation to sustainable development, and analyze them from a transdisciplinary point of view. The first problem is related to the incapacity of human talent to control and steer the actual complex world society. We suggest some solutions to this problem, ones that arise from a comparison of human society with the human mind and body as a multicellular entity, and analyze in which way human society could develop the capacity to see, think, evaluate and plan for the future as a semi-autonomous cooperative system. The second problem is related to the confusion and misunderstanding that exists in our actual society about the fundamental contradiction between potency and variety versus efficiency and yield. We show in which way this contradiction is a profound property of all layers of reality. By transdisciplinary reflection we investigate in which way solutions to this contradiction in old layers of reality can be applied to the present day. We show that very concrete problems, such as the problem of the acceleration of present-day society and the problem of peaceful collaboration, are due to the subtle confusion about this contradiction, and apply our analysis of it to the proposal of concrete solutions to these problems.- Aerts, D. (2001). Quantum Structures and their future importance.
Soft Computing,5,p. 131. doi: 10.1007/s005000100092. download pdf.- Aerts, D., Deses, D. and Van der Voorde, A. (2001). Connectedness applied to closure spaces and state property systems.
Journal of Electrical Engineering,52, pp. 18-21. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0205163. download pdf.

Abstract:In earlier work we gave a description of a physical entity by means of a state property system and we proved that any state property system is equivalent to a closure space. In the present paper we investigate the relations between classical properties and connectedness for closure spaces. The main result is a decomposition theorem, which allows us to split a state property system into a number of 'pure nonclassical state property systems' and a 'totally classical state property system'.## 2002

- Aerts, D. (2002). The unification of personal presents: a dialogue of different world views.
International Readings on Theory, History and Philosophy of Culture: Ontology of Dialogue,12, pp. 63-82. download pdf.

Abstract:We want to analyse in this article the process of ontological unification of personal world views to a common world view. The hypothesis that we want to put forward is that this process is badly understood and its misunderstandings are at the origin of some of the deep paradoxes about the nature of reality. The title might suggest that we will concentrate mostly on the process of unification that takes place within the psycho- cognitive regions of reality, namely how the psychological, moral, ethical, etc ... aspects of personal world views interact towards the formation of a common world view. This is however not true. We do not underestimate the importance of the process of unification in the psycho-cognitive region, but we will concentrate in this article on a more primitive region of reality, namely the physical region, where the process of unification takes place at early age, and we have mostly forgotten about its nature, which is at the origin of some of the misunderstandings that exist, and the paradoxes that are a consequence of these misunderstandings.- Aerts, D. (2002). Being and change: foundations of a realistic operational formalism. In D. Aerts, M. Czachor and T. Durt (Eds.),
Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics(pp. 71-110). Singapore: World Scientific. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0205164. download pdf.

Abstract: 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 has 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, but, 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 the categorical setting, a way that has been investigated extensively in earlier work, and hence, from a merological covariance principle, we derive the morphisms between state context property systems and introduce the categorySCOPwith elements the state context property systems and morphisms the ones that we derived from this merological covariance principle. We introduce property completeness and state completeness and study the operational foundation of the formalism.- Aerts, D. (2002). Reality and probability: introducing a new type of probability calculus. In D. Aerts, M. Czachor and T. Durt (Eds.),
Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics(pp. 205-229). Singapore: World Scientific. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0205165. download pdf.

Abstract: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 different 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 different 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 Geneva-Brussels approach to quantum mechanics, which was not possible with standard probability theory. We introduce the formal mathematical structure of a 'state experiment probability system', by using this new type of probability theory, as a general description of a physical entity by means of its states, experiments and probability. We derive the state property system as a special case of this structure, when we only consider the 'certain' aspects of the world. The categorySEPof state experiment probability systems and their morphisms is linked with the categorySPof state property systems and their morphisms, that has been studied in earlier articles in detail.- Aerts, D., Broekaert, J. and Gabora, L. (2002). Intrinsic contextuality as the crux of consciousness. In K. Yasue, M. Jibu and T. Della Senta (Eds.),
No Matter, Never Mind(pp. 173-181). Amsterdam: John Benjamins (Volume 33 of the seriesAdvances in Consciousness Research, ISSN 1381 -589X). download pdf.- Aerts, D., Colebunders, E., Van der Voorde, A. and Van Steirteghem, B. (2002). On the amnestic modification of the category of state property systems.
Applied Categorical Structures,10, pp. 469-480. doi: 10.1023/A:1020501521418. download pdf.

Abstract:State property systems were created on the basis of physical intuition in order to describe a mathematical model for physical systems. A state property system consists of a triple: a set of states, a complete lattice of properties and a specified function linking the other two components. The definition of morphisms between such objects was inspired by the physical idea of a subsystem. In this paper we give an isomorphic description of the category of state property systems, thus introducing a categorySP, which is concrete overSet. This isomorphic description enables us to investigate further categorical properties ofSP. It turns out that the categorySPis not amnestic. In our main theorem we prove that the amnestic modification ofSPis the constructClsof closure spaces and continuous maps. Moreover we observe that the categorical product and coproduct inClsfind, throughSP, application in physics.- Aerts, D., Czachor, M. and Durt, T. (Eds.) (2002).
Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics. Singapore: World Scientific.- Aerts, D., Czachor, M. and Durt, T. (2002). Probing the structure of quantum mechanics. In D. Aerts, M. Czachor and T. Durt (Eds.),
Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics(pp. 1-19). Singapore: World Scientific. download pdf.- Aerts, D. and Deses, D. (2002). State property systems and closure spaces: extracting the classical and nonclassical parts. In D. Aerts, M. Czachor and T. Durt (Eds.),
Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics(pp. 130-148). Singapore: World Scientific. Archive reference and link: http://uk.arxiv.org/pdf/quant-ph/0404070. download pdf.

Abstract:In earlier work an equivalence of the categories of state property systems and their morphisms and closure spaces and continuous maps was proven. It has been shown, using the equivalence between these two categories, that some of the axioms of quantum axiomatics are equivalent with separation axioms on the corresponding closure space. More particularly it was proven that the axiom of atomicity is equivalent to the T1 separation axiom. In the present article we analyze the intimate relation that exists between classical and nonclassical in the state property systems and disconnected and connected in the corresponding closure space. We introduce classical properties using the concept of super selection rule,i.e.two properties are separated by a superselection rule iff there do not exist 'superposition states' related to these two properties. Then we show that the classical properties of a state property system correspond exactly to the clopen subsets of the corresponding closure space. Thus connected closure spaces correspond precisely to state property systems for which the elements 0 and I are the only classical properties, the so called pure nonclassical state property systems. The main result is a decomposition theorem, which allows us to split a state property system into a number of 'pure nonclassical state property systems' and a 'totally classical state property system'. This decomposition theorem for a state property system is the translation of a decomposition theorem for the corresponding closure space into its connected components.- Aerts, D. and D'Hooghe B. (2002). Quantum computation: Towards the construction of a 'between quantum and classical computer'. In D. Aerts, M. Czachor and T. Durt (Eds.),
Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics(pp. 230-247). Singapore: World Scientific. download pdf.

Abstract:Using the 'between quantum and classical' models that have been constructed explicitly within the hidden measurement approach of quantum mechanics we investigate the possibility to construct a 'between quantum and classical' computer. In this view, the pure quantum computer and the classical Turing machine can be seen as two special cases of our general computer. We have shown in earlier research that the intermediate 'between quantum and classical' systems cannot be described within standard quantum theory. We argue that the general categoral approach of state property systems might provide a unified framework for the study of these 'between quantum and classical' models, and hence also for the study of classical and quantum computers as special cases.- Aerts, D. and Valckenborgh, F. (2002). The linearity of quantum mechanics at stake: the description of separated quantum entities. In D. Aerts, M. Czachor and T. Durt (Eds.),
Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics(pp. 20-46). Singapore: World Scientific. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0205161. download pdf.

Abstract:We concentrate on the situation of a physical entity that is the compound entity of two 'separated' quantum entities. In earlier work it has been proved by one of the authors that such a physical entity cannot be described by standard quantum mechanics. More specifically, it was shown that two of the axioms of traditional quantum axiomatics are at the origin of the impossibility for standard quantum mechanics to describe the compound entity of two separated quantum entities. One of these axioms is equivalent with the superposition principle, which means that separated quantum entities put the linearity of quantum mechanics at stake. We analyze the conceptual steps that are involved in this proof, and expose the necessary material of quantum axiomatics to be able to understand the argument.- Aerts, D. and Valckenborgh, F. (2002). Linearity and compound physical systems: the case of two separated spin 1/2 entities. In D. Aerts, M. Czachor and T. Durt (Eds.),
Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics(pp. 47-70). Singapore: World Scientific. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0205166. download pdf.

Abstract:We illustrate some problems that are related to the existence of an underlying linear structure at the level of the property lattice associated with a physical system, for the particular case of two explicitly separated spin 1/2 objects that are considered, and mathematically described, as one compound system. It is shown that Aerts' separated product of the property lattices corresponding with the two spin 1/2 objects does not have an underlying linear structure, although the property lattices associated with the subobjects in isolation manifestly do. This is related at a fundamental level with the fact that separated products do not behave well with respect to the covering law of elementary lattice theory. In addition, we discuss the orthogonality relation associated with the separated product in general and consider the related problem of the behaviour of the corresponding Sasaki projections.- Gabora, L. and Aerts, D. (2002). Contextualizing concepts. In
Proceedings of the 15th International FLAIRS Conference (Special Track 'Categorization and Concept Representation: Models and Implications'), Pensacola Florida, May 14-17, 2002, American Association for Artificial Intelligence. download pdf.

Abstract: The mathematics of quantum mechanics was developed to cope with problems arising in the description of (1) contextual interactions, and (2) the generation of new states with new properties when particles become entangled. Similar problems arise with concepts. This paper summarizes the rationale for and preliminary results of using a generalization of standard quantum mechanics based on the lattice formalism to describe the contextual manner in which concepts are evoked, used, and combined to generate meaning. Concepts are viewed not as fixed representations but dynamically Ôre-constructedŐ entities generated on the fly through interaction between cognitive state and situation or context.- Gabora, L. and Aerts, D. (2002). Contextualizing concepts using a mathematical generalization of the quantum formalism.
Journal of Experimental and Theoretical Artificial Intelligence,14, pp. 327-358. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0205161. doi: 10.1080/09528130210162253. download file.

Abstract: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 observable form as an 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.## 2003

- Aerts, D., Czachor, M., Gabora, L., Kuna, M., Posiewnik, A., Pykacz, J. and Syty, M. (2003). Quantum morphogenesis: A variation Thom's catastrophe theory,
Physical Review E,67, 051926. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0211105. doi: 10.1103/PhysRevE.67.051926. download pdf.

Abstract:Noncommutative propositions are characteristic of both quantum and nonquantum (sociological, biological, psychological) situations. In a Hilbert space model states, understood as correlations between all the possible propositions, are represented by density matrices. If systems in question interact via feedback with environment their dynamics is nonlinear. Nonlinear evolutions of density matrices lead to phenomena of morphogenesis which may occur in noncommutative systems. Several explicit exactly solvable models are presented, including 'birth and death of an organism' and 'development of complementary properties'.- Gershenson, C., Broekaert, J. and Aerts, D. (2003). Contextual random boolean networks.
Advances in Artificial Life. Lecture Notes in Artificial Intelligence,2801, pp. 615-624. Archive reference and link: nlin.AO/0303021. doi: 10.1007/978-3-540-39432-7_66. download pdf.

Abstract:We propose the use of Deterministic Generalized Asynchronous Random Boolean Networks as models of contextual deterministic discrete dynamical systems. We show that changes in the context have drastic effects on the global properties of the same networks, namely the average number of attractors and the average percentage of states in attractors. We introduce the situation where we lack knowledge on the context as a more realistic model for contextual dynamical systems. We notice that this makes the network non-deterministic in a specific way, namely introducing a non-Kolmogorovian quantum-like structure for the modelling of the network. In this case, for example, a state of the network has the potentiality (probability) of collapsing into different attractors, depending on the specific form of lack of knowledge on the context.## 2004

- Aerts, D. and Aerts. S. (2004). Towards a general operational and realistic framework for quantum mechanics and relativity theory. In A. C. Elitzur, S. Dolev and N. Kolenda (Eds.),
Quo Vadis Quantum Mechanics? Possible Developments in Quantum Theory in the 21st Century(pp. 153-208). New York: Springer. download pdf.

Abstract:We propose a general operational and realistic framework for a theory that generalizes 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 making specific reference to events to be experienced, and realistic, which means that it takes seriously the hypothesis of an independent existing reality. To come to this framework we present a detailed study of standard quantum mechanics within the axiomatic approach to quantum mechanics, more specifically the Geneva-Brussels 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, and that these 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 replacing the two failing axioms is a necessity to built a theory that can contain standard quantum mechanics as well as relativity theory as a special case. Next we analyze the nature of the quantum probability model and show that it can be interpreted as the consequence of the presence of a lack of knowledge on the interaction between the measurement apparatus and the physical entity under consideration. These two insights, the failing axioms and the nature of quantum probability, give rise to a very specific view on the quantum phenomenon of nonlocality. Nonlocality should be interpreted as nonspatiality. This means that an entity in a nonlocal state, like for example the typical EPR state, is 'not inside space'. As a consequence space is no longer the all embracing theatre of reality, but a structure that has emerged together with the macroscopic material entities that have emerged from the microworld. This clarifies why general relativity theory cannot be used as a basis for the elaboration of the new generalized theory, since in general relativity theory the set of events is takena priorito be the time-space continuum. Hence in general relativity theory time-space is a basic structure considered to capture all of reality. In our framework we introduce 'happenings' and the 'set of happenings' constituting reality. A happening is however not identified with a point of time-space, as this is the case for an events of general relativity theory. We analyze different aspects of the new framework, and list the most important problems to be investigated for an elaboration of this framework into a workable and as complete as possible theory.- Aerts, D. and Czachor, M. (2004). Quantum aspects of semantic analysis and symbolic artificial intelligence.
Journal of Physics A: Mathematical and Theoretical,37, pp. L123-L132. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0309022. doi: 10.1088/0305-4470/37/12/L01. download pdf.

Abstract:Modern approaches to semantic analysis if reformulated as Hilbert-space problems reveal formal structures known from quantum mechanics. A similar situation is found in distributed representations of cognitive structures developed for the purpose of neural networks. We take a closer look at similarities and differences between the above two fields and quantum information theory.- Aerts, D. and Valckenborgh, F. (2004). Failure of standard quantum mechanics for the description of compound quantum entities.
International Journal of Theoretical Physics,43, pp. 251-264. doi: 10.1023/B:IJTP.0000028862.91652.98. download pdf.

Abstract:We reformulate the 'separated quantum entities' theorem,i.e.the theorem that proves thattwo separated quantum entities cannot be described by means of standard quantum mechanics, within the fully elaborated operational Geneva-Brussels approach to quantum axiomatics, where the basic mathematical structure is that of a State Property System. We give arguments that show that the core of this result indicates a failure of standard quantum mechanics, and not just some peculiar shortcoming due to the axiomatic approach to quantum mechanics itself.- Note, N. and Aerts, D. (2004). The perception of the human self: A proposal for ethical adjustment.
Differentiation and Integration of Worldviews. International Readings on Theory, History and Philosophy of Culture,20, pp. 34-57. download pdf.

Abstract:The late-modern or post-modern era is facing many challenges. Environmental issues, the fragmentation and moral disintegration of society and increasing levels of aggression have developed into serious problems worldwide. But also the phenomena of alienation and unbridled individual autonomy have been recognised as requiring our attention. In this paper we will set out to argue that at least some of these issues have a common ground for being rooted in todayŐs perception of the Self. According to this perception, we think of ourselves as essentially Ôself-reliantŐ and surrounded by a world that is sheer concrete reality. This has made it possible Đ though not necessary! Đ for man to adopt a detached stance towards his social and physical environment and, on the personal level, has led to a decreased meaningfulness of life. In the first part of this article we will try to pinpoint the way in which todayŐs structuring blocks defining the human being have led man to believe that he is essentially self-reliant, and we will discuss the consequences of this belief. If we are to counteract these socio-political and individual challenges, we will have to move away from such a detached perception i.e. we will have to adjust the very concept of man. In an attempt to make a positive contribution towards a new perception and definition of manŐs Self, the second part of this paper suggests a number of (ethical) building blocks that may help achieve this.## 2005

- Aerts, D. (2005). Ceci n'est pas Heinz von Foerster.
Constuctivist Foundations,1, pp. 13-16. download pdf.- Aerts, D. (2005). Towards a new democracy: Consensus through quantum parliament. In D. Aerts, B. D'Hooghe and N. Note (Eds.),
Worldviews, Science and Us, Redemarcating Knowledge and its Social and Ethical Implications. Singapore: World Scientific. Archive reference and link: http://uk.arxiv.org/abs/physics/0503078. download pdf.

Abstract:We compare different actual forms of democracy and analyse in which way they are variations of a 'natural consensus decision process'. We analyse how 'consensus decision followed by majority voting' is open to 'false play' by the majority, and investigate how other types of false play appear in alternative types of democratic decision procedures.We introduce the combined notion of 'quantum parliament' and 'quantum decision procedure', and prove it to be the only one, when applied after consensus decision, that is immune to false play.- Aerts, S., Aerts, D. and Schroeck, F. E. (2005). Necessity of combining mutually incompatible perspectives in the construction of a global view: Quantum probability and signal analysis. In D. Aerts, B. D'Hooghe and N. Note (Eds.),
Worldviews, Science and Us: Redemarcating Knowledge and Its Social and Ethical Implications. Singapore: World Scientific. download pdf.

