What has changed since the Copenhagen interpretation?

[DarMM]

How does this require more than the well-known equivalence between the Heisenberg and Schrodinger pictures, given the algebra stuff is the Heisenberg observables?

Firstly I will say note they are only that equivalent in QM, in QFT the Schrodinger picture can require additional renormalizations, so they aren't unitarily equivalent as the Schrodinger picture will have a slightly different Hamiltonian due to the new counter terms.

Deutsch's algebra perspective moves beyond the equivalence of the Schrodinger and Heisenberg picture and has nothing to do with it really. That's just moving time evolution between the state and observables. He is staying that the quantum state could ultimately be eliminated from the theory as it has no ontic existence. When a measuring device and a particle interact, that is purely an interaction between the device's algebra-stuff and the particle's algebra-stuff, the state doesn't exist. It's only useful to convey constraints on measuring the algebra (i.e. extracting a real number from an algebra element) when you don't know the device algebra details, i.e. the state is epistemic.

This does make Deutsch's view local as the evolution of the algebra is provably locally, most generally in algebraic QFT where regional algebra are even called "local algebras". However it is typically viewed that the state is nonlocal, nobody questions the locality of the observable algebra. Another way of putting it is that the dynamics are thought to be local, but the states are not. Deustch avoids this by dropping the ontic status of the quantum state and thus all his ontic elements are local.

However the state not being ontic, but just an epistemic constraint, and the algebra being is very different from MWI or indeed any interpretation of QM, it's not just the Heisenberg picture. It's also a very undeveloped view.

If you retain the state as ontic, then Deutsch's proof is just a (interesting) Heisenberg picture demonstration of no-signalling, not locality.

You're saying that because when it's defined on an algebra without spacetime it's alocal by definition and when defined on an algebra with spacetime $\mathcal{M}^4$ as a primitive element that it contains global properties and becomes non-local.

The algebra doesn't have the spacetime as an element, it's a sheaf over the spacetime (and always a sheaf over a spacetime, can't be without it). The state is then nonlocal as it doesn't factorise across the algebras of regions.

This isn't the definition of *local I'm using. A theory is local if it possibly can be reformulated in terms of separate regions, so that actions on separated regions don't effect each other.

Before we continue and I think this might be core to the whole thing, I would add:
"can be reformulated in terms of ontic elements in separate regions, so that actions on separated regions don't effect each other".

If the restrictions to regions that don't effect each other are necessarily epistemic, then any locality demonstrations would only be non-signalling demonstrations, agreed?

 

[akvadrako]

Deutsch's algebra perspective moves beyond the equivalence of the Schrodinger and Heisenberg picture and has nothing to do with it really. That's just moving time evolution between the state and observables. He is staying that the quantum state could ultimately be eliminated from the theory as it has no ontic existence.

It doesn't seem like that's quite the argument...

It is true that such states – including |0> itself – like the laws of motion, the axioms of quantum mechanics, and the global topology of spacetime, are all elements of reality that do not have locations, and are ‘non-local’ in that sense – a sense in which all theories in physics are ‘non-local’. But they also have in common that they cannot be altered. Being universal constants, they are ‘independent of what is done to S1', and therefore their existence does not violate Einstein’s criterion of locality. To avoid confusion, they should be called ‘global’, not ‘non-local’.​

So if all the time evolution is moved to the observables and the state never changes, then the state doesn't provide any way for distant systems to effect each other. Would you call something like the constant global state, the laws of physics or other constants epistemic?

If you retain the state as ontic, then Deutsch's proof is just a (interesting) Heisenberg picture demonstration of no-signalling, not locality.

It's more than just no-signaling, it's Einstein's criterion of locality. Even taking the state as ontic, it's saying that all the ontic elements in a region are unaffected by events in distant regions. The local algebra can't be changed (by proof) and the state can't be changed (by definition).

The algebra doesn't have the spacetime as an element, it's a sheaf over the spacetime (and always a sheaf over a spacetime, can't be without it). The state is then nonlocal as it doesn't factorise across the algebras of regions.

