Nelson's Stochastic Mechanics, simple argument why interference exists?

  • #1
Spinnor
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Derivation of the Schrödinger Equation from Newtonian Mechanics​

Edward Nelson Phys. Rev. 150, 1079 – Published 28 October 1966​


Abstract,

"We examine the hypothesis that every particle of mass m is subject to a Brownian motion with diffusion coefficient ℏ2m and no friction. The influence of an external field is expressed by means of Newton's law F=ma, as in the Ornstein-Uhlenbeck theory of macroscopic Brownian motion with friction. The hypothesis leads in a natural way to the Schrödinger equation, but the physical interpretation is entirely classical. Particles have continuous trajectories and the wave function is not a complete description of the state. Despite this opposition to quantum mechanics, an examination of the measurement process suggests that, within a limited framework, the two theories are equivalent."

Can someone think of a simple argument why we should expect interference in Nelson's work above given the initial assumptions? For example consider a simple case of a particle in one dimensional box. Where does quantization of energy and momentum come from in what seem like simple assumptions of his work.

Thanks.
 
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  • #2
Nelson said:
The hypothesis leads in a natural way to the Schrödinger equation
This is the key. Since there is the Schrödinger equation in the theory, it implies interference. The interference can be thought of as a purely mathematical feature associated with any wave equation, including the Schrödinger equation, irrespective of the physical interpretation.
 
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  • #3
The "natural way" for Dr. Nelson was a bunch of mathematics which I am slowly trying to un-pack, I was hoping to understand in a simple way why from the simple assumptions Nelson starts with get you to the Schrödinger equation. How does a particle with some continuous but non-differentiable path give rise to interference? It seems pretty remarkable to me that you can get so close to quantum mechanics with continuous paths, I just don't see how you get interference. Maybe it won't seem so remarkable if I understand it better?

Thank you.
 
  • #4
Spinnor said:
The "natural way" for Dr. Nelson was a bunch of mathematics which I am slowly trying to un-pack, I was hoping to understand in a simple way why from the simple assumptions Nelson starts with get you to the Schrödinger equation.
Take a look at my https://arxiv.org/abs/quant-ph/0505143 Sec. 2, maybe you will understand it better then.
 
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  • #5
Spinnor said:
The "natural way" for Dr. Nelson was a bunch of mathematics which I am slowly trying to un-pack, I was hoping to understand in a simple way why from the simple assumptions Nelson starts with get you to the Schrödinger equation. How does a particle with some continuous but non-differentiable path give rise to interference? It seems pretty remarkable to me that you can get so close to quantum mechanics with continuous paths, I just don't see how you get interference. Maybe it won't seem so remarkable if I understand it better?

Thank you.
It might be worth noting that you can get interference from statistical considerations. What is required is violations of total probability (i.e. measurement probabilities depend on the context).

https://arxiv.org/abs/quant-ph/0106072
https://arxiv.org/abs/quant-ph/0403021

In fact, statistical interference is an established phenomena in psychology / social science now by this mechanism (the kind of phenomena you find in the late, great Daniel Kahneman's popular book Thinking Fast and Slow).

https://www.annualreviews.org/content/journals/10.1146/annurev-psych-033020-123501

And the relation to violations of total probability is the same in conventional quantum mechanics. Compatible observables produce no interference because there is no room for interference terms when the probabilities sum up correctly. Compatible observables do not disturb each other statistically.

If you think of it this way then it could be plausible that particle interference occurs just as a statistical effect even though they are from individual localized particles. Clearly if interference occurs in psychology for similar contextual reasons, we have a generic kind of statistical property on our hands, at least on some level.
 
  • #6
I took a quick look and wonder if and how the papers you linked would explain the simple quantum problem of a particle in a 1-D box? Thank you.
 
  • #7
Spinnor said:
I took a quick look and wonder if and how the papers you linked would explain the simple quantum problem of a particle in a 1-D box? Thank you.
Well, unfortunately none of these papers are even in the ballpark of what you are asking. I am just using them to highlight that interference may be a generic statistical effect that follows from violations of total probability as you find with incompatible observables in standard quantum mechanics. Why incompatibility? Uncertainty relations. It seems to me that that is what fundamentally stops position and momentum being "jointly measurable" as people say.
 
