I Confused by nonlocal models and relativity

[DarMM]

I'm also lost. Now the apparently clear notion of "physical degree of freedom" is also blurred by philsophical unclear redefinitions.

In physics there are two kinds of "degrees of freedom", applying either to point-particle mechanics, where it is described as a finite number of independent configuration-space variables $q^k$ with $k \in \{1,\ldots f \}$, or field-degrees of freedom. E.g., the electromagnetic field is described by 6 field-degrees of freedom, e.g., the 3 electric and the 3 magnetic field components.

Now they start to claim that the quantum mechanical wave function $\psi$ is a "degree of freedom". What should that mean?

Basically they mean it's like a field degree of freedom on a higher dimensional space (configuration space). As you said, yes this is basically what Schrodinger originally thought and yes like you I have literally no idea what it could mean in QFT.

 

[atyy]

I'm not sure I fully follow, but the wavefunction of minimal QM comes from the Bohmian epistemic $\psi$ not the fundamental $\Psi$ of Bohmian Mechanics. This distinction matters in Frauchiger-Renner type scenarios.

Ah that is an interesting point. I have never understood what the proper classification of the various Bohmian psis are. @Demystifier has explained this to me several times, but I have not understood it well enough to have the answer off the top of my head. I presume in this case you don't mean that the Bohmian epistemic psi is "psi-epistemic" in the Harrigan and Spekkens sense?

I don't agree with this. Bohr, Heisenberg, Pauli, Peierls and many others explicitly state the wave-function is just a catalogue of probabilities. More modern Copenhagen people say this. It's what's said about Copenhagen in the Perimeter Institute's Quantum Foundations courses no matter who teaches it each year.

Yes, but earlier you mentioned Dirac-von Neumann. Now I doubt that Dirac and von Neumann were any less sophisticated than Bohr or Peierls, but just to use your terminology, it does indicate that there is an idea that within the conventional formalism that the wave function cam be considered nonlocal. Apart from the Harrigan and Spekkens paper, one can see this attitude in https://arxiv.org/abs/0706.1232 (Fig. 1) and https://arxiv.org/abs/quant-ph/0209123 (p51) "In other words, even if one can discuss whether or not quantum mechanics is local or not at a fundamental level, it is perfectly clear that its formalism is not ..."

So what I understand this view to be is it is Copenhagen, and it starts off with the wave function is just a catalogue of probabilities, but it adds: "the wave function is just a catalogue of probabilities because of some obvious absurdities like collapse if it is real. Nonetheless, from an operational point of view we don't need to decide on the reality of the wave function to proceed with using the formalism, so we set up the cut and state and quantum formalism. At the point of usage, since we believe it makes no operational difference whether the wave function is real or not, we can use the mental picture of a real wave function to help with calculations, ie. reality is a tool to predict the probabilities of measurement outcomes." Since the reality of the wave function only enters at the last stage, I think it has enough of Copenhagen to be protected from the extended Wigner friend scenarios.

 

[DarMM]

Now I doubt that Dirac and von Neumann were any less sophisticated than Bohr or Peierls

It's not so much named after them to indicate it's their view, but that it's what one might naively think from reading the axioms that first appear in their texts. von Neumann didn't himself think of $\psi$ as an actual real field and Dirac is fairly explicit in thinking $\psi$ is just a collection of probabilities.
Regardless in modern usage in papers and books, Copenhagen is taken to have as a defining element the view that $\psi$ is not a real propogating field but just a collection of statistics.

I'll respond to the rest tomorrow.

 

[Demystifier]

Ah that is an interesting point. I have never understood what the proper classification of the various Bohmian psis are. @Demystifier has explained this to me several times, but I have not understood it well enough to have the answer off the top of my head. I presume in this case you don't mean that the Bohmian epistemic psi is "psi-epistemic" in the Harrigan and Spekkens sense?