Abstract:The scientific fields of quantum mechanics and signal-analysis originated within different settings, aimed at different goals and started from different scientific paradigms. Yet the development of the two subjects has become increasingly intertwined. We argue that these similarities are rooted in the fact that both fields of scientific inquiry had to deal with finding a single description for a phenomenon that yields complete information about itself only when we consider mutually incompatible accounts of that phenomenon.- Aerts, D. and Deses, D. (2005). State property systems and orthogonality.
International Journal of Theoretical Physics,44, pp. 919-929. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0211095. doi: 10.1007/s10773-005-706. download pdf.

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, and the category of state property systems was proven to be equivalence to the category of closure spaces. 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, 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. 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.- Aerts, D., D'Hondt, E. and D'Hooghe, B. (2005). A geometrical representation of entanglement as internal constraint.
International Journal of Theoretical Physics,44, pp. 897-907. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0211094. doi: 10.1007/s10773-005-7067-6. download pdf.

Abstract:We study a system of two entangled spin 1/2, were the spin's are represented by a sphere model developed within the hidden measurement approach which is a generalization of the Bloch sphere representation, such that also the measurements are represented. We show how an arbitrary tensor product state can be described in a complete way by a specific internal constraint between the ray or density states of the two spin 1/2. We derive a geometrical view of entanglement as a 'rotation' and 'stretching' of the sphere representing the states of the second particle as measurements are performed on the first particle. In the case of the singlet state entanglement can be represented by a real physical constraint, namely by means of a rigid rod.- Aerts, D. and D'Hooghe, B. (2005). The nature of time as a consequence of how we construct the world. In R. Buccheri, A. C. Elitzur and M. Saniga (Eds.),
Endophysics, Time, Quantum and the Subjective(pp. 113 - 130). Singapore: World Scientific. download pdf.

Abstract:In classical physics there was a clear understanding of what physical space and time are: physical space is the theatre of the collection of all events that are actual at a certain moment of time, and physical time is the parametrization of the flow of time. 3-dimensional space and 1-dimensional time have been substituted by 4-dimensional time-space in relativity theory. But if reality is the 4-dimensional time-space manifold of relativity theory, what is then the meaning of 'change in time'? We investigate this problem of relativity theory by following an operational approach originally elaborated for quantum mechanics. We show that the contradiction between a geometric view and process view of reality is due to a misconception in the interpretation of relativity theory. We argue that it is not time which is space-like, with the inevitable paradoxical situation of a block universe as result, but on the contrary, it is space which is time-like. This 'dynamic', 'time-like' conception of space answers the question of the meaning of 'change in time' within the 4-dimensional reality of relativity theory, and puts forward a new view on other aspects of the theory.- Aerts, D., D'Hooghe, B. and Note, N. (Eds.), (2005).
Worldviews, Science and Us: Redemarcating Knowledge and its Social and Ethical Implications. Singapore: World Scientific.- Aerts, D., D'Hooghe, B. and Note, N. (2005). Worldviews, sciences and us: Global perspectives. In D. Aerts, B. D'Hooghe and N. Note (Eds.),
Worldviews, Science and Us: Redemarcating Knowledge and Its Social and Ethical Implications. Singapore: World Scientific. download pdf.- Aerts, D. and Gabora, L. (2005). A theory of concepts and their combinations I: The structure of the sets of contexts and properties.
Kybernetes,34, pp. 167-191. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0402207. doi:10.1108/03684920510575799. download pdf.

Abstract:We propose a theory for modeling concepts that uses the state-context-property theory (SCOP), a generalization of the quantum formalism, whose basic notions are states, contexts and properties. This theory enables us to incorporate context into the mathematical structure used to describe a concept, and thereby model how context influences the typicality of a single exemplar and the applicability of a single property of a concept. We introduce the notion `state of a concept' to account for this contextual influence, and show that the structure of the set of contexts and of the set of properties of a concept is a complete orthocomplemented lattice. The structural study in this article is a preparation for a numerical mathematical theory of concepts in the Hilbert space of quantum mechanics that allows the description of the combination of concepts (see quant-ph/0402205).- Aerts, D. and Gabora, L. (2005). A theory of concepts and their combinations II: A Hilbert space representation.
Kybernetes,34, pp. 192-221. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0402205. doi: 10.1108/03684920510575807. download pdf.

Abstract:The sets of contexts and properties of a concept are embedded in the complex Hilbert space of quantum mechanics. States are unit vectors or density operators, and contexts and properties are orthogonal projections. The way calculations are done in Hilbert space makes it possible to model how context influences the state of a concept. Moreover, a solution to the combination of concepts is proposed. Using the tensor product, a procedure for describing combined concepts is elaborated, providing a natural solution to the pet fish problem. This procedure allows the modeling of an arbitrary number of combined concepts. By way of example, a model for a simple sentence containing a subject, a predicate and an object, is presented.- Gabora, L. and Aerts, D. (2005). Evolution as context-driven actualization of potential.
Interdisciplinary Science Reviews 30, pp. 69-88. Archive reference and link: http://uk.arxiv.org/abs/q-bio/0511007. doi: 10.1179/030801805X25873. download pdf.

Abstract:While natural selection is often viewed as synonymous with evolution, it is widely felt to be inadequate as a theory of biological evolution; moreover, historically the concept of evolution has not been limited to biology. We propose an integrative framework for characterizing how entities evolve, in which evolution is viewed as a process of context-driven actualization of potential (CAP). Processes of change differ according to the degree of nondeterminism, and the degree to which they are sensitive to, internalize, and depend upon a particular environment or context. The approach enables us to embed phenomena across multiple disciplines into a broader conceptual framework. It suggests that the dynamical evolution of a quantum entity as described by the Schršdinger equation is not fundamentally different from change provoked by a measurement often referred to as collapse but a limiting case, with only one way to collapse. The biological transition to coded replication is seen as a means of preserving structure in the face of context, and sexual replication as a means of increasing potentiality thus enhancing diversity through interaction with context. The integrative framework sheds light on biological concepts like selection and fitness, reveals how exceptional Darwinian evolution is as a means of Ôchange of stateŐ, and clarifies in what sense culture (and the creative process underlying it) is and is not Darwinian.- Gabora, L. and Aerts, D. (2005). Distilling the essence of an evolutionary process, and implications for a formal description of culture. In W. Kistler (Ed.),
Proceedings of Center for Human Evolution Workshop #5: Cultural Evolution, May 2000, Foundation for the Future, Seattle WA. download pdf.- Note, N., Pinxten, H. and Aerts, D. (2005). Towards a re-delineation of the Human Self-understanding within the western worldview: Its social and ethical implications. In D. Aerts, B. D'Hooghe and N. Note (Eds.),
Worldviews, Science and Us: Redemarcating Knowledge and Its Social and Ethical Implications. Singapore: World Scientific. download pdf.

Abstract:This article focuses on the relation between worldviews, sciences and us. Its point of departure is the significant mutual influence of the Western worldview and sciences. It shows how the intertwined construction of science and worldview has modelled our conceptual self-understanding, our being and our acting. The issue is considered from a philosophical-anthropological stance, with due attention being given to past delineations and future alternatives. It is argued that, within the framework of the Western worldview, self-realisation is considered essential for being a human self. There is a tacit, yet conscious, agreement that the way to attain self-realisation is through the gradual development of two potentials: the rational potential and the potential for self-expression. The authors recognise that both are indispensable in forming the human self, but point out that the nature of the development of these potentials can conceptually be misinterpreted, causing problems on the individual, societal and ecological levels. In order to prevent the development of the rational potential and the potential for self-expression from receiving undue emphasis, two more potentials are introduced on the conceptual level, to wit the ethical potential and the potential to be situated in and oriented towards a larger and meaningful whole. The assumption is that bringing these to the fore will also affect the very definition of self-realisation.## 2006

- Aerts, D. and Czachor, M. (2006). Abstract DNA-type systems.
Nonlinearity,19, pp. 575-589. Archive reference and link: http://uk.arxiv.org/abs/q-bio/0411031. doi: 10.1088/0951-7715/19/3/003. download pdf.

Abstract:An abstract DNA-type system is defined by a set of nonlinear kinetic equations with polynomial nonlinearities that admit soliton solutions associated with helical geometry. The set of equations allows for two different Lax representations: A von Neumann form and a Darboux-covariant Lax pair. We explain why non-Kolmogorovian probability models occurring in soliton kinetics are naturally associated with chemical reactions. The most general known characterization of soliton kinetic equations is given and a class of explicit soliton solutions is discussed. Switching between open and closed states is a generic behaviour of the helices. The effect does not crucially depend on the order of nonlinearity (i.e. types of reactions), a fact that may explain why simplified models possess properties occuring in realistic systems. We explain also why fluctuations based on Darboux transformations will not destroy the dynamics but only switch between a finite number of helical structures.- Aerts, D., Czachor, M. and D'Hooghe, B. (2006). Towards a quantum evolutionary scheme: violating Bell's inequalities in language. In N. Gontier, J. P. Van Bendegem and D. Aerts (Eds.),
Evolutionary Epistemology, Language and Culture - A non adaptationist systems theoretical approach. [Theory and Decision Library Series A: Philosophy and Methodology of the Social Sciences. Series editor: Julian Nida-Ruemelin]. Dordrecht: Springer. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0407150. download pdf.

Abstract:We show the presence of genuine quantum structures in human language. The neo-Darwinian evolutionary scheme is founded on a probability structure that satisfies the Kolmogorovian axioms, and as a consequence cannot incorporate quantum-like evolutionary change. In earlier research we revealed quantum structures in processes taking place in conceptual space. We argue that the presence of quantum structures in language and the earlier detected quantum structures in conceptual change make the neo-Darwinian evolutionary scheme strictly too limited for Evolutionary Epistemology. We sketch how we believe that evolution in a more general way should be implemented in epistemology and conceptual change, but also in biology, and how this view would lead to another relation between both biology and epistemology.- Aerts, D., Czachor, M., Gabora, L. and Polk, P. (2006). Soliton kinetic equations with non-Kolmogorovian structure: A new tool for biological modeling?,
AIP Conference Proceedings,810, pp. 19-33. download pdf.

Abstract:Non-commutative diagrams, where X goes to Y goes to Z is allowed and X goes Z goes to Y is not, may equally well apply to Malusian experiments with photons traversing polarizers, and to sequences of elementary chemical reactions. This is why non-commutative probabilistic, logical, and dynamical structures necessarily occur in chemical or biological dynamics. We discuss several explicit examples of such systems and propose an exactly solvable nonlinear toy model of a 'brainĐheart' system. The model involves non-Kolmogorovian probability calculus and soliton kinetic equations integrable by Darboux transformations.- Aerts, D., Czachor, M. and Pawlowski, M. (2006). Entangled-state cryptographic protocol that remains secure even if nonlocal hidden variables exist and can be measured with arbitrary precision,
Physical Review A,73, 034303. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0501003. doi: 10.1103/PhysRevA.73.034303. download pdf.

Abstract:Standard quantum cryptographic protocols are not secure if one assumes that nonlocal hidden variables exist and can be measured with arbitrary precision. The security can be restored if one of the communicating parties randomly switches between two standard protocols.- Aerts, D., Czachor, M. and Pawlowski, M. (2006). Entangled-state cryptographic protocol that remains secure even if nonlocal hidden variables exist and can be measured with arbitrary precision,
Erratum - Physical Review A,73, 059901(E).- Aerts, D. and Pulmannova, S. (2006). Representation of state property systems.
Journal of Mathematical Physics,47, 072105. doi: 10.1063/1.2217807. download pdf.

Abstract:A state property system is the mathematical structure which models an arbitrary physical system by means of its set of states, its set of properties, and a relation of 'actuality of a certain property for a certain state'. We work out a new axiomatization for standard quantum mechanics, starting with the basic notion of state property system, and making use of a generalization of the standard quantum mechanical notion of 'superposition' for state property systems.- Broekaert, J., Aerts, D. and D'Hooghe, B. (2006). The generalised Liar Paradox: A quantum model and interpretation.
Foundations of Science,11, pp. 399-418. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0404066. doi: 10.1007/s10699-004-6248-8. download pdf.

Abstract:The formalism of abstracted quantum mechanics is applied in a model of the generalized Liar Paradox. Here, the Liar Paradox, a consistently testable configuration of logical truth properties, is considered a dynamic conceptual entity in the cognitive sphere. Basically, the intrinsic contextuality of the truth-value of the Liar Paradox is appropriately covered by the abstracted quantum mechanical approach. The formal details of the model are explicited here for the generalized case. We prove the possibility of constructing a quantum model of the m-sentence generalizations of the Liar Paradox. This includes (i) the truth-falsehood state of the m-Liar Paradox can be represented by an embedded 2m-dimensional quantum vector in a 2m to the power m dimensional complex Hilbert space, with cognitive interactions corresponding to projections, (ii) the construction of a continuous 'time' dynamics is possible: typical truth and falsehood value oscillations are described by Schrodinger evolution, (iii) Kirchoff and von Neumann axioms are satisfied by introduction of 'truth-value by inference' projectors, (iv) time invariance of unmeasured state.- Gontier, N., Van Bendegem, J. P. and Aerts, D. (Eds.), (2006).
Evolutionary Epistemology, Language and Culture - A non adaptationist systems theoretical approach. [Theory and Decision Library Series A: Philosophy and Methodology of the Social Sciences. Series editor: Julian Nida-Ruemelin]. Dordrecht: Springer.## 2007

- Aerts, D. and Czachor, M. (2007). Cartoon computation: Quantum-like algorithms without quantum mechanics,
Journal of Physics A: Mathematical and Theoretical,40, F259-F266, Fast Track Communication. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0611279. doi: 10.1088/1751-8113/40/13/F01. download pdf.

Abstract: We present a computational framework based on geometric structures. No quantum mechanics is involved, and yet the algorithms perform tasks analogous to quantum computation. Tensor products and entangled states are not needed -- they are replaced by sets of basic shapes. To test the formalism we solve in geometric terms the Deutsch-Jozsa problem, historically the first example that demonstrated the potential power of quantum computation. Each step of the algorithm has a clear geometric interpetation and allows for a cartoon representation.- Aerts, D. and Czachor, M. (2007). Two-state dynamics for replicating two-strand systems.
Open Systems & Information Dynamics,14, 397-410. Archive reference and link: http://uk.arxiv.org/abs/q-bio/0512048. doi: 10.1007/s11080-007-9064-0. download pdf.

Abstract: We propose a formalism for describing two-strand systems of a DNA type by means of soliton von Neumann equations, and illustrate how it works on a simple example exactly solvably by a Darboux transformation. The main idea behind the construction is the link between solutions of von Neumann equations and entangled states of systems consisting of two subsystems evolving in time in opposite directions. Such a time evolution has analogies in realistic DNA where the polymerazes move on leading and lagging strands in opposite directions.- Aerts, D., Czachor, M., Dehaene, J., De Moor, B. and D'Hooghe, B. (2007). Macroscopic models for quantum systems and computers.
Journal of Physics: Conference Series,70, 012001. doi: 10.1088/1742-6596/70/1/012001. download pdf.

Abstract: We present examples of macroscopic systems entailing a quantum mechanical structure. One of our examples has a structure which is isomorphic to the spin structure for a spin 1/2 and another system entails a structure isomorphic to the structure of two spin 1/2 in the entangled singlet state. We elaborate this system by showing that an arbitrary tensor product state representing two entangled qubits can be described in a complete way by a specific internal constraint between the ray or density states of the two qubits, which describes the behavior of the state of one of the spins if measurements are executed on the other spin. Since any n-qubit unitary operation can be decomposed into 2-qubit gates and unary operations, we argue that our representation of 2-qubit entanglement contributes to a better understanding of the role of n-qubit entanglement in quantum computation. We illustrate our approach on two 2-qubit algorithms proposed by Deutsch, respectively Arvind et al. One of the advantages of the 2-qubit case besides its relative simplicity is that it allows for a nice geometrical representation of entanglement, which contributes to a more intuitive grasp of what is going on in a 2-qubit quantum computation.- Aerts, D., Czachor, M. and Pawlowski, M. (2007). Security in quantum cryptography vs. nonlocal hidden variables.
AIP Conference Proceedings,889, pp. 71-78. doi: 10.1063/1.2713448. download pdf.

Abstract: In order to prove equivalence of quantum mechanics with nonlocal hidden-variable theories of a Bohm type one assumes that all the possible measurements belong to a restricted class: (a) we measure only positions of particles and (b) have no access to exact values of initial conditions for Bohm's trajectories. However, in any computer simulation based on Bohm's equations one relaxes the assumption (b) and yet obtains agreement with quantum predictions concerning the results of positional measurements. Therefore a theory where (b) is relaxed, although in principle allowing for measurements of a more general type, cannot be experimentally falsified within the current experimental paradigm. Such generalized measurements have not been invented, or have been invented but the information is qualified, but we cannot exclude their possibility on the basis of known experimental data. Since the measurements would simultaneously allow for eavesdropping in standard quantum cryptosystems, the arguments for security of quantum cryptography become logically circular: Bohm-type theories do not allow for eavesdropping because they are fully equivalent to quantum mechanics, but the equivalence follows from the assumption that we cannot measure hidden variables, which would be equivalent to the possibility of eavesdropping... Here we break the vicious circle by a simple modification of entangled-state protocols that makes them secure even if our enemies have more imagination and know how to measure hidden-variable initial conditions with arbitrary precision.- Gabora, L. and Aerts, D. (2007). A cross-disciplinary framework for the description of contextually mediated change.
Electronic Journal of Theoretical Physics,4, 1-22.

