Bohmian Mechanics of DCES (swaps), featuring several papers

[Matterwave]

It's not that complicated as you might think it is. See my https://arxiv.org/abs/2205.05986.

I have added this to my reading list on BM. Next to Bohm and Hiley haha. I don't know when I'll get to it.. but I will try.

 

[Demystifier]

But... I really don't think my math skills are the issue here.

There is a simple test to see if someone has math skills appropriate to understand BM. Take the Norsen's Eq. (2) and, from it, prove mathematically that particle n' does not influence particle n if the two particles are not entangled. If you can prove it, then your math skills are probably OK. If you can't prove it, then they are not. If you haven't already done that proof, I strongly recommend you to do it, because it's essential for understanding BM.

 

[Demystifier]

I have added this to my reading list on BM. Next to Bohm and Hiley haha. I don't know when I'll get to it.. but I will try.

That book also studies a field version of BM, in Chapter 11. But of course, nobody presented Bohmian QFT in such a simple form as I did.

 

[DrChinese]

I don't pick and choose anything. It's the Bohmian equation of motion that determines when the position of one particle influences the motion of the other, and when it doesn't. But there is not a good way to explain it by words. You must read the math of the Bohmian equation of motion.

It's not that complicated of a Bohmian scenario. I have 2 entangled photons 1 and 2 as part of a Bell test, while photon 2 is still midflight.

i) What is the initial influence of photon 1 on photon 2 as to polarization and momentum, while photon 1 is midflight?
ii) What is sum influence of the atoms of the PBS measuring photon 1 on photon 2 as to polarization and momentum?
iii) What is sum influence of the atoms of the detector registering photon 1 on photon 2 as to polarization and momentum?
iv) What is influence of the rest of the universe on photon 2 as to polarization and momentum?

Norsen mentions i) in his discussion (Fig. 6), but does not really mention the others. But you have mentioned ii) iii) and iv) previously. If the influence of any of these 4 is negligible, then I'd be interested to know that. Presumably, distance is not a factor. Or, if distance is a factor in the Bohmian equation of motion, just say so.

I would claim oQM says:
i) there is no influence whatsoever of Photon 1 on photon 2 while they are both midflight.
ii) there is no influence whatsoever of the individual atoms of a PBS on photon 1 or photon 2 (only the orientation - a single variable - is relevant).
iii) there is no influence whatsoever of the individual atoms of a detector on photon 1 or photon 2.
iv) there is no influence whatsoever of the rest of the universe on photon 1 or photon 2.

So... in experiment after experiment, the statistical results seems to only depend on one single variable: polarizer orientation. None of the other items you input into "the math of the Bohmian equation of motion" seem to matter at all. Why is that? If you don't pick and choose, why is it that all these inputs always seem to vanish (i.e. net to zero) in polarization experiments?

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In other words: I know that we don't know the precise particle positions, point accepted. But if you specified them exactly in an even easier example, can you tell me what the polarization results would be? In other words: If BM has deterministic features, I would think you could fabricate an example where I start with a |+> polarized photons that I will run into a H/V oriented PBS (or some other theta); and you could tell me (using Bohmian math) which specific port it would exit and be detected (H or V). You can specify whatever initial conditions you like to create the example (apparati, source, etc).

In oQM the answer is plain: This problem cannot be solved exactly, and there is only 1 variable (theta). I don't think it can be solved in BM either, but I am trying to understand how many variables are involved "under the hood" so to speak.

 

[DrChinese]

Demystifier, referencing Norsen's (2): ...particle n' does not influence particle n if the two particles are not entangled. ...

That's a new one for me, have never seen that stated elsewhere. Not sure how I missed that in other Bohmian papers. I would have thought this point might have been repeated in almost any discussion of Bohmian entanglement.

Are we talking the same "entangled" usually discussed? If so, exactly how does an entire Beam Splitter become entangled with (or due to) a single photon? And what if 2 independent photons happen to be in a Beam Splitter at the same time? And is the Beam Splitter still entangled with the same photon as it enters the Detector? Or does the entanglement now move, transfer, dissipate, or ?

Huggett: "Therefore, the particles are in some sense “pushed” around by the wave function on the configuration space – hence BM is sometimes referred to as the ‘pilot wave’ theory". On the other hand, Huggett also mentions (as you do I believe) that: "... the particle’s wavefunction evolves according to the standard Schrödinger equation (2.1), and particles’ definite positions neither appear in ψ nor feature in the Schrödinger equation, so the position variable of the wavefunction has much the same significance as before – a variable in a probability function."