  • #8
iste said:
It seems to me that that is what fundamentally stops position and momentum being "jointly measurable"
What stops position and momentum from being "jointly measurable" is that they do not commute and so they do not have common eigenstates. The uncertainty relations between position and momentum are another consequence of them not commuting.
 
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  • #9
PeterDonis said:
What stops position and momentum from being "jointly measurable" is that they do not commute and so they do not have common eigenstates. The uncertainty relations between position and momentum are another consequence of them not commuting.
I understand that this is the general view. It just feels to me like uncertainty relations have an active role in why observables do not commute. I maybe have 3 points which make me lean a bit differently:

1) Work by Busch et al. seems to suggest that unsharp measurement functions can be co-measurable and even commute under the condition that they respect uncertainty relations.

https://arxiv.org/abs/quant-ph/0609185
https://arxiv.org/abs/1303.7101

2) Related to that, if you look at the Husimi distribution, a genuine joint probability distribution, you see that it approximates eigenstates (for position or momentum, not both) at mutually exclusive ends of its squeezing parameter which allocates uncertainty between observables of minimum uncertainty states. What it suggests to me is that the eigenstates are not co-measurable because sharp incompatible observables cannot satisfy uncertainty relations at the same time, hence why there is nowhere along the squeezing parameter settings that both incompatible observables can be sharp. But if observables do satisfy uncertainty relations, you can have a joint distribution, because the Husimi distribution is a genuine joint distribution, albeit for gaussian-broadened marginals. Incidentally, if you allow Wigner functions to respect uncertainty relations, they just become normal, genuine probability distributions like the Husimi.

[Mentor Note: link to archive.org deleted because of copyright violations] (Ballentine, 2014; phase space distributions chapter)
https://www.sciencedirect.com/science/article/abs/pii/037843717690145X

This kinda suggests to me that the reason why incompatible observables cannot share common eigenstates is because eigenstates are sharp and so they will naturally contradict uncertainty relations considered together. The fact that manipulating uncertainty seems to affect "joint measurability" seems to indicate that it isn't merely just a consequence, and this reads more intuitively to me.

3) In the path integral formulation, commutation relations can be derived from the continuous, non-differentiable paths. It seems you can also derive uncertainty relations generically for stochastic systems from continuous but non-differentiable trajectories:

https://www.sciencedirect.com/science/article/abs/pii/S0375960118303633

To me, this points perhaps to a deeper origin for both uncertainty and non-commutativity, or at least complicating the idea that its just as simple as "here is non-commutativity and uncertainty relations are caused by them".
 
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  • #10
iste said:
I understand that this is the general view.
It's not "the general view". It's math.

iste said:
It just feels to me like uncertainty relations have an active role in why observables do not commute.
Again, you're getting it backwards. The uncertainty relations are derived, mathematically, from the fact that the observables do not commute.

iste said:
1) Work by Busch et al. seems to suggest that unsharp measurement functions can be co-measurable and even commute under the condition that they respect uncertainty relations.

https://arxiv.org/abs/quant-ph/0609185
https://arxiv.org/abs/1303.7101
From the first paper (p. 7):

"Considering that the (sharp) position and momentum observables Q and P do not commute with each other, we recover immediately the well-known fact that these observables have no joint observable, that is, they are not jointly measurable."

This is just another way of stating what I have said: that from the fact that the observables do not commute, we derive the uncertainty relations.

However, if you pick different observables, i.e., ones that are not position or momentum (##P## or ##Q##), it is possible to pick ones that are jointly measurable (i.e., have common eigenstates), and which can still give some information about position and momentum. That is what the second paper is basically about. But what you can't do with any of this is get more joint information about position and momentum of the same quantum system, in the same state, than the uncertainty relations between position and momentum allow. None of this contradicts anything I've said.

iste said:
2) Related to that, if you look at the Husimi distribution, a genuine joint probability distribution, you see that it approximates eigenstates (for position or momentum, not both) at mutually exclusive ends of its squeezing parameter which allocates uncertainty between observables of minimum uncertainty states. What it suggests to me is that the eigenstates are not co-measurable because sharp incompatible observables cannot satisfy uncertainty relations at the same time, hence why there is nowhere along the squeezing parameter settings that both incompatible observables can be sharp.
Again, none of this contradicts anything I've said. The non-commutation of the observables is still behind all of it, because that is what requires there to be a "squeezing parameter" which can take a range of values, specifying how much of the joint uncertainty due to non-commutation of the observables you want to allocate to each one. If two observables commute, there is no "squeezing parameter" because there is no tradeoff to be made; the observables have joint eigenstates in which both can have sharp values.

