. Some remarks about Collapse Theories

A. Pais famously recalls in his biography of Einstein:

اقتباس :: We often discussed his notions on objective reality. I recall that during one walk Einstein suddenly stopped, turned to me and asked whether I really believed that the moon exists only when I look at it (Pais 1982, p. 5).

In the context of Einstein’s remarks in Albert Einstein, Philosopher-Scientist (Schilpp 1949), we can regard this reference to the moon as an extreme example of ‘a fact that belongs entirely within the sphere of macroscopic concepts’, as is also a mark on a strip of paper that is used to register the outcome of a decay experiment, so that

اقتباس :: as a consequence, there is hardly likely to be anyone who would be inclined to consider seriously […] that the existence of the location is essentially dependent upon the carrying out of an observation made on the registration strip. For, in the macroscopic sphere it simply is considered certain that one must adhere to the program of a realistic description in space and time; whereas in the sphere of microscopic situations one is more readily inclined to give up, or at least to modify, this program (p. 671).

However,

اقتباس :: the ‘macroscopic’ and the ‘microscopic’ are so inter-related that it appears impracticable to give up this program in the ‘microscopic’ alone (p. 674).

One might speculate that Einstein would not have taken the DRP seriously, given that it is a fundamentally indeterministic program. On the other hand, the DRP allows precisely for this middle ground, between giving up a ‘classical description in space and time’ altogether (the moon is not there when nobody looks), and requiring that it be applicable also at the microscopic level (as within some kind of ‘hidden variables’ theory). It would seem that the pursuit of ‘realism’ for Einstein was more a program that had been very successful rather than an a priori commitment, and that in principle he would have accepted attempts requiring a radical change in our classical conceptions concerning microsystems, provided they would nevertheless allow to take a macrorealist position matching our definite perceptions at this scale.
In the DRP, we can say of an electron in an EPR-Bohm situation that ‘when nobody looks’, it has no definite spin in any direction , and in particular that when it is in a superposition of two states localised far away from each other, it cannot be thought to be at a definite place (see, however, the remarks in Section 11). In the macrorealm, however, objects do have definite positions and are generally describable in classical terms. That is, in spite of the fact that the DRP program is not adding ‘hidden variables’ to the theory, it implies that the moon is definitely there even if no sentient being has ever looked at it. In the words of J. S. Bell, the DRP

اقتباس :: allows electrons (in general microsystems) to enjoy the cloudiness of waves, while allowing tables and chairs, and ourselves, and black marks on photographs, to be rather definitely in one place rather than another, and to be described in classical terms (Bell 1986, p. 364).

Such a program, as we have seen, is implemented by assuming only the existence of wave functions, and by proposing a unified dynamics that governs both microscopic processes and ‘measurements’. As regards the latter, no vague definitions are needed. The new dynamical equations govern the unfolding of any physical process, and the macroscopic ambiguities that would arise from the linear evolution are theoretically possible, but only of momentary duration, of no practical importance and no source of embarrassment.
We have not yet analyzed the implications about locality, but since in the DRP program no hidden variables are introduced, the situation can be no worse than in ordinary quantum mechanics: ‘by adding mathematical precision to the jumps in the wave function’, the GRW theory ‘simply makes precise the action at a distance of ordinary quantum mechanics’ (Bell 1987, p. 46). Indeed, a detailed investigation of the locality properties of the theory becomes possible as shown by Bell himself (Bell 1987, p. 47). Moreover, as it will become clear when we will discuss the interpretation of the theory in terms of mass density, the QMSL and CSL theories lead in a natural way to account for a behaviour of macroscopic objects corresponding to our definite perceptions about them, the main objective of Einstein’s requirements.
The achievements of the DRP which are relevant for the debate about the foundations of quantum mechanics can also be concisely summarized in the words of H.P. Stapp:

اقتباس :: The collapse mechanisms so far proposed could, on the one hand, be viewed as ad hoc mutilations designed to force ontology to kneel to prejudice. On the other hand, these proposals show that one can certainly erect a coherent quantum ontology that generally conforms to ordinary ideas at the macroscopic level (Stapp 1989, p. 157).

