PBR & Relativity: Wave Function Uniqueness?

[atyy]

Fine. Then I propose the following conjecture (as a generalization of the PBR theorem to multiple observers):
When there are multiple ontic observers $o_1,o_2,...$, then wave function $\psi_{o_k}$ (associated with any observer $o_k$) is ontic and uniquely defined by $o_k$ and other ontic stuff.

In some sense wouldn't a generalized theorem violate the spirit of the original PBR, since the observer dependence seems to still argue for an "epistemic" view of the wave function?

If by "answer" you mean probability of the measurement outcome, then yes.

Considering the different wave functions for different frames in relativity, would you add choice of reference frame to your conjectured generalization of PBR?

 

[Demystifier]

In some sense wouldn't a generalized theorem violate the spirit of the original PBR, since the observer dependence seems to still argue for an "epistemic" view of the wave function?

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

Considering the different wave functions for different frames in relativity, would you add choice of reference frame to your conjectured generalization of PBR?

I wouldn't, but perhaps someone would.

 

[Sunil]

What I'm thinking of is that in Copenhagen, the collapse is typically frame dependent. If there is emergent relativitvistic quantum theory in Bohmian mechanics, then wouldn't it be possible to derive the frame-dependent collapse of Copenhagen from Bohmian mechanics?

No. In Bohmian mechanics one needs a preferred frame in the relativistic context. As in every realist or causal interpretation. This is not in conflict with minimal relativity which talks only about observables, only with fundamental relativity which forbids hidden preferred frames.

All you can do is to use BM for different frames, which will show different and incompatible trajectories, and show that observable probabilities do not depend on this, so that one cannot tell by observation which of the many Bohmian versions is the correct one. And then try again the old positivist trick that once they are not observable they do not exist at all.

 

[atyy]

No. In Bohmian mechanics one needs a preferred frame in the relativistic context. As in every realist or causal interpretation. This is not in conflict with minimal relativity which talks only about observables, only with fundamental relativity which forbids hidden preferred frames.

All you can do is to use BM for different frames, which will show different and incompatible trajectories, and show that observable probabilities do not depend on this, so that one cannot tell by observation which of the many Bohmian versions is the correct one. And then try again the old positivist trick that once they are not observable they do not exist at all.

If I understood correctly, @Demystifier gave the opposite answer in post #27. In emergent relativity, there is a preferred frame due to the underlying Bohmian mechanics. In that preferred frame (invisible to the Copenhagen observer), one can derive the quantum formalism for a Copenhagen observer who happens to use the preferred frame. Because of emergent relativity, the quantum formalism will predict the same probabilities for measurement outcomes regardless of which frame the Copenhagen observer uses.

 

[Demystifier]

If I understood correctly, @Demystifier gave the opposite answer in post #27. In emergent relativity, there is a preferred frame due to the underlying Bohmian mechanics. In that preferred frame (invisible to the Copenhagen observer), one can derive the quantum formalism for a Copenhagen observer who happens to use the preferred frame. Because of emergent relativity, the quantum formalism will predict the same probabilities for measurement outcomes regardless of which frame the Copenhagen observer uses.

I don't see that as opposite. What @Sunil says is fully compatible with my claims. And I also agree with you. The only thing I don't understand is where is the conflict between your and Sunil's claims. It looks to me as if we are all saying the same thing, but from slightly different perspectives.

 

[Demystifier]

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

PBR = Proving Bohm Rationally

 

[atyy]

PBR = Proving Bohm Rationally

I wish there were a counterpart showing that the wave function must be epistemic. Then since the wave function must be real and must be epistemic, there must be two wave functions as in Bohmian mechanics.

 

[Demystifier]

I wish there were a counterpart showing that the wave function must be epistemic. Then since the wave function must be real and must be epistemic, there must be two wave functions as in Bohmian mechanics.

That would be nice, but I'm afraid that Everetians would say that this proves their interpretation too.

 

[atyy]

That would be nice, but I'm afraid that Everetians would say that this proves their interpretation too.

It's not yet a coherent interpretation (though I respect Wallace's work immensely), so that should be ok.

 

[Sunil]

If I understood correctly, @Demystifier gave the opposite answer in post #27. In emergent relativity, there is a preferred frame due to the underlying Bohmian mechanics. In that preferred frame (invisible to the Copenhagen observer), one can derive the quantum formalism for a Copenhagen observer who happens to use the preferred frame. Because of emergent relativity, the quantum formalism will predict the same probabilities for measurement outcomes regardless of which frame the Copenhagen observer uses.

And where do you see a difference?

This is a general property of emergent relativistic symmetry. Once it emerged, observers cannot see the differences between the frames. But once it is only emergent, and not fundamental, there is a preferred frame for the realists.

Maybe my "BM for different frames" was misleading. This should be understood like "the Lorentz ether theory for different frames". One may be a proponent of BM/Lorentz ether or whatever has a preferred frame but one may not know what is the preferred frame in the actual situation. So, one would have to make guesses, what if the preferred frame is CMBR with harmonic time, what if CMBR with comoving proper time, or whatever else could be a reasonable candidate for a preferred frame. These would be, essentially, different physical theories with different hypotheses about what is the preferred frame. For BM, they would predict different trajectories.

I wish there were a counterpart showing that the wave function must be epistemic. Then since the wave function must be real and must be epistemic, there must be two wave functions as in Bohmian mechanics.

No. PBR is simply a total failure, as explained and shown by Caticha's entropic dynamics. Which does not have two wave functions, its wave function is purely epistemic.

