Publications in chronological order

1978

Aerts, D. and Daubechies, I. (1978). Structure-preserving maps of a quantum mechanical propositional system. Helvetica Physica Acta, 51, pp. 637-660.
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.
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). Connection between propositional systems in Hilbert spaces and Von Neumann algebra's. Helvetica Physica Acta, 52, pp. 184-199.
Aerts, D. and Daubechies, I. (1979). Characterization of subsystems in physics. Letters in Mathematical Physics, 3, pp. 11-17.
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.
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). Role of modular pairs in the category of complete orthomodular lattices. Letters in Mathematical Physics, 3, pp. 1-10.
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.
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 royal de Belgique, Classes des Sciences, 66, pp. 705-714.
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. Dordrecht: Kluwer Academic.
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.
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.
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.
Aerts, D. (1983). The description of one and many physical systems. In C. Gruber (Ed.), Foundations of Quantum Mechanics (pp. 63-148). Lausanne: AVCP.
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.
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 makes it possible to describe the joint system of a classical and a quantum system. Reports in Mathematical Physics, 20, pp. 421-428.
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.
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.
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: Kluwer Academic.
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, 2, 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: Kluwer Academic.
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.
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.
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: Kluwer Academic.
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.
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.
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.
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., Van Landschoot, J. (1991). Wereldbeelden, Van Fragmentering naar Integratie,. Kapellen: Pelckmans.
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.
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.
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.
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). A mechanistic classical laboratory situation with a quantum logic structure. International Journal of Theoretical Physics, 31, pp. 1839-1848.
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.
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 �uctuations 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.
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., Van Bogaert, B. (1993). A physical example of quantum fuzzy sets and the classical limit. Tatra Mountains Mathematical Publications, 1, pp. 5-15.
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., 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.
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.
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.
Aerts, D. (1994). Quantummechanica. In L. Apostel and F. Verbeure (Eds.), Verwijdering of Ontmoeting? (pp. 123-142). Kapellen: Pelckmans.
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., Aerts, S. (1994). Applications of quantum statistics in psychological studies of decision processes. Foundations of Science, 1, pp. 85-97.
Aerts, D., Apostel, L., De Moor, B., Hellemans, S., Maex, E., Van Belle, H., Van der Veken, J. (1994). Worldviews, from Fragmentation towards Integration. Brussels: VUBPress.
Aerts, D., Apostel, L., De Moor, B., Hellemans, S., Maex, E., Van Belle, H., 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.
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.
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.
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.
Aerts, D., Apostel, L., De Moor, B., Hellemans, S., Maex, E., Van Belle, H., Van der Veken, J. (1995). Perspectives on the World, an Interdisciplinary Reflection. Brussels: VUBPress.
1996

Aerts, D. (1996). Framework for possible unification of quantum and relativity theories. International Journal of Theoretical Physics, 35, pp. 2399-2416.
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.
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.
Aerts, D. and D'Hooghe, B. (1996). Operator structure of a non-quantum and a non-classical system. International Journal of Theoretical Physics, 35, pp. 2285-2298.
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, Kluwer Academic, Dordrecht.
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: Kluwer Academic.
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: Kluwer Academic.
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.
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.
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.
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., Veretennicoff, I. (1997). Niet-ruimtelijkheid als werktuig. In J. Van Pelt, Grenzeloze Wetenschap: Dertig Gesprekken met Vlamingen over Onderzoek. Leuven-Apeldoorn: Garant.
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.
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.
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.
Aerts, D. (1998). Synthesis and analysis, interdisciplinarity and foundations. Foundations of Science, 3, pp. 203-206.
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.
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.
1999

