## 1. First steps

I arrived in the theoretical physics group at Brussels Free University as a doctoral student in 1976. My young enthusiastic fascination was with quantum mechanics. As a student the course on quantum mechanics had intrigued me very much and now I was eager to learn more about it. I also wanted to find out whether it would be possible to work on a doctorate by doing research on the theory of quantum mechanics itself. I was lucky that my thesis director, professor Jean Reignier, who was also the head of the theoretical physics group, was very open minded and supported the idea. Although he was not himself engaged in research on this topic, he was interested in the foundations of quantum mechanics. He warned me however that it is not obvious for a young student to engage in this type of research, and that a certain risk was involved concerning the possibility to make a doctorate. At the same time I could feel however that he would support me if I made this choice. He did tell me that I would have to look for a scientist working in this field that could be my scientific thesis director, and he allowed me to look all over the world, because there would be money to have me travelling and even staying some period of time abroad. This is the constellation in which I started my research as a young doctoral student.

When I started to read articles on the foundations of quantum mechanics, rather by accident, because I found it in a forgotten pile of papers at the mathematics department during tea break, I stumbled upon an article of Constantin Piron. The title of the article was: "Survey of Generalised Quantum Mechanics". I was highly intrigued by the content of the article. Piron explained how the standard theory of quantum mechanics could be derived from a more general approach, that he called 'generalised quantum mechanics' in an axiomatic way. I found it even more intriguing to learn that this generalised quantum mechanics was defined operationally, starting from very the simple physical objects of 'yes-no-experiments', and that standard quantum mechanics as well as classical mechanics arose both as special cases of the theory. Reading eagerly the article of Piron led me to more of his papers, more specifically the one of 1964, that contains the results of his doctoral thesis, and where he derives (almost - we will come to this later) the structure of the complex Hilbert space of standard quantum mechanics from the set of yes-no-experiments and axioms defined on this set.

I was completely taken by the approach to quantum mechanics that I came to detect and could not help to talk to all the members of the theoretical physics group about it. I bought the books of Jozef Maria Jauch and of George Mackey that also follow in great lines a similar approach and I started to read and study intensively.

## 2. The basic setting of operational axiomatic quantum mechanics

Standard quantum mechanics, mathematically rigorously defined in the book of John von Neumann, described the states of a quantum entity by means of unit vectors of a complex Hilbert space. Physical quantities or observables of a quantum entity are described as self-adjoint operators on this complex Hilbert space, where the outcomes of the observable are the numbers of the spectrum of the self-adjoint operator. Hence the most simple of all observables, an observable with only two possible outcomes, labelled 'yes' and 'no', is described by a self-adjoint operator with only two elements in its spectrum, and this is an orthogonal projection operator. The operational axiomatic approaches that I go into concentrated on the 'yes-no-experiments' as basic objects. Actually, this approach was originated by John von Neumann himself, in 1936, only a few years after he wrote his book, in an article that he wrote together with Garett Borkhoff entitled "The logic of quantum mechanics". The reason why 'yes-no-experiments' can be taken as basic objects for a general approach to quantum mechanics is because a general experiment with more than two outcomes can always be subdivided in different yes-no-experiments, considering each time again a subset and its complement of the outcome set. This means that if we develop a framework for the general description of yes-no-experiments we can reconstruct all the experiments by considering them as compostions of their underlying yes-no-experiments. Much in the same way as a self-adjoint operator is completely defined by means of its spectral resolution, which is the set of orthogonal projections corresponding to the yes-no-experiments that are sub experiments of the observable described by this self-adjoint operator.

So the focus of these general operational axiomatic approaches to quantum mechanics is on the set of orthogonal projection operators of the complex Hilbert space that corresponds to the quantum entity under consideration. There is a one to one correspondence between an orthogonal projection operator and the closed subspace of the Hilbert space that is the range of this projection operator, in the sense that if we know the projection operator we also know this subspace, and if we consider a closed subspace of the Hilbert space, this also defines the orthogonal projection operator on this subspace. That is the reason that more often it are the closed subspaces of the Hilbert space that are taken to represent the yes-no-experiments, and that is also what we will do here. Let us denote the complex Hilbert space corresponding to a quantum entity

Sby means ofH. Traditonally the set of closed subspaces of the complex Hilbert spaceHis then denoted byL(H).As we mentioned already, yes-no-experiments are the basic operational tools of the approach, and for a quantum entity described in a complex Hilbert space

Hthey are represented by the set of closed subspacesL(H)of the Hilbert space. Each yes-no-experiment tests a certain property of the quantum entity under consideration. That is the reason that often also the properties of the quantum entity under consideration are taken to be the basic objects. Of course, a yes-no-experment is not the same object as a property, and we will come back later to the more subtle aspects of the approach. For the purpose of introducing the approach, it is all right that we do not distuinguish yet all this, and consider a closed subspace of the Hilbert space to represent as well a yes-no-experiment as the property tested by this yes-no-experiment. The aim of a axiomatic approach to standard quantum mechanics is to put forward axioms on a basic structure that will lead to standard quantum mechanics once these axioms are fulfilled. To have an idea of what type of axioms have been investigated, let us put forward some of thye basic properties of the setL(H)of closed subsets of a complex Hilbert spaceH.