iste said:
Ballentine, 2014; phase space distributions chapter
Which chapter number is that?

iste said:
3) In the path integral formulation, commutation relations can be derived from the continuous, non-differentiable paths. It seems you can also derive uncertainty relations generically for stochastic systems from continuous but non-differentiable trajectories:

https://www.sciencedirect.com/science/article/abs/pii/S0375960118303633

To me, this points perhaps to a deeper origin for both uncertainty and non-commutativity, or at least complicating the idea that its just as simple as "here is non-commutativity and uncertainty relations are caused by them".
The paper you reference here is not limited to QM, but is about a generalized formulation that covers other models--models which are less fundamental than QM (and which ultimately have to be approximations to QM). I don't think you can conclude anything about fundamentals like the "origin" of something from this.
 
  • #11
Sorry, late reply.

PeterDonis said:
It's not "the general view". It's math.
PeterDonis said:
Again, you're getting it backwards. The uncertainty relations are derived, mathematically, from the fact that the observables do not commute.

Yes, but does deriving something necessarily mean it is directly caused by it? You could argue it doesn't imply much more than an inferential relationship where if you assume something, you can demonstrate that something else holds. Similar to how correlation doesn't imply causation. I am not disagreeing about uncertainty relations being derived from non-commuting nature of observables. I think my line on this, to clarify the original post of mine you replied to, will be something like the non-commutativity and uncertainty relations are more or less tapping into the same phenomena, pwrhaps from just a different kind of angle. But the angle from uncertainty relations personally has always seemed more explanatorily interesting to me.

I think I will have to re-emphasize the point of Busch's work. He is saying that if you loosen the constraints on measurements by allowing them to be noisy or unsharp, you can have "measurements" of position and momentum which minimally disturb each other, and the reason for this is that by injecting noise into the measurements, you are making them respect uncertainty relations.

Again, I am not saying that uncertainty relations are not derived from non-commutativity. But I am arguing that the relationship is maybe more nuanced and goes both ways if, by altering uncertainty, you can minimize disturbance and produce commuting functions which are "smeared" versions of position and momentum observables. What has been identified by Busch is also essentially the same as what is being said in my point about the Husimi distribution.

PeterDonis said:
Again, none of this contradicts anything I've said. The non-commutation of the observables is still behind all of it, because that is what requires there to be a "squeezing parameter" which can take a range of values, specifying how much of the joint uncertainty due to non-commutation of the observables you want to allocate to each one. If two observables commute, there is no "squeezing parameter" because there is no tradeoff to be made; the observables have joint eigenstates in which both can have sharp values.
I understand what you're saying but my concern is this: asked what it would take to get a distribution for both eigenstates, I don't see anything stopping me from just saying "take away the minimum uncertainty constraint". That seems to be the essence of the (in)ability for observables to share common eigenstates in the Husimi picture. There are precisely two limits of the parameter where we have sharp observables of one but not the other. If there was no minimum uncertainty limit, it seems there would be nothing stopping a joint probability distribution of both sharp observables simultaneously sharing common eigenstates. When you reintroduce the uncertainty constraint, it then becomes clear that there must be disturbances purely for the reason that the eigenstates can no longer exist in the same joint probability distributions. Instead, it is only logically possible to represent eigenstate statistics in two separate contexts, and you then can kind of see why the product rule and law of total probability breaks down in quantum mechanics. Valid marginalizations for an observable can only be made in one context, not the other, whereas classical probability allows you to find marginal probabilities from a single joint probability distribution.

Injecting noise into the measurements will then minimize the disturbance because the uncertainty isn't redundant, it has an active role in determining what is allowable. With enough uncertainty you can have genuine joint distributions for position and momentum without requiring disturbance, ones that exist among allowable Husimi distributions, but which just happen to not be the distributions physicists want.

So, from what I can see, it looks like uncertainty relations have significant explanatory value in regard to non-commutativity and incompatibility. I am obviously not making rigorous statements but it just looks like to me that the minimum uncertainty constraint is having an active role in making the difference for measurability, and so I am inclined to say that the uncertainty relations and non-commutativity are just different ways of tapping into what is fundamentally the same phenomena.