That does sound more precise that what I said and basically what I was thinking. And I see how it looks non-local, but that doesn't mean the situation it's representing is non-local. I know that's not a very clear way to phrase it, but I'm saying it could be a consequence of, for example, ignoring some information via the factorization. In this case, isn't it considering the overlap of all the worlds at each spacetime point? That is a very strange thing to do. Take worlds and regions $W_1\{r_1=\uparrow,r_2=\uparrow\}, W_2\{r_1=\downarrow,r_2=\downarrow\}$ and factor by region, then throw away the world labels. It take a perfectly sensible situation and gives you nonsense: $r_1\{\uparrow,\downarrow\} \times r_2\{\uparrow,\downarrow\}$.

Before we continue and I think this might be core to the whole thing, I would add:
"can be reformulated in terms of ontic elements in separate regions, so that actions on separated regions don't effect each other".

If the restrictions to regions that don't effect each other are necessarily epistemic, then any locality demonstrations would only be non-signalling demonstrations, agreed?

I'm having trouble parsing that question, so let me try by stating my own understanding of non-signaling and locality.

• non-signaling — observer A cannot communicate any information to distant observer B. This also implies system A can't do anything which will have measurable consequences on system B.
• locality — also called separability. Given a composite system A+B, the most complete description possible of the composite system can be written in terms of the most complete description possible of A & B separately. Also, as long as A & B don't interact, events at A can have no effect on the description of B.

This probably also gives a clue about my understanding of the terms ontic and epistemic. Ontic is the most complete description possible. It doesn't mean it's a fundamental description, but within the framework you're using, there is nothing you can learn which will increase your knowledge of the system. Thus it will give you maximum predictive power. Epistemic would then be about what partial knowledge, by say throwing aware a dimension to simplify a problem.

 

[DarMM]

It doesn't seem like that's quite the argument...

The state is a constraint in Deutsch's theory, a relation between the objects, it's an element of the theory in the same sense that the action is in classical mechanics, a constraint among the ontic objects. This is what he seems to be arguing to me when he says only the algebra exists, the state merely is a "law of physics" for them, but it's not an ontic element of the theory, i.e. in Classical Mechanics you could move between the Hamiltonian and the Lagrangian as ways of expressing the constraints on the ontic elements (the particles), but the ontic elements are just the particles.

At least that is how his argument seems to me.

So if all the time evolution is moved to the observables and the state never changes, then the state doesn't provide any way for distant systems to effect each other. Would you call something like the constant global state, the laws of physics or other constants epistemic?

I don't think the state is the same as "a law of physics" in Deutsch's paper, even though Deutsch claims that what it is. Basically I'm just not taking the claim that it's exactly the same as a law of physics at face value because Deutsch says so. If it were I would be unjustified in my claim, but again that's the issue here.

In Deutsch's view the state is a "law of physics" in a sense, as it expresses a constraint among the actual ontic elements, the algebra constituents. However what is the constraint? The average real number you will extract from an algebra element when you measure it (Deutsch himself says this). That is an epistemic concept to me as it attaches to repeated experimental runs. This is exactly the same point Timpson makes here: https://arxiv.org/abs/quant-ph/0312155

I agree with the vindication paper overall and he doesn't seem to deny Timpson on this point alone, i.e. the meaning of the state is the expected real number you'd extract from an algebra element.

Now maybe it is ontic, maybe you can show that interaction among algebra elements leads to Boolean subalgebra worlds splitting into ones where the element is effectively a real number and averaged across worlds that value agrees with the one you'd get from the state, so the state just encodes something you can prove about the evolution of the algebra. However that would need to be shown.

So I agree that it is a global constant, but it seems to be an epistemic one.

Deutsch's proof can be taken as proof of locality for an interpretation where the world is made of C*-algebra elements (that's pretty cool, it makes objects way more complex than the usual picture) and there is a primitive global notion of the average experimental value associated with each element. The latter seems to introduce measurement as a primitive (like Copenhagen).

To remove measurement as a primitive you'd need to show that average value represents a shorthand for interactions among a microscopic algebra and a large scale algebra.

