## The Born Rule

The Born rule (also called the Born law, Born's rule, or Born's law) is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of quantum mechanics. There have been many attempts to derive the Born rule from the other assumptions of quantum mechanics, with inconclusive results.

## The rule

*see*bra–ket notation), then

- the measured result will be one of the eigenvalues of , and
- the probability of measuring a given eigenvalue will equal , where is the projection onto the eigenspace of corresponding to .

- (In the case where the eigenspace of corresponding to is one-dimensional and spanned by the normalized eigenvector , is equal to , so the probability is equal to . Since the complex number is known as the
*probability amplitude*that the state vector assigns to the eigenvector , it is common to describe the Born rule as telling us that probability is equal to the amplitude-squared (really the amplitude times its own complex conjugate). Equivalently, the probability can be written as .)

- the probability that the result of the measurement lies in a measurable set will be given by .

###
Representation of The Born Rule, by Diederik Aerts, Massimiliano Sassoli de Bianchi:

“The
extended Bloch representation of quantum mechanics and the hidden-measurement
solution to the measurement problem.”

http://dx.doi.org/10.1016/j.aop.2014.09.020 0003-4916/©2014

Published by Elsevier Inc. This is an open access article under the CCBY-NC-ND license

(http:// creativecommons.org/licenses/by-nc-nd/3.0/).

Highlights

• An
extended Bloch representation of quantum measurements is given.

• Quantum
measurements are explained in terms of hidden-measurement interactions.

•
Quantum measurements are explained as tripartite processes.

•
The Born rule results from a universal average, over all possible measurement processes.”

Abstract

“A
generalized Bloch sphere, in which the states of a quantum entity of arbitrary
dimension are geometrically represented, is investigated and further extended,
to also incorporate the measurements.This extended representation constitutes a
general solution to the measurement problem, in as much it allows to derive the
Born rule as an average over hidden-variables, describing not the state of the quantum
entity, but its interaction with the measuring system. According to this
modelization, a quantum measurement is to be understood, in general, as a tripartite
process, formed by an initial deterministic decoherence-like process, a subsequent
indeterministic collapse-like process, and a final deterministic
purification-like process. We also show that quantum probabilities can be generally
interpreted as the probabilities of a first-order non-classical theory,
describing situations of maximal lack of knowledge regarding the process of
actualization of potential interactions, during a measurement.”

A simplified overview of the above paper can be viewed at:

A simplified overview of the above paper can be viewed at:

Using an extended Bloch representation of quantum mechanics, Diederik Aerts and Massimiliano Sassoli de Bianchi have developed a model that allows a visualisation of Born’s rule using an elastic connection between the alternatives:

either in the form of a band when n = 2, or a membrane when n>2.

The description works and allows a picture of the probabilities of the Born Rule up to n = 4 (the method works for n>4, but the visualisation becomes impossible. However, the elastic membrane has to be accepted as something forceful connecting the alternative wavefunctions onto which the originating wavefunction ψ

_{0}is deposited at a particular point, which allows the resultant wavefunction as the membrane deteriorates. There is no suggestion as to the physical nature of the membrane and the action that causes the membrane to begin to deteriorate in any particular position.
Taking the 3-Particle case:

Figure 1: The Membrane Between the Alternatives

The membrane can be divided as shown:
Figure 2

Diederik Aerts and Massimiliano Sassoli de Bianchi show the effect three-dimensionally:

Figure 3

There is no explanatory description as to why should appear where it does with respect to

_{ }ψ_{1 }ψ_{2 }ψ_{3}, or why the membrane deteriorates where it does.But applying Unified Absolute Relativity to Figure 3, as ψ

_{0}is never a particle, but the wave effect from an originating atom , it can be shown that the illustrated system works when the sphere is so positioned to share the wavefunction ψ

_{0 }between the three alternatives:

Figure 4

The probability distribution remains the same, describing the results of the Born Rule, just as if the same point-particle had been placed on the membrane. Now though, there is no need to look for a means by which the membrane breaks down; it is simply a question of which receiving particle, each with its attendant potential waveform, has the greatest share of the wavefunction ψ

_{0}.
From this, there is no breakdown, no fragmenting of the, each with it attendant potential waveform, has the greatest share of the waveform ψ0 distributed between the three receiving positions, which would be better, or more accurately described using eigenvalues λ1, λ2 or λ3.

Figure 5

We have shown that Unified Absolute Relativity work here, just as it explains and simplifies every aspect of Quantum Mechanics.

_{}

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