Abstract: We present a mathematical framework (referred to as Context-driven Actualization of Potential, or CAP) for describing how entities change over time under the influence of a context. The approach facilitates comparison of change of state of entities studied in different disciplines. Processes are seen to differ according to the degree of nondeterminism, and the degree to which they are sensitive to, internalize, and depend upon a particular context. Our analysis suggests that the dynamical evolution of a quantum entity described by the Schrodinger equation is not fundamentally different from change provoked by a measurement often referred to as collapse but a limiting case, with only one way to collapse. The biological transition to coded replication is seen as a means of preserving structure in the face of context, and sexual replication as a means of increasing potentiality thus enhancing diversity through interaction with context. The framework sheds light on concepts like selection and fitness, reveals how exceptional Darwinian evolution is as a means of 'change of state', and clarifies in what sense culture (and the creative process underlying it) are Darwinian.## 2008

- Aerts, D. (2008). De Potentie van Mens en Maatschappij. In J. Van der Veken and H. Van Belle (Eds.),
Nieuwheid denken: De wetenschappen en het creatieve aspect van de werkelijkheid. Leuven: Acco. download pdf.- Aerts, S. and Aerts, D. (2008). When can a data set be described by quantum theory? In P. Bruza , W. Lawless, K. van Rijsbergen, D. Sofge, B. Coecke and S. Clark (Eds.),
Proceedings of the Second Quantum Interaction Symposium, Oxford 2008, pp. 27-33. London: College Publications. download pdf.

Abstract: There have been recent claims in various fields of research that aim to show the presence of quantum structure outside the generally accepted domain of quantum theory. We will take a pragmatic and probabilistic perspective to answer the question when it makes sense to describe a given data set by means of quantum theory.- Aerts, D. and Czachor, M. (2008). Tensor-product vs. geometric-product coding.
Physical Review A,77, 012316. doi: 10.1103/PhysRevA.77.012316. download pdf.

Abstract: Quantum computation is based on tensor products and entangled states. We discuss an alternative to the quantum framework where tensor products are replaced by geometric products and entangled states by multivectors. The resulting theory is analogous to quantum computation but does not involve quantum mechanics. We discuss in detail similarities and differences between the two approaches and illustrate the formulas by explicit geometric objects where multivector versions of the Bell-basis, GHZ, and Hadamard states are visualized by means of colored oriented polylines.- Aerts, D., Czachor, M., Dehaene, J., De Moor, B., D'Hondt, E. and D'Hooghe, B. (2008). A macroscopic device for quantum computation.
International Journal of Theoretical Physics,47, 200-211. doi: 10.1007/s10773-007-9507-y. download pdf.

Abstract: We show how a compound system of two entangled qubits in a non-product state can be described in a complete way by extracting entanglement into an internal constraint between the two qubits. By making use of a sphere model representation for the spin 1/2, we derive a geometric model for entanglement. We illustrate our approach on 2-qubit algorithms proposed by Deutsch, respectively Arvind. One of the advantages of the 2-qubit case is that it allows for a nice geometrical representation of entanglement, which contributes to a more intuitive grasp of what is going on in a 2-qubit quantum computation.- Aerts, D., Dehaene, J., De Moor, B., D'Hooghe, B., Posiewnik, A. and Pykacz, J. (2008). How to play two-players restricted quantum games with 10 cards.
International Journal of Theoretical Physics,47, 61-68. doi: 10.1007/s10773-007-9466-3. download pdf.

Abstract: We show that it is possible to play 'restricted' two-players quantum games proposed originally by Marinatto and Weber by purely macroscopic means, in the simplest case having as the only equipment a pack of 10 cards. Our example shows also that some apparently 'genuine quantum' results, even those that emerge as a consequence of dealing with entangled states, can be obtained by suitable application of Kolmogorovian probability calculus and secondary-school mathematics, without application of the 'Hilbert space machinery'.- Gabora, L., Rosch, E. and Aerts, D. (2008). Toward an ecological theory of concepts.
Ecological Psychology,20, 84-116. download pdf.

Abstract: Psychology has had difficulty accounting for the creative, context-sensitive manner in which concepts are used. We believe this stems from the view of concepts as identifiers rather than bridges between mind and world that participate in the generation of meaning. This paper summarizes the history and current status of concepts research, and provides a non-technical summary of work toward an ecological approach to concepts. We outline the rationale for applying generalizations of formalisms originally developed for use in quantum mechanics to the modeling of concepts, showing how it is because of the role of context that deep structural similarities exist between the two. A concept is defined not just in terms of exemplary states and their features or properties, but also by the relational structures of these properties, and their susceptibility to change under different contexts. The approach implies a view of mind in which the union of perception and environment drives conceptualization, forging a web of conceptual relations or 'ecology of mind'.## 2009

- Aerts, D. (2009). Quantum structure in cognition.
Journal of Mathematical Psychology,53, 314-348. Archive reference and link: http://uk.arxiv.org/abs/0805.3850. doi:10.1016/j.jmp.2009.04.005 download pdf.

Abstract:The broader scope of our investigations is the search for the way in which concepts and their combinations carry and influence meaning and what this implies for human thought. More specifically, we examine the use of the mathematical formalism of quantum mechanics as a modeling instrument and propose a general mathematical modeling scheme for the combinations of concepts. We point out that quantum mechanical principles, such as superposition and interference, are at the origin of specific effects in cognition related to concept combinations, such as the guppy effect and the overextension and underextension of membership weights of items. We work out a concrete quantum mechanical model for a large set of experimental data of membership weights with overextension and underextension of items with respect to the conjunction and disjunction of pairs of concepts, and show that no classical model is possible for these data. We put forward an explanation by linking the presence of quantum aspects that model concept combinations to the basic process of concept formation. We investigate the implications of our quantum modeling scheme for the structure of human thought, and show the presence of a two-layer structure consisting of a classical logical layer and a quantum conceptual layer. We consider connections between our findings and phenomena such as the disjunction effect and the conjunction fallacy in decision theory, violations of the sure thing principle, and the Allais and Elsberg paradoxes in economics.- Aerts, D. (2009). Quantum axiomatics. In K. Engesser, D. Gabbay and D. Lehmann (Eds.),
Handbook of Quantum Logic and Quantum Structures,. Amsterdam: Elsevier. download pdf.

Abstract: We present an axiomatic and operational theory of quantum mechanics. The theory is founded on the axiomatic and operational approach started in Geneva mainly by Constantin Piron and his students and collaborators and developed further in Brussels by myself and different students and collaborators. A physical entity, which a priori can be a classical entity or a quantum entity or a combination of both, is described by means of its set of states, its set of properties and a physical notion of `actuality of a property the entity being in a state'. This leads to the mathematical structure of a state property space. We introduce seven axioms such that if satisfied the state property space can be represented by the direct union over a classical state space of irreducible state property spaces, where each one of the irreducible state property spaces is a Hilbert space state property space of standard quantum mechanics, over the real, complex or quaternionic numbers. The axioms are introduced in an as much as possible operational way, such that we can analyze their physical meaning.- Aerts, D. (2009). Operational quantum mechanics, quantum axiomatics and quantum structures. In D. Greenberger, K. Hentschel and F. Wienert (Eds.),
Compendium of Quantum Physics Concepts, Experiments, History and Philosophy(pp. 434-440). Berlin, Heidelberg: Springer. Archive reference and link: http://uk.arxiv.org/abs/0811.2516. doi: 10.1007/978-3-540-70626-7. download pdf.- Aerts, D. (2009). Quantum particles as conceptual entities: A possible explanatory framework for quantum theory.
Foundations of Science,14, 361-411. Archive reference and link: http://uk.arxiv.org/abs/1004.2530, doi: 10.1007/s10699-009-9166-y. download pdf.

Abstract: We put forward a possible new interpretation and explanatory framework for quantum theory. The basic hypothesis underlying this new framework is that quantum particles are conceptual entities. More concretely, we propose that quantum particles interact with ordinary matter, nuclei, atoms, molecules, macroscopic material entities, measuring apparatuses, in a similar way to how human concepts interact with memory structures, human minds or artificial memories. We analyze the most characteristic aspects of quantum theory, i.e. entanglement and non-locality, interference and superposition, identity and individuality in the light of this new interpretation, and we put forward a specific explanation and understanding of these aspects. The basic hypothesis of our framework gives rise in a natural way to a Heisenberg uncertainty principle which introduces an understanding of the general situation of Ôthe one and the manyŐ in quantum physics. A specific view on macro and micro different from the common one follows from the basic hypothesis and leads to an analysis of SchršdingerŐs Cat paradox and the measurement problem different from the existing ones. We reflect about the influence of this new quantum interpretation and explanatory framework on the global nature and evolutionary aspects of the world and human worldviews, and point out potential explanations for specific situations, such as the generation problem in particle physics, the confinement of quarks and the existence of dark matter.- Aerts, D., Aerts, S. and Gabora, L. (2009). Experimental evidence for quantum structure in cognition. In P. D. Bruza, D. Sofge, W. Lawless, C. J. van Rijsbergen and M. Klusch (Eds.),
Proceedings of QI 2009-Third International Symposium on Quantum Interaction, Book series: Lecture Notes in Computer Science,5494, pp. 59-70. Berlin, Heidelberg: Springer. Archice reference and link: http://uk.arxiv.org/abs/0810.5290. doi: 10.1007/978-3-642-00834-4_7. download pdf.- Aerts, D., Czachor, M. and De Moor, B. (2009). Geometric analogue of holographic reduced representation.
Journal of Mathematical Psychology,53, pp. 389-398. Archive reference and link: http://uk.arxiv.org/abs/0710.2611. doi: 10.1016/j.jmp.2009.02.005 download pdf.

Abstract: Holographic reduced representations (HRRs) are distributed representations of cognitive structures based on superpositions of convolution-bound n-tuples. Restricting HRRs to n-tuples consisting of ±1 one reinterprets the variable binding as a representation of the additive group of binary n-tuples with addition modulo 2. Since convolutions are not defined for vectors, the HRRs cannot be directly associated with geometric structures. Geometric analogues of HRRs are obtained if one considers a projective representation of the same group in the space of blades (geometric products of basis vectors) associated with an arbitrary n-dimensional Euclidean (or pseudo-Euclidean) space. Switching to matrix representations of Clifford algebras one can always turn a geometric analogue of HRR into a form of matrix distributed representation. In typical applications the resulting matrices are sparse, so that the matrix representation is less efficient than the representation directly employing the rules of geometric algebra. A yet more efficient procedure is based on 'projected products', a hierarchy of geometrically meaningful n-tuple multiplication rules obtained by combining geometric products with projections on relevant multivector sub-spaces. In terms of dimensionality the geometric analogues of HRRs are in between holographic and tensor-product representations.- Aerts, D., Czachor, M. and Orlowski, L. (2009). Teleportation of geometric structures in 3D,
Journal of Physics A: Mathematical and Theoretical,42, 135307. Archive reference and link: http://uk.arxiv.org/abs/0809.0579. doi: 10.1088/1751-8113/42/13/135307. download pdf.

Abstract: Simplest quantum teleportation algorithms can be represented in geometric terms in spaces of dimensions 3 (for real state-vectors) and 4 (for complex state-vectors). The geometric representation is based on geometric-algebra coding, a geometric alternative to the tensor-product coding typical of quantum mechanics. We discuss all the elementary ingredients of the geometric version of the algorithm: Geometric analogs of states and controlled Pauli gates. Fully geometric presentation is possible if one employs a nonstandard representation of directed magnitudes, formulated in terms of colors defined via stereographic projection of a color wheel, and not by means of directed volumes.

Abstract: We proof a theorem that shows that a collection of experimental data of membership weights of items with respect to a pair of concepts and its conjunction cannot be modeled within a classical measure theoretic weight structure in case the experimental data contain the effect called overextension. Since the effect of overextension, analogue to the well-known guppy effect for concept combinations, is abundant in all experiments testing weights of items with respect to pairs of concepts and their conjunctions, our theorem constitutes a no-go theorem for classical measure structure for common data of membership weights of items with respect to concepts and their combinations. We put forward a simple geometric criterion that reveals the non classicality of the membership weight structure and use experimentally measured membership weights estimated by subjects in experiments to illustrate our geometrical criterion. The violation of the classical weight structure is similar to the violation of the well-known Bell inequalities studied in quantum mechanics, and hence suggests that the quantum formalism and hence the modeling by quantum membership weights can accomplish what classical membership weights cannot do.- Aerts, D. and D'Hooghe, B. (2009). Classical logical versus quantum conceptual thought: Examples in economics, decision theory and concept theory. In P. D. Bruza, D. Sofge, W. Lawless, C. J. van Rijsbergen and M. Klusch (Eds.),
Proceedings of QI 2009-Third International Symposium on Quantum Interaction, Book series: Lecture Notes in Computer Science,5494, pp. 128-142. Berlin, Heidelberg: Springer. Archice reference and link: http://uk.arxiv.org/abs/0810.5332. doi: 10.1007/978-3-642-00834-4_12. download pdf.

Abstract: Inspired by a quantum mechanical formalism to model concepts and their disjunctions and conjunctions, we put forward in this paper a specific hypothesis. Namely that within human thought two superposed layers can be distinguished: (i) a layer given form by an underlying classical deterministic process, incorporating essentially logical thought and its indeterministic version modeled by classical probability theory; (ii) a layer given form under influence of the totality of the surrounding conceptual landscape, where the different concepts figure as individual entities rather than (logical) combinations of others, with measurable quantities such as 'typicality', 'membership', 'representativeness', 'similarity', 'applicability', 'preference' or 'utility' carrying the influences. We call the process in this second layer 'quantum conceptual thought', which is indeterministic in essence, and contains holistic aspects, but is equally well, although very differently, organized than logical thought. A substantial part of the 'quantum conceptual thought process' can be modeled by quantum mechanical probabilistic and mathematical structures. We consider examples of three specific domains of research where the effects of the presence of quantum conceptual thought and its deviations from classical logical thought have been noticed and studied, i.e. economics, decision theory, and concept theories and which provide experimental evidence for our hypothesis.- Gabora, L. and Aerts, D. (2009). A model of the emergence and evolution of integrated worldviews.
Journal of Mathematical Psychology,53, pp. 434-451. doi:10.1016/j.jmp.2009.06.004. download pdf.

Abstract: It is proposed that the ability of humans to flourish in diverse environments and evolve complex cultures reflects the following two underlying cognitive transitions. The transition from the coarse-grained associative memory of Homo habilis to the fine-grained memory of Homo erectus enabled limited representational redescription of perceptually similar episodes, abstraction, and analytic thought, the last of which is modeled as the formation of states and of lattices of properties and contexts for concepts. The transition to the modern mind of Homo sapiens is proposed to have resulted from onset of the capacity to spontaneously and temporarily shift to an associative mode of thought conducive to interaction amongst seemingly disparate concepts, modeled as the forging of conjunctions resulting in states of entanglement. The fruits of associative thought became ingredients for analytic thought, and vice versa. The ratio of associative pathways to concepts surpassed a percolation threshold resulting in the emergence of a self-modifying, integrated internal model of the world, or worldview.- Meurs, P., Note, N and Aerts, D. (2009). This world without another. On Jean-Luc Nancy and 'la mondialisation'.
Journal of Critical Globalization Studies,1, pp. 31-46. download pdf.

Abstract: In this paper, we turn to the philosophy of Jean-Luc Nancy. In his work 'La Creation du Monde ou la Mondialisation' of 2002 the French philosopher analyses the process of globalisation. Rather than denoting a new homogeneity, the term refers to a world horizon characterized in its inter- palpable multiplicity of cultural, socio-economical, ideological and politico-moral content. According to Nancy, globalisation refers to ag-glome-ration: the decay of what once was a globe and now nothing more than a glome. On the one hand, Nancy indicates that the world has changed by an unknown increase of techno-science, the worsening of inequalities between growing populations and by the changing and disappearing of given certainties, views and identities of the world and of man. On a large scale, this deformation is due to the relation between the capitalist evolution and the capitalising of worldviews. On the other hand, due to the inter-palpability of the multiplicity, this means that on our planet there is only space for one world. The world gradually becomes the only world. In this paper we will investigate what Nancy means with the becoming-world of the world and how this relates to our being in the world. For Nancy globalisation reveals two possible destinies of our relation with the world. In 'La Creation du Monde ou la Mondialisation' he discerns globalisation from mondialisation to analyze these two possibilities. We will investigate this distinction of Nancy and its consequences for everyday life.## 2010

- Aerts, D. (2010). Interpreting quantum particles as conceptual entities.
International Journal of Theoretical Physics,49, pp. 2950-2970. doi: 10.1007/s10773-010-0440-0. Archive reference and link: http://uk.arxiv.org/abs/1004.2531. download pdf.

Abstract:We elaborate an interpretation of quantum physics founded on the hypothesis that quantum particles are conceptual entities, playing the role of communication vehicles between material entities composed of ordinary matter which function as memory structures for these quantum particles. We show in which way this new interpretation gives rise to a natural explanation for the quantum e?ects of interference and entanglement by analyzing how interference and entanglement emerges for the case of human concepts. We put forward a scheme to derive a metric type of structure based on similarity as a predecessor for 'space, time, momentum, energy' and 'quantum particles interacting with ordinary matter' structure underlying standard quantum physics, within the new interpretation, and making use of aspects of traditional quantum axiomatics. More speciŢcally we analyze how the e?ect of non locality arises as a consequence of the confrontation of such an emerging metric type of structure and a remaining of the presence of basic conceptual structure on the fundamental level, with the potential of being revealed in specific situations.- Aerts, D. (2010). A potentiality and conceptuality interpretation of quantum physics.
Philosophica,83, pp. 15-52. Archive reference and link: http://uk.arxiv.org/abs/1005.3767. download pdf.