## 3. The lattice of closed subspaces of a complex Hilbert space

a) L(H), is a partially ordered set.There exists the structure of a partial order relation on

L(H). Let us explain what we mean by this. Consider two closed subspacesAandB, then it is possible thatAis contained inB. We denote this byAB. Clearly we have, forAL(H), thatAA. This shows that the relation is 'reflexive' (every element relates to itself). Furthermore, forA, BL(H), we have thatABandBAimpliesA = B. This shows that the relation is symmetric (if a first element relates to a second one, then this second element also relates to the first one). Finally, forA, B, CL(H), we have thatABandBC, impliesAC. This shows that the relation is transitive (if a first element relates to a second and this second to a third, then the first element also relates to the third). A relation that is reflexive, symmetric and transitive is called a 'partial order relation'. A set with a partial order relation is called a patially ordered set.

b) L(H), , , , is a complete lattice.If a set is partially ordered, for a mathematician a natural question arises: to investigate whether a collection of elements of this set has an infimum (greatest lower bound) or a supremum (least upper bound) for this partial order. Let us check this first for two elements of

L(H). So consider two closed subspacesA, BL(H). The intersectionA Bwould be a natural candidate for the infimum of the collection consisting of the two elementsAandB. This makes of course only sense if the intersection of two closed subspaces is again a closed subspace. It can be proven that indeed the intersection of any number of closed subspaces of a complex Hilbert space is again a closed subspace of this complex Hilbert space. We will not prove this, because it takes some Hilbert space mathematics to show this, and we do not want to elaborate in this here. But taken for granted this property of closed subspaces of a complex Hilbert space, it is easy to verify thatA Bis an infimum. First we have to verify that it is a lower bound, namely thatA B AandA B B. This follows immediately from the definition of the set theoretical intersection of sets. Second we have to verify that it is the greatest of all the lower bounds. This means that if we consider another arbitrary lower bound, lets call itD, henceD AandD B, that this implies thatD A B. Also this follows from the set theoretical definition of the intersection of sets, but lets prove it explicitly. To prove thatD A Bit is sufficient to show that every vector of the Hilbert space that is contained inDis also contained inA B. So consider a vectorx Hsuch thatx D. SinceD AandD Bwe havex Aandx B. But thenx A B. This proves thatA Bis an infimum of the collection consisting ofAandB. What we have shown now for a collection consisting of two elements is also true for a collection (A_{i})_{i}consisting of any number of closed subspaces. The intersection of these closed subspaces_{i}A_{i}is again a closed subspace and it is the infimum. A natural candidate for the supremum of a collection of closed subspaces would, at first sight perhaps, be the union of these closed subspaces. This is however not true. The union of closed subspaces is not even again a subspace in general. The fact that the union of closed subspaces is in general not again a subspace is directly linked to the superpostion principle in quantum mechanics. It can be seen already for the case of a Hilbert space of little dimensions, for example dimension 2. For a two dimensional complex Hilbert space, the only subspaces are the one dimensional subspaces (the rays), the subspace 0, and the whole space. Obviously the set theoretical union of two such one dimensional subspaces, in case they are different, is not a one dimensional subspace, and neither is it 0 nor the whole space. So it is not a subspace, because they are the only ones. Form the fact that for any collection of closed subspaces there exists an infimum, it can however be proven that for any collection of closed subspaces there also exist a supremum. This supremum is just the infimum of the collection of all upper bounds of the considered collection. This means that the supremum of a collection of closed subspaces is the smallest closed subspace that contains all of these. It can be proven, using again some Hilbert space mathematics, that this is the topological closure of the linear closure of all these closed subspaces. A partially ordered set such that for all collections of elements there exists an infimum and a supremum is called a complete lattice. For a collection (A_{i})_{i}of closed subspaces we denote the supremum of this collection by_{i}A_{i}.

c) L(H), , , , ', is a complete orthocomplemented lattice.In Hilbert space exists an orthogonality relation on the vectors. Two vectors are orthogonal if the inproduct between these two vectors is zero. This orthogonality relation on

Hintroduces the structure of an orthocomplementation onL(H). Let us explain what this structure is. Consider a closed subspaceA L(H). Let us defineA' to be the set of all vectors that are orthogonal to all the vectors contained inA. It can be proven in a Hilbert space thatA' is again a closed subspace. Moreover we have (A')' =A. If we consider two closed subspacesA, B L(H), such thatA B, it can be shown that this implies thatB'A'. No vector is orthogonal to itself, and this implies thatA A' = 0. A function ' , that maps each elementA L(H)to an elementA' L(H), such that (i) (A')' =A, (ii)A BgivesB'A', (iii)A A' = 0, is called an orthocomplementation, and a lattice with such a function on it, is called an orthocomplemented lattice. So we have shown thatL(H)is an orthocomplemented lattice.

d) L(H), , , , ', is an atomistic lattice.An atom of a lattice is a smallest element different from 0. In the case of the lattice of closed subspaces of a complex Hilbert space, it are the one dimensional subspaces that are atoms. A complete lattice is called atomistic if every element is the supremum of the atoms that are contained in this element. This is the case for

L(H). An arbitrary closed subspace is indeed the supremum of the one dimensional subspaces that are contained in this closed subspace.

e) L(H), , , , ', is weakly modular.Weak modularity is a somewhat more complex mathematical property. It means that for closed subspaces that are related by the partial order, there is distributivity between infimum and supremum. More specifically, suppose that

A, B, C P(H), such thatA B, then (A B)B' =A. It is possible to prove thatL(H)is weakly modular. In fact, the weak modularity is equivalent with the Cauchy completeness (every Cauchy sequence has a limit) of the complex Hilbert space.

f) L(H), , , , ', satisfies the covering law.The covering law is the property that makes an atomistic lattice into a projectice geometry, where the atoms of the lattice are the points of the geometry. To be able to put forward the covering law, we first have to say what is meant by an element of a lattice 'covering' another element. Suppose that we consider two elements

AandBof a lattice. We say thatBcoversAif it is such thatAis contained inBand there is no element in betweenAandB. With other words, fromA D Bfollows thatA=DorB=D.L(H)satisfies the covering law.

## 4. Piron's representation theorem

## 5. To be continued