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تاريخ التسجيل : 26/10/2009
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 | |  The Problem of the Tails of the Wave Function | |
In recent years, there has been a lively debate around a problem which has its origin, according to some of the authors which have raised it, in the fact that even though the localization process which corresponds to multiplying the wave function times a Gaussian and thus lead to wave functions strongly peaked around the position of the hitting, they allow nevertheless the final wavefuntion to be different from zero over the whole of space. The first criticism of this kind was raised by A. Shimony (1990) and can be summarized by his sentence, - اقتباس :
- one should not tolerate tails in wave functions which are so broad that their different parts can be discriminated by the senses, even if very low probability amplitude is assigned to them.
After a localization of a macroscopic system, typically the pointer of the apparatus, its centre of mass will be associated to a wave function which is different from zero over the whole space. If one adopts the probabilistic interpretation of the standard theory, this means that even when the measurement process is over, there is a nonzero (even though extremely small) probability of finding its pointer in an arbitrary position, instead of the one corresponding to the registered outcome. This is taken as unacceptable, as indicating that the DRP does not actually overcome the macro-objectification problem.Let us state immediately that the (alleged) problem arises entirely from keeping the standard interpretation of the wave function unchanged, in particular assuming that its modulus squared gives the probability density of the position variable. However, as we have discussed in the previous section, there are much more serious reasons of principle which require to abandon the probabilistic interpretation and replace it either with the ‘flash ontology’, or with the ‘ mass density ontology’ which we have discussed above.Before entering into a detailed discussion of this subtle point we need to focus better the problem. We cannot avoid making two remarks. Suppose one adopts, for the moment, the conventional quantum position. We agree that, within such a framework, the fact that wave functions never have strictly compact spatial support can be considered puzzling. However this is an unavoidable problem arising directly from the mathematical features (spreading of wave functions) and from the probabilistic interpretation of the theory, and not at all a problem peculiar to the dynamical reduction models. Indeed, the fact that, e.g., the wave function of the center of mass of a pointer or of a table has not a compact support has never been taken to be a problem for standard quantum mechanics. When, e.g., the center of mass of a table is extremely well peaked around a given point in space, it has always been accepted that it describes a table located at a certain position, and that this corresponds in some way to our perception of it. It is obviously true that, for the given wave function, the quantum rules entail that if a measurement were performed the table could be found (with an extremely small probability) to be kilometers far away, but this is not the measurement or the macro-objectification problem of the standard theory. The latter concerns a completely different situation, i.e., that in which one is confronted with a superposition with comparable weights of two macroscopically separated wave functions, both of which possess tails (i.e., have non-compact support) but are appreciably different from zero only in far-away narrow intervals. This is the really embarrassing situation which conventional quantum mechanics is unable to make understandable. To which perception of the position of the pointer (of the table) does this wave function correspond?The implications for this problem of the adoption of the QMSL theory should be obvious. Within GRW, the superposition of two states which, when considered individually, are assumed to lead to different and definite perceptions of macroscopic locations, are dynamically forbidden. If some process tends to produce such superpositions, then the reducing dynamics induces the localization of the centre of mass (the associated wave function being appreciably different from zero only in a narrow and precise interval). Correspondingly, the possibility arises of attributing to the system the property of being in a definite place and thus of accounting for our definite perception of it. Summarizing, we stress once more that the criticism about the tails as well as the requirement that the appearance of macroscopically extended (even though extremely small) tails be strictly forbidden is exclusively motivated by uncritically committing oneself to the probabilistic interpretation of the theory, even for what concerns the psycho-physical correspondence: when this position is taken, states assigning non-exactly vanishing probabilities to different outcomes of position measurements should correspond to ambiguous perceptions about these positions. Since neither within the standard formalism nor within the framework of dynamical reduction models a wave function can have compact support, taking such a position leads to conclude that it is just the linear character of the Hilbert space description of physical systems which has to be given up.It ought to be stressed that there is nothing in the GRW theory which forbids or makes problematic to assume that the localization function has compact support, but it also has to be noted that following this line would be totally useless: since the evolution equation contains the kinetic energy term, any function, even if