The numbers with the publications refer to the numbers that these publications have in the other lists (chronological and year by year) and also the numbers that are used to refer to publications in the research webpage.

While investigating in which way the satisfying of physically plausible coupling conditions gave rise to the tensor product procedure for the description of the joint quantum entity of two sub quantum entities (see section "Physical justification for the tensor product procedure to describe joint entities" for details), I became more and more aware of the fact that the problem of the description of the joint entity touched at some basic matters in quantum mechanics. Certainly also because meanwhile the research related to the Einstein Podolsky Rosen paradox and the violation of Bell inequalities was in the focus of attention of the foundations of quantum physics community (see sections "The Einstein Podolsky Rosen paradox", "Nonlocality, entanglement and the role of space", "The violation of Bell inequalities" for more details). I also have to mention that the axiomatic lattice theoretic approach that I had been looking to, together with Ingrid Daubechies, for the problem of the description of joint entities, had a firm operational foundation in its version developed in Geneva, meanly by Contantin Piron. I was studying more and more also this operational foundation of the axiomatic lattice theoretic quantum axiomatic. This came to my going to Geneva to work with Constantin Piron at the end of the seventies. Some years before, in 1976, Piron had published a book "Foundations of Quantum Physics", where his work on axiomatic quantum mechanics was carefully exposed. The Geneva approach proposed a description of a physical entity by means of its states and properties in a very general way. The operational foundation of the approach was that properties were introduced as equivalence classes of tests (also called questions or experimental projects), and states as sets of actual properties. At that time the operational aspects of the approach were however not very well developed. In his book, Piron introduced the properties by means of the tests (as equivalence classes), but then the whole enterprise forgot more or less about the tests, and went on to calculate in the mathematical structure of the properties. One axiom, (1) demanding the set of properties to be a complete lattice, had been operationally derived by Piron at that time, but the other axioms needed for the structure to become representable into a generalized Hilbert space, were more or less put ad hoc, mostly with the aim of getting to this generalized Hilbert space. These other axioms were: (2) an orthocomplementation for the lattice of properties, (3) weak modularity for the orthocomplemented lattice of properties, (4) atomicity for the lattice of properties, (5) the covering law to be satisfied. Indeed, Piron's representation theorem proved that a (1) complete, (2) orthocomplemented, (3) weakly modular, (4) atomic lattice, (5) satisfying the covering law, and being irreducuble, could be represented as the lattice of closed subspaces of a generalized Hilbert space (see the research webpage for more details). So this was the state of affairs when I started to study the general axiomatic approach developed in Geneva. Five axioms, completeness, orthocomplementation, weak modularity, atomicity, and the covering law, leading to standard quantum mechanics in a generalized Hilbert space. One of the axioms, the completeness, was operationally founded by means of the tests.

In the work that I did together with Ingrid Daubechies (see section "Physical justification for the tensor product procedure to describe joint entities" for details), we had in a certain way already used as a guiding spirit the Geneva approach, but all the time we had worked directly in the lattices of closed subspaces of a complex Hilbert space. At that time, it did not seem possible to attack the problem of the description of the joint entity of two entities within the general formalism. The idea came to me that perhaps however it would be possible to investigate the most simple of all situations, namely the situation where the sub entities forming the joint entity are 'separated' entities, within the general approach. With this idea in mind I started to work on the level of the tests instead of the level of the properties in the approach, because indeed, the concept of 'separated' could easily be expressed on this level of the tests. Two tests are 'separated tests' if and only if the performance of one of the tests does not influence the outcomes of the other test. I could develop a calculus of tests and in this way built 'by hand' the lattice of properties of the joint entity consisting of two separated sub entities. To my amazement the lattice that results for the properties of the joint entity consisting of two separeted sub entities showed 'not' to satisfy all the axioms needed for Piron's representation theorem. I could prove that completeness, orthocomplementation and atomicity do not pose a problem: if the sub entities satisfy these axioms also the joint entity satisfies them. But weak modularity and the covering law both pose a problem. It was possible to prove that if one demands the lattice of properties of the joint entity of two separated sub entities to be weakly modular or to satisfy the covering law, this implies that the lattice of one of the sub entities becomes a classical lattice of properties, and as a consequence one of the sub entities is a classical entity (see section "Classical and pure quantum as special cases: a general decomposition theorem" for more details on what we mean with a classical entity). A consequence of this is that the lattice of properties of the joint entity consisting of two separated quantum entities 'never' satisfies the axiom of weak modularity and never satisfies the axiom of the overing law. This also means that this lattice of properties of the joint entity of two separated entities cannot be described within a generalized Hilbert space. Standard quantum mechanics cannot describe separated quantum entities.

I decided to use these results for my doctoral dissertation that I meanwhile worked on under Constantin Piron at the university of Geneva. I was however eager to publish part of the results before I would defend my thesis and this can be found in publication [8]. My doctoral thesis, publication [10], contains all of the material. After I defended my thesis in september 1981, I wrote the material related to the finding of the failing quantum axioms for the description of separated quantum entities, down in publication [12]. Also the calculus of tests that I had to develop to be able to prove this result can be found in publication [12]. In 1983 I gave a course at a winter school on the foundations of quantum mechanics in Montana, Switserland, organised by 'l'Association Vaudoise de Chercheurs en Physique'. Publication [14] contains the texts of the courses that took place at this winter school.

