The Copenhagen Interpretation of quantum mechanics postulates the Born rule which gives the probability that an observer will measure a quantum system in a particular state. For example, if a photon is sent through a beam splitter, the Born rule says that the photon will be detected on the reflection path half the time and on the transmission path the other half of the time.

The Born rule gives the correct probabilistic predictions. But what explains the Born rule? And why are there probabilities at all? According to the Copenhagen interpretation (which rejects causality) there is no explanation - the probabilities are just a brute fact of the universe. Hence the Born rule must be postulated under that interpretation.

However for causal interpretations (such as Everett and de Broglie-Bohm), the probabilities must have an underlying causal explanation. So, in the case of Everett, the Born rule should be derivable from unitary quantum mechanics, not merely postulated. I'm going to outline a derivation below - note that it draws from Carroll's and Sebens' derivation.

The first issue is why we should expect probabilities at all. With the beam splitter example above, the Everett interpretation says that a photon is detected on both paths (as described by the wave function). But an observer only reports detecting a photon on one path. Therefore, on the Everett interpretation, the probabilities describe the self-locating uncertainty of the observer (or the observing system), not the probabilities that the photon will be detected exclusively on one path or the other.

The second issue is why the probability is that given by the Born rule and not by some other rule.

The Born rule states that the probability is given by squaring the magnitude of the amplitude for a particular state. In the beam splitter example above, the amplitude for each relative state is √(1/2). So the probability for each state is 1/2.

It appears that in these scenarios we should be indifferent about which relative state we will end up observing. So the simple rule here is to assign an equal probability to each state. If there are two states, then the probability of each is 1/2. If there are three states, then the probability of each is 1/3. And so on. This is often called branch-counting and it makes the correct predictions in cases where the amplitudes for each state are equal.

So far so good. But this approach doesn't work if the amplitudes are not equal. For example, suppose we have a beam splitter where the probability of reflection is 1/3 and the probability of transmission is 2/3.

In this scenario there are two states with amplitudes √(1/3) and √(2/3). If we are indifferent about which state we will end up observing, then we will wrongly assign a probability of 1/2 to each state. Branch counting doesn't work in this case.

So we have a simple indifference rule that only works in specific circumstances and seems inapplicable here. What can be done? One thing we can do is to transform the setup so that the rule does become applicable.

To do this, we can add a second beam splitter (with 1/2 probability of reflection and transmission) to the transmission path of the first beam splitter. When we send a photon through this setup, there are now three final states and they all have equal amplitudes of √(1/3) each (i.e., √(2/3) * √(1/2) = √(1/3)).

Now the indifference rule can be applied to get a probability of 1/3 for each final state. Since there are two final states on the first beam splitter's transmission path, their probabilities add to give a total probability for the first beam splitter's transmission path state of 2/3. This is the correct prediction as per the Born rule and it only assumes the indifference rule as required.

This works for any scenario where states of unequal amplitudes are factorable into states of equal amplitudes.

Does anyone see any problems with this?