Mangled Worlds Quantum Mechanics

This variation on the many worlds interpretation of quantum mechanics allows us to derive the Born probability rule via finite world counting and no new physics.

One of the deepest questions in physics is this: what exactly happens during a quantum measurement? Under the traditional (or "Copenhagen") view, quantum mechanics tells you how to calculate the probabilities of different measurement outcomes. You are to create a wave that describes your initial situation, and then have your wave evolve in time according to a certain linear deterministic rule until the time of a measurement. The equation that describes this rule is very much like the equations that govern the spread of waves over water, or of sound waves in the air.

At the time of a measuremment you are to use the "Born rule" to convert your wave into probabilities of seeing different outcomes. This rule says to break your wave into compoments corresponding to each measurement outcome, and that the probability of each outcome is the measure (or size) of the corresponding component. After a measurement, you can again continue to evolve your wave via the linear deterministic rule, starting with the wave component corresponding to the outcome that was seen.

The problem is, this procedure seems to say that during quantum measurements physical systems evolve according to a fundamentally different process. If, during a quantum measurement, you applied the usual wave propogation rule, instead of the Born probability rule, you would get a different answer. Now for generations students have been told not to worry about this, that the quantum wave doesn't describe what is really out there, but only what we know about what is out there. But when students ask what is really out there, they are told either that is one of the great mysteries of physics, or that such questions just do not make sense.

The many worlds view of quantum mechanics tries to resolve this puzzle by postulating that the apparent Born rule evolution can really be understood as the usual linear rule in disguise. The main idea is that under the linear rule the wave component corresponding to a particular measured outcome becomes decoupled from the components corresponding to the other outcomes, making its future evolution independent of those other outcomes. We might thus postulate that all of the measurement outcomes actually happen, but each happens in a different independent branch "world," split from the original pre-measurement world.

If this view is correct, the universe is far larger than you may have thought possible, and you will have to come to terms with having no obvious answer to the question of which future world "you" would live in after some future measurement. (All of them would contain a creature very much like you just before the measurement.) But these are not strong reasons to reject the many worlds view.

The big problem with the many worlds view is that no one has really shown how the usual linear rule in disguise can reproduce Born probability rule evolution. Many worlders who try to derive the Born rule from symmetry assumptions often forget that there is no room for "choosing" a probability rule to go with the many worlds view; if all evolution is the usual linear deterministic rule in disguise, then aside from unknown initial or boundary conditions, all experimentally verifiable probabilities must be calculable from within the theory. So what do theory calculations say? After a world splits a finite number of times into a large but finite number of branch worlds, the vast majority of those worlds will not have seen frequencies of outcomes near that given by the Born rule, but will instead have seen frequencies near an equal probability rule. If the probability of an outcome is the fraction of worlds that see an outcome, then the many worlds view seems to predict equal probabilities, not Born probabilities.

(Some philosophers say world counts are meaningless because exact world counts can depend sensitively on one's model and representation. But entropy, which is a state count, is similarly sensitive to the same sort of choices. The equal frequency prediction is robust to world count details, just as thermodynamic predictions are robust to entropy details.) That is, if the many worlds view is true, then you and I are right now together in some particular world. Because of previous measurement-like (i.e., "decoherence") processes, there are a googol or googolplex or more other worlds out there. Since others in our world have in the past done statistical tests of the Born rule, these many other worlds are in part distinguished by the results of those statistical tests. In some worlds, including our world, the tests were passed, while in other worlds the tests were failed. (And in far more worlds, the tests were never tried.)

We have done enough tests by now that if the many worlds view were right, the worlds where the tests were passed would constitute an infinitesimally tiny fraction of the set of all those worlds where the test was tried. So the key question is: how is it that we happen to be in one of those very rare worlds? Any classical statistical significance test would strongly reject the hypothesis that we are in a typical world.

It does no good to point to ambiguities in distinguishing worlds - you and I are now at least in a world clearly distinguished from others where Born rule tests failed. And the fact that most worlds see near equal frequencies is not sentitive to how we distinguish worlds. It also does no good to talk about what worlds we care more about - even if our ancestors should have cared a lot more about our world than those other worlds, we should still be surprised to be in one of those rare worlds which our ancestors should have cared more about. And introducing symmetry arguments, about which things "should" have the same probabilities, just avoids confronting the key problem.

