I would like to know what physicists think about the propensity viewpoint.If this latter one is in line with physics and especially Quantum Mechanics. Otherwise, what is the most coherent philosophical perspective on probability with Quantum Mechanics?
Most physicists don't care about the interpretation of probability. Nor do they care about the controversy over the "interpretation" of quantum mechanics.
The philosophy of probability is not in a good state. It is mostly divided between frequentists who think that probability is defined by long run frequency and Bayesians who think that probability has something or other to do with credence. No real experiment involves measuring an infinite set of systems, so the frequency interpretation is irrelevant to all real experiments. It's also difficult to see why credence should play any role at all in physics.
The propensity interpretation of probability has the merit of saying that a theory of probability has to be relevant to real experiments, but falls down when it comes to explaining that relevance.
There are various attempts to explain probability in quantum mechanics:
but most physicists don't seem to think there is any problem here worth considering. I think this is much to their discredit since having a theory where one of the most important concepts is unexplained makes properly understanding, testing, criticising and replacing that theory impossible. David Deutsch has made some interesting comments on these issues:
In short, propensities are, precisely, the quantum mechanical probabilities.
In a classical, deterministic, world we set up approximations to the ideal of a repeatable experiment being understood that each experimental run differs from the others in small mechanical variations. In a classical context propensities are thus extracted from the deterministic response of the system to small perturbations, in a setting in which such perturbations occur at random; so, a mechanical source of (pseudo)randomness is required (a stationary closed orbit, e.g., has no propensities), and probability distributions model our lack of knowledge.
In a quantum mechanical world, however, propensities (of pure states) are fundamental properties of the physical system, independent of the lack of knowledge we may have. Quantum Mechanics is consistent with exactly repeatable experiments, and thus provides a fundamental propensity theory, in which we compute relative weights of different histories with exactly the same initial state.
Therefore, although with both classes of theories propensities are measured from (classical) statistics, as relative frequencies, in classical mechanics such measurements only yield some cumbersome knowledge of the experiment, so that a propensity theory is hard to hold as something really different from relative frequency; while in Quantum Mechanics we deduce 'true' propensities and compare results with measured frequencies.
Consider a classical coin tossing theory. From the symmetry of the coin and the dynamics of a 'normal' tossing we naturally infer a fifty-fifty distribution, since we don't have any knowledge of the experiment from which to infer a higher propensity of heads or tails. However, precisely because we are based on lack of knowledge, a coin tosser may, in principle, temporarily show a significant deviation from the fifty-fifty distribution, drawing, say, heads at 60% rate; so, what's then the point in theorizing such rate as a propensity of the situation, when we can only guess? Measuring the mass distribution of the coin makes much more sense, because it may be done with very high precision and stability and yields basic dynamical constants. The same applies to weather physics and any classical approach, while in the quantum mechanical world we have propensities like particle decay rates (measurable with high precision) as basic dynamical constants.
Moreover, in classical statistical physics Boltzmann's equation holds with the assumption (questioned by Einstein) that all microstates have the same weight (probability); so, these basic relative weights are not measured but postulated as a flat (totally ignorant) distribution, which assumes there's no such thing as Maxwell's demon.
According to Richard Feynman this is exactly where the central mystery in QM lies.
Probability was implicated in QM right from the start. Recall, that probability in its usual sense - as frequentist is the technology that allowed Boltzmann to think of thermodynamics as the statistics of a large number of atoms and one of the central problems here was to work out the spectrum of black body radiation. A solution based on Boltzmanns theory was found by Wien & Raleigh but both were inaccurate in some domain, one in the low energy regime or the infrared and the other in the high energy or the ultraviolet.
It was Plancks achievement to come up with an accurate formula and he did this by postulating - as purely a mathematical fix - a discrete and indivisible amount of energy; this was reified by Einstein into quanta, the 'atoms' of energy.
It was Born that came up with the interpretation of the square of the quantum wave function as probability, but unlike Boltzmanns probability, which was epistemological, this was ontological.
The mystery is that physically speaking, quantum amplitudes are the square root of probabilities. There is no good explanation of this within the frequentist interpretation of why we can take the square root of probabilities and why this should make sense.
This suggests an analogy with the square root of -1 which was for centuries seen as a notion useful in helping to solve equations but had no ontological standing as some definite number until the 19th C when a geometric rather than arithmetic understanding was developed and then this became reified into a duality in mathematics between geometry and algebra, two perspectives on the same mathematical reality.
It may take likewise a similar amount of time before we can fully understand how probability is to be properly understood in QM.