We must admit that this opinion is not universally shared. According to various authors, the ‘rules of the game’ embodied in the precise formulation of the GRW and CSL theories represent all there is to say about them. However, this cannot be the whole story: stricter and more precise requirements than the purely formal ones must be imposed for a theory to be taken seriously as a fundamental description of natural processes (an opinion shared by J. Bell). This request of going beyond the purely formal aspects of a theoretical scheme has been denoted as (the necessity of specifying) the Primitive Ontology (PO) of the theory in an extremely interesting recent paper (Allori et al. 2008). The fundamental requisite of the PO is that it should make absolutely precise what the theory is fundamentally about.
This is not a new problem; as already mentioned it has been raised by J. Bell since his first presentation of the GRW theory. Let me summarize the terms of the debate. Given that the wavefunction of a many-particle system lives in a (high-dimensional) configuration space, which is not endowed with a direct physical meaning connected to our experience of the world around us, Bell wanted to identify the ‘local beables’ of the theory, the quantities on which one could base a description of the perceived reality in ordinary three-dimensional space. In the specific context of QMSL, he (Bell 1987 p. 45) suggested that the ‘GRW jumps’, which we called ‘hittings’, could play this role. In fact they occur at precise times in precise positions of the three-dimensional space. As suggested in (Allori et al. 2008) we will denote this position concerning the PO of the GRW theory as the ‘flashes ontology.’
However, later, Bell himself suggested that the most natural interpretation of the wavefunction in the context of a collapse theory would be that it describes the ‘density […] of stuff’ in the 3N-dimensional configuration space (Bell 1990, p. 30), the natural mathematical framework for describing a system of [ltr]NN[/ltr] particles. Allori et al. (2008) appropriately have pointed out that this position amounts to avoiding commitment about the PO ontology of the theory and, consequently, to leaving vague the precise and meaningful connections it permits to be established between the mathematical description of the unfolding of physical processes and our perception of them.
The interpretation which, in the opinion of the present writer, is most appropriate for collapse theories, has been proposed in (Ghirardi, Grassi and Benatti 1995) and has been referred in Alloriet al. 2008 as ‘the mass density ontology’. Let us briefly describe it.
First of all, various investigations (Pearle and Squires 1994) had made clear that QMSL and CSL needed a modification, i.e., the characteristic localization frequency of the elementary constituents of matter had to be made proportional to the mass characterizing the particle under consideration. In particular, the original frequency for the hitting processes [ltr]f=10[size=13]−16f=10−16[/ltr] sec[ltr]−1−1[/ltr] is the one characterizing the nucleons, while, e.g., electrons would suffer hittings with a frequency reduced by about 2000 times. Unfortunately we have no space to discuss here the physical reasons which make this choice appropriate; we refer the reader to the above paper, as well as to the recent detailed analysis by Peruzzi and Rimini (2000). With this modification, what the nonlinear dynamics strives to make ‘objectively definite’ is the mass distribution in the whole universe. Second, a deep critical reconsideration (Ghirardi, Grassi, and Benatti 1995) has made evident how the concept of ‘distance’ that characterizes the Hilbert space is inappropriate in accounting for the similarity or difference between macroscopic situations. Just to give a convincing example, consider three states [ltr]∣h⟩,∣h∗⟩∣h⟩,∣h∗⟩[/ltr] and [ltr]∣t⟩∣t⟩[/ltr] of a macrosystem (let us say a massive macroscopic bulk of matter), the first corresponding to its being located here, the second to its having the same location but one of its atoms (or molecules) being in a state orthogonal to the corresponding state in [ltr]∣h⟩∣h⟩[/ltr], and the third having exactly the same internal state of the first but being differently located (there). Then, despite the fact that the first two states are indistinguishable from each other at the macrolevel, while the first and the third correspond to completely different and directly perceivable situations, the Hilbert space distance between [ltr]∣h⟩∣h⟩[/ltr]and [ltr]∣h∗⟩∣h∗⟩[/ltr], is equal to that between [ltr]∣h⟩∣h⟩[/ltr] and [ltr]∣t⟩∣t⟩[/ltr].[/size]