Hugh Everett, who originally introduced the many worlds view, tried to resolve this problem by noting that worlds that see Born rule frequencies have much larger measures that most worlds, and then declaring that worlds whose relative measure goes to zero in the limit of an infinity of measurements do not count. But even a googolplex of decoherence events falls far short (actually infinitely short) of an infinity of such events. Without an infinity of decoherence events, we must wonder why we do not find ourselves in one of those very many very small worlds.

Others have tried to resolve this problem by postulating new non-linear processes, or an infinity of worlds per quantum state, that diverge for some unknown reason according to the Born rule. These approaches might work, but like the (promising) objective collapse approaches, they introduce new physics beyond the standard linear deterministic rule.

The mangled worlds approach to quantum mechanics is a variation on many worlds that tries to resolve the Born rule problem by resorting only to familiar probability concepts, standard linear physical processes, and a finite number of worlds. The basic idea is that while we have identified physical "decoherence" processes that seem to describe measurements, since they produce decoupled wave components corresponding to different measurement outcomes, these components are in fact not exactly decoupled. And while the deviations from exact decoherence might be very small, the relative size of worlds can be even smaller.

As a result, inexact decoherence can allow large worlds to drive the evolution of very small worlds, "mangling" those worlds. Observers in mangled worlds may fail to exist, or may remember events from larger worlds. In either case, the only outcome frequencies that would be observed would be those from unmangled worlds. Thus worlds that fall below a certain size cutoff would become mangled, and so should not count when calculating probabilities as the fraction of worlds that see an outcome.

This mangling process allows us to ignore the smaller worlds, but this by itself is not enough to produce the Born probability rule. To get that we also need the cutoff in size between mangled and unmangled worlds to be in the right place. Specifically, we need the cutoff to be much nearer to the median measure world size than to the median world size. The median measure is the world size where half of all measure is held by worlds larger that this size, and half is held by worlds smaller than this size. The median world size is the size where half of all worlds are larger, and half are smaller.

Now since it is actually the measure of some worlds that would allow them to mangle other worlds, it is not crazy to expect a cutoff near this position. But we will have to see if further theoretical analysis of familiar quantum systems supports this conjecture. Fortunately, the mangled worlds view can in principle be verified or refuted entirely via theoretical analysis, by seeing if theory predicts that familiar quantum systems evolving according to the usual linear rule behave as the mangled worlds view predicts.

To square the mangled worlds view with what we see, we also need to conjecture that world mangling is a relatively sudden process, and that it is thermodynamically irreversible. After all, we do not observe our world as partially mangled, or see historical records of a mangling period in our past. Finally, since the mangled worlds view predicts that worlds with a low rate of decoherence events will be selected, we must also conjecture that our world's rate of such events is nearly as low as possible.

These predictions may well turn out to be false, and we may need to resolve the Born rule puzzle via new fundamental physics. But for now, at least, a hope remains that it can be resolved using only familiar physical processes and standard logic, probability, and decision theory.

A powerpoint presenation on this subject is here.

Two academic papers on this topic are:

Robin Hanson, When Worlds Collide: Quantum Probability From Observer Selection?. *Foundations of Physics* 33(7):1129-1150, July 2003.

In Everett's many worlds interpretation, quantum measurements are considered to be decoherence events. If so, then inexact decoherence may allow large worlds to mangle the memory of observers in small worlds, creating a cutoff in observable world size. Smaller world are mangled and so not observed. If this cutoff is much closer to the median measure size than to the median world size, the distribution of outcomes seen in unmangled worlds follows the Born rule. Thus deviations from exact decoherence may allow the Born rule to be derived via world counting, with a finite number of worlds and no new fundamental physics.

Robin Hanson, Drift-Diffusion in Mangled Worlds Quantum Mechanics, *Proceedings of Royal Society A*, 462(2069):1619-1627, May 8, 2006.

In Everett's many worlds interpretation, where quantum measurements are seen as decoherence events, inexact decoherence may allow large worlds to mangle the memories of observers in small worlds, creating a cutoff in observable world size. This paper solves a growth-drift-diffusion-absorption model of such a mangled worlds scenario. Closed form expressions show that this model reproduces the Born probability rule closely, though not exactly. Thus deviations from exact decoherence can allow the Born rule to be derived in a many worlds approach via world counting, using a finite number of worlds and no new fundamental physics.

Three news articles on this topic are:

Bad News,BBC Focus, p. 23, April 2006.Andrea Moore, The End Is Coming - For One Of Yourselves,

All Headline News10:00 p.m. EST, February 23, 2006.Maggie McKee, Is our universe about to be mangled?,

NewScientist.com, 17:43, February 23, 2006.

A belorussian translation of this page here.