Quantum probabilities or frequencies? Derivation or a postulate

Question

The Born rule $$p=|\langle\psi|\phi\rangle|^2$$ defines the quantum probability and answers the question what is the probability that the measurement of $\psi$ will produce the outcome $\phi$? This is the standard interpretation of the formula but some authors, say, @Luboš Motl here and here apparently popularly claims just that “the word "interpretation" shoudn't be there at all”.

What is the generally accepted and experimentally supported reading of the formula? after all. What do experimentalists interpret in their labs? I've come across yet another derivation of the formula in the arXiv-preprint 1905.03332, but I'm sceptical about this because its author emphasizes at the end that even the probability is not needed here. There is only frequencies there. Is this yet another crazy?

It is common knowledge that the frequency interpretation of quantum probability is often criticized (Wallace, Zurek 2000's and others). On the other hand, additivity does really appear in many works on this topic. Even Gleason in the 1960s wrote on this, unless I'm mistaken. Could someone kindly (and precisely) comment on a relationship between the approaches? Probability vs frequencies, axiom vs derivation, and, preferably, the proposed `new derivation' of the rule through frequencies in the preprint above? (I'm not good in math).

PS. Does Zurek's derivation in the framework of decoherence solve the problem? My inclination is to believe that decoherence has already been confirmed in experiments many times and, perhaps, explains the transition under consideration because all the quantum systems are actually open at any one time.

Answer 1

Quantum mechanics is the underlying theory of all physical theories as far as we now know. It has been validated, i.e. prediction using quantum mechanical calculations have not be falsified, innumerable times. The theory of quantum mechanics relies on the mathematics of differential equations but in addition it has extra axioms,called postulates, that pick up the subset of the mathematical solutions that is relevant to the physics problems.

The Born rule in the list is the wave function postulate. third page:

It is one of the postulates of quantum mechanics that for a physical system consisting of a particle there is an associated wavefunction. This wavefunction determines everything that can be known about the system. The wavefunction is assumed here to be a single-valued function of position and time, since that is sufficient to guarantee an unambiguous value of probability of finding the particle at a particular position and time. The wavefunction may be a complex function, since it is its product with its complex conjugate which specifies the real physical probability of finding the particle in a particular state.

One does not interpret postulates that lead to a theory of physics, because changing their clear meaning makes nonsense of the theory, or a different theory.

Interpretations of quantum mechanics , as for example the Bohm mechanics, interpret the same data, i.e make the exact same predictions for the data.If the predictions differ between quantum mechanical predictions and "new theory" predictions, then it is not an interpretation but a new theory to be validated or falsified on its own.

Quantum probabilities or frequencies? Theory and experiment

Quantum mechanics has $Ψ$ as a function whose $Ψ^*Ψ$ gives the probability of the particle described by $Ψ$ to be measured at an (x,y,z,t). Since $Ψ$ is a solution of a wave function differential equation, there are frequencies, so the probability can show interferences and effects similar to water wave and sound wave interactions, but the frequency is related to the probability of detection in an experiment with specific coordinates or four vectors. The particle itself is a point particle or a system of point particles ( as the protons and neutrons and ...).