Demystifier said:
But in the quantum case, the phase function ##\phi## is single valued.
One physical consequence of this is quantization of angular momentum.
No. In QT, what is single-valued is only the wave function ##\psi = \sqrt{\rho}e^{i\phi}##. Once ##e^{i\phi} = e^{i\phi+2n\pi i}## this leads to quantization conditions. ##\phi## can have multiple values, but they have to fulfill some discrete condition to make ##\psi = \sqrt{\rho}e^{i\phi}## single-valued.
Such a multi-valued function ##\phi## cannot be continuous and defined everywhere, there will be places where it is not defined. But the wave function has to be defined there. That's possible if at those places ##\rho=0##. So, QT does not give any indication that ##\phi## has to be single-valued.
Demystifier said:
The requirement that ##\phi## must be single valued indicates that ##\phi## is ontic, unlike ##\phi_{\rm HJ}##.
This is obviously a weak place. Yes, one would expect that an ontic function would be single-valued. But epistemic functions can be single-valued too, and necessarily single-valued. In Caticha's entropic dynamics, ##\phi## is ##S - \ln\sqrt{\rho}##, and I see no way to define entropy as a multi-valued function.
Demystifier said:
But if ##\phi## is ontic, how can it be compatible with the fact that it is related to entropy in Caticha theory?
A variant of the PBR error. You see one property of ontic objects (single-valued , in PBR no overlaps) and conclude that a function with such properties has to be ontic. Entropy has the same properties, but is epistemic (or can be interpreted in this way - a lot of physicists seem to think that entropy is ontic).
But I see that you implicitly rely also on the entropy being somehow ontic:
Demystifier said:
Analogy with classical physics is useful again, in this case with classical statistical mechanics (CSM). In CSM, there are two different definitions of entropy: Gibbs entropy and Boltzmann entropy. In general they are inequivalent, but in thermal equilibrium they turn out to be numerically equal to each other. The most important thing here is that Gibbs entropy is a function of
probability density in the phase space, while the Boltzmann entropy is a function of
point in the phase space. This means that Gibbs entropy is naturally interpreted as epistemic quantity, while Boltzmann entropy is naturally interpreted as ontic quantity.(*) This demonstrates that entropy can be ontic even when it can be expressed by a formula that looks epistemic.
(*) For more details I refer to
https://arxiv.org/abs/1903.11870.
Following your source, Bolzmann entropy depends on
Γ(X) is the set of all phase points that “look macroscopically the same” as X.
This is clearly a function of incomplete knowledge about X, not of X itself. So I'm not impressed by the idea that entropy is ontic.
But this is certainly a question one can argue about. Last but not least, Jaynes together with his followers (Caticha is one of them) have, after they have essentially won the fight about the interpretation of probability itself (you may disagree but that's another question) started to reinterpret thermodynamics. This alone shows that they thought that there was some necessity of reinterpretation.
The line of argument to win the fight was also in some sense similar: The frequency interpretation has left no room for probabilistic considerations of theories (theories cannot be true with some frequency). But this was simply necessary if one wants to use statistical experiments to make choices between theories. So for this purpose was invented a different science, stochastics, which was essentially imprecise plausible reasoning. And Jaynes showed that the objective Bayesian approach covers all of stochastics, and allows to improve and correct a lot of things in stochastics.
Similarly, claims have been made that the Bayesian approach to thermodynamics gives new and better results for some non-equilibrium problems. Whatever, the question if the Bayesian or earlier interpretation of thermodynamics is correct is something which one has to consider separately. And one would have to consider the original sources for this to reach an own decision.
I have to admit that I have not checked those mathematical claims of superiority of the Bayesian approach. My preference for the objective Bayesian approach is based on its conceptual simplicity and consistency. And the failure of the other side to present something comparable. Moreover, a childhood memory influenced me here: I had thought quite early that all that mathematical logic is much less useful than claimed, given that all what we know is never completely certain. But what to do with this? To hope for some precise logical rules for vague, uncertain reasoning seemed hopeless.
And when I learned about the objective Bayesian approach, this impressed me a lot - what I thought is hopeless is not hopeless at all, but has a simple and beautiful solution, probability theory. And I think a similar error will be quite common - if there are precise, exact laws, we tend to think that this has to be something objective, something ontological. Instead, the rules which guide our imprecise knowledge have to be somehow imprecise too. But that's not the case. The rules of reasoning are pure mathematics, and that's the most precise thing which we have, and it does not matter at all if the reasoning itself is about certain or uncertain things.