A short reminder. Suppose that a closed system contains two particles. Then its wave function is $\Psi({\bf x}_1,{\bf x}_2,t)$ and always satisfies the Schrodinger equation. The wave function of the open subsystem, e.g. the wave function of the first particle, is then
$$\psi({\bf x}_1,t)=\Psi({\bf x}_1,{\bf X}_2(t),t)$$
where ${\bf X}_2(t)$ is the Bohmian trajectory of the second particle. The wave function $\psi({\bf x}_1,t)$ does not always satisfy Schrodinger equation.

 

[vanhees71]

What's the physical meaning of $\psi(\mathbf{x}_1,t)$ then?

 

[Demystifier]

What's the physical meaning of $\psi(\mathbf{x}_1,t)$ then?

Suppose that $\mathbf{X}_2$ is not a position of a single microscopic particle, but a collective position of a macroscopic pointer in the measuring apparatus. In other words, suppose that $\mathbf{X}_2$ is a position that we can directly perceive (a perceptible, in the language on my paper). Then $\psi(\mathbf{x}_1,t)$ is the wave function that we associate with the first particle when we know the position of the pointer. This is what, in standard QM, corresponds to the update of wave function (which some people like to call "collapse", but we both agree that there is no true collapse) given a knowledge provided by the measuring apparatus. In this way, Bohmian mechanics explains what standard QM takes for granted, namely that a macroscopic pointer has a position and that knowledge of this position can be used to update the wave function.

 

[vanhees71]

Interesting, and what's the formal justification for this interpretation?

 

[Demystifier]

Interesting, and what's the formal justification for this interpretation?

One justification is the fact that in this way one can reproduce the standard textbook "collapse" rule (which correctly predicts probabilities of subsequent measurements), without having an actual collapse. It is explained in more detail in the book on Bohmian mechanics by Durr et al that you read.

Or perhaps you would like a different type of justification?

 

[A. Neumaier]

Suppose that $\mathbf{X}_2$ is not a position of a single microscopic particle, but a collective position of a macroscopic pointer in the measuring apparatus. In other words, suppose that $\mathbf{X}_2$ is a position that we can directly perceive (a perceptible, in the language on my paper). Then $\psi(\mathbf{x}_1,t)$ is the wave function that we associate with the first particle when we know the position of the pointer. This is what, in standard QM, corresponds to the update of wave function (which some people like to call "collapse", but we both agree that there is no true collapse) given a knowledge provided by the measuring apparatus. In this way, Bohmian mechanics explains what standard QM takes for granted, namely that a macroscopic pointer has a position and that knowledge of this position can be used to update the wave function.

And what is the particle's wave function before the measurement?

 

[Demystifier]

And what is the particle's wave function before the measurement?

Before the measurement we usually assume that there is no entanglement, in which case there is no important difference between $\Psi$ and $\psi$. Does it answer your question?

 

[A. Neumaier]

Before the measurement we usually assume that there is no entanglement, in which case there is no important difference between $\Psi$ and $\psi$. Does it answer your question?

One cannot assume this when one only has the Bohmian dynamics, unless one assumes that the universe was created at the time of the preparation of the experiment. This can be seen by running the Bohmian mechanics backwards from your assumed condition - it will be unentangled only for an instant.

In contrast, Copenhagen does not claim to model the whole universe but only the small system between preparation and measurement, hence can make initial assumptions without running into a contradiction.

 

[vanhees71]

Usually you assume that before the measurement, i.e., the interaction between the measured system and the measurement device these are unentangled, which makes sense since the preparation of the system and the device before the measurement should be independent to have a defined distinction between them and a well-defined measurement to begin with.

 

[A. Neumaier]

Usually you assume that before the measurement, i.e., the interaction between the measured system and the measurement device these are unentangled, which makes sense since the preparation of the system and the device before the measurement should be independent to have a defined distinction between them and a well-defined measurement to begin with.

Yes, this is permitted in interpretations that do not claim to derive their dynamics from the dynamics of a bigger system that also involves the preparation and measurement procedure. But Bohmian mechanics is supposed to derive everything from a deterministic dynamics of the universe, hence has no room for additional assumptions unless these can be proved from the deterministic model.