Abstract:We elaborate on a new interpretation of quantum mechanics which we introduced recently. The main hypothesis of this new interpretation is that quantum particles are entities interacting with matter conceptually, which means that pieces of matter function as interfaces for the conceptual content carried by the quantum particles. We explain how our interpretation was inspired by our earlier analysis of non-locality as non-spatiality and a specific interpretation of quantum potentiality, which we illustrate by means of the example of two interconnected vessels of water. We show by means of this example that philosophical realism is not in contradiction with the recent findings with respect to Leggett's inequalities and their violations. We explain our recent work on using the quantum formalism to model human concepts and their combinations and how this has given rise to the foundational ideas of our new quantum interpretation. We analyze the equivalence of meaning in the realm of human concepts and coherence in the realm of quantum particles, and how the duality of abstract and concrete leads naturally to a Heisenberg uncertainty relation. We illustrate the role played by interference and entanglement and show how the new interpretation explains the problems related to identity and individuality in quantum mechanics. We put forward a possible scenario for the emergence of the reality of macroscopic objects.- Aerts, D., Czachor, M., D'Hooghe, B. and Sozzo, S. (2010). The Pet-Fish problem on the World-Wide Web.
Proceedings of the AAAI Fall Symposium (FS-10-08), Quantum Informatics for Cognitive, Social, and Semantic Processes, pp. 17-21. download pdf.

Abstract: We identify the presence of Pet-Fish problem situations and the corresponding Guppy effect of concept theory on the World-WideWeb. For this purpose, we introduce absolute weights for words expressing concepts and relative weights between words expressing concepts, and the notion of 'meaning bound' between two words expressing concepts, making explicit use of the conceptual structure of the World-Wide Web. The Pet-Fish problem occurs whenever there are exemplars - in the case of Pet and Fish these can be Guppy or Goldfish - for which the meaning bound with respect to the conjunction is stronger than the meaning bounds with respect to the individual concepts.- Aerts, D., Czachor, M. and Sozzo, S. (2010). A contextual quantum-based formalism for population dynamics.
Proceedings of the AAAI Fall Symposium (FS-10-08), Quantum Informatics for Cognitive, Social, and Semantic Processes, pp. 22-25. download pdf.

Abstract: Population ecology is mainly based on nonlinear equations of the Lotka-Volterra type, which provide mathematical models for describing the dynamics of interacting species. However, for many interacting populations, these equations entail complex dynamical behavior and unpredictability, generating such difficulties and problematical situations as illustrated by the "paradox of the plankton" and the "paradox of enrichment", for instance. A careful analysis shows that an ecosystem is a fundamentally contextual system, hence any formalism describing such systems should incorporate contextuality from the very beginning. But existing approaches are based on classical physics and probability theory, and introduce contextuality as an external effect, so that they cannot generally explain the main peculiarities of ecosystems. Basing ourselves on a contextual formalism elaborated to study microscopic systems in quantum mechanics and including appropriate nonlinear equations, we construct a generalization of the Lotka-Volterra equations for contextual systems, apply these equations to discuss some paradoxical situations encountered in ecology, and propound alternative solutions to those currently existing in the literature.- Aerts, D., D'Hooghe, B. and Haven, E. (2010). Quantum experimental data in psychology and economics.
International Journal of Theoretical Physics,49, pp. 2971-2990. Archive reference and link: http://uk.arxiv.org/abs/1004.2529. doi: 10.1007/s10773-010-0477-0. download pdf.

Abstract: We prove a theorem which shows that a collection of experimental data of probabilistic weights related to decisions with respect to situations and their disjunction cannot be modeled within a classical probabilistic weight structure in case the experimental data contain the effect referred to as the 'disjunction effect' in psychology. We identify different experimental situations in psychology, more specifically in concept theory and in decision theory, and in economics (namely situations where Savage's Sure-Thing Principle is violated) where the disjunction effect appears and we point out the common nature of the effect. We analyze how our theorem constitutes a no-go theorem for classical probabilistic weight structures for common experimental data when the disjunction effect is affecting the values of these data. We put forward a simple geometric criterion that reveals the non classicality of the considered probabilistic weights and we illustrate our geometrical criterion by means of experimentally measured membership weights of items with respect to pairs of concepts and their disjunctions. The violation of the classical probabilistic weight structure is very analogous to the violation of the well-known Bell inequalities studied in quantum mechanics. The no-go theorem we prove in the present article with respect to the collection of experimental data we consider has a status analogous to the well known no-go theorems for hidden variable theories in quantum mechanics with respect to experimental data obtained in quantum laboratories. For this reason our analysis puts forward a strong argument in favor of the validity of using a quantum formalism for modeling the considered psychological experimental data as considered in this paper.## 2011

- Aerts, D. (2011). Quantum interference and superposition in cognition: Development of a theory for the disjunction of concepts. In D. Aerts, J. Broekaert, B. D'Hooghe and N. Note (Eds.),
Worldviews, Science and Us: Bridging Knowledge and Its Implications for Our Perspectives of the World(pp. 169-211). Singapore: World Scientific. Archive reference and link: http://uk.arxiv.org/abs/0705.0975. download pdf.

Abstract:We elaborate a theory for the modeling of concepts using the mathematical structure of quantum mechanics. Items and concepts are represented by vectors in the complex Hilbert space of quantum mechanics and membership weights of items are modeled by quantum weights calculated following the quantum rules. We apply this theory to model the disjunction of concepts and show that the predictions of our theory for the membership weights of items with respect to the disjunction of concepts match with great accuracy the results of an experiment conducted by Hampton (1988b). It is the quantum effects of interference and superposition that are at the origin of the effects of overextension and underextension observed by Hampton as deviations from a classical use of the disjunction. We show that the complex numbers of the Hilbert space are essential to obtaining the experimental predictions, i.e. vector space models over real numbers do not provide predictions matching the experimental data. We put forward an explanation of the effects of overextension and underextension by interpreting the quantum model applied to the modeling of the disjunction of concepts.- Aerts, D. (2011). Measuring meaning on the World-Wide Web. In D. Aerts, J. Broekaert, B. D'Hooghe and N. Note (Eds.),
Worldviews, Science and Us: Bridging Knowledge and Its Implications for Our Perspectives of the World(pp. 304-313). Singapore: World Scientific. Archive reference and link: http://arxiv.org/abs/1006.1786. download pdf.

Abstract:We introduce the notion of the 'meaning bond' of a word with respect to another word by making use of the World-Wide Web as a conceptual environment for meaning. This meaning bond is calculated as a value from a formula that expresses the relative appearance of both words in each other neighborhood with respect to the absolute appearance of both words on the whole World-Wide Web. We derive this formula analyzing the comparison of relative appearance with respect to absolute appearance, and prove that the meaning bond is equal to the product of the number of webpages containing both words by the total number of webpages of the World-Wide Web, and dividing the result by the product of the number of webpages for each of the single words. We calculate the meaning bonds for several words and investigate different aspects by looking at specific examples.- Aerts, D., Broekaert, J., Czachor, M. and D'Hooghe, B. (2011). A quantum-conceptual explanation of violations of expected utility in economics.
Proceedings of QI2011-Fifth International Symposium on Quantum Interaction, Robert Gordon University, Aberdeen, Scotland, June 27-29, 2011.Quantum Interaction. Lecture Notes in Computer Science,7052, pp. 192-198. Archive reference and link: http://uk.arxiv.org/abs/1004.2525. doi: 10.1007/978-3-642-24971-6_19. download pdf.

Abstract: The expected utility hypothesis is one of the building blocks of classical economic theory and founded on Savage's Sure-Thing Principle. It has been put forward, e.g. by situations such as the Allais and Ellsberg paradoxes, that real-life situations can violate Savage's Sure-Thing Principle and hence also expected utility. We analyze how this violation is connected to the presence of the 'disjunction effect' of decision theory and use our earlier study of this effect in concept theory to put forward an explanation of the violation of Savage's Sure-Thing Principle, namely the presence of 'quantum conceptual thought' next to 'classical logical thought' within a double layer structure of human thought during the decision process. Quantum conceptual thought can be modeled mathematically by the quantum mechanical formalism, which we illustrate by modeling the Hawaii problem situation, a well-known example of the disjunction effect, and we show how the dynamics in the Hawaii problem situation is generated by the whole conceptual landscape surrounding the decision situation. We analyze the Ellsberg paradox situation, taking into account our hypothesis of the presence of quantum conceptual thought. We put forward a model for the conceptual landscape surrounding the decision situation of the Ellsberg paradox by making use of the data we gathered by conducting an experimental test of the Ellsberg paradox situation. We show how this gives rise to the violation of Savage's Sure-Thing Principle and explain its dynamics.- Aerts, D., Broekaert, J., D'Hooghe, B. and Note, N. (Eds.), (2011).
Worldviews, Science and Us: Bridging Knowledge and Its Implications for Our Perspectives of the World. Singapore: World Scientific.- Aerts, D., Broekaert, J., D'Hooghe, B. and Note, N. (2011). Interdisciplinarity and bridging knowledge. In D. Aerts, J. Broekaert, B. D'Hooghe and N. Note (Eds.),
- Aerts, D., Broekaert, J., D'Hooghe, B. and Sozzo, S. (2011). Quantum structure in economics: Risk versus ambiguity. In D. Aerts, J. Broekaert, B. D'Hooghe and N. Note (Eds.),
- Aerts, D., Broekaert, J. and Gabora, L. (2011). A case for applying an abstracted quantum formalism to cognition.
New Ideas in Psychology,29, pp. 136-146. doi: 10.1016/j.newideapsych.2010.06.002. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0404068. download pdf.

Abstract: The contextual nature of conscious experience suggests that in order to formally model it we should look to the domain of science where contextuality has been most seriously addressed: quantum mechanics. As in quantum mechanics, conscious experience consists of segments of dynamical evolution, which are not contextual and do not involve resolution of ambiguity or decision, and collapse events, which are context-dependent and involve a decision or the resolution of ambiguity. An abstracted quantum mechanical representation of the entity-context interaction, with its hidden creation of new states, is adapted to the description of the conceptualization process for various cognitive domains. We show that Bell inequalitiesŃthe definitive test for quantum structureŃare violated in the relationship between an abstract concept and instances or exemplars of that concept. We summarize work on a theory of concepts that focuses on how the potentiality of a concept gets actualized through interaction with a context, and how the mathematics of entanglement can be applied to concept combination. A full quantum model is presented for the description of (1) contextually-elicited opinions (as in an opinion pole situation), and (2) the alternating change of cognitive state in the liar paradox.- Aerts, D., Bruza, P., Hou, YX., Jose, J., Melucci, M., Nie, JY., Song, DW. (2011). Quantum theory-inspired search. In E. Giacobino, R. Pfeifer (Eds.),
Proceedings of the 2nd European Future Technologies Conference and Exhibition 2011 (FET 11). Book Series: Procedia Computer Science,7, pp. 278-280. doi: 10.1016/j.procs.2011.09.050.

Abstract: With the huge number and diversity of the users, the advertising products and services, the rapid growth of online multimedia resources, the context of information needs are even more broad and complex. Although research in search engine technology has led to various models over the past three decades, the investigation for effectively integrating the dimensions of context to deploy advanced search technology has been limited due to the lack of a unified modeling and evaluation framework. Quantum Theory (QT) has created new and unprecedented means for communicating and computing. Besides computer science, optics, electronics, physics, QT and search engine technology can be combined: interference in user interaction; entanglement in cognition; superposition in word meaning; non-classical probability in information ranking; complex vector spaces in multimedia search. This paper highlights our recent results on QT-inspired search engine technology.- Aerts, D., Bundervoet, S., Czachor, M., D'Hooghe, B., Gabora L. and Polk, P. (2011). On the foundations of the theory of evolution. In D. Aerts, J. Broekaert, B. D'Hooghe and N. Note (Eds.),

Abstract: In this paper we suggest an alternative to standard neodarwinian evolution theory. The problem is that Darwinism, which sees evolution as a consequence of random variation and natural selection is based on a materialistic - i.e. matter-based - view of science, while matter in itself is considered to be a very complex notion in modern physics. More specifically, on a microlevel, matter and energy are no longer retained within their simple form, and quantum mechanical models are proposed wherein potential form is considered in addition to actual form. We suggest that the starting point of evolution theory cannot be limited to actual variation whereupon is selected, but to variation in the potential of entities according to the context. We are developing formalism, referred to as 'Context driven Actualization of Potential' (CAP), which handles potentiality. CAP describes the evolution of entities as an actualization of potential which is defined by the context each instance of time. As in quantum mechanics, lack of knowledge of the entity, its context, or the interaction between context and entity leads to different forms of indeterminism in relation to the state of the entity. This indeterminism generates a - non-Kolmogorovian - distribution of probabilities that is different from the classical distribution of chance described by Darwinian evolution theory, which stems from a 'actuality focused', i.e. materialistic view of nature. In this paper we present a quantum evolution game that highlights the main differences, which stem from applying our new perspective. As a formal framework, CAP makes it possible to unite different aspects and perspectives on evolution. We conclude that it is more fundamental to consider evolution in general, and hence also biological evolution in specific, as a process of 'context driven actualization of potential', for which its material reduction is only a perspective.- Aerts, D., Czachor, M. and Sozzo, S. (2011). Quantum interaction approach in cognition, artificial intelligence and robotics.
In V. Privman and V. Ovchinnikov (Eds.), IARIA, Proceedings of the Fifth International Conference on Quantum, Nano and Micro Technologies, pp. 35-40. Archive reference and link: http://arxiv.org/abs/1104.3345. download pdf.

Abstract: The mathematical formalism of quantum mechanics has been successfully employed in the last years to model situations in which the use of classical structures gives rise to problematical situations, and where typically quantum effects, such as 'contextuality' and 'entanglement', have been recognized. This 'Quantum Interaction Approach' is briefly reviewed in this paper focusing, in particular, on the quantum models that have been elaborated to describe how concepts combine in cognitive science, and on the ensuing identification of a quantum structure in human thought. We point out that these results provide interesting insights toward the development of a unified theory for meaning and knowledge formalization and representation. Then, we analyze the technological aspects and implications of our approach, and a particular attention is devoted to the connections with symbolic artificial intelligence, quantum computation and robotics.- Aerts, D., de Ronde, C. and D'Hooghe, B. (2011). Compatibility and separability for classical and quantum entanglement. In D. Aerts, J. Broekaert, B. D'Hooghe and N. Note (Eds.),

Abstract: We study the concepts of compatibility and separability and their implications for quantum and classical systems. These concepts are illustrated on a macroscopic model for the singlet state of a quantum system of two entangled spin 1/2 with a parameter reflecting indeterminism in the measurement procedure. By varying this parameter we describe situations from quantum, intermediate to classical and study which tests are compatible or separated. We prove that for classical deterministic systems the concepts of separability and compatibility coincide, but for quantum systems and intermediate systems these concepts are generally different. More specifically, equal physical constraints in the model, which for classical deterministic measurements lead to non-separability 'and' non-compatibility of the measurements, give rise to compatible measurements that are non-separated whenever indeterminism is introduced. As a consequence, compatible measurements that are non-separated, and hence violate Bell's inequalities, can only be realized in the presence of indeterminism.- Aerts, D. and D'Hooghe, B. (2011). Potentiality states: Quantum versus classical emergence. In D. Aerts, J. Broekaert, B. D'Hooghe and N. Note (Eds.),

Abstract: We identify emergence with the existence of states of potentiality related to relevant physical quantities. We introduce the concept of 'potentiality state' operationally and show how it reduces to 'superposition state' when standard quantum mechanics can be applied. We consider several examples to illustrate our approach, and define the potentiality states giving rise to emergence in each example. We prove that Bell inequalities are violated by the potentiality states in the examples, which, taking into account Pitowsky's theorem, experimentally indicates the presence of quantum structure in emergence. In the first example emergence arises because of the many ways water can be subdivided into different vessels. In the second example, we put forward a full quantum description of the Liar paradox situation, and identify the potentiality states, which in this case turn out to be superposition states. In the example of the soccer team, we show the difference between classical emergence as stable dynamical pattern and emergence defined by a potentiality state, and show how Bell inequalities can be violated in the case of highly contextual experiments.- Aerts, D., D'Hooghe, B., Pinxten, H. and Wallerstein, I. (Eds.), (2011).
Worldviews, Science and Us: Interdisciplinary Perspectives on Worlds, Cultures and Society. Singapore: World Scientific.- Aerts, D., D'Hooghe, B., Pinxten, H. and Wallerstein, I. (2011). Interdisciplinary perspectives on worlds, cultures and society. In D. Aerts, B. D'Hooghe, R. Pinxten and I. Wallerstein (Eds.),
Worldviews, Science and Us: Interdisciplinary Perspectives on Worlds, Cultures and Society (pp 1-4). Singapore: World Scientific.- Aerts, D., D'Hooghe, B. and Sioen, M. (2011). Quantum axiomatics: Topological properties and classical properties of state property systems.
International Journal of Theoretical Physics,50, pp. 3635-3645. doi: 10.1007/s10773-011-0885-9.

Abstract: The definition of 'classical state' from (Aerts in K. Engesser, D. Gabbay and D. Lehmann (Eds.), Handbook of Quantum Logic and Quantum Structures. Elsevier, Amsterdam, 2009), used e.g. in Aerts et al. (http://arxiv.org/abs/quant-ph/0503083, 2010) to prove a decomposition theorem internally in the language of State Property Systems, presupposes as an additional datum an orthocomplementation on the property lattice of a physical system. In this paper we argue on the basis of the (epsilon,d)-model on the Poincare sphere that a notion of topologicity for states can be seen as an alternative (operationally foundable) classicality notion in the absence of an orthocomplementation, and compare it to the known and operationally founded concept of classicality.- Aerts, D., D'Hooghe, B. and Sozzo, S. (2011). A quantum cognition analysis of the Ellsberg paradox.
Proceedings of QI2011-Fifth International Symposium on Quantum Interaction, Robert Gordon University, Aberdeen, Scotland, June 27-29, 2011.Quantum Interaction. Lecture Notes in Computer Science,7052, pp. 95-104. Archive reference and link: http://arxiv.org/abs/1104.1459. doi: 10.1007/978-3-642-24971-6_10. download pdf.