If standard quantum mechanics cannot describe the most simple of all physical situations, namely the one of two separated quantum entities, this could mean only two things. (a) Separated quantum entities do not exist in nature. And then of course, it is not necessary for quantum mechanics to be able to describe separated quantum entities. (b) Quantum mechanics is not a complete theory, and does not deliver of description of all of nature, and worse, even one of the most simple situations, the one of separated quantum entities, is not in its realm of description. If (b) is true, we have detected a severe shortcoming of standard quantum mechanics. A shortcoming that is sufficient for the theory to have to be generalized in a fundamental way.

Let us explain why we personally believe that we are in case (b). As we defined it in our approach, entities are separated if a test performed on one of the entities does not provoke a change of state of the other entity, in the sense that this change of state gives rise to a change of outcome probabilities for tests performed on the other entity. This means that separated entities can have interaction between them going on, as long as this interaction is of the dynamical type, influencing the dynamical change of each others states. So the sun and the earth are separated entities in our approach. Two entangled quantum entities are not separated entities. But two quantum entities in a product state (non entangled state) are separated entities. This shows immediately that indeed separated quantum entities do exist in nature, and we describe them even by putting them in product states. The remark could now be made that, since we describe separated quantum entities by putting them in product states, a description does exist within standard quantum mechanics. This seems to be a contradiction with the result that we obtained, but it is not. The situation is subtle, and that is, in my opinion, the reason why the problem has not been identified earlier on. The failure of standard quantum mechanics to describe separated quantum entities does not appear on the level of the states. The tensor product procedure indeed delivers enough states to describe separated entities, namely the product states. What is lacking are properties: some properties, that are well identified and made explicit in the publications [10, 12, 14], cannot be described as closed subspaces (projection operators) of the tensor product Hilbert space. This has a repercussion on the dynamics and makes it so that also dynamical transformations (in standard quantum mechanics represented by unitary operators) are lacking in the tensor product procedure of standard quantum mechanics for the case of separated entities. The shortcoming has been noticed here, without knowing however that it is a shortcoming of the overall structure of standard quantum mechanics: we know that a Schrodinger equation of two separated but dynamically interacting quantum entities transforms the joint entity in such a way that even if the initial state is a product state, this product state evolves to an entangled state right away. There is no Schrodinger equation with dynamical interaction that conserves product states. If two neutrons interact in empty space only by gravitation, and hence in our approach are separated entities, they evolve immediately into entangled states when the Schrodinger equation is applied. Our result states that if we would describe the two neutrons in a formalism more general than standard quantum mechanics, where two of the five axioms, weak modularity and the covering law, are not satisfied -- which, by the way, would of course still be a quantum formalism, but non standard -- it would be possible to have a description of the two neutrons remaining separated in product states, while interacting dynamically through gravitation.

In publications [19,27] we put forward some ideas of how quantum mechanics would have to be changed towards a more general quantum-like theory to solve the problem of the description of separated quantum entities. It is important to know that one of the failing axioms, namely the covering law, gives rise to the linear structure of the set of states. This means that if we have to drop this axiom, the set of states within the more general theory will not be imbedded in a vector space, like this is the case for standard quantum mechanics. Hence the superposition principle will not hold any longer. Publications [37,43] put forward the problem in a general perspective.

Meanwhile the content of the original article and of my PhD thesis was critizised, correctly so I found after studying the critique, in an article: Cattaneo, G. and Nistico, G., 1990, "A note on Aerts' description of separated entities, Found. Phys., 20, 119. The main point of the critique was that some properties of the lattice that represents the properties of the joint entity consisting of separated sub entities are lacking because only tests (i.e. experiments with only two possible outcomes 'yes' and 'no') are considered in the operational derivation of the properties. We have responded to this critique by showing that the whole construction could be overdone by also considering experiments with more than two possible outcomes, and the result that we obtained in our original work remains valid. This is published in publication [41].

The problem of constructing a dynamics for the situation of separated quantum entities, taking into account our result, has recently been investigated by Boris Ichi for his doctoral thesis: "L'evolution des systemes separes en interaction". But still a lot of work has to be done to solve the problem. What is lacking is a concrete mathematical space that would substitute the Hilbert space of standard quantum mechanics. Indeed, we have identified the failing axioms, but the problem is that when we drop these two axioms (weak modularity and the covering law) the resulting structure (a complete orthocomplemented atomic lattice) is too general to do physics. So we need to find 'good' axioms that substitute for the two 'bad' axioms and that render our mathematical structure concrete enough to do physics. A small step in this direction has been made (perhaps) in our investigation of Soler's theorem (see section "Quantum axiomatics and Soler's theorem" for details).

[8]

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.

[10]

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.

[12]

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.

[14]

Aerts, D. (1983). The description of one and many physical systems. In C. Gruber (Ed.), Foundations of Quantum Mechanics (pp. 63-148). Lausanne: AVCP.

[19]

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.

[27]

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.

[37]

Aerts, D. (1993). De Muze van het Leven, Quantummechanica en de Aard van de Werkelijkheid, Kapellen: Pelckmans. Kampen: Agora Kok.

[41]

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.

[43]

Aerts, D. (1994). Quantummechanica. In L. Apostel and F. Verbeure (Eds.), Verwijdering of Ontmoeting? (pp. 123-142). Kapellen: Pelckmans.