Quantum mechanics, with its revolutionary implications, has posed innumerable problems to philosophers of science. In particular, it has suggested reconsidering basic concepts such as the existence of a world that is, at least to some extent, independent of the observer, the possibility of getting reliable and objective knowledge about it, and the possibility of taking (under appropriate circumstances) certain properties to be objectively possessed by physical systems. It has also raised many others questions which are well known to those involved in the debate on the interpretation of this pillar of modern science. One can argue that most of the problems are not only due to the intrinsic revolutionary nature of the phenomena which have led to the development of the theory. They are also related to the fact that, in its standard formulation and interpretation, quantum mechanics is a theory which is excellent (in fact it has met with a success unprecedented in the history of science) in telling us everything about
what we observe, but it meets with serious difficulties in telling us
what is. We are making here specific reference to the central problem of the theory, usually referred to as
the measurement problem, or, with a more appropriate term, as the
macro-objectification problem. It is just one of the many attempts to overcome the difficulties posed by this problem that has led to the development of
Collapse Theories, i.e., to the
Dynamical Reduction Program (DRP). As we shall see, this approach consists in accepting that the dynamical equation of the standard theory should be modified by the addition of stochastic and nonlinear terms. The nice fact is that the resulting theory is capable, on the basis of a single dynamics which is assumed to govern all natural processes, to account at the same time for all well-established facts about microscopic systems as described by the standard theory as well as for the so-called postulate of wave packet reduction (WPR). As is well known, such a postulate is assumed in the standard scheme just in order to guarantee that
measurements have outcomes but, as we shall discuss below, it meets with insurmountable difficulties if one takes the measurement itself to be a process governed by the linear laws of the theory. Finally, the collapse theories account in a completely satisfactory way for the classical behavior of macroscopic systems.
In spite of their phenomenological character, we think that Collapse Theories have a remarkable relevance, since they have made clear that there are new ways to overcome the difficulties of the formalism, to close the circle in the precise sense defined by Abner Shimony (1989), which until a few years ago were considered impracticable, and which, on the contrary, have been shown to be perfectly viable. Moreover, they have allowed a clear identification of the formal features which should characterize any unified theory of micro and macro processes. Last but not least, Collapse theories qualify themselves as rival theories of quantum mechanics and one can easily identify some of their physical implications which, in principle, would allow crucial tests discriminating between the two. To get really stringent indications from such tests requires experiments involving technological techniques which have been developed only very recently. Actually, it is just due to remarkable improvements in dealing with mesoscopic systems and to important practical steps forward, that some specific bounds have already been obtained for the parameters characterizing the theories under investigation, and, more important, precise families of physical processes in which a violation of the linear nature of the standard formalism might emerge have been clearly identified and are the subject of systematic investigations which might lead, in the end, to relevant discoveries.
[size=30]1. General Considerations
As stated already, a very natural question which all scientists who are concerned about the meaning and the value of science have to face, is whether one can develop a coherent worldview that can accommodate our knowledge concerning natural phenomena as it is embodied in our best theories. Such a program meets serious difficulties with quantum mechanics, essentially because of two formal aspects of the theory which are common to all of its versions, from the original nonrelativistic formulations of the 1920s, to the quantum field theories of recent years: the linear nature of the state space and of the evolution equation, i.e., the validity of the superposition principle and the related phenomenon of entanglement, which, in Schrödinger’s words:
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- اقتباس :
- is not one but the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought (Schrödinger, 1935, p. 807).
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These two formal features have embarrassing consequences, since they imply
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- objective chance in natural processes, i.e., the nonepistemic nature of quantum probabilities;
- objective indefiniteness of physical properties both at the micro and macro level;
- objective entanglement between spatially separated and non-interacting constituents of a composite system, entailing a sort of holism and a precise kind of nonlocality.
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For the sake of generality, we shall first of all present a very concise sketch of ‘the rules of the quantum game’.
