It was formulated and published by German physicist Max Born in July, 1926.
[1] The Born rule states that an observable, measured in a system with normalized wave function
is not wholly discrete, the spectral theorem proves the existence of a certain projection-valued measure (PVM)
In this case: For example, a single structureless particle can be described by a wave function
The Born rule implies that the probability density function
The Born rule can also be employed to calculate probabilities (for measurements with discrete sets of outcomes) or probability densities (for continuous-valued measurements) for other observables, like momentum, energy, and angular momentum.
In some applications, this treatment of the Born rule is generalized using positive-operator-valued measures (POVM).
A POVM is a measure whose values are positive semi-definite operators on a Hilbert space.
In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, a POVM is a set of positive semi-definite matrices
, such that the probability of obtaining it when making a measurement on the quantum state
The Born rule, together with the unitarity of the time evolution operator
being Hermitian), implies the unitarity of the theory: a wave function that is time-evolved by a unitary operator will remain properly normalized.
(In the more general case where one considers the time evolution of a density matrix, proper normalization is ensured by requiring that the time evolution is a trace-preserving, completely positive map.)
[4] In this paper, Born solves the Schrödinger equation for a scattering problem and, inspired by Albert Einstein and Einstein's probabilistic rule for the photoelectric effect,[5] concludes, in a footnote, that the Born rule gives the only possible interpretation of the solution.
(The main body of the article says that the amplitude "gives the probability" [bestimmt die Wahrscheinlichkeit], while the footnote added in proof says that the probability is proportional to the square of its magnitude.)
In 1954, together with Walther Bothe, Born was awarded the Nobel Prize in Physics for this and other work.
[5] John von Neumann discussed the application of spectral theory to Born's rule in his 1932 book.
[6] Gleason's theorem shows that the Born rule can be derived from the usual mathematical representation of measurements in quantum physics together with the assumption of non-contextuality.
Andrew M. Gleason first proved the theorem in 1957,[7] prompted by a question posed by George W.
[8][9] This theorem was historically significant for the role it played in showing that wide classes of hidden-variable theories are inconsistent with quantum physics.
[10] Several other researchers have also tried to derive the Born rule from more basic principles.
[15] In 2018, an approach based on self-locating uncertainty was suggested by Charles Sebens and Sean M. Carroll;[16] this has also been criticized.
[17] Simon Saunders, in 2021, produced a branch counting derivation of the Born rule.
The ratios of the numbers of branches thus defined give the probabilities of the various outcomes of a measurement, in accordance with the Born rule.
[18] In 2019, Lluís Masanes, Thomas Galley, and Markus Müller proposed a derivation based on postulates including the possibility of state estimation.
[19][20] It has also been claimed that pilot-wave theory can be used to statistically derive the Born rule, though this remains controversial.
[21] Within the QBist interpretation of quantum theory, the Born rule is seen as an extension of the normative principle of coherence, which ensures self-consistency of probability assessments across a whole set of such assessments.
It can be shown that an agent who thinks they are gambling on the outcomes of measurements on a sufficiently quantum-like system but refuses to use the Born rule when placing their bets is vulnerable to a Dutch book.