Abstract: The 'expected utility hypothesis' is one of the foundations of classical approaches to economics and decision theory and Savage's 'Sure-Thing Principle' is a fundamental element of it. It has been put forward that real-life situations exist, illustrated by the 'Allais' and 'Ellsberg paradoxes', in which the Sure-Thing Principle is violated, and where also the expected utility hypothesis does not hold. We have recently presented strong arguments for the presence of a double layer structure, a 'classical logical' and a 'quantum conceptual', in human thought and that the quantum conceptual mode is responsible of the above violation. We consider in this paper the Ellsberg paradox, perform an experiment with real test subjects on the situation considered by Ellsberg, and use the collected data to elaborate a model for the conceptual landscape surrounding the decision situation of the paradox. We show that it is the conceptual landscape which gives rise to a violation of the Sure-Thing Principle and leads to the paradoxical situation discovered by Ellsberg.- Aerts, D., Gabora, L., Sozzo, S. and Veloz, T. (2011). Quantum structure in cognition: Fundamentals and applications.
In V. Privman and V. Ovchinnikov (Eds.), IARIA, Proceedings of the Fifth International Conference on Quantum, Nano and Micro Technologies, pp. 57-62. Archive reference and link: http://arxiv.org/abs/1104.3344. download pdf.

Abstract: Experiments in cognitive science and decision theory show that the ways in which people combine concepts and make decisions cannot be described by classical logic and probability theory. This has serious implications for applied disciplines such as information retrieval, artificial intelligence and robotics. Inspired by a mathematical formalism that generalizes quantum mechanics the authors have constructed a contextual framework for both concept representation and decision making, together with quantum models that are in strong alignment with experimental data. The results can be interpreted by assuming the existence in human thought of a double-layered structure, a 'classical logical thought' and a 'quantum conceptual thought', the latter being responsible of the above paradoxes and nonclassical effects. The presence of a quantum structure in cognition is relevant, for it shows that quantum mechanics provides not only a useful modeling tool for experimental data but also supplies a structural model for human and artificial thought processes. This approach has strong connections with theories formalizing meaning, such as semantic analysis, and has also a deep impact on computer science, information retrieval and artificial intelligence. More specifically, the links with information retrieval are discussed in this paper.- Aerts, D. and Sozzo, S. (2011). Quantum structure in cognition: Why and how concepts are entangled.
Proceedings of QI2011-Fifth International Symposium on Quantum Interaction, Robert Gordon University, Aberdeen, Scotland, June 27-29, 2011.Quantum Interaction. Lecture Notes in Computer Science,7052, pp. 116-127. Archive reference and link: http://arxiv.org/abs/1104.1322. doi: 10.1007/978-3-642-24971-6_12. download pdf.

Abstract: One of us has recently elaborated a theory for modelling concepts that uses the state context property (SCoP) formalism, i.e. a generalization of the quantum formalism. This formalism incorporates context into the mathematical structure used to represent a concept, and thereby models how context influences the typicality of a single exemplar and the applicability of a single property of a concept. The notion of 'state of a concept' is introduced to account for this contextual influence, which proposes a solution for the 'Pet-Fish problem' and several difficulties occurring in concept theory. Then, a quantum model has been worked out which reproduces the membership weights of several exemplars of concepts and their combinations. We show in this paper that a further relevant effect appears in a natural way whenever two or more concepts combine, namely, 'entanglement'. The presence of entanglement is explicitly revealed by considering a specific example with two concepts, constructing some Bell's inequalities for this example, testing them in a real experiment with test subjects, and finally proving that Bell's inequalities are violated in this case. We show that the intrinsic and unavoidable character of entanglement can be explained in terms of the weights of the exemplars of the combined concept with respect to the weights of the exemplars of the component concepts, and elaborate a concrete quantum model for the proposed example.- Patyk, A., Czachor, M. and Aerts, D. (2011). A comparison of geometric analogues of holographic reduced representations, original holographic reduced representations and binary spatter codes.
Proceedings of the Federated Conference on Computer Science and Information Systems (FedCSIS), Szczecin, Poland, 18-21 September, 2011, pp. 221-228.

Abstract: Geometric Analogues of Holographic Reduced Representations (GA HRR) employ role-filler binding based on geometric products. Atomic objects are real-valued vectors in n-dimensional Euclidean space and complex statements belong to a hierarchy of multivectors. The paper reports a battery of tests aimed at comparison of GA HRR with Holographic Reduced Representation (HRR) and Binary Spatter Codes (BSC). Firstly, we perform a test of GA HRR which is analogous to the one proposed by Plate. Plate's simulation involved several thousand 512-dimensional vectors stored in clean-up memory. The purpose was to study efficiency of HRR but also to provide a counterexample to claims that role-filler representations do not permit one component of a relation to be retrieved given the others. We repeat Plate's test on a continuous version of GA HRR - GAc (as opposed to its discrete version) and compare the results with the original HRR and BSC. The object of the test is to construct statements concerning multiplication and addition. For example, "2á3 = 6" is constructed as times2,3 = times+operand*(num2 + num3)+result*num6. To look up this vector one then constructs a similar statement with one of the components missing and checks whether it points correctly to times2,3. We concentrate on comparison of recognition percentage for the three models for comparable data size, rather than on the time taken to achieve high percentage. Results show that the best models for storing and recognizing multiple similar statements are GAc and Binary Spatter Codes with recognition percentage highly above 90.- Patyk, A., Czachor, M. and Aerts, D. (2011). Distributed representations based on geometric algebra: The continuous model.
Informatica,35, pp. 407-417.

Abstract: Authors revise the concept of a distributed representation of data as well as two previously developed models: Holographic Reduced Representation (HRR) and Binary Spatter Codes (BSC). A Geometric Analogue (GAc Ń "c" stands for continuous as opposed to its discrete version) of HRR is introduced Đ it employs role-filler binding based on geometric products. Atomic objects are real-valued vectors in n-dimensional Euclidean space while complex data structures belong to a hierarchy of multivectors. The paper reports on a test aimed at comparison of GAc with HRR and BSC. The test is analogous to the one proposed by Tony Plate in the mid 90s. We repeat PlateŐs test on GAc and compare the results with the original HRR and BSCŃwe concentrate on comparison of recognition percentage for the three models for comparable data size, rather than on the time taken to achieve high percentage. Results show that the best models for storing and recognizing multiple similar structures are GAc and BSC with recognition percentage highly above 90. The paper ends with remarks on perspective applications of geometric algebra to quantum algorithms.- Veloz, T., Gabora, L., Eyjolfson, J. and Aerts, D. (2011). Toward a formal model of the shifting relationship between concepts and contexts during associative thought.
Proceedings of QI2011-Fifth International Symposium on Quantum Interaction, Robert Gordon University, Aberdeen, Scotland, June 27-29, 2011.Quantum Interaction. Lecture Notes in Computer Science,7052, pp. 25-34. doi: 10.1007/978-3-642-24971-6_4.

Abstract: The quantum inspired State Context Property (SCOP) theory of concepts is unique amongst theories of concepts in offering a means of incorporating that for each concept in each different context there are an unlimited number of exemplars, or states, of varying degrees of typicality. Working with data from a study in which participants were asked to rate the typicality of exemplars of a concept for different contexts, and introducing a state-transition threshold, we built a SCOP model of how states of a concept arise differently in associative versus analytic (or divergent and convergent) modes of thought. Introducing measures of expected typicality for both states and contexts, we show that by varying the threshold, the expected typicality of different contexts changes, and seemingly atypical states can become typical. The formalism provides a pivotal step toward a formal explanation of creative thought processes.## 2012

- Aerts, D., Broekaert, J., Gabora, L. and Veloz, T. (2012). The guppy effect as interference.
Quantum Interaction. Lecture Notes in Computer Science,7620, pp 36-47. download pdf. doi: 10.1007/978-3-642-35659-9_4.

Abstract:People use conjunctions and disjunctions of concepts in ways that violate the rules of classical logic, such as the law of compositionality. Specifically, they overextend conjunctions of concepts, a phenomenon referred to as the Guppy Effect. We build on previous efforts to develop a quantum model that explains the Guppy Effect in terms of interference. Using a well-studied data set with 16 exemplars that exhibit the Guppy Effect, we developed a 17-dimensional complex Hilbert space H that models the data and demonstrates the relationship between overextension and interference. We view the interference effect as, not a logical fallacy on the conjunction, but a signal that out of the two constituent concepts, a new concept has emerged.- Aerts, D., D'Hooghe, B. and Sozzo, S. (2012). A quantum-like approach to the stock market. In M. D'Ariano, S.-M. Fei, E. Haven, B. Hiesmayr, G. Jaeger, A. Khrennikov, J.-A. Larsson (Eds.), Foundations of Probability and Physics - 6, AIP Conference Proceedings,
1424, pp. 495-506. download pdf.

Abstract: Modern approaches to stock pricing in quantitative finance are typically founded on the 'Black-Scholes model' and the underlying 'random walk hypothesis'. Empirical data indicate that this hypothesis works well in stable situations but, in abrupt transitions such as during an economical crisis, the random walk model fails and alternative descriptions are needed. For this reason, several proposals have been recently forwarded which are based on the formalism of quantum mechanics. In this paper we apply the 'SCoP formalism', elaborated to provide an operational foundation of quantum mechanics, to the stock market. We argue that a stock market is an intrinsically contextual system where agents' decisions globally influence the market system and stocks prices, determining a nonclassical behavior. More specifically, we maintain that a given stock does not generally have a definite value, e.g., a price, but its value is actualized as a consequence of the contextual interactions in the trading process. This contextual influence is responsible of the non-Kolmogorovian quantum-like behavior of the market at a statistical level. Then, we propose a 'sphere model' within our 'hidden measurement formalism' that describes a buying/selling process of a stock and shows that it is intuitively reasonable to assume that the stock has not a definite price until it is traded. This result is relevant in our opinion since it provides a theoretical support to the use of quantum models in finance.- Aerts, D., Czachor, M., Kuna, M., Sinervo, B. and Sozzo, S. (2012). Quantum probabilistuc structures in competing lizard communities. Nature Precedings, hdl:10101/npre.2012.6954.1. download pdf.

Abstract: Almost two decades of research on the use of the mathematical formalism of quantum theory as a modeling tool for entities and their dynamics in domains different from the micro-world has now firmly shown the systematic appearance of quantum structures in aspects of human behavior and thought, such as in cognitive processes of decision-making, and in the way concepts are combined into sentences. In this paper, we extend this insight to animal behavior showing that a quantum probabilistic structure models the mating competition of three side-blotched lizard morphs. We analyze a set of experimental data collected from 1990 to 2011 on these morphs, whose territorial behavior follows a cyclic rock-paper-scissors (RPS) dynamics. Consequently we prove that a single classical Kolmogorovian space does not exist for the lizard's dynamics, and elaborate an explicit quantum description in Hilbert space faithfully modeling the gathered data. This result is relevant for population dynamics as a whole, since many systems, e.g. the so called plankton paradox situation, are believed to contain elements of cyclic competition.- Aerts, D. and Sozzo, S. (2012). A contextual risk model for the Ellsberg paradox.
Journal of Engineering Science and Technology Review,4, pp. 246-250. download pdf.

Abstract: The Allais and Ellsberg paradoxes show that the expected utility hypothesis and Savage's Sure-Thing Principle are violated in real life decisions. The popular explanation in terms of 'ambiguity aversion' is not completely accepted. On the other hand, we have recently introduced a notion of 'contextual risk' to mathematically capture what is known as 'ambiguity' in the economics literature. Situations in which contextual risk occurs cannot be modeled by Kolmogorovian classical probabilistic structures, but a non-Kolmogorovian framework with a quantum-like structure is needed. We prove in this paper that the contextual risk approach can be applied to the Ellsberg paradox, and elaborate a 'sphere model' within our 'hidden measurement formalism' which reveals that it is the overall conceptual landscape that is responsible of the disagreement between actual human decisions and the predictions of expected utility theory, which generates the paradox. This result points to the presence of a 'quantum conceptual layer' in human thought which is superposed to the usually assumed 'classical logical layer'.- Aerts, D. and Sozzo, S. (2012). Contextual risk and its relevance in economics.
Journal of Engineering Science and Technology Review,4, pp. 241-245. download pdf.

Abstract: Uncertainty in economics still poses some fundamental problems illustrated, e.g., by the Allais and Ellsberg paradoxes. To overcome these difficulties, economists have introduced an interesting distinction between 'risk' and 'ambiguity' depending on the existence of a (classical Kolmogorovian) probabilistic structure modeling these uncertainty situations. On the other hand, evidence of everyday life suggests that 'context' plays a fundamental role in human decisions under uncertainty. Moreover, it is well known from physics that any probabilistic structure modeling contextual interactions between entities structurally needs a non-Kolmogorovian quantum-like framework. In this paper we introduce the notion of 'contextual risk' with the aim of modeling a substantial part of the situations in which usually only 'ambiguity' is present. More precisely, we firstly introduce the essentials of an operational formalism called 'the hidden measurement approach' in which probability is introduced as a consequence of fluctuations in the interaction between entities and contexts. Within the hidden measurement approach we propose a 'sphere model' as a mathematical tool for situations in which contextual risk occurs. We show that a probabilistic model of this kind is necessarily non-Kolmogorovian, hence it requires either the formalism of quantum mechanics or a generalization of it. This insight is relevant, for it explains the presence of quantum or, better, quantum-like, structures in economics, as suggested by some authors, and can serve to solve the aforementioned paradoxes.- Aerts, D. and Sozzo, S. (2012). Quantum interference in cognition: Structural aspects of the brain. In V. Ovchinnikov and P. Dini (Eds.),
IARIA, Proceedings of the Sixth International Conference on Quantum, Nano and Micro Technologies, pp. 33-41. download pdf.

Abstract: We identify the presence of typically quantum effects, namely 'superposition' and 'interference', in what happens when human concepts are combined, and provide a quantum model in complex Hilbert space that represents faithfully experimental data measuring the situation of combining concepts. Our model shows how 'interference of concepts' explains the effects of underextension and overextension when two concepts combine to the disjunction of these two concepts. This result supports our earlier hypothesis that human thought has a superposed two-layered structure, one layer consisting of 'classical logical thought' and a superposed layer consisting of 'quantum conceptual thought'. Possible connections with recent findings of a 'grid-structure' for the brain are analyzed, and influences on the mind/brain relation, and consequences on applied disciplines, such as artificial intelligence and quantum computation, are considered.- Aerts, D. and Sozzo, S. (2012). Quantum Model Theory (QMod): Modeling contextual emergent entangled interfering entities.
Quantum Interaction. Lecture Notes in Computer Science,7620, pp 126-137. download pdf. doi: 10.1007/978-3-642-35659-9_12.

Abstract: In this paper we present 'Quantum Model Theory' (QMod), a theory we developed to model entities that entail the typical quantum effects of 'contextuality', 'superposition', 'interference', 'entanglement' and 'emergence'. The aim of QMod is to put forward a theoretical framework that is more general than standard quantum mechanics, in the sense that, for its complex version it only uses this quantum calculus locally, i.e. for each context corresponding to a measurement, and for its real version it does not need the property of 'linearity of the set of states' to model the quantum effect. In this sense, QMod is a generalization of quantum mechanics, similar to how the general relativity manifold mathematical formalism is a generalization of special relativity. We prove by means of a representation theorem that QMod can be used for any entity entailing the typical quantum effects mentioned above. Some examples of application of QMod in concept theory and macroscopic physics are also considered.- Aerts, D. and Sozzo, S. (2012). Entanglement of conceptual entities in Quantum Model Theory (QMod).
Quantum Interaction. Lecture Notes in Computer Science,7620, pp 114-125. download pdf. doi: 10.1007/978-3-642-35659-9_11.

Abstract: We have recently elaborated 'Quantum Model Theory' (QMod) to model situations where the quantum effects of contextuality, interference, superposition, entanglement and emergence, appear without the entities giving rise to these situations having necessarily to be of microscopic nature. We have shown that QMod models without introducing linearity for the set of the states. In this paper we prove that QMod, although not using linearity for the state space, provides a method of identification for entangled states and an intuitive explanation for their occurrence. We illustrate this method for entanglement identification with concrete examples.- Aerts, D. and Sozzo, S. (2012). Quantum structure in economics: The Ellsberg paradox. In M. D'Ariano, S.-M. Fei, E. Haven, B. Hiesmayr, G. Jaeger, A. Khrennikov, J.-A. Larsson (Eds.), Foundations of Probability and Physics - 6, AIP Conference Proceedings,
1424, pp. 487-494. download pdf.

Abstract: The expected utility hypothesis and Savage's Sure-Thing Principle are violated in real life decisions, as shown by the Allais and Ellsberg paradoxes. The popular explanation in terms of ambiguity aversion is not completely accepted. As a consequence, uncertainty is still problematical in economics. To overcome these difficulties a distinction between risk and ambiguity has been introduced which depends on the existence of a Kolmogorovian probabilistic structure modeling these uncertainties. On the other hand, evidence of everyday life suggests that context plays a fundamental role in human decisions under uncertainty. Moreover, it is well known from physics that any probabilistic structure modeling contextual interactions between entities structurally needs a non-Kolmogorovian framework admitting a quantum-like representation. For this reason, we have recently introduced a notion of contextual risk to mathematically capture situations in which ambiguity occurs. We prove in this paper that the contextual risk approach can be applied to the Ellsberg paradox, and elaborate a sphere model within our hidden measurement formalism which reveals that it is the overall conceptual landscape that is responsible of the disagreement between actual human decisions and the predictions of expected utility theory, which generates the paradox. This result points to the presence of a quantum conceptual layer in human thought which is superposed to the usually assumed classical logical layer, and conceptually supports the thesis of several authors suggesting the presence of quantum structure in economics and decision theory.- Aerts, D., Sozzo, S. and Tapia, J. (2012). A quantum model for the Ellsberg and Machina paradoxes.
Quantum Interaction. Lecture Notes in Computer Science,7620, pp 48-59. download pdf. doi: 10.1007/978-3-642-35659-9_5.