I'm having trouble parsing that question...
This probably also gives a clue about my understanding of the terms ontic and epistemic. Ontic is the most complete description possible

Probably my bad phrasing. Let me be clearer, in your view, in a subregion of spacetime $\mathcal{A}$, what mathematical objects are the ontic elements that don't interact with similar elements at another spacetime region $B$? The local density matrices? The algebra elements?

 

[DarMM]

A way of seeing this is that even in a modestly sized region of spacetime a typical algebra element will be millions of dimensions in size. The state then associates a single real number with each element. I would expect that to be shown from interactions between the system algebra and the device algebra, having an ontic global "average value extracted from algebra element" is a bit strange to me.

If we have a region $\mathcal{O}$, then the subalgebra of a system $\mathcal{E} \subset \mathcal{A}(\mathcal{O})$ and that of a device $\mathcal{F} \subset \mathcal{A}(\mathcal{O})$ apparently interact in some way that embed $G \in \mathcal{E}$ not in its true algebraic form but as a collection of real numbers in $\mathcal{F}$, each real number being a "world" and $\mathcal{F}$ gains the effective form of a boolean algebra (device decoheres). Then the average value attached, weighted by the manner $\mathcal{F}$ turns Boolean is $\rho(G)$.

This needs to be shown, also this is obviously not just trivially the same thing as Everettian Many Worlds where the state is real.

 

[akvadrako]

So I agree that it is a global constant, but it seems to be an epistemic one.

I can see now how the state is epistemic. It seems like indexical information, akin to the particle location in dBB. Timpson's mentions this:

Now the qi(t) capture the effects of our sequence of unitary operations for all initial states. Thus their time evolution can be said to depict the histories of the entire set of possible worlds; whilst the world from amongst these that is realized is determined by which initial state is chosen. However, when we move to the ontological view, the very same structure (the sequence of time evolving qi(t)) only represents a single world, as the choice of initial state is a fixed part of the formalism. What seems like it can represent a range of possible worlds, we are to suppose, can only represent a single one; and conversely, the structure being used to describe a single world in the ontological Deutsch-Hayden picture is one we know in fact to be adequate to describe a whole set of possible worlds in quantum mechanics. Thus the Deutsch-Hayden picture, taken ontologically, would seem to be extremely, perhaps implausibly, extravagant in the structure it uses to depict a single world.​

This would indeed imply the state is arbitrarily chosen. If we are talking about the universal wavefunction, then the basis is completely arbitrary, so an initial state like $|0\rangle$ would be quite unnatural. Though when talking about specific situations like our pocket universe, then it seems like the state should be at least partially specified; at least all the dimensions that have affected measurement results.

Well, this changes my view. Now I understand that what their analysis is showing is that a fixed state leads to locality. And the fixed state corresponds to one world-line. I will have to re-read some of the Deutsch-Hayden papers with this in mind. Perhaps what they're saying isn't that the state is fixed, just constant.

I would expect that to be shown from interactions between the system algebra and the device algebra, having an ontic global "average value extracted from algebra element" is a bit strange to me.

Does this average value effect the evolution of the system? It can't change the algebra or the state. It looks like it should be given by the Born rule and be a measure of world density.

Probably my bad phrasing. Let me be clearer, in your view, in a subregion of spacetime $\mathcal{A}$, what mathematical objects are the ontic elements that don't interact with similar elements at another spacetime region $B$? The local density matrices? The algebra elements?

My answer will be basically the same as before, but I'll try again. In my view, the local information must include both the algebra and the state vector, though either can be considered constant. If we are only considering one copy (one world) of $\mathcal{A}$, then all the local qubits must be entangled, $\mathcal{A}$ is pure, containing all of $\mathcal{A}$'s information about $B$. If we ignore some local information then $\mathcal{A}$'s view of $B$ would be mixed, but that's not an ontic operation.

However, due to self-locating uncertainty, we should consider all worlds containing $\mathcal{A}$. I suppose that makes it a mixed state, but then we aren't really anymore talking about just that one universe with $\mathcal{A}$ and $B$. And if we consider the global wavefunction, I don't know if it begins to look like a pure state again.