When the localization frequency is related to the mass of the constituents, then, in completely generality (i.e., even when one is dealing with a body which is not almost rigid, such as a gas or a cloud), the mechanism leading to the suppression of the superpositions of macroscopically different states is fundamentally governed by the the integral of the squared differences of the mass densities associated to the two superposed states. Actually, in the original paper the mass density at a point was identified with its average over the characteristic volume of the theory, i.e., [ltr]10[size=13]−1510−15[/ltr] cm[ltr]33[/ltr] around that point. It is however easy to convince oneself that there is no need to do so and that the mass density at any point, directly identified by the statevector (see below), is the appropriate quantity on which to base an appropriate ontology. Accordingly, we take the following attitude: what the theory is about, what is real ‘out there’ at a given space point [ltr]xx[/ltr], is just a field, i.e., a variable [ltr]m(x,t)m(x,t)[/ltr] given by the expectation value of the mass density operator [ltr]M(x)M(x)[/ltr] at [ltr]xx[/ltr]obtained by multiplying the mass of any kind of particle times the number density operator for the considered type of particle and summing over all possible types of particles which can be present:[/size]

Here [ltr]∣F,t⟩∣F,t⟩[/ltr] is the statevector characterizing the system at the given time, and [ltr]a[size=13]∗(k)(x)a(k)∗(x)[/ltr] and [ltr]a(k)(x)a(k)(x)[/ltr] are the creation and annihilation operators for a particle of type [ltr]kk[/ltr] at point [ltr]xx[/ltr]. It is obvious that within standard quantum mechanics such a function cannot be endowed with any objective physical meaning due to the occurrence of linear superpositions which give rise to values that do not correspond to what we find in a measurement process or what we perceive. In the case of GRW or CSL theories, if one considers only the states allowed by the dynamics one can give a description of the world in terms of [ltr]m(x,t)m(x,t)[/ltr], i.e., one recovers a physically meaningful account of physical reality in the usual 3-dimensional space and time. To illustrate this crucial point we consider, first of all, the embarrassing situation of a macroscopic object in the superposition of two differently located position states. We have then simply to recall that in a collapse model relating reductions to mass density differences, the dynamics suppresses in extremely short times the embarrassing superpositions of such states to recover the mass distribution corresponding to our perceptions. Let us come now to a microsystem and let us consider the equal weight superposition of two states [ltr]∣h⟩∣h⟩[/ltr] and [ltr]∣t⟩∣t⟩[/ltr] describing a microscopic particle in two different locations. Such a state gives rise to a mass distribution corresponding to 1/2 of the mass of the particle in the two considered space regions. This seems, at first sight, to contradict what is revealed by any measurement process. But in such a case we know that the theory implies that the dynamics running all natural processes within GRW ensures that whenever one tries to locate the particle he will always find it in a definite position, e.g., one and only one of the Geiger counters which might be triggered by the passage of the proton will fire, just because a superposition of ‘a counter which has fired’ and ‘one which has not fired’ is dynamically forbidden.[/size]
This analysis shows that one can consider at all levels (the micro and the macroscopic ones) the field [ltr]m(x,t)m(x,t)[/ltr] as accounting for ‘what is out there’, as originally suggested by Schrödinger with his realistic interpretation of the square of the wave function of a particle as representing the ‘fuzzy’ character of the mass (or charge) of the particle. Obviously, within standard quantum mechanics such a position cannot be maintained because ‘wavepackets diffuse, and with the passage of time become infinitely extended … but however far the wavefunction has extended, the reaction of a detector … remains spotty’, as appropriately remarked in (Bell 1990). As we hope to have made clear, the picture is radically different when one takes into account the new dynamics which succeeds perfectly in reconciling the spread and sharp features of the wavefunction and of the detection process, respectively.
It is also extremely important to stress that, by resorting to the quantity (7) one can define an appropriate ‘distance’ between two states as the integral over the whole 3-dimensional space of the square of the difference of [ltr]m(x,t)[/ltr]