PeterDonis said:
Which chapter number is that?

15.

PeterDonis said:
The paper you reference here is not limited to QM, but is about a generalized formulation that covers other models--models which are less fundamental than QM (and which ultimately have to be approximations to QM). I don't think you can conclude anything about fundamentals like the "origin" of something from this.

Well, the quantum mechanical case exists within the generalized case so it would make it seem valid to me. This generalized formulation may be applied to descriptions of systems less fundamental physically  but if quantum mechanical relations are also a special case then I think arguably it could be talking about something which is mathematically more fundamental.
 
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  • #12
iste said:
does deriving something necessarily mean it is directly caused by it?
"Caused" doesn't even apply here since we're talking about mathematical properties of models constructed by humans. I'm talking about logical derivation.

iste said:
If there was no minimum uncertainty limit, it seems there would be nothing stopping a joint probability distribution of both sharp observables simultaneously sharing common eigenstates.
But if there were no non-commutation of observables, there would be no minimum uncertainty limit.
 
  • #13
iste said:
it looks like uncertainty relations have significant explanatory value in regard to non-commutativity and incompatibility
Again you're getting it backwards. The non-commutativity is what explains the uncertainty relations and the "incompatibility".
 
  • #14
PeterDonis said:
"Caused" doesn't even apply here since we're talking about mathematical properties of models constructed by humans. I'm talking about logical derivation.

Okay, yes.
PeterDonis said:
But if there were no non-commutation of observables, there would be no minimum uncertainty limit.

PeterDonis said:
Again you're getting it backwards. The non-commutativity is what explains the uncertainty relations and the "incompatibility".

Yes, uncertainty relations are derived from the non-commutativity in quantum mechanics but from my perspective, maybe uncertainty relations can be talked about in terms of how observables do not commute which adds additional explanatory value rather than just being about a single logical derivation that just relates two components from a wider theory; I think explanatory value is broader than that. I mean, clearly a single derivation like that only has a limited purview on what's going on; for instance, with regard to third factors they both may be related to.
 
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  • #15
iste said:
from my perspective, maybe uncertainty relations can be talked about in terms of how observables do not commute
That's exactly how the derivation works: the minimum uncertainty in the uncertainty relation is computed from the commutator of the observables.

iste said:
with regard to third factors they both may be related to.
Do you have any reference to back this up? Personal speculation is off limits here.
 
  • #16
PeterDonis said:
Do you have any reference to back this up? Personal speculation is off limits here.

I didn't give any references because I already mentioned it and gave a reference earlier.
 
  • #17
iste said:
I didn't give any references because I already mentioned it and gave a reference earlier.
What "third factors" do those references give?
 
  • #18
PeterDonis said:
What "third factors" do those references give?

Sorry, I was referring to the sciencedirect reference I gave earlier about uncertainty relations in stochastic systems from sciencedirect. So the third factor is non-differentiability of the paths in path integral formulation.
 
  • #19
iste said:
the third factor is non-differentiability of the paths in path integral formulation.
But that's not true in standard QM, correct?

Discussion of alternate theories really belongs in the Beyond the Standard Models forum.
 
  • #20
PeterDonis said:
But that's not true in standard QM, correct?

Discussion of alternate theories really belongs in the Beyond the Standard Models forum.

Aha, you made me examine this more closely.

Right, It is true in standard quantum mechanics but I am not entirely sure about the specific paper. The non-differentiable paths that the uncertainty relations are derived from are an object in standard quantum mechanics and their relation to non-commutativity is even on the path integral wikipedia page, dating back to Feynmann.

I always looked at the paper I linked as just a method for deriving uncertainty relations that applies generically, but maybe it does insert some assumptions which are not strictly in standard quantum mechanics.

However, I did find an alternative paper:

https://arxiv.org/abs/math/0211071

Which does seem to detail a direct bi-directional link between Heisenberg uncertainty and non-differentiable paths. Maybe what the paper itself cannot be seen as standard quantum mechanics, but the result seems to be a section which is about standard quantum mechanics which they then want to use to prove something else.

They give other references for it in addition, and following them, this relationship seems to date back to a paper from Abbott and Wise in 1981.

https://scholar.google.co.uk/scholar?cluster=9621050886572313269&hl=en&as_sdt=0,5&as_vis=1
 

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