Abstract
We share today’s widespread opinion that standard quantum mechanics (SQM), in spite of its enormous successes, has failed in giving a satisfactory picture of the world, as we perceive it. The difficulties about the conceptual foundations of the theory arising, as is well known, from the so-called objectification problem, have stimulated various attempts to overcome them. Among these one should mention the search for a deterministic completion of the theory, the many worlds and many minds interpretations, the so called environment-induced superselection rules, the quantum histories approach and the dynamical reduction program. In this paper we will focus our attention on the only available and precisely formulated examples of a deterministic completion and of a stochastic and nonlinear modification of SQM (i.e., Bohm’s theory and the spontaneous reduction models, respectively). It is useful to stress that while the first theory is fully equivalent, from a predictive point of view, to SQM, the second one qualifies itself as a rival of SQM, but with empirical divergence so small that it can claim all the same experimental support. Accordingly, they represent explicit answers to the conclusion reached by Bell (Bell 1987b) that either the wave function as given by the Schrödinger equation is not everything or it is not right.
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Notes
It is useful to point out that at least two different attitudes can be taken about the interpretation of the theory. They have been focused in a recent paper by Bas van Fraassen (1994) and have been denoted as the Cartesian and the Newtonian construals, respectively. According to the first attitude, what is real of the theory are only the positions of the particles. Consequently, since equal positions at a given time can evolve into different positions at a subsequent time, under this construal the theory is not deterministic. In the Newtonian construal, besides the position of the particles, also the wave function is considered as real. The theory is then strictly deterministic. We will take the Newtonian attitude in what follows.
Recently, some beautiful results about this point (Dürr et al. 1992a ) have been derived. The authors shows that typical configurations of the universe as a whole imply that the positions of the particles of an ensemble associated to a given wave function 1p are distributed according to 111,12 with overwhelming probability. We cannot discuss this important point here. We refer the reader to the above paper.
In Eq. (3.8) and following we have changed the notation for the state vector from I’IÇ(t)) used in Eq. (3.2) to 1 WB(t)) to stress the fact that, under our assumptions, the state at time t does not depend on the specific sample function w(t) in the interval (0, t) but only on its integral B(t) of Eq. (3.9).
Note that, even though the spread ’/lit tends to œ for t —.0, its ratio to the distance 2(a — ß)yt between the two considered peaks of the distribution tends to zero.
In a previous paper (Ghirardi et al. 1995 ) we have simply used the expression “objective” to denote what we call here “objectively possessed or accessible”. We had a vague feeling that the term we were using was not completely pertinent and could give rise to misunderstandings. Professor S. Goldstein has appropriately called our attention to the fact that both usual meanings of that term (i.e., “real” or “opposite to subjective”) do not fit with the sense which emerges for it from our work and has suggested the expression “accessible”. In the search for an expression which would embody precisely what we had in mind, we have considered also the possibility of resorting to the expression “empirically adequate”. However, this term reminds one directly of the “empirical reality” concept introduced by B. d’Espagnat (d’Espagnat 1990 ). Because of this fact, the use of such term could seriously mislead the reader. In fact, according to the definition of empirical reality, due to the practical impossibility of distinguishing, at the macroscopic level, a pure state from a statistical mixture, replacing one with the other would be empirically adequate, which is completely at odds with the meaning we want to give to the expression objectively possessed or accessible.
We point out that here we assume that the measurement can be chosen by free will. In any case, even if one gives up such an assumption, it turns out to be impossible to attribute to the system considered by itself objective properties (i.e., properties which do not depend on the overall experimental set up). Accordingly, property attribution becomes a relational feature.
We note that if the “pointer particle” would be on the negative semiaxis at t = 0, it would be shifted by a negative quantity of the same order, independently of the sign of c. Thus, the outcome of the measurement of p, given a complete specification at t = 0 of the state of the measured particle, depends both on the sign of the coupling constant (which can be chosen at free will by the observer) and on the initial position of the “pointer particle” (a parameter which, however, cannot be controlled by the experimenter).
A typical example is represented by an intergalactic cloud with a very low mass density and wave functions for its particles with a large spread. According to our strict criterion (5.8), the mass density of the cloud is nonobjective, while, if we adopt the physically meaningful criterion put forward at the beginning of this section and to test it we resort to consider its effects (e.g., on the trajectory of a far asteroid), we would be led to state that our claims about the mass density are objectively true since they agree with the “physical outcome”. Here, an important distinction is appropriate. The theory contains a specific dynamical mechanism and specific parameters which allow a precise mathematical formulation of the objectivity criterion. Obviously, when one is interested in a particular class of physical processes, the objectivity requirement at the beginning of this section could be satisfied even though the strict mathematical one is not. This does not raise any problem. It finds its counterpart in the fact that to analyze a specific physical situation one can use a coarser graining than the one corresponding to the fundamental characteristic length of the theory. Accordingly, if we adapt the graining to the problem of interest, we can assert that the mass of the cloud is objective. Obviously, while for practical purposes one can change the graining according to the problem he is interested in, adopting a graining which is finer than the fundamental one of the theory makes no sense because, if one takes the theory seriously as describing the laws governing the evolution of the universe, then mass density would almost never become objective for such a graining.
For a more precise definition of such a set see (Ghirardi et al. 1995).
This attitude has the same logical status as the assertion that the probabilities concerning physical processes have an epistemic character and are not due to the presence of genuine elements of chance in nature.
In Eq. (7.8) and following we make, for the same reasons discussed there, a change of notation analogous to the one made for (3.8).
See below for the precise meaning we attribute to the expression in quotation marks.
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© 1996 Springer Science+Business Media Dordrecht
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Ghirardi, G., Grassi, R. (1996). Bohm’s Theory Versus Dynamical Reduction. In: Cushing, J.T., Fine, A., Goldstein, S. (eds) Bohmian Mechanics and Quantum Theory: An Appraisal. Boston Studies in the Philosophy of Science, vol 184. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8715-0_25
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DOI: https://doi.org/10.1007/978-94-015-8715-0_25
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