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Interpreting Quantum Theory Without Losing Your Marbles:
A Look at the Implications of Spontaneous Localization Theories on Physical Reality

As a predictive theory, quantum mechanics has been extraordinarily successful; the successful prediction of the electron's anomalous magnetic moment, for example, is considered one of the greatest triumphs of theoretical physics. However, the nondeterminism at the heart of quantum theory raises some troubling philosophical questions; after all, quantum mechanics does not explicitly prohibit a macroscopic system from being in a superposition of two states, but we never actually see such a macroscopic superposition. Physicists tend to dismiss such concerns; after all, quantum mechanics makes excellent predictions, and physics is, in the end, the business of making predictions about the world. Ghirardi, Rimini, and Weber have proposed a mechanism for avoiding macroscopic superposition states, which they term Spontaneous Localization (SL) theory. It, and its refinements such as Continuous Spontaneous Localization, offer an interesting way of dealing with the problem of macroscopic superpositions, and may lead to other interesting insights into the foundations of quantum reality.

### I. The Macro-objectification Problem

The state-vector formalism in quantum mechanics allows for superpositions of states, such that the state vector |Ψ> describing a particle in a box could be, for example, an admixture of two energy eigenstates: |Ψ> = a|ψ_{a}> + b|ψ_{b}>. While this feature of quantum mechanics undeniably allows accurate predictions on the energy and distance scales where classical mechanics inevitably fails, it also allows, on first glance, a potentially embarassing situation: nothing in quantum theory explicitly prohibits macroscopic objects from being placed in superpositions of two or more perceptually different states. This leads to two separate problems: first, what meaning can we attach to a situation in which an object is in a superposition of perceptually different states; and second, assuming that meaning can be reasonably attached to such a situation, why are these superpositions never observed? After all, it is said that everything not forbidden by the laws of nature is mandatory. This second problem is known as the macro-objectification problem, and it is one that generally does not bother physicists.

Physicists, on the whole, tend to look to the Copenhagen interpretation for a resolution to the macro-objectification problem; this states that, for example, upon measurement of the energy of a particle in the previously-discussed state |Ψ> = a|ψ_{a}> + b|ψ_{b}>, the particle will be found to have either energy E_{a} or E_{b}, with probabilities |a|^{2} and |b|^{2}, respectively. Further, the act of measurement "collapses" the state vector, so that the particle assumes the pure state associated with the measured energy. But what, exactly, constitutes a "measurement," and what is the mechanism by which the state-vector collapses when one occurs? On this point, standard quantum theory is resolutely silent. Something about this situation seems to have bothered Schrödinger, who put this question in the form of his (in)famous cat paradox:

A cat is placed in a steel chamber, together with the following hellish contraption… in a Geiger counter there is a tiny amount of radioactive substance, so tiny that maybe within an hour one of the atoms decays, but equally probably none of them decays. If one decays then the counter triggers and via a relay activates a little hammer which breaks a container of cyanide. If one has left this entire system for an hour, then one would say the cat is living if no atom has decayed. The first decay would have poisoned it. The wave function of the entire system would express this by containing equal parts of the living and dead cat.^{1}

In the situation described, the cat would be neither alive nor dead until the chamber was opened, and it would be the opening of the chamber which forced the cat to become either alive or dead. Schrödinger felt that this situation was absurd, and not many people would argue with him on that point. The problem lies in the

Copenhagen interpretation's use of the word "measurement" – what, exactly, constitutes a measurement? What sort of interactions cause the wavefunction to collapse, and why do no macroscopic superposition states persist? D. J. Griffiths' view on this matter is, I think, representative of the mainstream of the scientific community:

Of course, in some ultimate sense the macroscopic system is itself described by the laws of quantum mechanics. But wave functions, in the first instance, describe individual elementary particles; the wave function of a macroscopic object would be a monstrously complicated composite, built out of all the wave functions of its 10^{23} constituent particles. Presumably somewhere in the statistics of large numbers macroscopic linear combinations become extremely improbable.^{2}

In other words, the physics community is largely inclined to take quantum mechanics as a ssuccessful predictive theory and leave the

macro-objectification problem alone for the time being; the problem, however remains: how to account for the absence of observed macroscopic superposition states, given a quantum theory which does not prohibit their existence?

