The Many-Worlds Interpretation and the Born Rule

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The many-worlds interpretation faces a major challenge: by itself, it has not yet fully explained the
Born rule — the rule that the probability of an observation is proportional to the squared modulus of the wave function.
Below, I summarize the problem and introduce a few ideas that may point toward a derivation of the Born rule.

What Is the Born Rule?

The Born rule is the empirical rule that the probability of an observation is proportional to the squared modulus of the wave function. It was inferred from experimental results rather than derived from a deeper principle.

Most textbooks introduce the Born rule as a basic postulate, so it is rarely questioned. I believe, however, that it remains one of the fundamental mysteries of quantum mechanics.

Many-Worlds and the Born Rule

Some papers argue that the Born rule arises naturally within the many-worlds interpretation, but I do not think the issue is so simple.

In his 1957 paper, Hugh Everett introduced the idea of a measure and stated that it is proportional to the squared modulus of the wave function. However, he did not clearly identify this measure with “the number of worlds.”

If we adopt the many-worlds interpretation, it seems natural to think of probability in terms of “how many worlds” correspond to each outcome. If the measure is really proportional to the number of worlds, then the measure itself must be something physically meaningful. The wave function is also physically meaningful, since it gives rise to interference.
How are these two quantities connected? I believe the key to deriving the Born rule lies in that connection.

Four Clues Toward a Derivation

I searched for pairs of quantities in which one is proportional to the square of the other, hoping that such examples might suggest a relationship between the wave function and the measure.

1. The length of a spring and its elastic energy
2. The radius of a circle and its area
3. The radius of a sphere and its surface area
4. The number of points and the number of paths connecting them

The first three are geometric or mechanical analogies, but I find the fourth — points and paths — the most promising. For example, three points can be connected by nine directed paths. If the number of points is N, the number of such paths is N², so a square law appears naturally.

I hope this article encourages further thought about the problem of probability in the many-worlds interpretation.

• The Many-Worlds Interpretation of Quantum Mechanics (2013/6/22)
• Questions About the Many-Worlds Interpretation (2014/1/11)
• The Equivalence Principle and the Origin of Gravity (2014/2/11)

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