There is something else important about my view that I'm not sure I need to clarify. The quantum description exists on the boundary between systems and is symmetric, $\mathcal{A} \rightarrow \psi \leftarrow B$. The boundary doesn't contain complete information about the other system, so they will evolve in different but overlapping worlds. So the description of region $\mathcal{A}$ from $\mathcal{C}$'s viewpoint is different than from $B$'s. $B$'s model of $\mathcal{A}$ is actually stored in $B$'s region (or the B/A boundary).

The ontic description of $\mathcal{A}$ would then be the union of all other viewpoints. The important part is normally we don't consider all viewpoints from all worlds (though I suspect that's $\mathcal{A}$'s experience) but just $\mathcal{C}$'s view of them, which is where that important global state comes from.

 

[DarMM]

Thanks @akvadrako , I'll just need some time to think carefully, I don't want to just blurt out a response. I think I appreciate the locality now in terms of the state being regarded as a world index, in fact I think this may link back to our discussion about the backward state in the TSV formalism. Funny this was all in Timpson's paper which I read, but somehow didn't click until I read your post.

Just need to digest some facts.

 

[akvadrako]

I think I appreciate the locality now in terms of the state being regarded as a world index, in fact I think this may link back to our discussion about the backward state in the TSV formalism. Funny this was all in Timpson's paper which I read, but somehow didn't click until I read your post.

That just goes to show if I say enough things I'm bound to stumble upon something useful. :) The connection with TSV is suggestive — I wonder how similar the world-lines of the two formalisms are and if the results within each apply to both.

It's also interesting that this paragraph from Timpson seems to be the only time this claim, that a fixed state leads to a single world in the DH picture, is mentioned. At least I can't find it.

 

[DarMM]

In my view, the local information must include both the algebra and the state vector, though either can be considered constant.

Okay so taking this slow so I don't mess things up.

Assume a typical Bell set-up, or whatever you want.

You say, the local information at the region $\mathcal{A}$ is all the algebra elements contained within $\mathcal{A}$ and the state vector. The state vector is the part confusing me. Let's say you don't restrict to just one world, you're looking at everything at $\mathcal{A}$, i.e. let's take the "god's eye view" of $\mathcal{A}$.

What is this state vector? What explicit mathematical object is the state vector in $\mathcal{A}$? The restriction of the global state vector to the region $\mathcal{A}$, which will be a density matrix, or something else? I considering this from the most general point of view (algebraic QFT) and I'm having a hard time figuring out what aspect of the state can be considered to live purely in $\mathcal{A}$ other than a density matrix.

 

[akvadrako]

You say, the local information at the region $\mathcal{A}$ is all the algebra elements contained within $\mathcal{A}$ and the state vector. The state vector is the part confusing me. Let's say you don't restrict to just one world, you're looking at everything at $\mathcal{A}$, i.e. let's take the "god's eye view" of $\mathcal{A}$.

What is this state vector? What explicit mathematical object is the state vector in $\mathcal{A}$? The restriction of the global state vector to the region $\mathcal{A}$, which will be a density matrix, or something else? I considering this from the most general point of view (algebraic QFT) and I'm having a hard time figuring out what aspect of the state can be considered to live purely in $\mathcal{A}$ other than a density matrix.

I'm just trying to picture it and reason through it: to even define $\mathcal{A}$ you need to assume some shared past, since there is no way to say two regions from different worlds are the same region, unless they are connected. So we start by considering a single world or even the entire universe, which surely can be described by a pure state.

Yet assuming Timpson is right about a fixed state vector corresponding to one world-line, if we want to consider all the branches from that point on, we should only be considering part of the state vector, that corresponding to our region (especially time-wise), and leave the rest undefined. Something like $| 0, 0, ?, ?, ...\rangle$.

Given an initial state, $| 0, a, b, ...\rangle$, we can pick any branch, say $| 0, +, 1, ...\rangle$ and use the current algebra+state to compute the probability of it being observed. I suppose what we can do by focusing on a region is to restrict our algebra. Say $\mathcal{A}$ if fully defined by the value of $b$ and we trace over $|0,a|$. It seems that would result in a mixed state.