[size=30]2. The Formalism: A Concise Sketch[/size]
Let us recall the axiomatic structure of quantum theory:
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[*]States of physical systems are associated with normalized vectors in a Hilbert space, a complex, infinite-dimensional, complete and separable linear vector space equipped with a scalar product. Linearity implies that the superposition principle holds: if
[ltr]∣f⟩∣f⟩[/ltr] is a state and
[ltr]∣g⟩∣g⟩[/ltr] is a state, then (for
[ltr]aa[/ltr] and
[ltr]bb[/ltr] arbitrary complex numbers) also
[ltr]∣K⟩=a∣f⟩+b∣g⟩∣K⟩=a∣f⟩+b∣g⟩[/ltr]
is a state. Moreover, the state evolution is linear, i.e., it preserves superpositions: if
[ltr]∣f,t⟩∣f,t⟩[/ltr]and
[ltr]∣g,t⟩∣g,t⟩[/ltr] are the states obtained by evolving the states
[ltr]∣f,0⟩∣f,0⟩[/ltr] and
[ltr]∣g,0⟩∣g,0⟩[/ltr], respectively, from the initial time
[ltr]t=0t=0[/ltr] to the time
[ltr]tt[/ltr], then
[ltr]a∣f,t⟩+b∣g,t⟩a∣f,t⟩+b∣g,t⟩[/ltr] is the state obtained by the evolution of
[ltr]a∣f,0⟩+b∣g,0⟩a∣f,0⟩+b∣g,0⟩[/ltr]. Finally, the completeness assumption is made, i.e., that the knowledge of its statevector represents, in principle, the most accurate information one can have about the state of an individual physical system.
[*]The observable quantities are represented by self-adjoint operators
[ltr]BB[/ltr] on the Hilbert space. The associated eigenvalue equations
[ltr]B∣b[size=13]k⟩=bk∣bk⟩B∣bk⟩=bk∣bk⟩[/ltr][/size] and the corresponding eigenmanifolds (the linear manifolds spanned by the eigenvectors associated to a given eigenvalue, also called eigenspaces) play a basic role for the predictive content of the theory. In fact:
- The eigenvalues [ltr]bkbk[/ltr] of an operator [ltr]BB[/ltr] represent the only possible outcomes in a measurement of the corresponding observable.
- The square of the norm (i.e., the length) of the projection of the normalized vector (i.e., of length 1) describing the state of the system onto the eigenmanifold associated to a given eigenvalue gives the probability of obtaining the corresponding eigenvalue as the outcome of the measurement. In particular, it is useful to recall that when one is interested in the probability of finding a particle at a given place, one has to resort to the so-called configuration space representation of the statevector. In such a case the statevector becomes a square-integrable function of the position variables of the particles of the system, whose modulus squared yields the probability density for the outcomes of position measurements.
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We stress that, according to the above scheme, quantum mechanics makes only conditional probabilistic predictions (conditional on the measurement being actually performed) for the outcomes of prospective (and in general incompatible) measurement processes. Only if a state belongs already before the act of measurement to an eigenmanifold of the observable which is going to be measured, can one predict the outcome with certainty. In all other cases—if the completeness assumption is made—one has objective nonepistemic probabilities for different outcomes.
The orthodox position gives a very simple answer to the question: what determines the outcome when different outcomes are possible? Nothing—the theory is complete and, as a consequence, it is illegitimate to raise any question about possessed properties referring to observables for which different outcomes have non-vanishing probabilities of being obtained. Correspondingly, the referent of the theory are the results of measurement procedures. These are to be described in classical terms and involve in general mutually exclusive physical conditions.
As regards the legitimacy of attributing properties to physical systems, one could say that quantum mechanics warns us against requiring too many properties to be actually possessed by physical systems. However—with Einstein—one can adopt as a sufficient condition for the existence of an objective individual property that one be able (without in any way disturbing the system) to predict with certainty the outcome of a measurement. This implies that, whenever the overall statevector factorizes into the product of a state of the Hilbert space of the physical system
[ltr]SS[/ltr] and of the rest of the world,
[ltr]SS[/ltr] does possess some properties (actually a complete set of properties, i.e., those associated to appropriate maximal sets of commuting observables).[/size]
Before concluding this section we must add some comments about the measurement process. Quantum theory was created to deal with microscopic phenomena. In order to obtain information about them one must be able to establish strict correlations between the states of the microscopic systems and the states of objects we can perceive. Within the formalism, this is described by considering appropriate micro-macro interactions. The fact that when the measurement is completed one can make statements about the outcome is accounted for by the already mentioned WPR postulate (Dirac 1948): a measurement always causes a system to jump in an eigenstate of the observed quantity. Correspondingly, also the statevector of the apparatus ‘jumps’ into the manifold associated to the recorded outcome.
Collapse Theories
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