 

[Demystifier]

One cannot assume this when one only has the Bohmian dynamics, unless one assumes that the universe was created at the time of the preparation of the experiment.

One can assume this, of course only in a certain approximative sense. The other important concept in Bohmian mechanics that you are missing is the effective wave function, which explains why most of the entanglement that the subsystem has with the rest of the universe can be neglected.

 

[Demystifier]

hence has no room for additional assumptions unless these can be proved from the deterministic model.

They can be proved, but perhaps not rigorously proved as you would like. See https://arxiv.org/abs/quant-ph/0308039
especially Sec. 5.

 

[A. Neumaier]

One can assume this, of course only in a certain approximative sense.

But you need to show why, in some approximate sense, this assumption is justified by the deterministic theory!

The other important concept in Bohmian mechanics that you are missing is the effective wave function, which explains why most of the entanglement that the subsystem has with the rest of the universe can be neglected.

No. my question was aimed at this, and you had answered with separability. I don't see how the deterministic theory implies separability for the effective wave function.

They can be proved, but perhaps not rigorously proved as you would like. See https://arxiv.org/abs/quant-ph/0308039
especially Sec. 5.

(5.6) assumes that the universal wave function factorizes, an assumption completely unstable under small temporal changes, hence not warranted. They mention this after (5.11); it is strange that they discuss this unrealistic case at all. After (5.8), another unwarranted assumption is made ''that the interaction between the x-system and its environment can be ignored''. But the bigger a system the more entangled it is with the remainder of the universe! And you invoked a big system (consisting of a measured system and its measuring device)!

(5.12) doesn't help as it is questionable whether the entanglement can be described by a few separable summands with essentially disjoint support. That the form (5.15) ultimately assumed is typical for is an unproved assumption that I find quite unlikely to be derivable from the dynamics.

The only mathematically natural definition of an effective wave function for a subsystem consisting of particles in a set S and the complementary set E of particles in the environment is to take the universal wave function $\Psi(x_S,x_E,t)$ and substitute for all environmental particles their Bohmian position $x_E(t)$, i.e., $\psi(x_S,t):=\Psi(x_S,x_E(t),t)$. But is there any theory that this effective wave function would behave in the
Copenhagen way - i.e., according to the Schrödinger dynamics before a measurement is made, and collapsing after the measurement is made? I doubt it...

 

[Demystifier]

That the form (5.15) ultimately assumed is typical for is an unproved assumption that I find quite unlikely to be derivable from the dynamics.

I think it's (non-rigorously) proved from dynamics in the literature on decoherence theory. See e.g. the book by Schlosshauer, Sec. 2.8.4.

 

[A. Neumaier]

That the form (5.15) ultimately assumed is typical is an unproved assumption that I find quite unlikely to be derivable from the dynamics.

I think it's (non-rigorously) proved from dynamics in the literature on decoherence theory. See e.g. the book by Schlosshauer, Sec. 2.8.4.

The subsection ''Selection of Quasiclassical Properties''? This is 2 1/2 pages of prose arguing for preferred emerging pointer states in traditional quantum mechanics (not Bohmian mechanics). I don't see there anything related to (5.15).

Checking back it seems that whatever is done in Section 2.8 is based on the nondemolition assumption - (2.84) on p. 90 - which we had discussed before as being a nontypical special case.

 

[Demystifier]

The subsection ''Selection of Quasiclassical Properties''? This is 2 1/2 pages of prose

This prose contains references to the literature with more details. The purpose of prose in physics, of course, is not to "prove" something but to give intuitive undertstanding.

arguing for preferred emerging pointer states in traditional quantum mechanics (not Bohmian mechanics).

Exactly! The questions you are asking are not related exclusively to Bohmian mechanics. They are equally related to many worlds, decoherence, von Neumann measurement theory, and all related approaches.

I don't see there anything related to (5.15).