 

[A. Neumaier]

a fixed state vector corresponding to one world-line

Whose world line?

we should only be considering part of the state vector, that corresponding to our region (especially time-wise),

The region would probably be a 4-dimensional tube? Again, who decides which tube? That formed by our solar system?

 

[akvadrako]

Whose world line?

The region would probably be a 4-dimensional tube? Again, who decides which tube? That formed by our solar system?

I was assuming we consider the world-line of $\mathcal{A}$ and the region would include at most it's past light-cone.

 

[DarMM]

I'm just trying to picture it and reason through it: to even define $\mathcal{A}$ you need to assume some shared past, since there is no way to say two regions from different worlds are the same region, unless they are connected. So we start by considering a single world or even the entire universe, which surely can be described by a pure state.

Sorry just to be clear, $\mathcal{A}$ is an arbitrary region of spacetime, it may not include the complete lightcone of points within it. Although the standard example would be some region of space with its future and past up to a finite proper time included. I would be wondering what lives in this regions (to be observer independent as such). So $\mathcal{A}$ is just a region of spacetime, it can be defined without reference to QM. If this makes no difference to the rest of your arguments in your opinion, I'll carry on with them.

Currently I can only understand that density matrices would be defined as states on these regions (due to algebraic QFT).

 

[A. Neumaier]

I was assuming we consider the world-line of $\mathcal{A}$ and the region would include at most it's past light-cone.

How is the worldline of a region $\cal A$ defined? The union of all past and future light cones of points in $\cal A$?

 

[akvadrako]

Sorry just to be clear, $\mathcal{A}$ is an arbitrary region of spacetime, it may not include the complete lightcone of points within it.

That's clear. And I can't think of an argument against using mixed states and there has been some work done to extend the DH view to use them in Hewitt-Horsman and Vedral (2007), section 6. That only continues to use the constant $|0\rangle$ state though.

How is the worldline of a region $\cal A$ defined? The union of all past and future light cones of points in $\cal A$?

All the past light cones, though I wasn't considering the future ones. The DH paper doesn't talk about spacetime much at all, just that the information needed to describe a qubit comes totally from its past interactions.

 

[akvadrako]

I think this may link back to our discussion about the backward state in the TSV formalism. Funny this was all in Timpson's paper which I read, but somehow didn't click until I read your post.

Just need to digest some facts.

Sorry for bringing in even more literature on the subject, but I came across a recent paper which is quite relevant and you might find interesting. It claims to provide an alternative proof of the Deutsch-Hayden result, "that all no-signalling operational theories with a reversible dynamics, including finite-dimensional unitary quantum mechanics, are local-realistic". It uses some atypical terminology and contains a fair bit of philosophy, but it's from a respectable name in quantum information, Gilles Brassard.

 

[akvadrako]

A quick summary of what it shows:

1. First, he defines general requirements for local-realistic (ontic) and no-signaling operational (phenomenal) theories.
2. Then, he shows that unitary QM meets the requirements for a no-signaling operational theory. States are density matrixes and operations are unitary transforms up to a phase factor. No underlying reality is assumed.
3. Finally he shows it's possible to construct a LRT for every finite-dimensional NSOT if all operations are reversible.

It just briefly mentions that the local ontic states are based on Hilbert spaces and matrixes, but the full description is delegated to an unpublished paper.

 

[stevendaryl]

A quick summary of what it shows:

1. First, he defines general requirements for local-realistic (ontic) and no-signaling operational (phenomenal) theories.
2. Then, he shows that unitary QM meets the requirements for a no-signaling operational theory. States are density matrixes and operations are unitary transforms up to a phase factor. No underlying reality is assumed.
3. Finally he shows it's possible to construct a LRT for every finite-dimensional NSOT if all operations are reversible.

It just briefly mentions that the local ontic states are based on Hilbert spaces and matrixes, but the full description is delegated to an unpublished paper.

I haven't read the paper, yet. But the conclusion seems a little paradoxical.

• Bell proved that QM cannot be explained by a local realistic theory.
• QM does not have FTL signalling.
• The "collapse" interpretation of QM measurements is not reversible, but you can imagine that something reversible might replace the collapse.