### II. The GRW Spontaneous Localization Theory

Fortunately, several alternatives to the Copenhagen interpretation exist which purport to solve the macro-objectification problem in various ways; perhaps the most interesting of these is the collapse model proposed by Ghirardi, Rimini, and Weber. This model, termed Spontaneous Localization (SL) theory, proposes a modification to the evolution of quantum states described by the Schrödinger equation: every so often, particles experience something called "Gaussian hits," which involve multiplying the wavefunction of the particle by a Gaussian and renormalizing; this has the effect of locating the particle fairly precisely. In this sense, it is analogous to the effect of a position measurement under the Copenhagen interpretation.

Under the GRW spontaneous localization theory, a particle will experience a hit at randomly distributed times according to a Poisson distribution with mean frequency w; upon a hit at location a, the wavefunction is multiplied by the Gaussian G(x) = exp(-(½ d^{2})(x-a)^{2}) and then normalized. The parameter d describes how closely the particle is localized, and both d and w can be chosen to agree with observations; GRW suggest the values w = 10^{-15} s^{-1} and d = 10^{-5} cm.^{3} The probability that a hit occurs at location a is proportional to the norm of the wavefunction at a. Since w is miniscule, it is highly improbable that a single particle will experience a hitting within a reasonable time of observation, so the quantum properties of microscopic systems remain intact. In a macroscopic system, however, such as a marble containing on the order of 10^{23} particles, one of the constituent particles will be hit, on average, 10^{8} times in a second. Since the particles in the marble interact with each other and maintain their average separations, a hit on any particle localizes the entire marble – this leads to the familiar classical behavior for macroscopic systems without recourse to the ill-defined "collapse rule" of standard quantum theory.

SL theory is promising for several reasons. First and foremost, it yields the familiar behavior of objects at the micro- and macroscopic levels; objects containing only a few thousand particles show the behavior standard quantum theory would lead us to expect, and objects containing large numbers of particles behave classically. Second, it is empirically falsifiable; this is of paramount importance for physical theories. SL makes explicit predictions which differ from those of standard quantum mechanics, and tells us where to look for the anomalies. If SL holds, one would expect to see anomalous results in systems comprising on the order of 10^{15} particles, so that localizations could be expected to happen occasionally on reasonable experimental time scales. Two-slit diffraction experiments have been carried out using objects such as buckyballs, which are still too small to see anomalies related to SL; still on the horizon is a proposed diffraction experiment using virus particles. The virus diffraction experiment has the potential to shed some light on the behavior of systems which lie between the domains in which quantum and classical physics dominate. Finally, although SL appears at first glance to introduce two new physical constants, which is always grounds for skepticism, it has been recently proposed that the gravitational interaction could lead to the localizations, and that the parameters d and w are of the right order of magnitude for a gravitational interaction. If this turns out to be a viable proposition, it would certainly lend weight to the SL hypothesis.

### III. The Problem of the "Tails"

SL purports to solve the macro-objectification problem by multiplying the wavefunction of a particle with a Gaussian at randomly determined times. The astute observer will note that, even after localization, the wavefunction of any particle will still have infinite support – that is, there will still be a nonzero probability of finding the particle an arbitrary distance from the position at which it was "localized"! SL, then, avoids the most embarassing sort of superposition, the wavefunctions with two peaks of roughly equal probability at perceptually different locations – after all, the narrow Gaussian profile ensures that there can be only one peak – but it still leaves open the possibility that we could leave a table at location a, look for it, and find it in an arbitrarily distant position. The question is now whether SL has solved the macro-objectification problem at all, or merely swept it under the rug, so to speak.

What meaning can we assign to a wavefunction with tails, in terms of observable reality? After all, we certainly do not see many macroscopic objects spontaneously teleporting themselves around! Part of the reason for this is that the probability for this sort of teleportation is unfathomably small; Pearle gives the following illustrative example: "Let V be a sphere of radius 10^{-6} cm, and suppose that a hydrogen atom lies with its nucleus at the center of the sphere. Let us ask the question: is the electron in V? We readily calculate that <ψ|P_{1}V|ψ> ? 1- 10^{-169}."^{4} In other words, you are about 10^{160} times more likely to win the lottery than to make a position measurement and find your electron anywhere outside that sphere. In this case, it is probably fair to say that the electron is located within that sphere. Similarly, if you place a coin in a jar, the probability of later finding it elsewhere is even smaller than 10^{-169}, and it seems fair to say that the coin is in the jar, the tails of its wavefunction notwithstanding. Unfortunately, "it seems fair" is far from an acceptable argument in this context; electrons, for example, regularly show behavior that would be absurd from a classical viewpoint. We need some justification for our claim that the coin is located in the jar.