Abstract: The Ellsberg and Machina paradoxes reveal that expected utility theory is problematical when real subjects take decisions under uncertainty. Suitable generalizations of expected utility exist which attempt to solve the Ellsberg paradox, but none of them provides a satisfactory solution of the Machina paradox. In this paper we elaborate a quantum model in Hilbert space describing the Ellsberg situation and also the Machina situation, and show that we can model the specific aspect of the Machina situation that is unable to be modeled within the existing generalizations of expected utility.- Vanderbeeken, R., Le Roy, F., Stalpaert, C. and Aerts, D. (Eds.), (2011).
Drunk on Capitalism: An Interdisciplinary Reflection on Market Economy, Art and Science. Dordrecht: Springer.## 2013

- Aerts, D. (2013). La mecànica cuántica y la conceptualidad: Sobre materia, historias, semántica y espacio-tiempo.
Scientiae Studia,11, pp. 75-100. download pdf. doi: 10.1590/S1678-31662013000100004. (translated from: Aerts, D. (2012). Quantum theory and conceptuality: Matter, stories, semantics and space-time. Archive reference and link: http://arxiv.org/abs/1110.4766. download pdf.)

Abstract:We elaborate the new interpretation of quantum theory that we recently proposed, according to which quantum particles are considered conceptual entities mediating between pieces of ordinary matter which are considered to act as memory structures for them. Our aim is to identify what is the equivalent for the human cognitive realm of what physical space-time is for the realm of quantum particles and ordinary matter. For this purpose, we identify the notion of 'story' as the equivalent within the human cognitive realm of what ordinary matter is in the physical quantum realm, and analyze the role played by the logical connectives of disjunction and conjunction with respect to the notion of locality. Similarly to what we have done in earlier investigations on this new quantum interpretation, we use the specific cognitive environment of the World-Wide Web to elucidate the comparisons we make between the human cognitive realm and the physical quantum realm.- Aerts, D., Broekaert, J., Gabora, L. and Sozzo, S. (2013). Quantum structure and human thought.
Behavioral and Brain Sciences,36, pp. 274-277. doi: 10.1017/S0140525X12002841. download pdf.

Abstract:We support the authors' claims, except that we point out that also quantum structure different from quantum probability abundantly plays a role in human cognition. We put forward several elements to illustrate our point, mentioning entanglement, contextuality, interference, and emergence as effects, and states, observables, complex numbers, and Fock space as specific mathematical structures.- Aerts, D., Broekaert, J, Sozzo, S. and Veloz, T. (2013). Meaning-focused and quantum-inspired information retrieval. In H. Atmanspacher, E. Haven, K. Kitto and D. Raine (Eds),
Quantum Interaction. QI 2013. Lecture Notes in Computer Science,8369, pp. 71-83. Berlin, Heidelberg: Springer. Archive reference and link: http://arxiv.org/abs/1304.0104. download pdf.

Abstract: In recent years, quantum-based methods have promisingly integrated the traditional procedures in information retrieval (IR) and natural language processing (NLP). Inspired by our research on the identification and application of quantum structures in cognition, we put forward a new 'quantum meaning based' framework for structured query retrieval in text corpora and standardized testing corpora. This scheme for IR rests on 'entities of meaning', e.g., concepts and their combinations, and 'documents', i.e. traces of entities of meaning, as basic entities. The meaning content of the latter is then reconstructed in terms of the meaning content of the former through an 'inverse problem'. The advantages with respect to traditional approaches, such as Latent Semantic Analysis (LSA), are finally discussed by means of concrete examples.- Aerts, D., Broekaert, J, Sozzo, S. and Veloz, T. (2013). The quantum challenge in concept theory and natural language processing. In E. G. Lasker (Ed.),
Proceedings of the 25th International Conference on System Research, Informatics & Cybernetics, pp. 13-17, IIAS. Archive reference and link: http://arxiv.org/abs/1306.2838. download pdf.

Abstract: The mathematical formalism of quantum theory has been successfully used in human cognition to model decision processes and to deliver representations of human knowledge. As such, quantum cognition inspired tools have improved technologies for Natural Language Processing and Information Retrieval. In this paper, we overview the quantum cognition approach developed in our Brussels team during the last two decades, specifically our identification of quantum structures in human concepts and language, and the modeling of data from psychological and corpus-text-based experiments. We discuss our quantum-theoretic framework for concepts and their conjunctions/disjunctions in a Fock-Hilbert space structure, adequately modeling a large amount of data collected on concept combinations. Inspired by this modeling, we put forward elements for a quantum contextual and meaning-based approach to information technologies in which 'entities of meaning' are inversely reconstructed from texts, which are considered as traces of these entities' states.- Aerts, D., Czachor, M., Kuna, M. and Sozzo, S. (2013). Systems, environments, and soliton rate equations: A non-Kolmogorovian framework for population dynamics.
Ecological Modelling,267, pp. 80-92. doi: 10.1016/j.ecolmodel.2013.07.010. Archive reference and link: http://arxiv.org/abs/1303.0281. download pdf.

Abstract:Soliton rate equations are based on non-Kolmogorovian models of probability and naturally include autocatalytic processes. The formalism is not widely known but has great unexplored potential for applications to systems interacting with environments. Beginning with links of contextuality to non-Kolmogorovity we introduce the general formalism of soliton rate equations and work out explicit examples of subsystems interacting with environments. Of particular interest is the case of a soliton autocatalytic rate equation coupled to a linear conservative environment, a formal way of expressing seasonal changes. Depending on strength of the system-environment coupling we observe phenomena analogous to hibernation or even complete blocking of decay of a population.- Aerts, D., Gabora, L. and Sozzo, S. (2013). Concepts and their dynamics: A quantum-theoretic modeling of human thought. Topics in Cognitive Science,
5, pp. 737-772. doi: 10.1111/tops.12042. Archive reference and link: http://arxiv.org/abs/1206.1069. download pdf.

Abstract: We analyze different aspects of our quantum modeling approach of human concepts, and more specifically focus on the quantum effects of contextuality, interference, entanglement and emergence, illustrating how each of them makes its appearance in specific situations of the dynamics of human concepts and their combinations. We point out the relation of our approach, which is based on an ontology of a concept as an entity in a state changing under influence of a context, with the main traditional concept theories, i.e. prototype theory, exemplar theory and theory theory. We ponder about the question why quantum theory performs so well in its modeling of human concepts, and shed light on this question by analyzing the role of complex amplitudes, showing how they allow to describe interference in the statistics of measurement outcomes, while in the traditional theories statistics of outcomes originates in classical probability weights, without the possibility of interference. The relevance of complex numbers, the appearance of entanglement, and the role of Fock space in explaining contextual emergence, all as unique features of the quantum modeling, are explicitly revealed in this paper by analyzing human concepts and their dynamics.- Aerts, D. and Sozzo, S. (2013). Entanglement zoo I: Foundational and structural aspects. In H. Atmanspacher, E. Haven, K. Kitto and D. Raine (Eds),
Quantum Interaction. QI 2013. Lecture Notes in Computer Science,8369, pp. 84-96. Berlin, Heidelberg: Springer. Archive reference and link: http://arxiv.org/abs/1304.0100. download pdf.

Abstract: We put forward a general classification for a structural description of the entanglement present in compound entities experimentally violating Bell's inequalities, making use of a new entanglement scheme that we developed recently. Our scheme, although different from the traditional one, is completely compatible with standard quantum theory, and enables quantum modeling in complex Hilbert space for different types of situations. Namely, situations where entangled states and product measurements appear ('customary quantum modeling'), and situations where states and measurements and evolutions between measurements are entangled ('nonlocal box modeling', 'nonlocal non-marginal box modeling'). The role played by Tsirelson's bound and marginal distribution law is emphasized. Specific quantum models are worked out in detail in complex Hilbert space within this new entanglement scheme.- Aerts, D. and Sozzo, S. (2013). Entanglement zoo II: Examples in physics and cognition. In H. Atmanspacher, E. Haven, K. Kitto and D. Raine (Eds),
Quantum Interaction. QI 2013. Lecture Notes in Computer Science,8369, pp. 97-109. Berlin, Heidelberg: Springer. Archive reference and link: http://arxiv.org/abs/1304.0102. download pdf.

Abstract: We have recently presented a general scheme enabling quantum modeling of different types of situations that violate Bell's inequalities. In this paper, we specify this scheme for a combination of two concepts. We work out a quantum Hilbert space model where 'entangled measurements' occur in addition to the expected 'entanglement between the component concepts', or 'state entanglement'. We extend this result to a macroscopic physical entity, the 'connected vessels of water', which maximally violates Bell's inequalities. We enlighten the structural and conceptual analogies between the cognitive and physical situations which are both examples of a nonlocal non-marginal box modeling in our classification.- Aerts, D. and Sozzo, S. (2013). General quantum Hilbert Space modeling scheme for entanglement. In V. Ovchinnikov and P. Dini (Eds.),
Proceedings of the Seventh International Conference on Quantum, Nano and Micro Technologies (IARIA, 2013), pp. 25-31. Archive reference and link: http://arxiv.org/abs/1304.3733. download pdf.

Abstract: We work out a classification scheme for quantum modeling in Hilbert space of any kind of composite entity violating Bell's inequalities and exhibiting entanglement. Our theoretical framework includes situations with entangled states and product measurements ('customary quantum situation'), and also situations with both entangled states and entangled measurements ('nonlocal box situation', 'nonlocal non-marginal box situation'). We show that entanglement is structurally a joint property of states and measurements. Furthermore, entangled measurements enable quantum modeling of situations that are usually believed to be 'beyond quantum'. Our results are also extended from pure states to quantum mixtures.## 2014

- Aerts, D. (2014). Quantum theory and human perception of the macro-world.
Frontiers of Psychology,5, 554. Archive reference and link: https://arxiv.org/abs/1403.4307. download pdf.

Abstract:We investigate the question of 'why customary macroscopic entities appear to us humans as they do, i.e. as bounded entities occupying space and persisting through time', starting from our knowledge of quantum theory, how it affects the behavior of such customary macroscopic entities, and how it influences our perception of them. For this purpose, we approach the question from three perspectives. Firstly, we look at the situation from the standard quantum angle, more specifically the de Broglie wavelength analysis of the behavior of macroscopic entities, indicate how a problem with spin and identity arises, and illustrate how both play a fundamental role in well-established experimental quantum-macroscopical phenomena, such as Bose-Einstein condensates. Secondly, we analyze how the question is influenced by our result in axiomatic quantum theory, which proves that standard quantum theory is structurally incapable of describing separated entities. Thirdly, we put forward our new 'conceptual quantum interpretation', including a highly detailed reformulation of the question to confront the new insights and views that arise with the foregoing analysis. At the end of the final section, a nuanced answer is given that can be summarized as follows. The specific and very classical perception of human seeing -- light as a geometric theory -- and human touching -- only ruled by Pauli's exclusion principle -- plays a role in our perception of macroscopic entities as ontologically stable entities in space. To ascertain quantum behavior in such macroscopic entities, we will need measuring apparatuses capable of its detection. Future experimental research will have to show if sharp quantum effects -- as they occur in smaller entities -- appear to be ontological aspects of customary macroscopic entities.- Aerts, D. (2014). Quantum and concept combination, entangled measurements and prototype theory.
Topics in Cognitive Science,6, pp. 129-137. Archive reference and link: http://arxiv.org/abs/1303.2430. download pdf.

Abstract:We extend a quantum model in Hilbert space developed in Aerts (2007a) into a quantum field theoric model in Fock space for the modeling of the combination of concepts. Items and concepts are represented by vectors in Fock space and membership weights of items are modeled by quantum probabilities. We apply this theory to model the disjunction of concepts and show that the predictions of our theory for the membership weights of items regarding the disjunction of concepts match with great accuracy the complete set of results of an experiment conducted by Hampton (1988b). It are the quantum effects of interference and superposition of that are at the origin of the effects of overextension and underextension observed by Hampton as deviations from a classical use of the disjunction. It is essential for the perfect matches we obtain between the predictions of the quantum field model and Hampton's experimental data that items can be in superpositions of `different numbers states' which proves that the genuine structure of quantum field theory is needed to match predictions with experimental data.- Aerts, D., Czachor, M., Kuna, M., Sinervo, B. and Sozzo, S. (2014). Quantum probabilistic structures in competing lizard communities.
Ecological Modelling,281, pp. 38-51. Archive reference and link: http://arxiv.org/abs/1212.0109. download pdf.

Abstract:Almost two decades of research on the use of the mathematical formalism of quantum theory as a modeling tool for entities and their dynamics in domains different from the micro-world has now firmly shown the systematic appearance of quantum structures in aspects of human behavior and thought, such as in cognitive processes of decision-making, and in the way concepts are combined into sentences. In this paper, we extend this insight to animal behavior showing that a quantum probabilistic structure models the mating competition of three side-blotched lizard morphs. We analyze a set of experimental data collected from 1990 to 2011 on these morphs, whose territorial behavior follows a cyclic rock-paper-scissors (RPS) dynamics. Consequently we prove that a single classical Kolmogorovian space does not exist for the lizard's dynamics, and elaborate an explicit quantum description in Hilbert space faithfully modeling the gathered data. This result is relevant for population dynamics as a whole, since many systems, e.g. the so called plankton paradox situation, are believed to contain elements of cyclic competition.- Aerts, D. and Sassoli de Bianchi, M. (2014). Solving the hard problem of Bertrand's paradox.
Journal of Mathematical Physics,55, 083503. Archive reference and link: https://arxiv.org/abs/1403.4139. download pdf.

Abstract: Bertrand's paradox is a famous problem of probability theory, pointing to a possible inconsistency in Laplace's principle of insufficient reason. In this article we show that Bertrand's paradox contains two different problems: an "easy" problem and a "hard" problem. The easy problem can be solved by formulating Bertrand's question in sufficiently precise terms, so allowing for a non ambiguous modelization of the entity subjected to the randomization. We then show that once the easy problem is settled, also the hard problem becomes solvable, provided Laplace's principle of insufficient reason is applied not to the outcomes of the experiment, but to the different possible "ways of selecting" an interaction between the entity under investigation and that producing the randomization. This consists in evaluating a huge average over all possible "ways of selecting" an interaction, which we call a 'universal average'. Following a strategy similar to that used in the definition of the Wiener measure, we calculate such universal average and therefore solve the hard problem of Bertrand's paradox. The link between Bertrand's problem of probability theory and the measurement problem of quantum mechanics is also briefly discussed.- Aerts, D. and Sassoli de Bianchi, M. (2014). The extended Bloch representation of quantum mechanics and the hidden-measurement solution to the measurement problem.
Annals of Physics,351, pp. 975-1025. Archive reference and link: https://arxiv.org/abs/1404.2429. download pdf.

Abstract: A generalized Bloch sphere, in which the states of a quantum entity of arbitrary dimension are geometrically represented, is investigated and further extended, to also incorporate the measurements. This extended representation constitutes a general solution to the measurement problem, inasmuch it allows to derive the Born rule as an average over hidden-variables, describing not the state of the quantum entity, but its interaction with the measuring system. According to this modelization, a quantum measurement is to be understood, in general, as a tripartite process, formed by an initial deterministic decoherence-like process, a subsequent indeterministic collapse-like process, and a final deterministic purification-like process. We also show that quantum probabilities can be generally interpreted as the probabilities of a first-order non-classical theory, describing situations of maximal lack of knowledge regarding the process of actualization of potential interactions, during a measurement.- Aerts, D. and Sozzo, S. (2014). Quantum entanglement in concept combinations.
International Journal of Theoretical Physics,53, pp 3587-3603. Archive reference and link: http://arxiv.org/abs/1302.3831. download pdf.

Abstract: Research in the application of quantum structures to cognitive science confirms that these structures quite systematically appear in the dynamics of concepts and their combinations and quantum-based models faithfully represent experimental data of situations where classical approaches are problematical. In this paper, we analyze the data we collected in an experiment on a specific conceptual combination, showing that Bell's inequalities are violated in the experiment. We present a new refined entanglement scheme to model these data within standard quantum theory rules, where 'entangled measurements and entangled evolutions' occur, in addition to the expected 'entangled states', and present a full quantum representation in complex Hilbert space of the data. This stronger form of entanglement in measurements and evolutions might have relevant applications in the foundations of quantum theory, as well as in the interpretation of nonlocality tests. It could indeed explain some non-negligible 'anomalies' identified in EPR-Bell experiments.- Aerts, D. and Sozzo, S. (2014). What is quantum? Unifying Its micro-physical and structural appearance. In H. Atmanspacher, C. Bergomi, T. Filk and K. Kitto (Eds),
Quantum Interaction. QI 2014. Lecture Notes in Computer Science,8951, pp. 12-23. Berlin: Springer, Cham. Archive reference and link: https://arxiv.org/abs/1405.7572. download pdf.

