I PBR & Relativity: Wave Function Uniqueness?

[Demystifier]

The Schrödinger equation is derived. Essentially in a similar way as in Nelsonian stochastics, except that what is used is the scheme of entropic inference developed by the objective Bayesians.

This gives an equation for probability density $\rho(q,t)$ and the phase $\Phi(q,t)$. Then, one can see that $\psi(q,t) = \sqrt{\rho}\exp(\frac{i}{\hbar}\phi)$ fulfills the Schrödinger equation.

I have not seen a place where mixed states are considered, but I think that they are different follows from standard QM mathematics.

OK, now I read arXiv:1908.04693 and I think I understand the main ideas of ED. I think I understand its advantages and disadvantages.

In my opinion, the most problematic part is the drift potential constraint, Eq. (5). It is introduced in an ad hoc manner, just to reproduce quantum mechanics. There is no any other deeper argument for why should this constraint be true. Moreover, this constraint involves a quantity called drift potential, that later is associated with the phase of the wave function. But from the point of view of ED itself, where the wave function should be derived from something more fundamental, it is not clear where does the drift potential come from. It seems to be a fundamental primitive quantity. In fact, even though ED seems to be claiming that the drift potential is just epistemic, I don't see how can that be true. It seems to me that drift potential must be more than just epistemic, in the same sense in which wave function is more than just epistemic in Bohmian mechanics.

To further justify my claims, it's useful to consider an analogy with classical statistical mechanics. There, in a canonical ensemble, one starts from the Hamiltonian constraint. But we know where does the Hamiltonian constraint come from - it comes from the underlying classical deterministic mechanics where the Hamiltonian is conserved. By analogy one would expect something similar for ED regarding the drift potential constraint, but ED offers nothing of this sort. Moreover, the Hamiltonian itself in classical mechanics is something more than just epistemic (in the same sense in which wave function is more than just epistemic in Bohmian mechanics). Just because the Hamiltonian in a canonical ensemble defines an epistemic probability distribution does not imply that the Hamiltonian itself is just epistemic. It is more than that. By analogy, the drift potential seems to be more than just epistemic in ED.

So to conclude, in my opinion, ED is not a convincing example of the idea that wave function can be purely epistemic. Nevertheless, ED is an interesting reformulation of quantum mechanics that eventually may lead to something deeper.

 

[Sunil]

OK, now I read arXiv:1908.04693 and I think I understand the main ideas of ED. I think I understand its advantages and disadvantages.
In my opinion, the most problematic part is the drift potential constraint, Eq. (5). It is introduced in an ad hoc manner, just to reproduce quantum mechanics. There is no any other deeper argument for why should this constraint be true.
...
So to conclude, in my opinion, ED is not a convincing example of the idea that wave function can be purely epistemic.

If this is the result of reading arXiv:1908.04693, I think it would be better to use the original paper
Caticha, A. (2011). Entropic Dynamics, Time and Quantum Theory, J. Phys. A 44 , 225303, arxiv:1005.2357.

We have the configuration space of the system, $x\in \mathcal{X}$, together with other unspecified variables $y\in\mathcal{Y}$. Once we have incomplete knowledge, this knowledge defines a probability distribution $\rho(x,y)dx dy$. This defines, for each x, a probability $\rho(x) = \int_\mathcal{Y}\rho(x,y) dy$ and an entropy $S(x) = -\int_\mathcal{Y}\rho(y|x) \ln \rho(y|x)dy$. It is this entropy which is used to define the phase and then the wave function
$$ \psi = \sqrt{\rho}e^{i\phi} \qquad \phi = S - \ln \sqrt{\rho}$$
So, the wave function is defined by the probability distribution $\rho(x,y)dx dy$ and therefore as epistemic as imaginable.
Given $\rho$ and $S$, the continuity equation is the straightforward combination of diffusion and increasing entropy. The only problem is to derive the equation for $S$.

Caticha obviously tries a lot of different things to derive it. If you find what he has tried in arXiv:1908.04693 not satisfactory, ok. It is clear that something more has to be postulated, and the equation to be derived is sufficiently simple - Hamilton-Jacobi with Bohm's quantum potential, so that it is not easy to be much simpler, and if it is not much simpler the derivation is not worth much.

But that the wave function is epistemic follows essentially from the definition.

 

[Demystifier]

But that the wave function is epistemic follows essentially from the definition.