The standard solution to the tails problem is unfortunately not much clearer than our claim that "it seems fair" to say that the coin is in the jar, given the miniscule probability of finding it elsewhere. Albert and Loewer propose the following criterion, which they call PosR, for stating that a particle is in a certain location:

"Particle x is in region R" if and only if the proportion of the total squared amplitude of x's wave function which is associated with points in R is greater than or equal to 1-p.^{5}

Any value of p less than 0.5 suffices to prevent us from saying that a particle lies in disjoint regions, but it is clear from the examples above that we can use values of p which are much closer to 0. Essentially, p can be chosen arbitrarily to fit those situations in which we consider the particle localized. Clifton and Monton generalize PosR to multi-particle systems in the following manner, which they term the "

fuzzy link":

"Particle x lies in region R_{x} and y lies in R_{y} and z lies in R_{z} and…" if and only if the proportion of the total squared amplitude of y(t, r_{I}, …, r_{n}) that is associated with points in R_{x}, R_{y}, R_{z} … is greater than or equal to 1-p.^{6}

They then show that, given a large enough number of particles, it is possible to create a situation wherein, by PosR, it is possible to say that x lies in region R

_{x}, that y lies in R

_{y}, that z lies in R

_{z}, and so forth, but that according to the fuzzy link, the proposition "Particle x lies in region R

_{x} and y lies in R

_{y} and z lies in R

_{z} and…" is false; in other words, it is possible for counting to fail with macroscopic objects!

Additional arguments related to the failure of the enumeration principle under SL will be discussed below, but one might wonder whether it is possible to avoid the problem entirely by eliminating the tails of the wavefunction. Perhaps the "hits" could multiply the wavefunction by a triangular or boxcar window, rather than by a Gaussian – then the probability of finding the particle outside the region to which it is localized would be zero. Unfortunately, under SL theory, the Schrödinger equation is unmodified, and under the Schrödinger equation, any wavefunction with finite support in position space has an infinite spread in momentum. This means that even if the wavefunction has finite support at the time t of a hit, it will again have infinite support at any time t' > t. Even if a particle is perfectly localized at (x,t), there will be a nonzero probability of finding it arbitrarily far from x at any time afterward. This is no mere quirk of the Schrödinger equation, either – tails form just as quickly in the relativistic Dirac equation. In fact, according to Pearle, a wavefunction without tails is physically meaningless in a relativistic theory:

If you have a tail, no matter how small, and you know the field w(x, t) which the state vector evolved under, you can run the evolution equation backwards and recover the statevector at any earlier time. If, on the other hand, the tail was completely cut off, you get a nonsensical irrelevant earlier statevector, even in standard quantum theory… I cannot see how you could get sensible results in another Lorentz frame without having the tail to tell you how to do it.^{4}

The "tails" of the wavefunction, then, are necessitated by quantum theory – there is no way to avoid having them.

### IV. Are Tails Really a Problem?

Since it seems that tails are here to stay, at least until someone comes up with a replacement for the Schrödinger equation, we are forced to consider whether their presence is as problematic as it seems to be. What meaning can we attach, perceptually, to an object whose wavefunction is spread out in space? Fortunately, SL theory eliminates the possibility of macroscopic objects which are spread out by more than ~10^{-5} cm, so we need not consider situations in which a chair, for example, has equal probabilities of being in two positions 1 meter apart. We need only consider situations wherein a macroscopic object is well-localized, such that its probability of being found very far away from its expected position is miniscule. In this case, the answer is literally right in front of us; after all, Schrödinger tells us that the wavefunctions of everyday objects are spread out in space, but we percieve them as having definite positions. There is no reason to expect our perceptions to be any different now that we have a mechanism by which they could be localized.