Abstract: We can recognize two modes in which 'quantum appears' in macro domains: (i) a 'micro-physical appearance', where quantum laws are assumed to be universal and they are transferred from the micro to the macro level if suitable 'quantum coherence' conditions (e.g., very low temperatures) are realized, (ii) a 'structural appearance', where no hypothesis is made on the validity of quantum laws at a micro level, while genuine quantum aspects are detected at a structural-modeling level. In this paper, we inquire into the connections between the two appearances. We put forward the explanatory hypothesis that, 'the appearance of quantum in both cases' is due to 'the existence of a specific form of organisation, which has the capacity to cope with random perturbations that would destroy this organisation when not coped with'. We analyse how 'organisation of matter', 'organisation of life', and 'organisation of culture', play this role each in their specific domain of application, point out the importance of evolution in this respect, and put forward how our analysis sheds new light on 'what quantum is'.- Aerts, D., Sozzo, S. and Tapia, J. (2014). Identifying quantum structures in the Ellsberg paradox.
International Journal of Theoretical Physics,53, pp. 3666-3682. Archive reference and link: http://arxiv.org/abs/1302.3850. download pdf.

Abstract: Empirical evidence has confirmed that quantum effects systematically occur also outside the microscopic domain, while quantum structures satisfactorily model various situations in several areas of science, including biological, cognitive and social processes. In this paper, we elaborate a quantum mechanical model which faithfully describes the Ellsberg paradox in economics, showing that the mathematical formalism of quantum mechanics is capable to represent the ambiguity present in this kind of situations, because of the presence of contextuality. Then, we analyze the data collected in a concrete experiment we performed on the Ellsberg paradox and work out a complete representation of them in complex Hilbert space. We prove that the self-adjoint operators representing the two processes of decision of our test subjects related to the two bets cannot be commuting operators. This shows that the presence of quantum structure is genuine, and due to the statistics following from our experimental data. It also proves that no classical probabilistic model is possible for these collected experimental data on the Ellsberg conceptual situation. Our approach sheds light on `ambiguity laden' decision processes in economics and decision theory, and allows to deal with different Ellsberg-type generalizations, e.g., the Machina paradox.## 2015

- Aerts, D. and Sassoli de Bianchi, M. (2015). The unreasonable success of quantum probability I: Quantum measurements as uniform fluctuations.
Journal of Mathematical Psychology,67, pp. 51-75. Archive reference and link: http://arxiv.org/abs/1401.2647. download pdf.

Abstract:We introduce a model which allows to represent the probabilities associated with an arbitrary measurement situation and use it to explain the emergence of quantum probabilities (the Born rule) as 'uniform' fluctuations on this measurement situation. The model exploits the geometry of simplexes to represent the states, in a way that the measurement probabilities can be derived as the 'Lebesgue measure' of suitably defined convex subregions of the simplexes. We consider a very simple and evocative physical realization of the abstract model, using a material point particle which is acted upon by elastic membranes, which by breaking and collapsing produce the different possible outcomes. This easy to visualize mechanical realization allows one to gain considerable insight into the possible hidden structure of an arbitrary measurement process. We also show that the Lebesgue-model can be further generalized into a model describing conditions of lack of knowledge generated by 'non-uniform' fluctuations. In this more general framework we define and motivate a notion of 'universal measurement', describing the most general possible condition of lack of knowledge in a measurement, emphasizing that the uniform fluctuations characterizing quantum measurements can also be understood as an average over all possible forms of non-uniform fluctuations which can be actualized in a measurement context. This means that the Born rule of quantum mechanics can be understood as a first order approximation of a more general non-uniform theory, thus explaining part of the great success of quantum probability in the description of different domains of reality. This is the first part of a two-part article. In the second part, the proof of the equivalence between universal measurements and uniform measurements, and its significance for quantum theory as a first order approximation, is given and further analyzed.- Aerts, D. and Sassoli de Bianchi, M. (2015). The unreasonable success of quantum probability II: Quantum measurements as universal measurements.
Journal of Mathematical Psychology,67, pp. 76-90. Archive reference and link: http://arxiv.org/abs/1401.2650. download pdf.

Abstract:We introduce a model which allows to represent the probabilities associated with an arbitrary measurement situation and use it to explain the emergence of quantum probabilities (the Born rule) as 'uniform' fluctuations on this measurement situation. The model exploits the geometry of simplexes to represent the states, in a way that the measurement probabilities can be derived as the 'Lebesgue measure' of suitably defined convex subregions of the simplexes. We consider a very simple and evocative physical realization of the abstract model, using a material point particle which is acted upon by elastic membranes, which by breaking and collapsing produce the different possible outcomes. This easy to visualize mechanical realization allows one to gain considerable insight into the possible hidden structure of an arbitrary measurement process. We also show that the Lebesgue-model can be further generalized into a model describing conditions of lack of knowledge generated by 'non-uniform' fluctuations. In this more general framework we define and motivate a notion of 'universal measurement', describing the most general possible condition of lack of knowledge in a measurement, emphasizing that the uniform fluctuations characterizing quantum measurements can also be understood as an average over all possible forms of non-uniform fluctuations which can be actualized in a measurement context. This means that the Born rule of quantum mechanics can be understood as a first order approximation of a more general non-uniform theory, thus explaining part of the great success of quantum probability in the description of different domains of reality. This is the first part of a two-part article. In the second part, the proof of the equivalence between universal measurements and uniform measurements, and its significance for quantum theory as a first order approximation, is given and further analyzed.- Aerts, D. and Sassoli de Bianchi, M. (2015). Many-Measurements or Many-Worlds? A Dialogue.
Foundations of Science,20, pp. 399-427. Archive reference and link: https://arxiv.org/abs/1406.0620. download pdf.

Abstract:Many advocates of the Everettian interpretation consider that theirs is the only approach to take quantum mechanics really seriously, and that this approach allows to deduce a fantastic scenario for our reality, one that consists of an infinite number of parallel worlds that branch out continuously. In this article, written in dialogue form, we suggest that quantum mechanics can be taken even more seriously, if the 'many-worlds' view is replaced by a 'many-measurements' view. This allows not only to derive the Born rule, thus solving the measurement problem, but also to deduce a one-world 'non-spatial' reality, providing an even more fantastic scenario than that of the multiverse.- Aerts, D. and Sozzo, S. (2015). Modeling concept combinations in a quantum-theoretic framework. In H. Liljenstr&?ouml;m H. (Eds),
Advances in Cognitive Neurodynamics (IV). Advances in Cognitive Neurodynamics (pp. pp 393-399). Dordrecht: Springer. Archive reference and link: http://arxiv.org/abs/1401.3626. download pdf.

Abstract: We present modeling for conceptual combinations which uses the mathematical formalism of quantum theory. Our model faithfully describes a large amount of experimental data collected by different scholars on concept conjunctions and disjunctions. Furthermore, our approach sheds a new light on long standing drawbacks connected with vagueness, or fuzziness, of concepts, and puts forward a completely novel possible solution to the 'combination problem' in concept theory. Additionally, we introduce an explanation for the occurrence of quantum structures in the mechanisms and dynamics of concepts and, more generally, in cognitive and decision processes, according to which human thought is a well structured superposition of a `logical thought' and a 'conceptual thought', and the latter usually prevails over the former, at variance with some widespread beliefs.- Aerts, D., Sozzo, S. and Veloz, T. (2015). The quantum nature of identity in human thought: Bose-Einstein statistics for conceptual indistinguishability.
International Journal of Theoretical Physics,54, pp 4430-4443. Archive reference and link: https://arxiv.org/abs/1410.6854. download pdf.

Abstract:Increasing experimental evidence shows that humans combine concepts in a way that violates the rules of classical logic and probability theory. On the other hand, mathematical models inspired by the formalism of quantum theory are in accordance with data on concepts and their combinations. In this paper, we investigate a novel type of concept combination were a number is combined with a noun, e.g., `Eleven Animals. Our aim is to study 'conceptual identity' and the effects of 'indistinguishability' - in the combination 'Eleven Animals', the 'animals' are identical and indistinguishable - on the mechanisms of conceptual combination. We perform experiments on human subjects and find significant evidence of deviation from the predictions of classical statistical theories, more specifically deviations with respect to Maxwell-Boltzmann statistics. This deviation is of the 'same type' of the deviation of quantum mechanical from classical mechanical statistics, due to indistinguishability of microscopic quantum particles, i.e we find convincing evidence of the presence of Bose-Einstein statistics. We also present preliminary promising evidence of this phenomenon in a web-based study.- Aerts, D., Sozzo, S. and Veloz, T. (2015). Quantum structure in cognition and the foundations of human reasoning.
International Journal of Theoretical Physics,54, pp 4557-4569. Archive reference and link: https://arxiv.org/abs/1412.8704. download pdf.

Abstract:Traditional cognitive science rests on a foundation of classical logic and probability theory. This foundation has been seriously challenged by several findings in experimental psychology on human decision making. Meanwhile, the formalism of quantum theory has provided an efficient resource for modeling these classically problematical situations. In this paper, we start from our successful quantum-theoretic approach to the modeling of concept combinations to formulate a unifying explanatory hypothesis. In it, human reasoning is the superposition of two processes -- a conceptual reasoning, whose nature is emergence of new conceptuality, and a logical reasoning, founded on an algebraic calculus of the logical type. In most cognitive processes however, the former reasoning prevails over the latter. In this perspective, the observed deviations from classical logical reasoning should not be interpreted as biases but, rather, as natural expressions of emergence in its deepest form.- Aerts, D., Sozzo, S. and Veloz, T. (2015). Quantum structure of negation and conjunction in human thought.
Frontiers in Psychology,6, 1447. Archive reference and link: https://arxiv.org/abs/1503.04260. download pdf.

Abstract:We analyse in this paper the data collected in a set of experiments performed on human subjects on the combination of natural concepts. We investigate the mutual influence of conceptual conjunction and negation by measuring the membership weights of a list of exemplars with respect to two concepts, e.g., 'Fruits' and 'Vegetables', and their conjunction 'Fruits And Vegetables', but also their conjunction when one or both concepts are negated, namely, 'Fruits And Not Vegetables', 'Not Fruits And Vegetables' and 'Not Fruits And Not Vegetables'. Our findings sharpen existing analysis on conceptual combinations, revealing systematic and remarkable deviations from classical (fuzzy set) logic and probability theory. And, more important, our results give further considerable evidence to the validity of our quantum-theoretic framework for the combination of two concepts. Indeed, the representation of conceptual negation naturally arises from the general assumptions of our two-sector Fock space model, and this representation faithfully agrees with the collected data. In addition, we find a further significant deviation and a priori unexpected from classicality, which can exactly be explained by assuming that human reasoning is the superposition of an 'emergent reasoning' and a 'logical reasoning', and that these two processes can be successfully represented in a Fock space algebraic structure.- Aerts, D., Sozzo, S. and Veloz, T. (2015). A new fundamental evidence of non-classical structure in the combination of natural concepts.
Philosophical Transactions of the Royal Society A,374, 20150095. Archive reference and link: https://arxiv.org/abs/1505.04981. download pdf.

Abstract:We recently performed cognitive experiments on conjunctions and negations of two concepts with the aim of investigating the combination problem of concepts. Our experiments confirmed the deviations (conceptual vagueness, underextension, overextension, etc.) from the rules of classical (fuzzy) logic and probability theory observed by several scholars in concept theory, while our data were successfully modeled in a quantum-theoretic framework developed by ourselves. In this paper, we isolate a new, very stable and systematic pattern of violation of classicality that occurs in concept combinations. In addition, the strength and regularity of this non-classical effect leads us to believe that it occurs at a more fundamental level than the deviations observed up to now. It is our opinion that we have identified a deep non-classical mechanism determining not only how concepts are combined but, rather, how they are formed. We show that this effect can be faithfully modeled in a two-sector Fock space structure, and that it can be exactly explained by assuming that human thought is the supersposition of two processes, a 'logical reasoning', guided by 'logic', and a 'conceptual reasoning' guided by 'emergence', and that the latter generally prevails over the former. All these findings provide a new fundamental support to our quantum-theoretic approach to human cognition.## 2016

- Aerts, D., Broekaert, J., Gabora, L. and Sozzo, S. (2016). Generalizing Prototype Theory: A formal quantum framework.
Frontiers in Psychology,7, 418. Archive reference and link: https://arxiv.org/abs/1601.06610. download pdf.

Abstract:Theories of natural language and concepts have been unable to model the flexibility, creativity, context-dependence, and emergence, exhibited by words, concepts and their combinations. The mathematical formalism of quantum theory has instead been successful in capturing these phenomena such as graded membership, situational meaning, composition of categories, and also more complex decision making situations, which cannot be modeled in traditional probabilistic approaches. We show how a formal quantum approach to concepts and their combinations can provide a powerful extension of prototype theory. We explain how prototypes can interfere in conceptual combinations as a consequence of their contextual interactions, and provide an illustration of this using an intuitive wave-like diagram. This quantum-conceptual approach gives new life to original prototype theory, without however making it a privileged concept theory, as we explain at the end of our paper.- Aerts, D., Czachor, M. and Kuna, M. (2016). Crystallization of space: Space-time fractals from fractal arithmetic.
Chaos, Solitons and Fractals,83, pp. 201-211. Archive reference and link: https://arxiv.org/abs/1506.00487. download pdf.

Abstract:Fractals such as the Cantor set can be equipped with intrinsic arithmetic operations (addition, subtraction, multiplication, division) that map the fractal into itself. The arithmetics allows one to define calculus and algebra intrinsic to the fractal in question, and one can formulate classical and quantum physics within the fractal set. In particular, fractals in space-time can be generated by means of homogeneous spaces associated with appropriate Lie groups. The construction is illustrated by explicit examples.- Aerts, D., Czachor, M. and Kuna, M. (2016). Fourier transforms on Cantor sets: A study in non-Diophantine arithmetic and calculus.
Chaos, Solitons and Fractals,91, pp. 461-468. Archive reference and link: https://arxiv.org/abs/1603.05471. download pdf.

Abstract:Fractals equipped with intrinsic arithmetic lead to a natural definition of differentiation, integration and complex numbers. Applying the formalism to the problem of a Fourier transform on fractals we show that the resulting transform has all the expected basic properties. As an example we discuss a sawtooth signal on the ternary middle-third Cantor set. The formalism works also for fractals that are not self-similar.- Aerts, D. and Sassoli de Bianchi, M. (2016). The extended Bloch representation of quantum mechanics. Explaining superposition, interference and entanglement.
Journal of Mathematical Physics,57, 122110. Archive reference and link: https://arxiv.org/abs/1504.04781. download pdf.

Abstract:The extended Bloch representation of quantum mechanics was recently derived to offer a (hidden-measurement) solution to the measurement problem. In this article we use it to investigate the geometry of superposition and entangled states, explaining the interference effects, and the entanglement correlations, in terms of the different orientations that a state-vector can take within the generalized Bloch sphere. We also introduce a tensorial determination of the generators of SU(N), particularly suitable to describe multipartite systems, from the viewpoint of the sub-entities. We then use it to show that non-product states admit a general description in which the sub-entities can always remain in well-defined states, even when they are entangled. Therefore, the completed version of quantum mechanics provided by the extended Bloch representation, in which the density operators are also representative of pure states, allows to solve not only the well-known measurement problem, but also the lesser-known entanglement problem. This because we no longer need to give up the general physical principle saying that a composite entity exists, and therefore is in a pure state, if and only if its components also exist, and therefore are in well-defined pure states.- Aerts, D. and Sassoli de Bianchi, M. (2016). The GTR-model: a universal framework for quantum-like measurements. In D. Aerts, C. De Ronde, H. Freytes and R. Giuntini (Eds),
Probing the Meaning of Quantum Mechanics. Superpositions, Dynamics, Semantics and Identity, pp. 91-140. Singapore: World Scientific Publishing Company. Archive reference and link: https://arxiv.org/abs/1512.00880. download pdf.

Abstract:We present a very general geometrico-dynamical description of physical or more abstract entities, called the 'general tension-reduction' (GTR) model, where not only states, but also measurement-interactions can be represented, and the associated outcome probabilities calculated. Underlying the model is the hypothesis that indeterminism manifests as a consequence of unavoidable fluctuations in the experimental context, in accordance with the 'hidden-measurements interpretation' of quantum mechanics. When the structure of the state space is Hilbertian, and measurements are of the 'universal' kind, i.e., are the result of an average over all possible ways of selecting an outcome, the GTR-model provides the same predictions of the Born rule, and therefore provides a natural completed version of quantum mechanics. However, when the structure of the state space is non-Hilbertian and/or not all possible ways of selecting an outcome are available to be actualized, the predictions of the model generally differ from the quantum ones, especially when sequential measurements are considered. Some paradigmatic examples will be discussed, taken from physics and human cognition. Particular attention will be given to some known psychological effects, like question order effects and response replicability, which we show are able to generate non-Hilbertian statistics. We also suggest a realistic interpretation of the GTR-model, when applied to human cognition and decision, which we think could become the generally adopted interpretative framework in quantum cognition research.- Aerts, D. and Sassoli de Bianchi, M. (2016). A possible solution to the second entanglement paradox. In D. Aerts, C. De Ronde, H. Freytes and R. Giuntini (Eds),
Probing the Meaning of Quantum Mechanics. Superpositions, Dynamics, Semantics and Identity, pp. 351-360. Singapore: World Scientific Publishing Company. Archive reference and link: https://arxiv.org/abs/1502.06249. download pdf.

Abstract:Entangled states are in conflict with a general physical principle which expresses that a composite entity exists if and only if its components also exist, and the hypothesis that pure states represent the actuality of a physical entity, i.e., its 'existence'. A possible way to solve this paradox consists in completing the standard formulation of quantum mechanics, by adding more pure states. We show that this can be done, in a consistent way, by using the extended Bloch representation of quantum mechanics, recently introduced to provide a possible solution to the measurement problem. Hence, with the solution proposed by the extended Bloch representation of quantum mechanics, the situation of entangled states regains full intelligibility.- Aerts, D., Sassoli de Bianchi, M. and Sozzo, S, (2016). On the foundations of the Brussels operational-realistic approach to cognition.
Frontiers in Physics,417. Archive reference and link: https://arxiv.org/abs/1512.08710. download pdf.