If you can read between the lines, you can find it in item 1 at page 84.

Checking back it seems that whatever is done in Section 2.8 is based on the nondemolition assumption - (2.84) on p. 90 - which we had discussed before as being a nontypical special case.

In that discussion before, it seems that you missed my post #70, where I explained mathematically why the ideas work even without the nondemolition assumption.

 

[bhobba]

Basically they mean it's like a field degree of freedom on a higher dimensional space (configuration space). As you said, yes this is basically what Schrodinger originally thought and yes like you I have literally no idea what it could mean in QFT.

Nor do I - but I know of a book with the view QFT can be presented in a 'realistic' way:
https://www.amazon.com/dp/9812381767/?tag=pfamazon01-20

Here is the issue. We believe things like mass, momentum, and energy (remember mass is a form of energy) are real - without going into the issue what real is. IMHO its one of those fundamental things you can't really pin down - others of course will disagree. Anyway by Noether fields are real because they have energy and momentum. But when quantised the field is described by a very abstract thing - quantum operators. Note this is not saying reality is quantum operators - just its described by it. Since classical fields are real, and they are simply a limit of quantum fields, quantum fields must be real. Normal QM is QFT when particles are dilute - but since the fields are real that means normal QM is real. Now since everything is a quantum field the issue is looking at QFT in a way that seems reasonable - and that is what the book attempts to do.

Thanks
Bill

 

[DarMM]

I've read the book and don't really agree with it, but that would be another thread.

However note the object I'm talking about is not the quantum field $\hat{\phi}\left(x,t\right)$ but rather the wave functional $\Psi\left[\phi\left(x\right),t\right)$. It's the latter that has to be a real propagating object in a Many Worlds view of QFT.

Since classical fields are real, and they are simply a limit of quantum fields, quantum fields must be real

That's a much more subtle issue than one would think. There's no real limit where a quantum field becomes a classical field. Classical fields are more a certain limit of a quantum field's expectation values in certain states.

Since classical fields are real, and they are simply a limit of quantum fields, quantum fields must be real. Normal QM is QFT when particles are dilute - but since the fields are real that means normal QM is real

Again it's a complex issue the exact form the relation between QM and QFT takes. Again it's a limit along a certain subfamily of states. Complicated by the fact that in QFT we have that particle number $N$ is not well defined in general.

Also generally actual operators like $\phi$ are taken as "real" even in QM, since they can be measured, it's more a question about the "reality" of the quantum state $\psi$. Even for operators though one has the interesting effect that most POVMs don't correspond to the quantization of any classical variable.

 

[A. Neumaier]

If you can read between the lines, you can find it in item 1 at page 84.

I cannot read between the lines. Eigenstates of position form a continuum and cannot be split in the way (5.17) requires; supports are closed sets and disjoint supports therefore require a gap in the spectrum!

In that discussion before, it seems that you missed my post #70, where I explained mathematically why the ideas work even without the nondemolition assumption.

Yes, I had missed that. But your analysis there is not convincing; see my answer in that thread.

 

[DarMM]

Just to come back to this.

So what I understand this view to be is it is Copenhagen, and it starts off with the wave function is just a catalogue of probabilities, but it adds: "the wave function is just a catalogue of probabilities because of some obvious absurdities like collapse if it is real. Nonetheless, from an operational point of view we don't need to decide on the reality of the wave function to proceed with using the formalism, so we set up the cut and state and quantum formalism. At the point of usage, since we believe it makes no operational difference whether the wave function is real or not, we can use the mental picture of a real wave function to help with calculations, ie. reality is a tool to predict the probabilities of measurement outcomes." Since the reality of the wave function only enters at the last stage, I think it has enough of Copenhagen to be protected from the extended Wigner friend scenarios.

I haven't been able to find much on this view. Typically the cut is taken to be a subjective. This idea that the wave function is real, but there is also an objective cut doesn't seem to have much discussion in the literature. It seems hard to see how such an objective cut could be Lorentz invariant. It seems to be MWI, but with a length scale where the worlds "stop".

