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By Root 2028 0

means that, in the stated limit, the fractional number of times an observer measuring A will find αk is —which looks like the most straightforward derivation of the famous Born rule for quantum mechanical probability. From the Many Worlds perspective, it suggests that those worlds in which the fractional number of times that αk is observed fails to agree with the Born rule have zero Hilbert space norm in the limit of arbitrarily large n. In this sense, it seems as though quantum mechanical probability has a direct interpretation in the Many Worlds approach. All observers in the Many Worlds will see results with frequencies that match those of standard quantum mechanics, except for a set of observers whose Hilbert space norm becomes vanishingly small as n goes to infinity. As promising as this seems, on reflection it is less convincing. In what sense can we say that an observer with a small Hilbert space norm, or a norm that goes to zero as n goes to infinity, is unimportant or doesn’t exist? We want to say that such observers are anomalous or “unlikely,” but how do we draw a link between a vector’s Hilbert space norm and these characterizations? An example makes the issue manifest. In a two-dimensional Hilbert space, say with states spin-up , and spin-down , consider a state . This state yields the probability for measuring spin-up of about .98 and for measuring spin-down to be about .02. If we consider n copies of this spin system, , then as n goes to infinity, the vast majority of terms in the expansion of this vector have roughly equal numbers of spin-up and spin-down states. So from the standpoint of observers (copies of the experimenter) the vast majority would see spin-ups and spin-downs in a ratio that does not agree with the quantum mechanical predictions. Only the very few terms in the expansion of that have 98 percent spin-ups and 2 percent spin-downs are consistent with the quantum mechanical expectation; the result above tells us that these states are the only ones with nonzero Hilbert space norm as n goes to infinity. In some sense, then, the vast majority of terms in the expansion of (the vast majority of copies of the experimenter) need to be considered as “non existent.” The challenge lies in understanding what, if anything, that means.

I also independently found the mathematical result described above, while preparing lectures for a course on quantum mechanics I was teaching. It was a notable thrill to have the probabilistic interpretation of quantum mechanics seemingly fall out directly from the mathematical formalism—I would imagine the list of physicists (on this page) who found this result before me had the same experience. I’m surprised at how little known the result is among mainstream physics. For instance, I don’t know of any standard quantum physics textbook that includes it. My take on the result is that it is best thought of as (1) a strong mathematical motivation for the Born probability interpretation of the wavefunction—had Born not “guessed” this interpretation, the math would have led someone there eventually; (2) a consistency check on the probability interpretation—had this mathematical result not held, it would have challenged the internal sensibility of the probability interpretation of the wavefunction.

10. I’ve been using the phrase “Zaxtarian-type reasoning” to denote a framework in which probability enters through the ignorance of each inhabitant of the Many Worlds as to which particular world he or she inhabits. Lev Vaidman has suggested taking more of the particulars of the Zaxtarian scenario to heart. He argues that probability enters the Many Worlds approach in the temporal window between an experimenter completing a measurement and reading the result. But, skeptics counter, this is too late in the game: it’s incumbent on quantum mechanics, and science more generally, to make predictions about what will happen in an experiment, not what did happen. What’s more, it seems precarious for the bedrock of quantum probability to rely on what seems to be an avoidable time delay: if