Abstract:The scientific community is becoming more and more interested in the research that applies the mathematical formalism of quantum theory to model human decision-making. In this paper, we provide the theoretical foundations of the quantum approach to cognition that we developed in Brussels. These foundations rest on the results of two decade studies on the axiomatic and operational-realistic approaches to the foundations of quantum physics. The deep analogies between the foundations of physics and cognition lead us to investigate the validity of quantum theory as a general and unitary framework for cognitive processes, and the empirical success of the Hilbert space models derived by such investigation provides a strong theoretical confirmation of this validity. However, two situations in the cognitive realm, 'question order effects' and 'response replicability', indicate that even the Hilbert space framework could be insufficient to reproduce the collected data. This does not mean that the mentioned operational-realistic approach would be incorrect, but simply that a larger class of measurements would be in force in human cognition, so that an extended quantum formalism may be needed to deal with all of them. As we will explain, the recently derived 'extended Bloch representation' of quantum theory (and the associated 'general tension-reduction' model) precisely provides such extended formalism, while remaining within the same unitary interpretative framework.- Aerts, D. and Sozzo, S. (2016). From ambiguity aversion to a generalized expected Utility. Modeling preferences in a quantum probabilistic framework.
Journal of Mathematical Psychology,74, pp. 117-127. Archive reference and link: https://arxiv.org/abs/1510.09058. download pdf.

Abstract: Ambiguity and ambiguity aversion have been widely studied in decision theory and economics both at a theoretical and an experimental level. After Ellsberg's seminal studies challenging subjective expected utility theory (SEUT), several (mainly normative) approaches have been put forward to reproduce ambiguity aversion and Ellsberg-type preferences. However, Machina and other authors have pointed out some fundamental difficulties of these generalizations of SEUT to cope with some variants of Ellsberg's thought experiments, which has recently been experimentally confirmed. Starting from our quantum modeling approach to human cognition, we develop here a general probabilistic framework to model human decisions under uncertainty. We show that our quantum theoretical model faithfully represents different sets of data collected on both the Ellsberg and the Machina paradox situations, and is flexible enough to describe different subjective attitudes with respect to ambiguity. Our approach opens the way toward a quantum-based generalization of expected utility theory (QEUT), where subjective probabilities depend on the state of the conceptual entity at play and its interaction with the decision-maker, while preferences between acts are determined by the maximization of this 'state-dependent expected utility'.## 2017

- Aerts, D. (2017). Relativity theory refounded.
Foundations of Science, Online First https://doi.org/10.1007/s10699-017-9538-7. Archive reference and link: https://arxiv.org/abs/1511.08735. download pdf.

Abstract:We put forward a new view of relativity theory that makes the existence of a flow of time compatible with the four-dimensional block universe. To this end, we apply the creation-discovery view elaborated for quantum mechanics to relativity theory and in such a way that time and space become creations instead of discoveries and an underlying non temporal and non spatial reality comes into existence. We study the nature of this underlying non temporal and non spatial reality and reinterpret many aspects of the theory within this new view. We show that data of relativistic measurements are sufficient to derive the three-dimensionality of physical space. The nature of light and massive entities is reconsidered, and an analogy with human cognition is worked out.- Aerts, D., Aerts Argu&?euml;lles, J., Beltran, L., Beltran, L., Sassoli de Bianchi, M., Sozzo, S. and Veloz, T. (2017). Context and interference effects in the combinations of natural concepts. In P. Br\'ezillon, R. Turner and C. Penco. (Eds),
Modeling and Using Context. CONTEXT 2017. Lecture Notes in Computer Science,10257. Berlin: Springer, Cham. Archive reference and link: https://arxiv.org/abs/1612.06038. download pdf.

Abstract: The mathematical formalism of quantum theory exhibits significant effectiveness when applied to cognitive phenomena that have resisted traditional (set theoretical) modeling. Relying on a decade of research on the operational foundations of micro-physical and conceptual entities, we present a theoretical framework for the representation of concepts and their conjunctions and disjunctions that uses the quantum formalism. This framework provides a unified solution to the 'conceptual combinations problem' of cognitive psychology, explaining the observed deviations from classical (Boolean, fuzzy set and Kolmogorovian) structures in terms of genuine quantum effects. In particular, natural concepts 'interfere' when they combine to form more complex conceptual entities, and they also exhibit a 'quantum-type context-dependence', which are responsible of the 'over- and under-extension' that are systematically observed in experiments on membership judgments.- Aerts, D., Aerts Argu&?euml;lles, J., Beltran, L., Beltran, L., Sassoli de Bianchi, M., Sozzo, S. and Veloz, T. (2017). Testing quantum models of conjunction fallacy on the World Wide Web.
International Journal of Theoretical Physics,56, pp. 3744-3756. Archive reference and link: https://arxiv.org/abs/1609.07721. download pdf.

Abstract: The 'conjunction fallacy' has been extensively debated by scholars in cognitive science and, in recent times, the discussion has been enriched by the proposal of modeling the fallacy using the quantum formalism. Two major quantum approaches have been put forward: the first assumes that respondents use a two-step sequential reasoning and that the fallacy results from the presence of 'question order effects'; the second assumes that respondents evaluate the cognitive situation as a whole and that the fallacy results from the 'emergence of new meanings', as an 'effect of overextension' in the conceptual conjunction. Thus, the question arises as to determine whether and to what extent conjunction fallacies would result from 'order effects' or, instead, from 'emergence effects'. To help clarify this situation, we propose to use the World Wide Web as an 'information space' that can be interrogated both in a sequential and non-sequential way, to test these two quantum approaches. We find that 'emergence effects', and not 'order effects', should be considered the main cognitive mechanism producing the observed conjunction fallacies.- Aerts, D., Aerts Argu&?euml;lles, J., Beltran, L., Geriente, S., Sassoli de Bianchi, M., Sozzo, S. and Veloz, T. (2017). Spin and wind directions I: Identifying entanglement in nature and cognition.
Foundations of Science, Online First https://link.springer.com/article/10.1007/s10699-017-9528-9. Archive reference and link: https://arxiv.org/abs/1508.00434. download pdf.

Abstract: We present a cognitive psychology experiment where participants were asked to select pairs of spatial directions that they considered to be the best example of 'Two Different Wind Directions'. Data are shown to violate the CHSH version of Bell's inequality with the same magnitude as in typical Bell-test experiments with entangled spins. Wind directions thus appear to be conceptual entities connected through meaning, in human cognition, in a similar way as spins appear to be entangled in experiments conducted in physics laboratories. This is the first part of a two-part article. In the second part we present a symmetrized version of the same experiment for which we provide a quantum modeling of the collected data in Hilbert space.- Aerts, D., Aerts Argu&?euml;lles, J., Beltran, L., Geriente, S., Sassoli de Bianchi, M., Sozzo, S. and Veloz, T. (2017). Spin and wind directions II: A Bell state quantum model.
Foundations of Science, Online First https://link.springer.com/article/10.1007/s10699-017-9530-2. Archive reference and link: https://arxiv.org/abs/1706.01188. download pdf.

Abstract: In the first half of this two-part article, we analyzed a cognitive psychology experiment where participants were asked to select pairs of directions that they considered to be the best example of 'Two Different Wind Directions', and showed that the data violate the CHSH version of Bell's inequality, with same magnitude as in typical Bell-test experiments in physics. In this second part, we complete our analysis by presenting a symmetrized version of the experiment, still violating the CHSH inequality but now also obeying the marginal law, for which we provide a full quantum modeling in Hilbert space, using a singlet state and suitably chosen product measurements. We also address some of the criticisms that have been recently directed at experiments of this kind, according to which they would not highlight the presence of genuine forms of entanglement. We explain that these criticisms are based on a view of entanglement that is too restrictive, thus unable to capture all possible ways physical and conceptual entities can connect and form systems behaving as a whole. We also provide an example of a mechanical model showing that the violations of the marginal law and Bell inequalities are generally to be associated with different mechanisms.- Aerts, D., Aerts Argu&?euml;lles, J., Beltran, L., Geriente, S., Sassoli de Bianchi, M., Sozzo, S. and Veloz, T. (2017). Towards a Quantum World Wide Web.
Foundations of Science, Online First https://www.sciencedirect.com/science/article/pii/S0304397518301804. Archive reference and link: https://arxiv.org/abs/1703.06642. download pdf.

Abstract: We elaborate a quantum model for the meaning associated with corpora of written documents, like the pages forming the World Wide Web. To that end, we are guided by how physicists constructed quantum theory for microscopic entities, which unlike classical objects cannot be fully represented in our spatial theater. We suggest that a similar construction needs to be carried out by linguists and computational scientists, to capture the full meaning carried by collections of documental entities. More precisely, we show how to associate a quantum-like 'entity of meaning' to a 'language entity formed by printed documents', considering the latter as the collection of traces that are left by the former, in specific results of search actions that we describe as measurements. In other words, we offer a perspective where a collection of documents, like the Web, is described as the space of manifestation of a more complex entity - the QWeb - which is the object of our modeling, drawing its inspiration from previous studies on operational-realistic approaches to quantum physics and quantum modeling of human cognition and decision-making. We emphasize that a consistent QWeb model needs to account for the observed correlations between words appearing in printed documents, e.g., co-occurrences, as the latter would depend on the 'meaning connections' existing between the concepts that are associated with these words. In that respect, we show that both 'context and interference (quantum) effects' are required to explain the probabilities calculated by counting the relative number of documents containing certain words and co-ocurrrences of words.- Aerts, D., Beltran, L., Sassoli de Bianchi, M., Sozzo, S. and Veloz, T. (2017). Quantum cognition beyond Hilbert space: Fundamentals and applications. In J. de Barros, B. Coecke, E. Pothos (Eds),
Quantum Interaction. QI 2016. Lecture Notes in Computer Science,10106,, pp. 81-98. Berling: Springer, Cham. Archive reference and link: https://arxiv.org/abs/1604.08268. download pdf.

Abstract: The 'quantum cognition' paradigm was recently challenged by its proven impossibility to simultaneously model 'question order effects' and 'response replicability'. In the present article we describe sequential dichotomic measurements within an operational and realistic framework for human cognition, and represent them in a quantum-like 'extended Bloch representation', where the Born rule of quantum probability does not necessarily hold. We then apply this mathematical framework to successfully model question order effects, response replicability and unpacking effects, thus opening the way toward 'quantum cognition beyond Hilbert space'.- Aerts, D., Haven, E. and Sozzo, S. (2017). A proposal to extend expected utility in a quantum probabilistic framework.
Economic Theory, Online First https://link.springer.com/article/10.1007/s00199-017-1051-2#citeas. Archive reference and link: https://arxiv.org/abs/1612.08583. download pdf.

Abstract: Expected utility theory (EUT) is widely used in economic theory. However, its subjective probability formulation, first elaborated by Savage, is linked to Ellsberg-like paradoxes and ambiguity aversion. This has led various scholars to work out non-Bayesian extensions of EUT which cope with its paradoxes and incorporate attitudes toward ambiguity. A variant of the Ellsberg paradox, recently proposed by Mark Machina and confirmed experimentally, challenges existing non-Bayesian models of decision-making under uncertainty. Relying on a decade of research which has successfully applied the formalism of quantum theory to model cognitive entities and fallacies of human reasoning, we put forward a non-Bayesian extension of EUT in which subjective probabilities are represented by quantum probabilities, while the preference relation between acts depends on the state of the situation that is the object of the decision. We show that the benefits of using the quantum theoretical framework enables the modeling of the Ellsberg and Machina paradoxes, as the representation of ambiguity and behavioral attitudes toward it. The theoretical framework presented here is a first step toward the development of a 'state-dependent non-Bayesian extension of EUT' and it has potential applications in economic modeling.- Aerts, D. and Sassoli de Bianchi, M. (2017). Do spins have directions?
Soft Computing,21, pp. 1483-1504. Archive reference and link: https://arxiv.org/abs/1501.00693. download pdf.

Abstract:The standard Bloch sphere representation was recently generalized to the 'extended Bloch representation' describing not only systems of arbitrary dimension, but also their measurements. This model solves the measurement problem and is based on the 'hidden-measurement interpretation', according to which the Born rule results from our lack of knowledge about the interaction between measuring apparatus and the measured entity. We present here the extended Bloch model and use it to investigate the nature of quantum spin and its relation to our Euclidean space. We show that spin eigenstates cannot generally be associated with directions in the Euclidean space, but only with generalized directions in the Blochean space, which generally is a space of higher dimension. Hence, spin entities have to be considered as genuine non-spatial entities. We also show, however, that specific vectors can be identified in the Blochean theater that are isomorphic to the Euclidean space directions, and therefore representative of them, and that spin eigenstates always have a predetermined orientation with respect to them. We use the details of our results to put forward a new view of realism, that we call 'multiplex realism', providing a specific framework with which to interpret the human observations and understanding of the component parts of the world. Elements of reality can be represented in different theaters, one being our customary Euclidean space, and another one the quantum realm, revealed to us through our sophisticated experiments, whose elements of reality, in the quantum jargon, are the eigenvalues and eigenstates. Our understanding of the component parts of the world can then be guided by looking for the possible connections, in the form of partial morphisms, between the different representations, which is precisely what we do in this article with regard to spin entities.- Aerts, D. and Sassoli de Bianchi, M. (2017). Beyond-quantum modeling of question order Effects and response replicability in psychological measurements.
Journal of Mathematical Psychology,79, pp. 104-120. Archive reference and link: https://arxiv.org/abs/1508.03686. download pdf.

Abstract:A general tension-reduction (GTR) model was recently considered to derive quantum probabilities as (universal) averages over all possible forms of non-uniform fluctuations, and explain their considerable success in describing experimental situations also outside of the domain of physics, for instance in the ambit of quantum models of cognition and decision. Yet, this result also highlighted the possibility of observing violations of the predictions of the Born rule, in those situations where the averaging would not be large enough, or would be altered because of the combination of multiple measurements. In this article we show that this is indeed the case in typical psychological measurements exhibiting question order effects, by showing that their statistics of outcomes are inherently non-Hilbertian, and require the larger framework of the GTR-model to receive an exact mathematical description. We also consider another unsolved problem of quantum cognition: response replicability. It is has been observed that when question order effects and response replicability occur together, the situation cannot be handled anymore by quantum theory. However, we show that it can be easily and naturally described in the GTR-model. Based on these findings, we motivate the adoption in cognitive science of a hidden-measurements interpretation of the quantum formalism, and of its GTR-model generalization, as the natural interpretational framework explaining the data of psychological measurements on conceptual entities.- Aerts, D. and Sassoli de Bianchi, M. (2017). Quantum measurements as weighted symmetry breaking processes: the hidden measurement perspective.
International Journal of Quantum Foundations,3, pp. 1-16. Archive reference and link: https://arxiv.org/abs/1601.05222. download pdf.

Abstract:The purpose of the present note is twofold. Firstly, we highlight the similarities between the ontologies of Kastner's possibilist transactional interpretation (PTI) of quantum mechanics - an extension of Cramer's transactional interpretation - and the authors' hidden-measurement interpretation (HMI). Secondly, we observe that although a weighted symmetry breaking (WSB) process was proposed in the PTI, to explain the actualization of incipient transactions, no specific mechanism was actually provided to explain why the weights of such symmetry breaking are precisely those given by the Born rule. In other terms, PTI, similarly to decoherence theory, doesn't explain a quantum measurement in a complete way, but just the transition from a pure state to a fully reduced density matrix state. On the other hand, the recently derived extended Bloch representation (EBR) - a specific implementation the HMI - precisely provides such missing piece of explanation, i.e., a qualitative description of the WSB as a process of actualization of hidden measurement-interactions and, more importantly, a quantitative prediction of the values of the associated weights that is compatible with the Born rule of probabilistic assignment. Therefore, from the PTI viewpoint, the EBR provide the missing link for a complete description of a quantum measurement. However, EBR is in a sense more general than PTI, as it does not rely on the specific notion of transaction, and therefore remains compatible with other physical mechanisms that could be at the origin of the measurement-interactions.- Aerts, D. and Sassoli de Bianchi, M. (2017). The extended Bloch representation of entanglement and measurement in quantum mechanics.
International Journal of Theoretical Physics,56, pp. 3727-3739. Archive reference and link: https://arxiv.org/abs/1609.02658. download pdf.

Abstract:The quantum formalism can be completed by assuming that a density operator can also represent a pure state. An 'extended Bloch representation' (EBR) then results, in which not only states, but also the measurement-interactions can be represented. The Born rule is obtained as an expression of the subjective lack of knowledge about the measurement-interaction that is each time actualized. Entanglement can also be consistently described in the EBR, as it remains compatible with the principle according to which a composite entity exists only if its components also exist, and therefore are in pure states.- Aerts, D. and Sozzo, S. (2017). Quantum structure in cognition, origins, developments, successes and expectations. In E. Haven and A. Khrennikov (Eds),
The Palgrave Handbook of Quantum Models in Social Science. London: Palgrave Macmillan. Archive reference and link: https://arxiv.org/abs/1503.02994. download pdf.

Abstract: We provide an overview of the results we have attained in the last decade on the identification of quantum structures in cognition and, more specifically, in the formalization and representation of natural concepts. We firstly discuss the quantum foundational reasons that led us to investigate the mechanisms of formation and combination of concepts in human reasoning, starting from the empirically observed deviations from classical logical and probabilistic structures. We then develop our quantum-theoretic perspective in Fock space which allows successful modeling of various sets of cognitive experiments collected by different scientists, including ourselves. In addition, we formulate a unified explanatory hypothesis for the presence of quantum structures in cognitive processes, and discuss our recent discovery of further quantum aspects in concept combinations, namely, 'entanglement' and 'indistinguishability'. We finally illustrate perspectives for future research.