Do you have a link to a paper that lays out this view?

 

[vanhees71]

So far there is no objective cut found. Just recently the limits were pushed even further into the "mesoscopic" realm of very large molecules:

https://doi.org/10.1038/s41567-019-0663-9

 

[jk22]

Coming back to Bell's CHSH inequality, I'm confused that the formula : $$\int A(\vec{a},\lambda)B(\vec{b},\lambda)\rho(\lambda)$$ is considered local. Like how can the "element" lambda be at the same time both in A and B ? It seems lambda is non local there, at the measurement time. Or am I misunderstanding the physical setup described by the formula ?

 

[PeterDonis]

how can the "element" lambda be at the same time both in A and B ?

$\lambda$ refers to what Bell calls "hidden variables" that are in the causal past of both A and B. "Locality" in this context includes things in the causal past. It just doesn't include things outside the causal past, i.e., outside the past light cone. A and B are outside each other's past light cone, which is why, to satisfy locality, measurement A can only depend on the settings $\vec{a}$, not on $\vec{b}$, and vice versa. But both can depend on $\lambda$ because $\lambda$ is in the causal past of both.

 

[jk22]

$\lambda$ refers to what Bell calls "hidden variables" that are in the causal past of both A and B. "Locality" in this context includes things in the causal past. It just doesn't include things outside the causal past, i.e., outside the past light cone. A and B are outside each other's past light cone, which is why, to satisfy locality, measurement A can only depend on the settings $\vec{a}$, not on $\vec{b}$, and vice versa. But both can depend on $\lambda$ because $\lambda$ is in the causal past of both.

That's ok if lambda is information, but let suppose it is a physical particle. In a spacetime diagram lambda is first at the source, then it goes with at most speed of light in both directions ? For then arriving in A and B ?

This seems physically impossible, or that formula is a mixture of local a and nonlocal lambda, like a sweet-sour tasting dish.

I would suppose an example of completely local formula $\frac{1}{\lambda_1}\int_0^{\lambda_1} \cos(a-s_1)ds_1\frac{1}{\lambda_2}\int_0^{\lambda_2} \cos(a-s_2)ds_2$

Then by noting that : $\max_x\{A(x)+B(x)\}\leq\max_x\{A(x)\}+\max_x\{B(x)\}$ and computing the maximum of each covariance completely locally, the value 3.16 should appear ?

 

[PeterDonis]

That's ok if lambda is information, but let suppose it is a physical particle. In a spacetime diagram lambda is first at the source, then it goes with at most speed of light in both directions ? For then arriving in A and B ?

This seems physically impossible

Of course it is. And Bell's formulation rules this out anyway. $\lambda$ has to be something that can causally affect both measurements. Obviously it can't if it is a particle that can only go in one direction.

 

[PeterDonis]

I would suppose an example of completely local formula $\frac{1}{\lambda_1}\int_0^{\lambda_1} \cos(a-s_1)ds_1\frac{1}{\lambda_2}\int_0^{\lambda_2} \cos(a-s_2)ds_2$

Then by noting that : $\max_x\{A(x)+B(x)\}\leq\max_x\{A(x)\}+\max_x\{B(x)\}$ and computing the maximum of each covariance completely locally, the value 3.16 should appear ?

I don't understand what you're trying to do here.

 

[jk22]

I don't understand what you're trying to do here.

I'm just considering that a particle s1 is going to A and another particle s2 to B measurement places. But I'm still mixing information (math) and the physics described by the symbols. This I call local in the sense that "every element of physical reality (particles 1 and 2, each flying in one direction) has a counterpart in the theory (s1,s2)"[EPR quote].

Nevertheless, the particle s1 carries an information like a polarisation angle and arrives at A and interacts with the measurement apparatus which in turn contains information about the angle of measurement "a". The hidden variable is here lambda1, the bound of integration. It is found by extremising the value of the average. This is to say that extremal values are like stable measurement results.