 

[PeterDonis]

Caticha's entropic dynamics

Is this what you are referring to?

https://arxiv.org/abs/1908.04693

 

[Morbert]

No. PBR seems to imply this but fails. The problem is that knowledge of the preparation procedure is also part of reality. Last but not least, the measurement devices and the record about the preparation procedure are part of reality. Even more, the mind having that incomplete knowledge is also part of reality too. So, if reality is fixed completely, the incomplete information about the system in that mind is fixed too, thus, the corresponding pure state of the quantum system is fixed too. So, it is psi-ontological by definition of psi-ontology.

This argument would move us away from PBR and towards a more generic debate. See for example section II "Exactly How Quantum States Fail to Exist" in Fuch's paper https://arxiv.org/pdf/1612.07308.pdf

 

[atyy]

And where do you see a difference?

This is a general property of emergent relativistic symmetry. Once it emerged, observers cannot see the differences between the frames. But once it is only emergent, and not fundamental, there is a preferred frame for the realists.

Maybe my "BM for different frames" was misleading. This should be understood like "the Lorentz ether theory for different frames". One may be a proponent of BM/Lorentz ether or whatever has a preferred frame but one may not know what is the preferred frame in the actual situation. So, one would have to make guesses, what if the preferred frame is CMBR with harmonic time, what if CMBR with comoving proper time, or whatever else could be a reasonable candidate for a preferred frame. These would be, essentially, different physical theories with different hypotheses about what is the preferred frame. For BM, they would predict different trajectories.

Thanks, I understood you the second time round.

 

[Sunil]

Is this what you are referring to?

https://arxiv.org/abs/1908.04693

Yes.

 

[atyy]

No. PBR is simply a total failure, as explained and shown by Caticha's entropic dynamics. Which does not have two wave functions, its wave function is purely epistemic.

In Caticha's Entropic Dynamics, it seems the observer retains a special status? Is it similar in spirit to the derivations of quantum mechanics given by Hardy or by Chiribella and colleagues?

https://arxiv.org/abs/quant-ph/0101012
Quantum Theory From Five Reasonable Axioms
Lucien Hardy

https://arxiv.org/abs/1011.6451
Informational derivation of Quantum Theory
G. Chiribella, G. M. D'Ariano, P. Perinotti

It's rather amazing to me that Bohmian mechanics is supposed to be some limit of Entropic Dynamics.

 

[Demystifier]

PBR is simply a total failure, as explained and shown by Caticha's entropic dynamics.

I think it's an overstatement. PBR rules out a large class of theories that naively someone might consider reasonable. But on the other hand, it's good to know that there are also other definitions of "epistemic" theories that are not ruled out by PBR.

 

[Sunil]

I think it's an overstatement. PBR rules out a large class of theories that naively someone might consider reasonable. But on the other hand, it's good to know that there are also other definitions of "epistemic" theories that are not ruled out by PBR.

Sorry, no, this is what is done by Bell's theorem or by Kochen-Specker. They rule out certain interesting classes of theories, namely theories with Einstein causality and theories without contextuality. These would be features many people would like to have, so even if we have counterexamples with BM and other realist theories, the theorems remain useful.

But Caticha's entropic dynamics is not some somehow extravagant "other definition" of a psi-epistemic theory, it is psi-epistemic in the most natural, straightforward and beautiful way, there is nothing to object for a proponent of psi being epistemic. Nobody would try to construct something "more epistemic" or "epistemic in a more natural meaning" or so. The problem with PBR is simply a misguided definition of psi-ontology, and this makes the theorems (there are already many variants) worthless.

(Thinking about the difference between objective and subjective Bayesian - no. An objective Bayesian interpretation is by definition psi-ontological - the incomplete information is objective, means, part of reality. And if we have an objective Bayesian interpretation, we can also reinterpret this as subjective, so PBR cannot be used to forbid subjective Bayesian too.)

 

[Demystifier]

But Caticha's entropic dynamics is not some somehow extravagant "other definition" of a psi-epistemic theory, it is psi-epistemic in the most natural, straightforward and beautiful way, there is nothing to object for a proponent of psi being epistemic.

OK, some basic questions on ED. Is Schrödinger equation postulated or derived from something more fundamental? Is there a conceptual difference between pure and mixed states?

 

[Sunil]

In Caticha's Entropic Dynamics, it seems the observer retains a special status? Is it similar in spirit to the derivations of quantum mechanics given by Hardy or by Chiribella and colleagues?

No, Hardy as well as Chiribella et al. use a non-realistic approach, that means, the axioms are only about preparations/observables without any reference to something really existing inside the quantum system.

Instead, Caticha assumes that the real state of the quantum system is defined by some configuration, and presupposes a configuration space.

The observer plays a less important role than one may think, because behind this is the objective Bayesian interpretation. The question of the objective Bayesian interpretation is what is the rational choice given a certain amount of information. Subjective Bayesians care only about the internal consistency of their choices of probability. So, if there is no information about a dice, the objective Bayesian interpretation prescribes 1/6 for all, while the subjective prescribes nothing.

It's rather amazing to me that Bohmian mechanics is supposed to be some limit of Entropic Dynamics.

A nice result, but not that surprising, because all the realist interpretations use the same probability flow on the configuration space. So the average velocity in the stochastic variants is the same Bohmian velocity.

That's why I tend to think that what really matters is the relation to fundamental relativity. You cannot give the probability flow as defined by the Schrödinger equation for $|\psi|^2$ a physical interpretation without throwing away Einstein causality and returning to a preferred frame. This is the most important splitting line between different interpretations. If the Bohmian velocity is only an average or a deterministic velocity already does not change much - Einstein causality forbids this.

 

[Sunil]

OK, some basic questions on ED. Is Schrödinger equation postulated or derived from something more fundamental? Is there a conceptual difference between pure and mixed states?

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.

 

[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.