 

[Demystifier]

That's a new one for me, have never seen that stated elsewhere. Not sure how I missed that in other Bohmian papers. I would have thought this point might have been repeated in almost any discussion of Bohmian entanglement.

You don't need to read this explicitly. You have to derive it by yourself from Eq. (2) in Norsen's paper. If you can't derive it, then you don't understand BM, period. And no matter how much text about BM you read, if you don't see how this text is related to the equations, then you don't understand BM.

 

[DrChinese]

You don't need to read this explicitly. You have to derive it by yourself from Eq. (2) in Norsen's paper. If you can't derive it, then you don't understand BM, period. And no matter how much text about BM you read, if you don't see how this text is related to the equations, then you don't understand BM.

I really enjoy your comments about my understanding level. Really, I do, they don't bother me. Some of what you say is accurate, some is not.

So I come here asking questions to boost that understanding level. You say "textual" things I've never read anywhere else that I don't follow. Then you basically say "the mathematical proof is left to the reader". And you have yet to provide an actual quote of any of the usual Bohmian writers (Goldstein, Durr, etc.) that match your textual storyline. And you expect me to conclude the hole in my understanding of Bohmian math is insurmountable to gaining further understanding.

Well, no, I don't agree that " ...particle n' does not influence particle n if the two particles are not entangled. ..." is a true statement, and I question whether that is a general assertion of most Bohmians. Why? Because Bohmians reference Norsen's (1) frequently and say the extension (2) can be ignored because of the QEH of Norsen's (3). And in a system of N particles, N>2 per Norsen's (2), there are obvious issues regarding the amount of entanglement possible between any 2 particles - if what you say is correct.

 

[Demystifier]

And you have yet to provide an actual quote of any of the usual Bohmian writers (Goldstein, Durr, etc.) that match your textual storyline.

Here is an excerpt from the book by Bohm and Hiley, The Undivided Universe, page 59:

It says that when the wave function of two systems is a product (which means that they are not entangled), then the two systems behave independently. Is that explicit enough?

The book Durr and Teufel, Bohmian Mechanics, is even more explicit:

 

[DrChinese]

Here is an excerpt from the book by Bohm and Hiley, The Undivided Universe, page 59:
View attachment 374601
It says that when the wave function of two systems is a product (which means that they are not entangled), then the two systems behave independently. Is that explicit enough?

The book Durr and Teufel, Bohmian Mechanics, is even more explicit:
View attachment 374602

Perfect. Yes, that is exactly what I am asking for. Thanks.

And your reference exactly highlights why I was asking in the first place. Please note that I am NOT arguing or disagreeing with you with the below. As I have said many times, I will accept any statement you make as representing BM. But I have often pointed out that the applications of Bohmian concepts seems to be inconsistent not only between authors; but also within writings by the same author. So I think my questions fall in the area of "fair game".

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Now in my post #37 that you generously responded to, I qualified: "And in a system of N particles, N>2 per Norsen's (2), there are obvious issues regarding the amount of entanglement possible between any 2 particles..." So your reference specifically presents the N=2 case. That is not necessarily a problem, but I clearly anticipated this as a point we'd be discussing about right now - I presume you did too. Now if I understand correctly the use of the word "entanglement" here - nonseparable per the general equation - then you might say (I am attempting to paraphrase some of your ideas in my lingo):

As systems get larger in particle number N, there is a degree of entanglement being spread among those particles. That total entanglement does not violate any rules around monogamy of entanglement (MoE). (After all, there are plenty of variations on N>2 entanglement possible in oQM. Not all entanglement needs to be maximal, nor shared equally. As example for N=3, you have GHZ and W states.)

If I am "close" on the above paragraph, it would help me to understand some of your comments about entanglement when you apply them to laboratory apparati and in some other situations. Thoughts? Or do you think that MoE does not apply here?

 

[DrChinese]

Yes there is, it's formula (2) in the Norsen's paper. Roughly speaking, if one already knows standard QM, then all one needs to know about BM is encoded in this formula, for everything else can be reproduced from it, provided that one possesses appropriate math skills.

That's a big statement, you know!

1. Is Norsen's (3) - QEH - important too? Just kidding... I had to say that since you mentioned (2)...


2. So you have repeatedly made comments around the determinism of BM. So I'd like to get more clarity around that.
a. Norsen: "...one also has definite positions for each particle in the system." I think you have been pretty clear about that. What about momentum? Is that definite at all times too?
c. I'm still curious about spin in BM. Would you refer to it as an artifact of other quantum properties? If so, which ones? Norsen mentions "critical trajectory" as being the key element for particles going through a Stern-Gerlach apparatus. He makes it seem simple. Does polarization work equally simply, is there a critical trajectory to a photon?
d. If other particles must be entangled with a particle N to affect it: Does the effect change N's momentum? Is the effect mutual? In Norsen's example, it seems to be.