So what gives? Does that mean that a time-reversible version of QM is not possible? Or does it mean that a time-reversible version of QM would allow FTL signalling? Or does Bell's theorem implicitly depend on irreversibility?

 

[akvadrako]

So what gives? Does that mean that a time-reversible version of QM is not possible? Or does it mean that a time-reversible version of QM would allow FTL signalling? Or does Bell's theorem implicitly depend on irreversibility?

The authors are not sure if time-reversibility is needed, stating:

We need to require that all operations be reversible: the set of operations must be a group. It might be possible to achieve the same goal without a group structure, which is the subject of current research, but this would most likely come at the cost of significant loss in mathematical elegance.
​

and they have this to say about Bell's theorem:

The explanation for this conundrum is that there are more general ways for a world to be local-realistic than having to be ruled by local hidden variables, which is the only form of local realism considered by Bell in his paper [2]. We expound on the local construction of “nonlocal” boxes in a companion paper [5].
​

I have no idea how hidden variables differ from realism.

 

[zonde]

and they have this to say about Bell's theorem:

The explanation for this conundrum is that there are more general ways for a world to be local-realistic than having to be ruled by local hidden variables, which is the only form of local realism considered by Bell in his paper [2]. We expound on the local construction of “nonlocal” boxes in a companion paper [5].

There are couple of things that can be said about that quote.

Of course world is not ruled by variables. Variables are components of our models and not the world itself. Bell in his first paper about Bell inequalities is not considering experimental findings but rather Quantum theory and possible extensions of QM i.e. he speaks about hypothetical models of reality without considering how good they are in describing reality. Taking into account that we talk about models rather than reality itself and restricting our interests to models that satisfy locality condition, it can be seen that local hidden variables is the only choice for hypothetical local model that reproduces perfect correlations (QM prediction). The term "correlation" is defined using concept of "variables". So there is really no alternative to "variables" if we want to model "correlations".

Another thing that can be said is that besides the first paper on Bell inequalities there are other alternative proofs of Bell type inequalities. Among them are proofs that do not rely on perfect correlations and hidden variables and instead they use only observable variables - measurement results. One such informal proof is here - https://www.physicsforums.com/threads/a-simple-proof-of-bells-theorem.417173/#post-2817138
Another formal proof is here - https://www.physicsforums.com/threa...y-on-probability-concept.944672/#post-5977632
It is part of this paper https://journals.aps.org/pra/abstract/10.1103/PhysRevA.47.R747

 

[DarMM]

The term "correlation" is defined using concept of "variables". So there is really no alternative to "variables" if we want to model "correlations"

How does this relate to what Brassard and Raymond-Robichaud are saying? I would have taken their sentence to mean there the observable variables might not only find their explanation as coarse grained statistics of more fundamental variables.

 

[zonde]

I would have taken their sentence to mean there the observable variables might not only find their explanation as coarse grained statistics of more fundamental variables.

I'm not sure what you are talking about. Statistics enter the picture after coincidences from individual detection events are determined. And individual detection events are not considered hidden. How did you managed to tie explanation of observable variables with statistics?

 

[DarMM]

And individual detection events are not considered hidden. How did you managed to tie explanation of observable variables with statistics?

I'm trying to understand the relation of your comment to their statement.

Often in interpretations of QM one tries to explain the statistics of observables via underlying hidden variables. The difference between the statistics of the (noncommutative) probability theory given by QM and the theory of statistics from Kolmogorov probability is that the existence of an underlying explanation seems to blocked in many cases. This is unlike classical statistics where even if you aren't looking at the explanation and only looking at the statistics, the statistics are still compatible with an underlying explanation.

The "tie" here is that they are commenting on the relationship between the explanation and the statistics, hence me trying to understand your comment in relation to that, in brief the "tie" comes from that being what their comment seems to be about, not directly to your last post.

 

[zonde]

I'm trying to understand the relation of your comment to their statement.

Often in interpretations of QM one tries to explain the statistics of observables via underlying hidden variables.