9. Relativistic Dynamical Reduction Models

As soon as the GRW proposal appeared and attracted the attention of J.S. Bell it also stimulated him to look at it from the point of view of relativity theory. As he stated subsequently (Bell 1989a):

اقتباس :: When I saw this theory first, I thought that I could blow it out of the water, by showing that it was grossly in violation of Lorentz invariance. That’s connected with the problem of ‘quantum entanglement’, the EPR paradox.

Actually, he had already investigated this point by studying the effect on the theory of a transformation mimicking a nonrelativistic approximation of a Lorentz transformation and he arrived (Bell 1987) at a surprising conclusion:

اقتباس :: … the model is as Lorentz invariant as it could be in its nonrelativistic version. It takes away the ground of my fear that any exact formulation of quantum mechanics must conflict with fundamental Lorentz invariance.

What Bell had actually proved by resorting to a two-times formulation of the Schrödinger equation is that the model violates locality by violating outcome independence and not, as deterministic hidden variable theories do, parameter independence.
Indeed, with reference to this point we recall that, as is well known, (Suppes and Zanotti 1976; van Fraassen 1982; Jarrett 1984; Shimony 1983; see also the entry on Bell’s Theorem), Bell’s locality assumption is equivalent to the conjunction of two other assumptions, viz., in Shimony’s terminology, parameter independence and outcome independence. In view of the experimental violation of Bell’s inequality, one has to give up either or both of these assumptions. The above splitting of the locality requirement into two logically independent conditions is particularly useful in discussing the different status of CSL and deterministic hidden variable theories with respect to relativistic requirements. Actually, as proved by Jarrett himself, when parameter independence is violated, if one had access to the variables which specify completely the state of individual physical systems, one could send faster-than-light signals from one wing of the apparatus to the other. Moreover, in Ghirardi and Grassi (1996) it has been proved that it is impossible to build a genuinely relativistically invariant theory which, in its nonrelativistic limit, exhibits parameter dependence. Here we use the term genuinely invariant to denote a theory for which there is no (hidden) preferred reference frame. On the other hand, if locality is violated only by the occurrence of outcome dependence then faster-than-light signaling cannot be achieved (Eberhard 1978; Ghirardi, Rimini, and Weber 1980). Few years after the just mentioned proof by Bell, it has been shown in complete generality (Ghirardi, Grassi, Butterfield, and Fleming 1993) that the GRW and CSL theories, just as standard quantum mechanics, exhibit only outcome dependence. This is to some extent encouraging and shows that there are no reasons of principle making unviable the project of building a relativistically invariant DRM.
Let us be more specific about this crucial problem. P. Pearle was the first to propose (Pearle 1990) a relativistic generalization of CSL to a quantum field theory describing a fermion field coupled to a meson scalar field enriched with the introduction of stochastic and nonlinear terms. A quite detailed discussion of this proposal was presented in (Ghirardi et al. 1990a) where it was shown that the theory enjoys of all properties which are necessary in order to meet the relativistic constraints. Pearle’s approach requires the precise formulation of the idea of stochastic Lorentz invariance. The proposal can be summarized in the following terms:
One considers a fermion field coupled to a meson field and puts forward the idea of inducing localizations for the fermions through their coupling to the mesons and a stochastic dynamical reduction mechanism acting on the meson variables. In practice, one considers Heisenberg evolution equations for the coupled fields and a Tomonaga-Schwinger CSL-type evolution equation with a skew-hermitian coupling to a c-number stochastic potential for the state vector. This approach has been systematically investigated by Ghirardi, Grassi, and Pearle (1990), to which we refer the reader for a detailed discussion. Here we limit ourselves to stressing that, under certain approximations, one obtains in the non-relativistic limit a CSL-type equation inducing spatial localization. However, due to the white noise nature of the stochastic potential, novel renormalization problems arise: the increase per unit time and per unit volume of the energy of the meson field is infinite due to the fact that infinitely many mesons are created. This point has also been lucidly discussed by Bell (1989b) in the talk he delivered at Trieste on the occasion of the 25th anniversary of the International Centre for Theoretical Physics. This talk appeared under the title The Trieste Lecture of John Stewart Bell. For these reasons one cannot consider this as a satisfactory example of a relativistic reduction model.
