It is shown that the quadratic measure (Born's rule) is not an independent postulate, but is derived from the compatibility of two structural features of physical processes: linear reversible evolution before stable records emerge, and multiplicative composition of outcome weights after they form. Reversible evolution combines configurations additively via a compatibility parameter, while record formation gives weights a multiplicative structure. The requirement of consistency between these modes restricts the permissible assignment of weights to a quadratic dependence on amplitude. Thus, Born's rule emerges as the only measure compatible with reversible linear evolution and irreversible record formation, without assumptions about probabilistic interpretation or the use of specific quantum formalism.
Before measurement, a quantum object lives in a world of possibilities, where wave amplitudes add up like sound waves in a studio before pressing the “record” button. Crests amplify each other, troughs cancel out. But as soon as an irreversible record of an event appears—and this is always an increase in entropy (a measure of disorder)—the rules change. The weights of outcomes begin to multiply, and to reconcile this jump with the smooth wave dance, nature is forced to square the amplitude.
It was in this way, not by whim, that the famous Born rule, proposed by Max Born in 1926, arose. It underpins the standard model of physics. And the quadratic relation between wave and energy is familiar to everyone: for instance, in spectroscopy (the analysis of light), the brightness of a line is also given by the square of the amplitude.
Amazingly, if we abandon the square, any record of events loses meaning: the past ceases to be unambiguous. Without this simple exponentiation, our world of facts would crumble.
🎯 Max Born received the Nobel Prize for this rule only in 1954—almost 30 years after his discovery.
🎬 In the series “Dark”, quantum probabilities give rise to parallel realities—and thus, forks in fate.