Physicist and spy Klaus Fuchs has expressed the opinion that Born's rule (squared complex amplitudes are interpreted as probabilities or probability densities) could be derived from something deeper. I think that this wishful thinking is demonstrably impossible. Why? We just don't have any method or theorem in mathematics or physics that could allow us not to assume any statement of the sort

the probability is \(f(\theta)\)

and deduce the conclusion of the form

the probability is \(f(\theta)\)

For example, think about an electron whose spin is prepared to be aligned "up" with respect to an axis, and then measure the projection of the spin \(j_z\) with respect to the \(z\)-axis. The angle between the two axes is \(\theta\), the amplitude is \(\cos(\theta/2)\), up to a phase, and the probability to get "up" again is therefore \(\cos^2(\theta/2)\).

How could you possibly derive that from something "deeper"? We don't have anything "deeper" than probabilities that probabilities could be constructed from. At most, we may define probabilities as \(N/N_{\rm total}\), the frequentist formula by which we measure it – which would give us rational numbers if \(N_{\rm total}\) were some "fundamentally real" options. And we may deduce that the probability is \(p=1/N\) if \(N\) options are related by a symmetry. Or we may say that each state on a "shell of the phase space" – quantum mechanically, a subspace of the Hilbert space – has the probability \(p=1/N\) to be realized during a random evolution as envisioned by the ergodic theorem.

None of those

*Ansätze* can produce the statement "the probability is \(\cos^2(\theta/2)\)" and there are no other candidates of the "methods" in mathematics and physics. So I find it rather clear that unless someone finds a totally new mathematics that finds completely new definitions or laws for probabilities, and e.g. calculates probabilities from Bessel's function of the number of Jesus' disciples (which seems like a quantity of a different type than probability, and that's the main reason why this example should sound ludicrous), it is clearly impossible to derive statements like "the probability of 'up' is \(\cos^2(\theta/2)\)" from something that says nothing about the values of probabilities.

The people saying "Born's rule smells like it's derived" never respond to the argument above – which I consider a proof of a sort. I think that if one carefully looks at the task, he will agree that the only way to deduce that

*the probability is a continuous function of some variables* is to make at least some assumptions that

*the probability is a continuous function of some variables*. Quantum mechanics including Born's rule is making statements about Nature of the form

*the probability is a continuous function of some variables*. But if you have nothing like that as a fundamental law of physics, you just can't possibly derive any conclusion like that.

Quantum mechanics and its statistical character can't be "emergent". The statements about the values of probabilities have to appear

*somewhere in our derivations* for the first time. So the only way how a physical theory may make predictions of probabilities at all is that it

*contains an axiom with the formula telling us what the probabilities are*, namely (in the case of quantum mechanics) Born's rule. Such a rule can't be born out of nothing or out of something unrelated to probabilities, it's that simple.