But one does not try to explain statistics directly. Explanations tell how individual detection events can be produced that in turn obey predicted statistics.
These individual detection events are considered experimental facts and they have to appear in explanation. The next step - calculating statistics from these events is not hidden. Experimentalists calculate these statistics in experiments. And there is only one accepted interpretation about that step.

 

[DarMM]

But one does not try to explain statistics directly. Explanations tell how individual detection events can be produced that in turn obey predicted statistics.

Well yes, you have an explanation of the individual events in terms of your hidden variable theory let's say and then you must prove the individual events obey the statistics observed. All of that is true and I don't disagree with it. It just seems to be an obvious truism though, how does it relate to their paper?

 

[Auto-Didact]

Note that the fractal nature in the Abbott & Wise case is caused by measurement. On the other hand, unmeasured BM trajectories do not have a fractal nature.

I was going over the literature on this again and spotted one paper in particular, which caught my eye; I'm linking it here on the off-chance you haven't seen it before:

Sanz 2005, A Bohmian approach to quantum fractals

A quantum fractal is a wavefunction with a real and an imaginary part continuous everywhere, but differentiable nowhere. This lack of differentiability has been used as an argument to deny the general validity of Bohmian mechanics (and other trajectory--based approaches) in providing a complete interpretation of quantum mechanics. Here, this assertion is overcome by means of a formal extension of Bohmian mechanics based on a limiting approach. Within this novel formulation, the particle dynamics is always satisfactorily described by a well defined equation of motion. In particular, in the case of guidance under quantum fractals, the corresponding trajectories will also be fractal.

The paper is well worth a readthrough but for the convenience of discussion, here is the conclusion:

Conclusion

The consistent picture of quantum motion provided by Bohmian mechanics relies on a translation of the physics contained within the Schrödinger equation into a classical like theory of motion. This transformation from one theory to the other is based on the regularity or differentiability of wavefunctions. Therefore, it does not hold for quantum fractals, non-regular solutions of the Schrödinger equation. A priori, this seems to be a failure of Bohmian mechanics in providing a complete explanation of quantum phenomena, since quantum fractals would not have a trajectory based representation within its framework. However, taking into account the fact that Bohmian mechanics is formally equivalent to the standard quantum mechanics, this incompleteness results quite “suspicious".

By carefully studying the nature of quantum fractals, one can understand the source of such an incompatibility. These wavefunctions obey the Schrödinger equation in a weak sense, i.e., given the wavefunction as a linear superposition of eigenvectors of the Hamiltonian, the Schrödinger equation is satisfied by each eigenvector, but not by the wavefunction as a whole. This is because the eigenvectors are always continuous and differentiable everywhere, unlike quantum fractals, which are continuous everywhere, but differentiable nowhere. Taking this into account, a convenient way to express any arbitrary wavefunction, regular or fractal, is in terms of a superposition of eigenvectors of the Hamiltonian. This procedure is particularly important in those circumstances where the differentiability of the wavefunction is going to be invoked, like in the formulation of trajectory based quantum theories like Bohmian mechanics.

In order to have a truly consistent particle equation of motion, Bohmian mechanics must be then reformulated in terms of an eigenvector decomposition of the wavefunction instead of considering the latter as a whole (as happens in standard Bohmian mechanics). The resulting generalized equation of motion, defined by a (convergent) limiting process, is valid for any arbitrary wavefunction, and provides the correct Bohmian trajectories. In the case of quantum fractals, one obtains the desired trajectory based picture at the corresponding limit. Whereas, if the wavefunction is regular, the trajectories determined by means of this procedure will coincide with those given by the standard Bohmian equation of motion. This novel generalization thus proves the formal and physical completeness of Bohmian mechanics as a trajectory-based approach to quantum mechanics.

The trajectories associated to quantum fractals are also fractal. This explains both the formation of fractal quantum carpets and the unbounded expected value of the energy for quantum fractals. Although the example of a particle in a box has been used here to illustrate the peculiarities of quantum fractals. the analysis can be straightforwardly extended to continuum states or other trajectory based approaches to quantum mechanics, like Nelson‘s theory of quantum Brownian motion. Moreover, this kind of analysis can be of practical interest in the study of properties related to realistic systems, like those suggested by Wócik et al. and Amanatidis et al., providing moreover a causal insight on their physics.