3. If BM has deterministic features, I would think one could fabricate an example where I start with a |+> polarized photon that I will run into a H/V oriented PBS (45 degrees or some other theta); and then you could tell me (using Bohmian math) which specific port it would exit and be detected (H or V). I'm trying to understand what the inputs would need to be in order to solve this problem. Does the initial |+> polarized photon make a difference, or are all such photons considered identical? What about the PBS, what attributes does it have in addition to its orientation (theta)? What about measurement apparati, maybe the lab, since you have mentioned those in various posts. Norsen writes as if the spin case is simple, so am I overthinking this?

In oQM the answer is plain: This example cannot be solved exactly (outcomes are random), and there is only 1 variable (theta). I don't think it can be solved in BM either, but I am trying to understand how many variables are involved "under the hood" so to speak. I realize the values for those variables are unknowable in practice, I'm not questioning that or the QEH. And this is not a "trick" question: I am asking these for the exact purpose of discussing the Entanglement Swapping protocols of our references. My plan is to start with the initial pair of photons 1 and 2 and work forward from there. And I realize well that photon polarization and spin 1/2 particle spin are not the same thing, so if that difference becomes sufficiently relevant, just say so.

---

I realize there are a lot of questions in this post, and I certainly don't expect you or anyone to cover them all at one shot. Just know that I appreciate the time you are giving to me. Again, I'll accept whatever you tell me. I'm trying to apply those BM tenets.

Thanks,

-DrC

 

[Demystifier]

As systems get larger in particle number N, there is a degree of entanglement being spread among those particles. That total entanglement does not violate any rules around monogamy of entanglement (MoE). (After all, there are plenty of variations on N>2 entanglement possible in oQM. Not all entanglement needs to be maximal, nor shared equally. As example for N=3, you have GHZ and W states.)

If I am "close" on the above paragraph, it would help me to understand some of your comments about entanglement when you apply them to laboratory apparati and in some other situations.

Yes, that's correct. The entanglement is spread, and so is the influence of other particles on the motion of one partticle.

 

[Demystifier]

That's a big statement, you know!

I know.

1. Is Norsen's (3) - QEH - important too? Just kidding... I had to say that since you mentioned (2)...

Well, in a sense, (3) can be derived from (2). Not rigorously derived, but there are plausible arguments explaining why (3) should be valid if (2) is valid.

2. So you have repeatedly made comments around the determinism of BM. So I'd like to get more clarity around that.
a. Norsen: "...one also has definite positions for each particle in the system." I think you have been pretty clear about that. What about momentum? Is that definite at all times too?

Strictly speaking, Bohmian particles don't have momentum. They have velocity $v$, but to have the momentum $p=mv$, they would need to have the mass $m$, which they don't. The mass is not ontological in BM in the sense in which the position and velocity are ontological. I have explained it in much more detail in https://arxiv.org/abs/2308.10500 Sec. 5.2.

c. I'm still curious about spin in BM. Would you refer to it as an artifact of other quantum properties? If so, which ones?

As I said many times, it's an artifact of positions of particles constituting the apparatus.

Norsen mentions "critical trajectory" as being the key element for particles going through a Stern-Gerlach apparatus. He makes it seem simple. Does polarization work equally simply, is there a critical trajectory to a photon?

Norsen oversimplifies a bit. It sort of works for a simplified description of Stern-Gerlach apparatus, but in a more realistic description of a measuring apparatus that would not be so simple.

d. If other particles must be entangled with a particle N to affect it: Does the effect change N's momentum? Is the effect mutual? In Norsen's example, it seems to be.

As I said above, Bohmian particles don't have momentum. What can change are the velocities, but depending on details of pilot waves, sometimes even the velocities don't change. The effect does not need to be mutual.

3. If BM has deterministic features, I would think one could fabricate an example where I start with a |+> polarized photon that I will run into a H/V oriented PBS (45 degrees or some other theta); and then you could tell me (using Bohmian math) which specific port it would exit and be detected (H or V). I'm trying to understand what the inputs would need to be in order to solve this problem. Does the initial |+> polarized photon make a difference, or are all such photons considered identical? What about the PBS, what attributes does it have in addition to its orientation (theta)? What about measurement apparati, maybe the lab, since you have mentioned those in various posts. Norsen writes as if the spin case is simple, so am I overthinking this?