it has compact support at a given time, will instantaneously spread acquiring a tail extending over the whole of space. If one sticks to the probabilistic interpretation and one accepts the completeness of the description of the states of physical systems in terms of the wave function, the tail problem cannot be avoided.The solution to the tails problem can only derive from abandoning completely the probabilistic interpretation and from adopting a more physical and realistic interpretation relating ‘what is out there’ to, e.g., the mass density distribution over the whole universe. In this connection, the following example will be instructive. Take a massive sphere of normal density and mass of about 1 kg. Classically, the mass of this body would be totally concentrated within the radius of the sphere, call it [ltr]rr[/ltr]. In QMSL, after the extremely short time interval in which the collapse dynamics leads to a ‘regime’ situation, and if one considers a sphere with radius [ltr]r+10[size=13]−5r+10−5[/ltr] cm, the integral of the mass density over the rest of space turns out to be an incredibly small fraction (of the order of 1 over 10 to the power [ltr] 1015)1015)[/ltr] of the mass of a single proton. In such conditions, it seems quite legitimate to claim that the macroscopic body is localised within the sphere.[/size] However, also this quite reasonable conclusion has been questioned and it has been claimed (Lewis 1997), that the very existence of the tails implies that the enumeration principle (i.e., the fact that the claim ‘particle 1 is within this box & particle 2 is within this box & … & particle [ltr]nn[/ltr]is within this box & no other particle is within this box’ implies the claim ‘there are [ltr]nn[/ltr] particles within this box’) does not hold, if one takes seriously the mass density interpretation of collapse theories. This paper has given rise to a long debate which would be inappropriate to reproduce here.We conclude this brief analysis by stressing once more that, in the opinion of the present writer, all the disagreements and the misunderstandings concerning this problem have their origin in the fact that the idea that the probabilistic interpretation of the wave function must be abandoned has not been fully accepted by the authors who find some difficulties in the proposed mass density interpretation of the Collapse Theories. For a recent reconsideration of the problem we refer the reader to the paper by Lewis (2003).13. The Status of Collapse Models and Recent Positions about themWe recall that, as stated in Section 3, the macro-objectification problem has been at the centre of the most lively and most challenging debate originated by the quantum view of natural processes. According to the majority of those who adhere to the orthodox position such a problem does not deserve a particular attention: classical concepts are a logical prerequisite for the very formulation of quantum mechanics and, consequently, the measurement process itself, the dividing line between the quantum and the classical world, cannot and must not be investigated, but simply accepted. This position has been lucidly summarized by J. Bell himself (1981): - اقتباس :
- Making a virtue of necessity and influenced by positivistic and instrumentalist philosophies, many came to hold not only that it is difficult to find a coherent picture but that it is wrong to look for one—if not actually immoral then certainly unprofessional
The situation has seen many changes in the course of time, and the necessity of making a clear distinction between what is quantum and what is classical has given rise to many proposals for ‘easy solutions’ to the problem which are based on the possibility, for all practical purposes(FAPP), of locating the splitting between these two faces of reality at different levels.Then came Bohmian mechanics, a theory which has made clear, in a lucid and perfectly consistent way, that there is no reason of principle requiring a dichotomic description of the world. A universal dynamical principle runs all physical processes and even though ‘it completely agrees with standard quantum predictions’ it implies wave-packet reduction in micro-macro interactions and the classical behaviour of classical objects.As we have mentioned, the other consistent proposal, at the nonrelativistic level, of a conceptually satisfactory solution of the macro-objectification problem is represented by the Collapse Theories which are the subject of these pages. Contrary to bohmian mechanics, they are rival theory of quantum mechanics, since they make different predictions (even though quite difficult to put into evidence) concerning various physical processes.Let us now analyze other recent critical positions concerning the two just mentioned approaches (in what follows I will take advantage of the nice analysis of a paper which I have been asked to referee and of which I do not know the author). Various physicists have criticized Bohm approach on the basis that, being empirically indistinguishable from quantum mechanics, such an approach is an example of ‘bad science’ or of ‘a degenerate research program’. Useless to say, I do not consider such criticisms as appropriate; the conceptual advantages and the internal consistency of the approach render it an extremely appealing theoretical scheme (incidentally, one should not forget that it has been just the critical investigations on such a theory which have led Bell to derive his famous and conceptually extremely relevant inequality). On the contrary, I am fully convinced that to consider as acceptable a theory like the standard one, which is incapable of accounting for the way in which it assumes the measurement apparatuses to work, and to deal with them introduces a postulate which