## GGG

"Sometimes attaining the deepest familiarity with a question is our best substitute for actually having the answer." - Brian Greene, The Elegant Universe

In that sentence, I think that Brian Greene has summarized the attitude of many physicists, especially those whose training and experience has driven them into an over-familiarity with all the unanswered questions in modern particle physics. Quantum Mechanics has proposed too many inexplicable phenomena, with which too many physicists have become too complacently over-familiar. Quantum mechanics and classical physics just cannot be fully reconciled, too many say. Which would be fine if quantum mechanics then went on to explain everything that classical physics could not, but it doesn't. It attains the deepest familiarity with the questions and presents that as an alternative to having any answers.

But there are answers to all of the questions. If you would prefer a more simple explanation of all of the unanswered questions thrown up by Quantum Mechanics and other modern Quantum Physics concepts, along with their solutions, please do consider getting a copy of my book, which is detailed at the bottom of this page, after some very brief information about me. It is not necessary, however, to make any purchase in order to understand my Theory of Absolute Relativity; the following pages each take an important aspect of Quantum Physics, often referring to information published on the Net, scientific papers or other publications, and then applies my theory. Wherever I look in Quantum Physics, the Theory of Absolute Unified Relativity simplifies and explains, solving the problems that have confounded physicists since Einstein and Bohr.

So please do study the following pages, even if you do not choose to purchase a copy of my book; and do feel free to leave comments, as long as they are inoffensive and constructive.

## 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

The Born rule states that if an observable corresponding to a Hermitian operator ${\displaystyle A}$ with discrete spectrum is measured in a system with normalized wave function ${\displaystyle \scriptstyle |\psi \rangle }$ (see bra–ket notation), then

• the measured result will be one of the eigenvalues ${\displaystyle \lambda }$ of ${\displaystyle A}$, and
• the probability of measuring a given eigenvalue ${\displaystyle \lambda _{i}}$ will equal ${\displaystyle \scriptstyle \langle \psi |P_{i}|\psi \rangle }$, where ${\displaystyle P_{i}}$ is the projection onto the eigenspace of ${\displaystyle A}$ corresponding to ${\displaystyle \lambda _{i}}$.

(In the case where the eigenspace of ${\displaystyle A}$ corresponding to ${\displaystyle \lambda _{i}}$ is one-dimensional and spanned by the normalized eigenvector ${\displaystyle \scriptstyle |\lambda _{i}\rangle }$, ${\displaystyle P_{i}}$ is equal to ${\displaystyle \scriptstyle |\lambda _{i}\rangle \langle \lambda _{i}|}$, so the probability ${\displaystyle \scriptstyle \langle \psi |P_{i}|\psi \rangle }$ is equal to ${\displaystyle \scriptstyle \langle \psi |\lambda _{i}\rangle \langle \lambda _{i}|\psi \rangle }$. Since the complex number ${\displaystyle \scriptstyle \langle \lambda _{i}|\psi \rangle }$ is known as the probability amplitude that the state vector ${\displaystyle \scriptstyle |\psi \rangle }$ assigns to the eigenvector ${\displaystyle \scriptstyle |\lambda _{i}\rangle }$, 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 ${\displaystyle \scriptstyle |\langle \lambda _{i}|\psi \rangle |^{2}}$.)
In the case where the spectrum of ${\displaystyle A}$ is not wholly discrete, the spectral theorem proves the existence of a certain projection-valued measure ${\displaystyle Q}$, the spectral measure of ${\displaystyle A}$. In this case,

• the probability that the result of the measurement lies in a measurable set ${\displaystyle M}$ will be given by ${\displaystyle \scriptstyle \langle \psi |Q(M)|\psi \rangle }$.
If we are given a wave function ${\displaystyle \scriptstyle \psi }$ for a single structureless particle in position space, this reduces to saying that the probability density function ${\displaystyle p(x,y,z)}$ for a measurement of the position at time ${\displaystyle t_{0}}$ will be given by

${\displaystyle p(x,y,z)=}$${\displaystyle |\psi (x,y,z,t_{0})|^{2}.}$

### 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

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:

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 ψ0 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  ψ1   ψ2  ψ3 , or why the membrane deteriorates where it does.

But applying Unified Absolute Relativity to Figure 3, as ψ0 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 ψ0 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 ψ0.
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.