I What has changed since the Copenhagen interpretation?

[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/