plainly contradicts the other assumption of the theory, is not a scientifically tenable position.This being the situation, one would think that theories like the GRW model would be exempt from an analogous charge, since they actually are (in principle) empirically different from the standard theory. For instance they disagree from such a theory since they forbid the occurrence of macroscopic massive entangled states. In spite of this, they have been the object of an analogous attack by the adherents to the ‘new orthodoxy’ (Bub 1997; Joos et al. 1996; Zurek, 1993) pointing out that environmental induced decoherence shows that, FAPP, collapse theories are simply phenomenological accounts of the reduced state to which one has to resort since one has no control of the degrees of freedom of the environment. When one takes such a position, one is claiming that, essentially, GRW cannot be taken as a fundamental description of nature, mainly because it suffers from the limitation of being empirically indistinguishable from the standard theory, provided such a theory is correctly applied taking into account the actual physical situation. Also in this case, and even at the level at which such an analysis is performed, the practical indistinguishability from the standard approach should not be regarded as a sufficient reason to not take seriously collapse models. In fact, there are many very well known and compelling reasons (see, e.g., Bassi and Ghirardi 2000; Adler 2003) to prefer a logically consistent unified theory to one which makes sense only due to the alleged practicalimpossibility of detecting the superpositions of macroscopically distinguishable states. At any rate, in principle, such theories can be tested against the standard one and it seems that such a challenge is already under investigation. .But this is not the whole story. Another criticism, aimed to ‘deny’ the potential interest of collapse theories makes reference to the fact that within any such theory the ensuing dynamics for the statistical operator can be considered as the reduced dynamics deriving from a unitary (and, consequently, essentially a standard quantum) dynamics for the states of an enlarged Hilbert space of a composite quantum system [ltr]S+ES+E[/ltr] involving, besides the physical system [ltr]SS[/ltr] of interest, an ancilla [ltr]EE[/ltr] whose degrees of freedom are completely unaccessible: due to the quantum dynamical semigroup nature of the evolution equation for the statistical operator, any GRW-like model can always be seen as a phenomenological model deriving from a standard quantum evolution on a larger Hilbert space. In this way, the unitary deterministic evolution characterizing quantum mechanics would be fully restored.Apart from the obvious remark that such a critical attitude completely fails to grasp—and indeed, purposefully ignores—the most important feature of collapse theories, i.e., of dealing with individual quantum systems and not with statistical ensembles and of yielding a perfectly satisfactory description, matching our perceptions concerning individual macroscopic systems, invoking an unaccessible ancilla to account for the nonlinear and stochastic character of GRW-type theories is once more a purely verbal way of avoiding facing the real puzzling aspects of the quantum description of macroscopic systems. This is not the only negative aspect of such a position; any attempt considering legitimate to introduce unaccessible entities in the theory, when one takes into consideration that there are infinitely possible and inequivalent ways of doing this, amounts really to embarking oneself in a ‘degenerate research program’.Other reasons for ignoring the dynamical reduction program have been put forward recently by the community of scientists involved in the interesting and exciting field of quantum information. We will not spend too much time in analyzing and discussing the new position about the foundational issues which have motivated the elaboration of collapse theories. The crucial fact is that, from this perspective, one takes the theory not to be about something real ‘occurring out there’ in a real word, but simply about information. This point is made extremely explicit in a recent paper (Zeilinger 2005): - اقتباس :
- information is the most basic notion of quantum mechanics, and it is information about possible measurement results that is represented in the quantum state. Measurement results are nothing more than states of the classical apparatus used by the experimentalist. The quantum system then is nothing other than the consistently constructed referent of the information represented in the quantum state.
It is clear that if one takes such a position almost all motivations to be worried by the measurement problem disappear, and with them the reasons to work out what Bell has denoted as ‘an exact version of quantum mechanics’. The most appropriate reply to this type of criticisms is to recall that J. Bell (1990) has included ‘information’ among the words which must have no place in a formulation with any pretension to physical precision. In particular he has stressed that one cannot even mention information unless one has given a precise answer to the two following questions: Whose information? and Information about what?A much more serious attitude is to call attention, as many serious authors do, to the fact that since collapse theories represent rival theories with respect to standard quantum mechanics they lead to the identification of experimental situations which would allow, in principle, crucial tests to discriminate between the two. As we have discussed above, presently, fully discriminating tests seem not to be completely out of reach. | |
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