Every experiment to date in quantum mechanics has proved consistent with the Born rule proposed in a 1926 paper written by Max Born. The idea is pretty simple, any act of measuring in quantum mechanics returns a number called an eigenvalue and the square of the amplitude gives us the probability of getting that eigenvalue, when that measurement is performed and the wave function collapses to that state. At all other times the wave function is a vector in Hilbert space, that evolves according to the Schrodinger equation.
Everett’s interpretation is radically different to the standard formulation of quantum mechanics. According to Everett the wave function of the universe is unitary and always evolves deterministically, it never collapses, and as a consequence no one can say one particular eignestate is favoured over any other. What gives us the impression of "collapse" is when the wave function is decohered after entangled states interact with their environment.
The Born rule presents a challenge for Everett. How does one derive $Pr\left ( x \right ) = \left | amplitude \right |^{2}$ from the only two postulates that Everett’s interpretation entails (a) that quantum states are represented by vectors in Hilbert space and (b) that the wave function evolves at all times, according to a second order differential equation?
Probability in quantum mechanics isn’t like probability in classical mechanics. For a classic process, you survey all the events leading from A to B, using our known understanding of the laws of physics you calculate the probability of every event contained in that history. Then you multiply the probability of each of those events together. If there’s more than one history from A to B, you add the probabilities, and the total probability should be exactly one.
In quantum mechanics things are different. Each event has an amplitude which is a complex number, whose total value is less than one. It has two components it’s "absolute value" given by a real number and it’s "phase" given by an imaginary number. The probability is then calculated by taking the absolute value and squaring it, this is the Born rule and it’s the predictive part of quantum mechanics.
There is a caveat however, you have to take account of the phase of each complex number as well. If for example in Young’s slits experiment, the two waves interfere constructively (that’s to say in the same phase), then their probability will be greater. Whereas if they interfere destructively, (that’s out of phase) their probability will be less.
So, similarly to classical histories, if there’s more than one way to get from A to B we add up all the probabilities of each of the histories. Then we square their value to get their absolute probabilities, but unlike the amplitude this will always give us a real number. Their values are unitary meaning they should all add up to one. According to quantum mechanics $U(t)$ is an N x N matrix with a complex number as each component, no matter what the initial state is it will always preserve unitarity.
Things are a little more complicated when we ask what the probability of finding a particle in two different states simultaneously, this argument is courtesy of Lubos Motl. Any question like this that we can formulate using quantum mechanics, like the expectation value of an electron’s spin in two different states can be given an answer. Simply we can state it as being
P is the projection operator so that it is a product of two other projection operators on the state of an electron's spin. Such that
These are both projection operators but on a subspace. So that the expectation value of the electron in state "up" corresponding to the first projection operator and the expectation value of the electron in the state "down" corresponding to the second projection operator, are both one if the projection operator P is one. So that we have
Given that
We should see that
We then have an answer for the probability of finding the electron in both states simultaneously
The probability of finding the electron in both the spin up and the spin down state is zero. By contrast the Everett or Many Worlds Interpretation, if we assume it's a standard quantum state then both states of the electron are equally real in parallel branches of the wave function which would surely correspond to a value of one not zero.
Proponents of Everett's interpretation will probably argue that after a measurement the particle is entangled with the measuring device, so that it can't be described using a quantum state like the one I've given. Before the measurement we can describe the observer and the particle set up using a pure state, pure states don't evolve into mixed state, and so it's a very simple set up. If the argument is that the wave function of the universe has been evolving for billions of years, with increasingly complex and entangled superpositions then we should be able to describe the system prior to measurement using a mixed state as well. So that this would entail that you can't understand the initial state with any kind of certainty, so whether you use a pure state or a density matrix, the MWI gives you an answer which is wrong.
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