In principle, you would need to know initial positions of all particles involved, of the apparatus, of the environment, etc. But it is possible to analyze in detail some simplified models, which Norsen does.

I am trying to understand how many variables are involved "under the hood" so to speak.

A lot.

And I realize well that photon polarization and spin 1/2 particle spin are not the same thing, so if that difference becomes sufficiently relevant, just say so.

It is relevant. Polarization is not measured by the Stern-Gerlach apparatus, so there is no such a simple model of the measurement of polarization. In particular, the detection of photon destroys the photon, which means that a realistic Bohmian description complicates a lot.

Again, I'll accept whatever you tell me.

Really?

 

[DrChinese]

I know.

Well, in a sense, (3) can be derived from (2). Not rigorously derived, but there are plausible arguments explaining why (3) should be valid if (2) is valid.

...

Sorry it's taking a few days to properly respond to your comprehensive post. Will be back soon, I have tax return deadline I'm facing LOL.

 

[DrChinese]

I know.

1. Well, in a sense, (3) can be derived from (2). Not rigorously derived, but there are plausible arguments explaining why (3) should be valid if (2) is valid.

2. Strictly speaking, Bohmian particles don't have momentum. They have velocity $v$, but to have the momentum $p=mv$, they would need to have the mass $m$, which they don't. The mass is not ontological in BM in the sense in which the position and velocity are ontological. I have explained it in much more detail in https://arxiv.org/abs/2308.10500 Sec. 5.2.


3. As I said many times, it's an artifact of positions of particles constituting the apparatus.

Norsen oversimplifies a bit. It sort of works for a simplified description of Stern-Gerlach apparatus, but in a more realistic description of a measuring apparatus that would not be so simple.

As I said above, Bohmian particles don't have momentum. What can change are the velocities, but depending on details of pilot waves, sometimes even the velocities don't change. The effect does not need to be mutual.

3. In principle, you would need to know initial positions of all particles involved, of the apparatus, of the environment, etc. But it is possible to analyze in detail some simplified models, which Norsen does.

A lot.

4. It is relevant. Polarization is not measured by the Stern-Gerlach apparatus, so there is no such a simple model of the measurement of polarization. In particular, the detection of photon destroys the photon, which means that a realistic Bohmian description complicates a lot.

5. Really?

Well, my tax deadline turned into another deadline, and another... so again, my apologies for delaying this reply. Your efforts deserved better, @Demystifier ...

1.


2. So Bohmian particles lack "ontological" mass, I guess in the same way they lack spin?


3. So you are going with "deterministic in principle", right? Again, the Bohmian perspective has solidly been: BM is deterministic, and particles have definite positions at all times, which in principle could be determined. Because that stands starkly in contrast to the orthodox view of QM.


4. So what happens when a photon passes through a polarizer before it reaches a detector? I mean, what on earth does the detector possible have to do with which way the photon goes? Surely you don't mean that is a *relevant* factor. I say we forget the detector as part of this equation, I just don't see how it relates. But if you think I'm wrong, and it's an important factor, go ahead and say so.

A PBS sends a photon one way or the other. That photon has a Bohmian position and a Bohmian velocity, right? But no Bohmian spin. And assuming it doesn't: exactly how does the photon "know" to turn (or not) if it's "non-existent" polarization is one way or the other? What is it sensing? It's not like a PBS has a bunch of molecules lined up one way for one polarization (P), one way for the other (S). There is just a gradient at a coating.

I'm not asking how a PBS works, I'm asking how a Bohmian photon (polarized say |V> but with no "ontological" polarization attribute) knows which way to go.


5. For purposes of this discussion, sure.

 

[Demystifier]

I'm not asking how a PBS works, I'm asking how a Bohmian photon (polarized say |V> but with no "ontological" polarization attribute) knows which way to go.

I agree with other notes in your last post here, so I will only answer your question above. As long as we don't study photon creation or destruction, we can use a naive picture of Bohmian photon as a thing with a particle trajectory. In this picture the Bohmian photon does not have polarization, but the wave that guides it has. The Bohmian photon senses the properties of the wave, because the wave determines it's velocity. But it does not sense the whole wave, at each time it only senses the properties of the wave at one point where the photon currently is. So the Bohmian photon knows which way to go because the wave tells him so. And since the wave has the polarization, the position of the photon is related to the polarization of the wave. So even though the Bohmian photon does not have a polarization, it's behavior looks as it does, because its motion reflects local properties of the wave which has polarization.