9.2 The Born Rule as Harmonic Measure

The Born rule is quantum mechanics’s most mysterious postulate. It says the probability of a measurement outcome is proportional to the square of the wavefunction’s amplitude.

It works perfectly. Nobody knows why it is the square and not something else.

Everett’s Many Worlds interpretation[everett] makes this more acute: if all branches are real, why do we experience some branches more than others? The probability has to come from somewhere.

The speculation: the Mandelbrot set’s branching tree has a natural probability measure on its boundary called the harmonic measure[douady]. It assigns higher probability to simpler, more consonant branches, the large bulbs, the shallow Farey fractions. This measure is not imposed, it falls out of the geometry.

If the universe’s wavefunction branches according to a similar tree structure, with branching numbers determined by local algebraic structure rather than arbitrary choice, then the Born rule probabilities may fall out of the harmonic measure of that tree. Not a postulate. A consequence of the branching geometry.

This is speculative. But it is the right kind of speculative, it connects two things that are separately mysterious (why the Born rule, why the Mandelbrot measure) and asks if they are the same mystery.