In the years following the just mentioned attempts there has been a flourishing of researches aimed at getting the desired result. Let us briefly comment about them. As already mentioned, the source of the divergences is the assumption of point interactions between the quantum field operators in the dynamical equation for the statevector, or, equivalently, the white character of the stochastic noise. Having this aspect in mind P. Pearle (1989), L. Diosi (1990) and A. Bassi and G.C. Ghirardi (2002) reconsidered the problem from the beginning by investigating nonrelativistic theories with nonwhite Gaussian noises. The problem turns out to be very difficult from the mathematical point of view, but steps forward have been made. In recent years, a precise formulation of the nonwhite generalization (Bassi and Ferialdi 2009) of the so-called QMUPL model, which represents a simplified version of GRW and CSL, has been proposed. Moreover, a perturbative approach for the CSL model has been worked out (Adler and Bassi 2007, 2008). Further work is necessary. This line of thought is very interesting at the nonrelativistic level; however, it is not yet clear whether it will lead to a real step forward in the development of relativistic theories of spontaneous collapse.
In the same spirit, Nicrosini and Rimini (Nicrosini 2003) tried to smear out the point interactions without success because, in their approach, a preferred reference frame had to be chosen in order to circumvent the nonintegrability of the Tomonaga-Schwinger equation
Also other interesting and different approaches have been suggested. Among them we mention the one by Dove and Squires (Dove 1996) based on discrete rather than continuous stochastic processes and those by Dawker and Herbauts (Dawker 2004a) and Dawker and Henson (Dawker 2004b) formulated on a discrete space-time.
Before going on we consider it important to call attention to the fact that precisely in the same years similar attempts to get a relativistic generalization of the other existing ‘exact’ theory, i.e., Bohmian Mechanics, were going on and that they too have encountered some difficulties. Relevant steps are represented by a paper (Dürr 1999) resorting to a preferred spacetime slicing, by the investigations of Goldstein and Tumulka (Goldstein 2003) and by other scientists (Berndl et. al 1996). However, we must recognize that no one of these attempts has led to a fully satisfactory solution of the problem of having a theory without observers, like Bohmian mechanics, which is perfectly satisfactory from the relativistic point of view, precisely due to the fact that they are not genuinely Lorentz invariant in the sense we have made precise before. Mention should be made also of the attempt by Dewdney and Horton (Dewdney 2001) to build a relativistically invariant model based on particle trajectories.
Let us come back to the relativistic DRP. Some important changes have occurred quite recently. Tumulka (2006a) succeeded in proposing a relativistic version of the GRW theory for N non-interacting distinguishable particles, based on the consideration of a multi-time wavefunction whose evolution is governed by Dirac like equations and adopts as its Primitive Ontology (see the next section) the one which attaches a primary role to the space and time points at which spontaneous localizations occur, as originally suggested by Bell (1987). To my knowledge this represents the first proposal of a relativistic dynamical reduction mechanism which satisfies all relativistic requirements. In particular it is divergence free and foliation independent. However it can deal only with systems containing a fixed number of noninteracting fermions.
At this point explicit mention should be made of the most recent steps which concern our problem. D. Bedingham (2011) following strictly the original proposal by Pearle (1990) of a quantum field theory inducing reductions based on a Tomonaga-Schwinger equation, has worked out an analogous model which, however, overcomes the difficulties of the original model. In fact, Bedingham has circumvented the crucial problems deriving from point interactions by (paying the price of) introducing, besides the fields characterizing the Quantum Field Theories he is interested in, an auxiliary relativistic field that amounts to a smearing of the interactions whilst preserving Lorentz invariance and frame independence. Adopting this point of view and taking advantage also of the proposal by Ghirardi (2000) concerning the appropriate way to define objective properties at any space-time point [ltr]xx[/ltr], he has been able to work out a fully satisfactory and consistent relativistic scheme for quantum field theories in which reduction processes may occur.
It has also to be mentioned that, taking once more advantage of the ideas of the paper by Ghirardi (2000), various of the just quoted authors (see Bedingham et al. 2013), have been able to prove that it is possible to work out a relativistic generalization of Collapse models when their primitive ontology is taken to be the one given by the mass density interpretation for the nonrelativistic case we will present in what follows.
In view of these results and taking into account the interesting investigations concerning relativistic Bohmian-like theories,the conclusions that Tumulka has drawn concerning the status of attempts to account for the macro-objectification process from a relativistic perspective are well-founded:

