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Spontaneous Localization


The spontaneous localization (SL) approach, initiated by Philip Pearle around 1970, may be regarded as concerned with a minimal modification of the Schrödinger evolution into one in which wave functions of macroscopic systems behave in a sensible way. This goal proved elusive, but in 1985 a breakthrough [20] occurred: GianCarlo Ghirardi, Alberto Rimini, and Tulio Weber (GRW), by appreciating the privileged role somehow played by positions and thus focusing on the possibility of spatial localization, showed how to combine the Schrödinger evolution with spontaneous random collapses--given by ``Gaussian hits'' centered at random positions tex2html_wrap_inline1124 occurring at random times t--to obtain an evolution for wave functions that reproduces the Schrödinger evolution on the atomic level while avoiding the embarrassment of macroscopic superpositions.

Thus [1, page 204,] ``any embarrassing macroscopic ambiguity in the usual theory is only momentary in the GRW theory. The cat is not both dead and alive for more than a split second.'' Similarly, measurement pointers quickly point. Moreover, it is a more or less immediate consequence of the GRW dynamics that when a macroscopic superposition collapses under the GRW evolution to one of its terms, the probability that tex2html_wrap_inline1130 is the term that survives is tex2html_wrap_inline1132 , precisely as demanded by the collapse postulate of standard quantum theory.

It is tempting to say that with the SL approach, quantum mechanics is indeed fundamentally about the behavior of wave functions. I believe, however, that this is not quite right. The problem is that the purpose of any physical theory is to account for a pattern of events occurring in (ordinary 3-dimensional) space and time. But the behavior of a wave function of a many (N) particle universe, a field on an abstract (3N-dimensional) configuration space, has in and of itself no implications whatsoever regarding occurrences in physical space, however sensible its behavior may otherwise be. As Bell [1, page 204,] has noted ``it makes no sense to ask for the amplitude or phase or whatever of the wave function at a point in ordinary space. It has neither amplitude nor phase nor anything else until a multitude of points in ordinary three-space are specified.''

Therefore Ghirardi [21, page 8,] rightly emphasizes the importance of specifying what he calls ``the physical reality of what exists out there.'' For this he chooses the mass density function, which for the simple GRW theory described here can be identified with the mass weighted sum tex2html_wrap_inline1138 , over all particles, of the one-particle densities tex2html_wrap_inline1140 arising from integrating over the coordinates of all but one of the particles.gif

Bell [1, page 205,] has proposed a strikingly different possibility,gif that the space-time points tex2html_wrap_inline1144 at which the hits are centered (which are determined by the wave function trajectory) should themselves serve as the ``local beables of the theory. These are the mathematical counterparts in the theory to real events at definite places and times in the real world (as distinct from the many purely mathematical constructions that occur in the working out of physical theories, as distinct from things which may be real but not localized, and as distinct from the `observables' of other formulations of quantum mechanics, for which we have no use here.) A piece of matter then is a galaxy of such events.''

One can imagine, of course, many other choices, some better than others. The point I wish to emphasize here, however, is that if we are to have a well-defined physical theory at all, some such choice must be made. Indeed, any quantum theory without observers, and arguably any physical theory with any pretense to precision, requires as part of its formulation a specification of the ``local beables,'' of ``what exists out there,'' of what the theory is fundamentally about--which I would prefer to call the primitive ontology of the theory. (It might be argued that the unease sometimes expressed about DH arises from the obscurity of its primitive ontology--or from its failure to commit in this regard.) Moreover, we must also specify, for a quantum theory, the relationship between the wave function and this primitive ontology, which for SL will be provided by a mapping or code connecting the evolution of the wave function to a story in space and time.

Different such specifications define different theories. They might also have different observable consequences. Moreover, the symmetries of the theory may depend critically on this specification. For example with Bell's rather surprising choice the GRW theory obeys a certain ``relative time translation invariance'' and becomes [1, page 209,] ``as Lorentz invariant as it could be in the nonrelativistic version.'' Thus a careful analysis of the symmetries of a theory demands a careful specification of its primitive ontology.

As a matter of fact, one would have to make a rather perverse choice to arrive at any empirical disagreement with the predictions arising from the choices of Ghirardi or Bell. It is clear, however, because of its abrogation of the Schrödinger evolution, that SL (in whatever version and with whatever choice of primitive ontology) must disagree somewhat with the predictions of orthodox quantum theory. In fact, by the uncertainty principle, the wave function localizations will increase the momentum space spread in the wave function and hence energy will tend to increase at a very small rate--so small in fact that this effect may be rather difficult to observe.

The last version of quantum theory without observers that I shall describe agrees completely with orthodox quantum theory in its predictions. Precise and simple, it involves an almost obvious incorporation of Schrödinger's equation into an entirely deterministic reformulation of quantum theory.

next up previous
Next: Bohmian Mechanics Up: Quantum Theory Without Observers Previous: Decoherent Histories

Shelly Goldstein
Wed Aug 13 17:22:41 EDT 1997