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.

When I started to do research for my PhD, there existed a well established axiomatic formulation of standard quantum mechanics formulated mathematically within a lattice theoretic framework. It had been possible to regain all of standard quantum mechanics by a mixture of 'physically plausible axioms' and 'mathematical axioms directed towards the structure of a complex Hilbert space' in this lattice theoretic framework. This resulted in the celebrated representation theorem of Piron, where it proven that a complete, orthocomplemented, weakly modular, irreducible lattice satisfying the covering law, can be represented as the lattice of 'closed' subspaces of a 'generalised' Hilbert space (see research webpage for a detailed explanation). All this was however for the description of 'one' quantum entity. No research had been done on the description of joint quantum entities within this axiomatic approach. In the standard Hilbert space quantum mechanics a la von Neumann a well defined procedure existed for the description of joint quantum entities. The joint entity is described in the Hilbert space that is the tensor product of the two Hilbert spaces describing the sub entities. But this tensor product procedure had not been investigated in the axiomatic lattice theoretic approach.

The study of the tensor product procedure within the axiomatic approach to quantum mechanics was the problem that I started to work on together with Ingrid Daubechies. When we formulated the problem, it became clear that we would have to study the structure of general morphisms between Hilbert space lattices. The structure of isomorphisms between Hilbert space lattices had been investigated by Eugene Wigner in the beginning years of quantum mechanics, and lead to a deep and beautiful theorem (called Wigner's theorem) that shows that each isomorphism can be represented by a unitary or anti-unitary map between the Hilbert spaces. By making use of the projective geometry structure of a Hilbert space lattice we proved a non trivial generalisation of Wigner's theorem for the case of m-morphisms (general morphisms preserving modular pairs). This is the content of publication [1]. We proved that such an m-morphism is generated by a family of unitary and anti-unitary maps between the corresponding Hilbert spaces.

This generalisation of Wigner's theorem for m-morphisms made it possible for us to study the situation of a joint quantum entity composed of two sub quantum entities all described by complex Hilbert spaces in all its generality within the axiomatic lattice theoretic approach. We proved that when physically plausible 'coupling conditions' are satisfied, there are two solutions for the description of the joint entity. Or it is described by the tensor product of the Hilbert spaces describing the sub entities, or by the tensor product of one of the Hilbert spaces describing the sub entities with the dual Hilbert space of the other entity. This result can be found in publication [2]. We want to mention that the two solutions that we had found are two genuinely different solutions, because they are not canonically isomorphic within the 'universal problem set-up' given by the coupling conditions. Till now we do not have a good physical understanding why two solutions appear of which only one is used systematically in standard quantum mechanics, but, as can be seen in detail in my research webpages, and also in the section "The general description of joint entities", the existence of two solutions was the announcement of a deep problem with the general description of joint entities that we would start to investigate later.

What we did in publication [1] and [2] is formulated in the lattice theoretic approach to quantum mechanics. We had been looking also the the von Neumann algebra approach and our results made it possible to study also there the situation of sub entities. This is the content of publications [3] and [4].

The coupling conditions, since formulated in the category of lattices of closed subspaces of a Hilbert space, were general enough to study the situation of the joint entity in a more general setting, namely the setting where Piron's representation theorem is valid, and hence the quantum entities under consideration are described in generalised Hilbert spaces instead of complex Hilbert spaces. This was the subject of investigation of publication [4]. It was possible to prove that the division rings of these general Hilbert spaces have to be commutative and that a bi-linear map exists generating the morphisms of the coupling conditions. A summary of all these results can be found in publication [9].

Meanwhile Ingrid Daubechies had started to work on another scientific subject and I did not really use the material that we worked on together for my PhD thesis, because also meanwhile I had been investigating the problem of the description of the joint entity of two 'separated' quantum entities (see section "The description of separated entities and failing quantum axioms"). The results that I obtained there I used for my doctoral thesis, but at Brussels university each doctoral thesis is accompanied by what is called a 'second thesis', which has to be a result independent of the main thesis. Ingrid agreed that I would use the generalisation of Wigner's theorem that we proved together for my second thesis. That is the reason that we started to look again to the mathematics of the situation and got the intuition that perhaps our theorem could be proved for general morphisms (and not only for m-morphisms as we had done). Our intuition was true, and we had to use Gleason's theorem to make the generalisation for our original theorem, which makes it clear why this had not been obvious from the start. This is the content of publication [15].

[1]

Aerts, D. and Daubechies, I. (1978). Structure-preserving maps of a quantum mechanical propositional system. Helvetica Physica Acta, 51, pp. 637-660.

[2]

Aerts, D. and Daubechies, I. (1978). Physical justification for using the tensor product to describe two quantum systems as one joint system. Helvetica Physica Acta, 51, pp. 661-675.

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.

[3]

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.

[4]

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.

[7]

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.

[9]

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.

[15]

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.