Since there is no reason to expect any perceptual problem, we turn to the enumeration problem. As noted earlier, it is possible, given enough marbles, to be able to say with virtual certainty that each marble is in a box, and for the probability that all the marbles are in the box to be close to zero. This would seem to violate the enumeration principle which states that if marble 1 is in the box and marble 2 is in the box and … and marble n is in the box and no marbles are in the box, then there are n marbles in the box. A violation of the enumeration principle would have the extremely unfortunate consequence that arithmetic would not apply to macroscopic objects, and that consequence would be a fatal blow to a theory purporting to describe the macroscopic world. Clearly we should take a closer look at the details of this situation. The argument comes from Peter Lewis and involves a set of n non-interacting marbles in the state

|ψ_{all}> = (a|in>_{1} + b|out>_{1}) r (a|in>_{2} + b|out>_{2}) r … r (a|in>_{n} + b|out>_{n})

where |in>_{i} refers to the i^{th} marble being in the box, and |out>_{i} refers to the i^{th} marble being out of the box. Although PosR may require a to be very close to 1 before we can say that the marbles are in the box, a will never be exactly 1 because of the tails. Since the probability of finding any given marble in the box is a^{2}, the probability of finding all n marbles in the box is a^{2n} – and here is where the trouble begins. For sufficiently large n, a^{2n} << 1, and there is a very good chance that if we look, not all of the marbles will be in the box! Clearly something is not right here.

The string of replies and counter-replies on this topic is beyond the scope of this paper, but Clifton and Monton have argued that while it is in principle possible to violate the enumeration principle under SL theory, the interaction of the counter with the marbles guarantees that the enumeration principle can never be experimentally falsified, and thus that arithmetic continues to apply to everyday objects:

To manifest a failure of conjunction introduction, one has to get an … apparatus which measures the system as a whole appropriately correlated with the system, and one has to get … apparatuses which measure the location of each marble appropriately correlated with each marble. Once all this is done, the requisite entanglement between the marbles (or particles) will be established and the dynamics of the GRW theory will guarantee that the system will either be in, or almost instantaneously evolve to, a state where the various apparatuses are in agreement and no failure of arithmetic is ever manifest.^{6}

Thus GRW dodges another bullet; counting can never actually fail when carried out, since the process of counting entangles all of the marbles and ensures that they do not stay in a potentially dangerous state for more than a split second.

### V. Conclusions

As we have seen, GRW's spontaneous localization theory offers an attractive solution to the macro-objectification problem. Under SL, objects such as electrons, which familiarly exhibit quantum behavior, continue to do so; likewise macroscopic objects such as tables and chairs continue to behave classically. In short, SL predicts that most things will behave exactly as we see them behave; nevertheless, it represents enough of an alteration to standard quantum mechanics that we might expect to see the difference in objects of sizes intermediate between atoms and marbles. Although SL cannot eliminate wavefunctions' "tails," it is clear under the Schrödinger evolution that even if they were eliminated, they would return after a split second; furthermore, it is not at all clear that the tails involved in SL present any sort of problem, either perceptually or philosophically. In the end, SL and its more complex refinements present an intriguing solution to the macro-objectification problem, and it should be interesting to see whether these collapse theories stand up to the experimental scrutiny which is sure to be forthcoming.

### References

- E. Schrödinger, quoted in Griffiths, David J. Introduction to Quantum Mechanics, p. 382. (Upper Saddle River: Prentice-Hall, 1995)
- Griffiths, David J. Introduction to Quantum Mechanics, p. 383. (Upper Saddle River: Prentice-Hall, 1995)
- Ghirardi, G.C., Rimini, A., and Weber, T. 1986, ‘Unified dynamics for microscopic and macroscopic systems’, Physical Review, D 34, 470
- Pearle, P. "Tales and Tails and Stuff and Nonsense". Published in R. S. Cohen, M. A. Horne, and J. S. Stachel, ed. Experimental Metaphysics – Quantum Mechanical Studies in Honor of Abner Shimony, volume 1 (Dordrecht: Kluwer, 1997)
- Albert, D. and Loewer, B. "Tails of Schrödinger’s Cat". Published in R. Clifton, ed. Perspectives on Quantum Reality, (Dordrecht: Kluwer, 1996)
- Clifton, R. and Monton, B. "Losing Your Marbles in Wavefunction Collapse Theories". The British Journal for Philosophy of Science, December 1999.