اقتباس :: A somewhat surprising feature of the present situation is that we seem to arrive at the following alternative: Bohmian mechanics shows that one can explain quantum mechanics, exactly and completely, if one is willing to pay with using a preferred slicing of spacetime; our model suggests that one should be able to avoid a preferred slicing of spacetime if one is willing to pay with a certain deviation from quantum mechanics,

a conclusion that he has rephrased and reinforced in (Tumulka 2006c):

اقتباس :: Thus, with the presently available models we have the alternative: either the conventional understanding of relativity is not right, or quantum mechanics is not exact.

Very recently, a thorough and illuminating discussion of the important approach by Tumulka has been presented by Tim Maudlin (2011) in the third revised edition of his book Quantum Non-Locality and Relativity. Tumulka’s position is perfectly consistent with the present ideas concerning the attempts to transform relativistic standard quantum mechanics into an ‘exact’ theory in the sense which has been made precise by J. Bell. Since the only unified, mathematically precise and formally consistent formulations of the quantum description of natural processes are Bohmian mechanics and GRW-like theories, if one chooses the first alternative one has to accept the existence of a preferred reference frame, while in the second case one is not led to such a drastic change of position with respect to relativistic concepts but must accept that the ensuing theory disagrees with the predictions of quantum mechanics and acquires the status of a rival theory with respect to it.
In spite of the fact that the situation is, to some extent, still open and requires further investigations, it has to be recognized that the efforts which have been spent on such a program have made possible a better understanding of some crucial points and have thrown light on some important conceptual issues. First, they have led to a completely general and rigorous formulation of the concept of stochastic invariance. Second, they have prompted a critical reconsideration, based on the discussion of smeared observables with compact support, of the problem of locality at the individual level. This analysis has brought out the necessity of reconsidering the criteria for the attribution of objective local properties to physical systems. In specific situations, one cannot attribute any local property to a microsystem: any attempt to do so gives rise to ambiguities. However, in the case of macroscopic systems, the impossibility of attributing to them local properties (or, equivalently, the ambiguity associated to such properties) lasts only for time intervals of the order of those necessary for the dynamical reduction to take place. Moreover, no objective property corresponding to a local observable, even for microsystems, can emerge as a consequence of a measurement-like event occurring in a space-like separated region: such properties emerge only in the future light cone of the considered macroscopic event. Finally, recent investigations (Ghirardi and Grassi 1996; Ghirardi 2000) have shown that the very formal structure of the theory is such that it does not allow, even conceptually, to establish cause-effect relations between space-like events.
The conclusion of this section, is that the question of whether a relativistic dynamical reduction program can find a satisfactory formulation seems to admit a positive answer.
A last comment. Recently, a paper by Conway and Kochen (Conway 2006, 2006b), which has raised a lot of interest, has been published. A few words about it are in order, to clarify possible misunderstandings. The first and most important aim of the paper is the derivation of what the authors have called The Free Will Theorem, putting forward the provocative idea that if human beings are free to make their choices about the measurements they will perform on one of a pair of far-away entangled particles, then one must admit that also the elementary particles involved in the experiment have free will. One might make several comments on this statement. For what concerns us here the relevant fact is that the authors claim that their theorem implies, as a byproduct, the impossibility of elaborating a relativistically invariant dynamical reduction model. A lively debate has arisen. At the end, Goldstein et al (Goldstein 2010) have made clear why the argument of Conway and Kochen is not pertinent. We may conclude that nothing in principle forbids a perfectly satisfactory relativistic generalization of the GRW theory, and, actually, as repeatedly stressed, there are many elements which indicate that this is actually feasible.

. Some remarks about Collapse Theories

9. Relativistic Dynamical Reduction Models

. Some remarks about Collapse Theories :: تعاليق

. Some remarks about Collapse Theories