With you being the resident expert on BM, I'm very curious to hear your opinion about this paper and how the conclusions presented by the author, in particular the bolded part quoted above, would relate to your fundamentally non-relativistic BM theory.

 

[Demystifier]

With you being the resident expert on BM, I'm very curious to hear your opinion about this paper

I believe that wave functions which are nowhere differentiable do not appear in nature.

 

[martinbn]

I believe that wave functions which are nowhere differentiable do not appear in nature.

Interesting! I thought that physicists felt differently. What about all the elements in a typical $L^2$ space? I thought those were essential for QM.

 

[Demystifier]

Interesting! I thought that physicists felt differently. What about all the elements in a typical $L^2$ space? I thought those were essential for QM.

They exist mathematically, but I don't see how they are essential. Perhaps they are needed for a rigorous mathematical formulation of QM, but it doesn't mean that they appear in nature.

 

[martinbn]

They exist mathematically, but I don't see how they are essential. Perhaps they are needed for a rigorous mathematical formulation of QM, but it doesn't mean that they appear in nature.

None of the functions appear in nature, they appear in the mathematical description of nature. What I find interesting is that you have a preference on which functions should be used in the models. My impression was physicists are not that committed. After all, you all like things as the Dirac delta function.

 

[Demystifier]

After all, you all like things as the Dirac delta function.

I like it only as an idealization with which it is easy to make explicit computations.

 

[martinbn]

I like it only as an idealization with which it is easy to make explicit computations.

Hm, some would say that for the differentiable functions.

 

[Demystifier]

Hm, some would say that for the differentiable functions.

But if the wave function is not differentiable, then it cannot satisfy the Schrodinger equation (which is a partial differential equation).

 

[Auto-Didact]

I believe that wave functions which are nowhere differentiable do not appear in nature.

Why exactly? I'm assuming you are arguing based on the adherence of some physical principle (or else based on some aesthetic criteria as Sabine might put it).

How would their existence be precluded in terms of physics?

They exist mathematically, but I don't see how they are essential. Perhaps they are needed for a rigorous mathematical formulation of QM, but it doesn't mean that they appear in nature.

By this argument no actual fractals (of an infinite path length or equivalent criteria) exist in nature. I would assume you mean then that all actually occurring fractals in nature actually only scale up/down up to some limit.

Speaking not only as a physicist, but from the perspective of canonical classical physics, how would you explain the occurrence of strange attractors in phase space then? Are these not physical objects?

None of the functions appear in nature, they appear in the mathematical description of nature. What I find interesting is that you have a preference on which functions should be used in the models. My impression was physicists are not that committed. After all, you all like things as the Dirac delta function.

Yes, this also surprises me somewhat, maybe even very much. I was under the suspicion that other physicists today, especially after the Dirac delta function issue and the subsequent discovery in later decades of hyperfunction/distribution theory, more openly embraced what were once, for very good reasons, seen as mathematical pathological functions.

If physics can not allow non-differentiable things, then all of nonlinear dynamics (chaos, turbulence, catastrophe, Feigenbaum universality, etc) directly goes out of the window. To me, because of experimental facts, this position is clearly untenable.

But if the wave function is not differentiable, then it cannot satisfy the Schrodinger equation (which is a partial differential equation).

As the article puts it, non-differentiability doesn't hold for $$i\hbar \partial_t\Psi_t(x)=\hat H \Psi_t (x)$$but it does hold for $$[i\hbar \partial_t - \hat H] \Psi_t (x)=0$$The above is called a weak solution in PDE and has been studied extensively.

 

[martinbn]

But if the wave function is not differentiable, then it cannot satisfy the Schrodinger equation (which is a partial differential equation).

It can in a weak sense.

 

[atyy]

If physics can not allow non-differentiable things, then all of nonlinear dynamics (chaos, turbulence, catastrophe, Feigenbaum universality, etc) directly goes out of the window. To me, because of experimental facts, this position is clearly untenable.

http://blog.jessriedel.com/2017/07/...cs-part-7-quantum-chaos-and-linear-evolution/