OK, this now looks much more epistemic. But as he discusses in Sec. 9.3, the problem is the Wallstrom objection: If the fundamental (though epistemic) quantity is $\phi$ rather than $e^{i\phi}$, why should $e^{i\phi}$ be single valued? The answers that he proposes are not very convincing to me. It's my feeling that whatever reason you choose for $e^{i\phi}$ being single valued, it indirectly and tacitly assumes that $e^{i\phi}$ is somehow more than just epistemic.

 

[Sunil]

But as he discusses in Sec. 9.3, the problem is the Wallstrom objection: If the fundamental (though epistemic) quantity is $\phi$ rather than $e^{i\phi}$, why should $e^{i\phi}$ be single valued? The answers that he proposes are not very convincing to me. It's my feeling that whatever reason you choose for $e^{i\phi}$ being single valued, it indirectly and tacitly assumes that $e^{i\phi}$ is somehow more than just epistemic.

Yes, the Wallstrom objection is rather serious in this case.

But it shows only that single-valued theory is different from QM. If this difference becomes observable or not is another question. There should not be zeros of the wave function. But only in the configuration space representation. If there are no zeros, then you can take the logarithm.

Your problem I don't understand. The fundamental thing is $\phi$. Then, $e^{i\phi}$ is a well-defined function, which is automatically single-valued. The problem may be empirical, if there are states which can be created in experiments so that the wave functions necessarily have stable zeros. Schmelzer argues in https://arxiv.org/abs/1905.03075 that this is quite plausible in a particle ontology but not in a field ontology.

 

[Demystifier]

Your problem I don't understand. The fundamental thing is $\phi$. Then, $e^{i\phi}$ is a well-defined function, which is automatically single-valued.

You are right, at this point my explanation of the Wallstrom objection was incorrect. But as you said, the Wallstrom objection is still serious.

 

[Demystifier]

You are right, at this point my explanation of the Wallstrom objection was incorrect.

After some thought, let me now present an improved version of my argument.

Consider first a classical analogon of the phase function $\phi$. It is the classical Hamilton-Jacobi function $\phi_{\rm HJ}$. A particularly interesting case is circular motion of a particle (e.g. due to a central potential), in which case
$$\phi_{\rm HJ}(\varphi)=L\varphi$$
where $L$ is the angular momentum and $\varphi$ is the angular variable. The classical angular momentum is not quantized, which means that $L$ can take any value. The consequence is that $\phi_{\rm HJ}$ is not single valued, in the sense that
$$\phi_{\rm HJ}(\varphi+2n\pi)\neq\phi_{\rm HJ}(\varphi)$$
for integer $n$. But that's not a problem, because $\phi_{\rm HJ}$ is not an ontic quantity in classical mechanics. It is just an auxiliary tool to compute the velocity, so it's OK if it is not single valued, as long as the computed velocity is single valued.

But in the quantum case, the phase function $\phi$ is single valued. One physical consequence of this is quantization of angular momentum. The requirement that $\phi$ must be single valued indicates that $\phi$ is ontic, unlike $\phi_{\rm HJ}$.

But if $\phi$ is ontic, how can it be compatible with the fact that it is related to entropy in Caticha theory? 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.

@Sunil your possible comments would be very appreciated.

 

[martinbn]

.
.
The consequence is that $\phi_{\rm HJ}$ is not single valued, in the sense that
$$\phi_{\rm HJ}(\varphi+2n\pi)\neq\phi_{\rm HJ}(\varphi)$$
for integer $n$.
.
.

This is a very strange choice of words. For the function $f(x)=x^3$ it is also true that $f(x+2n\pi)\not=f(x)$, but no one would say that it is not a single valued function. It is not $2\pi$ periodic.

 

[Demystifier]

This is a very strange choice of words. For the function $f(x)=x^3$ it is also true that $f(x+2n\pi)\not=f(x)$, but no one would say that it is not a single valued function. It is not $2\pi$ periodic.

I would say that it depends on the domain for the variable $x$. If the domain is $\mathbb{R}$, then you are right. But if the domain is $\mathbb{R}\;{\rm mod}\; 2\pi$, then I think it makes sense to say that the function is not single valued.

 

[martinbn]

I would say that it depends on the domain for the variable $x$. If the domain is $\mathbb{R}$, then you are right. But if the domain is $\mathbb{R}\;{\rm mod}\; 2\pi$, then I think it makes sense to say that the function is not single valued.