Aerts, D. (Ed.) (1999). Einstein meets Magritte: An Interdisciplinary Reflection about Science, Nature, Art, Human Action and Society. Dordrecht: Kluwer Academic, series of eight volumes.
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.
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: Kluwer Academic.
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: Kluwer Academic.
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: Kluwer Academic. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0107044.
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: Kluwer Academic.
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: Kluwer Academic.
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: Kluwer Academic.
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: Kluwer Academic. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0106132.
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.
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.
Aerts, D., Broekaert, J. and Gabora, L. (1999). Formal and informal representations of science. Foundations of Science, 4, 1-2.
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: Kluwer Academic.
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: Kluwer Academic.
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.
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., 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.
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., Weyns, W. (Eds.) (1999). A World in Transition, Humankind and Nature. Dordrecht: Kluwer Academic.
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.
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: Kluwer Academic.
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: Kluwer Academic.
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.
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., Gutwirth, S., Smets, S. and Van Langehove, L. (Eds.) (1999). Science, Technology and Social Change. Dordrecht: Kluwer Academic.
Aerts, D., Mathijs, E. and Mosselmans, B. (Eds.) (1999). Science and Art. Dordrecht: Kluwer Academic.
Aerts, D., Pykacz, J., (Eds.) (1999). Quantum Structures and the Nature of Reality. Dordrecht: Kluwer Academic.
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: Kluwer Academic.
Aerts, D., Van Belle, H. and Van der Veken, J. (Eds.) (1999). Worldviews and the Problem of Synthesis. Dordrecht: Kluwer Academic.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 category SCOP with 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.
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 category SEP of state experiment probability systems and their morphisms is linked with the category SP of 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 series Advances in Consciousness Research, ISSN 1381 -589X).
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.
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.
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.
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.
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.
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.
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.
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 �re-constructed� 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.
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.
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.
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.
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 taken a priori to 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 General, 37, pp. L123-L132. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0309022.
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.
Abstract: We reformulate the 'separated quantum entities' theorem, i.e.the theorem that proves that two 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.
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 today�s perception of the Self. According to this perception, we think of ourselves as essentially �self-reliant� and surrounded by a world that is sheer concrete reality. This has made it possible � though not necessary! � 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 today�s 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 man�s 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.
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.
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, 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 Schr�dinger 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 �change of state�, 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.
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.
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.
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.
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.
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 'brain�heart' 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.
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, pp. 1-18.
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.
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.
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.
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.
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.
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.
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., 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.
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'.
Aerts, D. and Czachor, M. (2008). Tensor-product vs. geometric-product coding. Physical Review A, 77, 012316.
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.
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. (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.
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.
Gabora, L., Rosch, E. and Aerts, D. (2008). Toward an ecological theory of concepts. Ecological Psychology, 20, 84-116.
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
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.
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.
Aerts, D. (2009). Quantum particles as conceptual entities: A possible explanatory framework for quantum theory. Foundations of Science, 14, 361-411. doi: 10.1007/s10699-009-9166-y.
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 �the one and the many� 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 Schr�dinger�s 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.
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.
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.
Aerts, D., Czachor, M. and De Moor, B. (2009). Geometric analogue of holographic reduced representation. Journal of Mathematical Psychology, 53, 389-398. Archive reference and link: http://uk.arxiv.org/abs/0710.2611. doi: 10.1016/j.jmp.2009.02.005
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.
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.
Gabora, L. and Aerts, D. (2009). A model of the emergence and evolution of integrated worldviews. Journal of Mathematical Psychology, 53, 434-451. doi:10.1016/j.jmp.2009.06.004.
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, 31-46.
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 (in press or accepted for publication)

Aerts, D. (2010). Quantum interference and superposition in cognition: Development of a theory for the disjunction of concepts. In D. Aerts, B. D'Hooghe and N. Note (Eds.), Worldviews, Science and Us: Bridging Knowledge and Its Implications for Our Perspectives of the World. Singapore: World Scientific. Archive reference and link: http://uk.arxiv.org/abs/0705.0975 (in press).
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. (2010). General quantum modeling of combining concepts: A quantum field model in Fock space. Foundations of Science. Archive reference and link: http://uk.arxiv.org/abs/0705.1740.
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. (2010). Interpreting quantum particles as conceptual entities. International Journal of Theoretical Physics.
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 speci�cally 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., Broekaert, J. and Gabora, L. (2010). A case for applying an abstracted quantum formalism to cognition. New Ideas in Psychology. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0404068 (accepted).
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 inequalities�the definitive test for quantum structure�are 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., Bundervoet, S., Czachor, M., D'Hooghe, B., Gabora L. and Polk, P. (2010). On the foundations of the theory of evolution. In D. Aerts, B. D'Hooghe and N. Note (Eds.), Worldviews, Science and Us: Bridging Knowledge and Its Implications for Our Perspectives of the World. Singapore: World Scientific (in press).
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., de Ronde, C. and D'Hooghe, B. (2010). Compatibility and separability for classical and quantum entanglement. In D. Aerts, B. D'Hooghe and N. Note (Eds.), Worldviews, Science and Us: Bridging Knowledge and Its Implications for Our Perspectives of the World. Singapore: World Scientific (in press).
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. (2010). A quantum-conceptual explanation of violations of expected utility in economics. International Journal of Theoretical Physics.
Aerts, D., D'Hooghe, B. and Haven, E. (2010). Quantum experimental data in psychology and economics. International Journal of Theoretical Physics.
Aerts, D., D'Hooghe, B. and Note, N. (Eds.), (2010). Worldviews, Science and Us: Bridging Knowledge and Its Implications for Our Perspectives of the World. Singapore: World Scientific (in press).
Aerts, D., D'Hooghe, B. and Note, N. (2010). 'Interdisciplinarity and bridging knowledge. In D. Aerts, B. D'Hooghe and N. Note (Eds.), Worldviews, Science and Us: Bridging Knowledge and Its Implications for Our Perspectives of the World. Singapore: World Scientific (in press).
Submitted and in preparation

Patyk, A., Czachor, M. and Aerts, D. (2010). A geometric algebra based distributed (GAs) representation model. Artificial Intelligence (submitted).
Abstract: Authors present a new distributed representation model (GAc) that is based on geometric algebra. The model uses geometric product as a means of binding and ordinary addition as superposition. The projected product is used to cut down on computing unnecessary noisy parts of a sentence. Further, scaling properties of the GA, GAc, HRR and BSC models are studied. Authors concentrated on comparing the recognition percentage for the four models for comparable data size, rather than on the time taken to achieve high percentage.
Patyk, A., Czachor, M. and Aerts, D. (2010). Results on preserving information in speci�c order in the geometric algebra (GAs) and the holographic reduced (HRR) representations. Artificial Intelligence (submitted).
Abstract: Authors investigate the trajectory association technique used to store pieces of information in a given order. To solve this problem, the capacity of GAc and HRR models is studied. Four experiment results are described: template matching, finding correct item or correct place of an item in a sequence and finding the alignment of items in a sequence without the precise knowledge of trajectory vectors.

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