No, then the function is not well defined. It is not a function with that domain.

 

[Demystifier]

No, then the function is not well defined. It is not a function with that domain.

A mathematical term, I think, would be that the function has many branches. But in physics it's common to call it a multiply defined function.

 

[martinbn]

A mathematical term, I think, would be that the function has many branches. But in physics it's common to call it a multiply defined function.

Can you give an example? Where is it called that?

 

[Demystifier]

Can you give an example? Where is it called that?

https://en.wikipedia.org/wiki/Multivalued_function

 

[martinbn]

https://en.wikipedia.org/wiki/Multivalued_function

I am familiar with this, but it is not what you are trying to say. If you have a multivalued function $f$ defined on $\mathbb R$ modulo $2\pi$, then the value $f(x)$ will be a set of numbers (to be specific) but it still makes no sense to say that $f(x+2n\pi)\not=f(x)$. The values $f(x+2n\pi)$ and $f(x)$ may be sets of numbers, not just a single number, but they are the same set, because $x$ and $x+2n\pi$ are the same element of the domain of the function.

 

[Demystifier]

I am familiar with this, but it is not what you are trying to say. If you have a multivalued function $f$ defined on $\mathbb R$ modulo $2\pi$, then the value $f(x)$ will be a set of numbers (to be specific) but it still makes no sense to say that $f(x+2n\pi)\not=f(x)$. The values $f(x+2n\pi)$ and $f(x)$ may be sets of numbers, not just a single number, but they are the same set, because $x$ and $x+2n\pi$ are the same element of the domain of the function.

Well, the purpose of mathematicians is to put the imprecise statements by physicists into a precise form.

 

[Sunil]

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.

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.

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:

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.

 

[atyy]

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.

I intend to read Caticha's work more when I have time, but I confess I dislike objective Bayesian theory (not a fan of Jaynes). One reason I don't like it is I don't see why the Shannon entropy is unique, as there are other Renyi entropies.

 

[Demystifier]

@Sunil and @martinbn sorry for being (me) mathematically sloppy and unprecise! When I referred to $\phi$ as a "phase" function, I was implicitly assuming that it is physically dimensionless and obeys the equivalence relation
$$\phi\sim\phi+2\pi$$
It would take some effort to put all this into a precise form, but Sunil of course understood what I meant and responded to ideas that were essential to my argument. For martinbn I will quote Feynman: "Don't listen to what I say, listen to what I mean!"

 

[Demystifier]

If the ultimate goal of PBR theorem is to prove that Bohmian mechanics is the only interpretation that makes sense, then no.

Today appeared a paper https://arxiv.org/abs/2105.06445 pointing out that PBR theorem assumes that $\psi$ is not nomological, while in Bohmian interpretation $\psi$ is nomological. (Nomological means that it defines a law of motion.) See the paragraph around Eq. (4) and the two paragraphs after that. Hence it seems that PBR theorem is in fact irrelevant to Bohmian mechanics.

 

[Fra]

I intend to read Caticha's work more when I have time, but I confess I dislike objective Bayesian theory (not a fan of Jaynes). One reason I don't like it is I don't see why the Shannon entropy is unique, as there are other Renyi entropies.

I read a few of Ariel Catichas papers years ago, I if you aren't coming from there, they are very inspirational papers many of them! His mantra is "physics from inference", and it is right in line with my own thinking, so I can recommend it as well!

But that said, after some progress in my own work, I object to his approaches in several ways. First of all the objective path does not appeal to me either. Let's just say I have totally conviced myself that it's the wrong path; mainly because there are too many "choices", that are not natural, and that gives a bias. It's pretty obvious that you can "choose" the stochastic flow by choosing the measure. But the point is, that it's nature that must choose this measure. All that is similar to fine tuning in disguise. I also have a serious problem with his and Cox ideas to reconstruct probability theory, one big issue is that they introduce uncountable numbers too lightly. After lots of thinking I also convinced myself that this also isnt' the right path. Uncountable sets set loose, just do not belong in a fundamental reconstruction, it is more likely to appear as an approximation in the large scale. The proble is that once the limit is take, you loose track of the threads, so one can not start out with loose threads.

I like a lot of Ariels ambition, but a bit into the process I prefers a different path. The objective inference path violates also the intrinsic perspectve that I consider central. The intrisic or subjective path, is necessarly always incomplete, so a perspective of self organised evolution (not just entropic flow) seems required.

/Fredrik