Can a Point-Like Particle Defy the Uncertainty Principle?

[mikeyork]

Imagine a spatial frame of reference attached to a point-like particle. It has x=0 since it is at the origin and p=0 since it is at rest. Having definite position and momentum is normally considered a violation of the uncertainty principle. How would you resolve this paradox?

1. Position frames and momentum frames are not the same. I.e. there is no such thing as a common "spatial" frame, because such a frame would imply a common eigenstate for both position and momentum representations in the frame indicated above.
2. This special case is an exception.
3. The uncertainty principle applies only to actual measurements.
4. There is no such thing as a point particle.
5. Other.

I have my own resolution of this paradox, centered on (3), but the implications are many and complex and would probably be considered speculative (though I think they are obvious and focused on the distinction between translations and boosts and their unitary representations*) so I won't relate it here, but I'd love to hear what others think.

*If anyone wants to know my resolution, pm me.

 

[Arman777]

I don't think its possible to choose such frame of referance. You can't set a point which point particle is rest. Its just not possible due to QM affect so I don't think its a paradox.

 

[DrClaude]

It has x=0 since it is at the origin

This doesn't make sense. You define a frame such that the particle is at rest in that frame. It doesn't mean that the quantum particle is found at a single point in that frame.

 

[Demystifier]

1. is correct.
3. and 4. are correct in some interpretations of QM, but not all.

 

[mikeyork]

This doesn't make sense. You define a frame such that the particle is at rest in that frame. It doesn't mean that the quantum particle is found at a single point in that frame.

I defined it as attached to the particle (at the origin).

 

[Nugatory]

I defined it as attached to the particle (at the origin).

A frame of reference is a rule for assigning coordinates to events in spacetime; it's not "attached" to anything. We can perform a measurement on the particle, get some position out of that measurement, and declare that position at the moment of measurement to be the origin of our coordinate system. However, that doesn't attach the frame to the particle and there's no reason why a subsequent measurement of the particle should find it at the origin.

 

[mikeyork]

A frame of reference is a rule for assigning coordinates to events in spacetime; it's not "attached" to anything. We can perform a measurement on the particle, get some position out of that measurement, and declare that position at the moment of measurement to be the origin of our coordinate system. However, that doesn't attach the frame to the particle and there's no reason why a subsequent measurement of the particle should find it at the origin.

If you don't like the word "attached", think of it as a location of the origin. An observer has a frame in which they are at the origin and at rest in that frame. That is what I mean by attached. But a spatial frame of reference, does not need an observer, it just needs a location and a path for that location. If we think of the particle as having its own frame of reference (as if it were its own "observer"), then it is both at the origin and at rest in that frame and we can think of the frame as moving with the particle so that the statements "at the origin" and "at rest" are time independent.

 

[Ben Wilson]

The paradox is resolved by not breaking it in the first place. You began with "I have broken the uncertainty principle" and end with "now how does this reconcile with the uncertainty principle?". No such frame exists for electrons, which is the meaning of the principle.

By which i mean, when you say "p=0" - p does not equal 0. I think the big issue missing here is how momenta and space are linked - TIME.

 

[mikeyork]

No such frame exists for electrons, which is the meaning of the principle.

So, you are saying that an electron (1) cannot have its own frame of reference and (2) that there is no transformation (translation, boost or acceleration) relating it to the frame of its observer?

 

[Ben Wilson]

So, you are saying that an electron (1) cannot have its own frame of reference and (2) that there is no transformation (translation, boost or acceleration) relating it to the frame of its observer?

generally, yes. Take the double slit experiment: Which slit did the electron's frame of reference move through?

 

[mikeyork]

generally, yes. Take the double slit experiment: Which slit did the electrons frame of reference move through?

Only the electron and the slit know that. There is no other observer unless you place one at the slit.

 

[Ben Wilson]

Only the electron and the slit know that. There is no other observer unless you place one at the slit.

I must correct you in that the slit and electron do not know either, hence the interference pattern (see 2:10 in vid).

'Attatched to", "located at", "has a frame orginating from" etc. are different terminologies you are using for essentially the same thing. The point of unc princ is that you can't do any of those things if you want to have knowledge of electron momentum p no matter how you choose to paramaterise the spatial geometry.

 

[mikeyork]

I must correct you in that the slit and electron do not know either, hence the interference pattern.

The interference pattern results from the projection of the prepared state onto the position basis of the screen. This takes place in the frame of the screen. The electron always knows exactly where it is in its own frame (i..e. at the origin). (Remember that a frame transformation is a unitary transformation and therefore can convert an eigenstate into a superposition or vice versa. So the unitary transformation representing the frame change from electron to screen will actually transform an eigenstate of position in the electron's frame into a superposition of positions in the screen's frame and therefore generating the observable interference pattern.)

 

[Ben Wilson]

The interference pattern results from the projection of the prepared state onto the position basis of the screen. This takes place in the frame of the screen. The electron always knows exactly where it is in its own frame (i..e. at the origin). (Remember that a frame transformation is a unitary transorfmation and therefore can convert an eigenstate into a superposition or vice versa. So the unitary transformation representing the frame change from electron to screen will actually transform an eigenstate of position in the electron's frame into a superposition of positions in the screen's frame and therefore generating the observable interference pattern.)

Apologies for the patronising childishness of the vid, i just wanted the pattern haha. I don't think that's correct but there's a lot of jargon to break through.

 

[mikeyork]

Ben, I think your position is option (1) -- that there is no "spatial" fame as such and that position and momentum frames are distinct. This seems to be the popular position of others who have responded.

My criticism of this position is as follows:

Position and momentum eigenstates are related by unitary transformations. But so are frame transformations. If position representations and momentum representations both span the entire Hilbert space, this suggests that there is also a corresponding frame transformation that takes a position frame into a momentum frame. So what is this physically?

 

[Ben Wilson]

I don't see the relevance of that math, i apologise. the way i see it is you're trying to model electrons as little flying axes, but whether you label it as a point particle or flying axes you come unstuck because the position IS undefined, not 0. This is because if I'm in a different frame, and want to transform to the electron frame, then I'm going to need an operator that transforms [undefined] to [0] which i can't see having any nontrivial use? or maybe I'm mistaken

Maybe you can assign a co moving reference frame centered on a quantum wave packet, but not a point electron at position x=R.

 

[mikeyork]

I'm going to need an operator that transforms [undefined] to [0]

No. You need an operator that transforms a superposition to an eigenstate. This is a unitary operator.

 

[Nugatory]

So, you are saying that an electron (1) cannot have its own frame of reference and (2) that there is no transformation (translation, boost or acceleration) relating it to the frame of its observer?

"The frame of its observer" and "its own frame of reference" are common but sloppy ways of saying "a frame in which the electron is at rest" and "a frame in which the observer is at rest", respectively. It's easier to speak this way and the sloppiness generally doesn't matter, which is why we often use these not quite precise terms, but sometimes it can mislead - and this is one of those times.

There is no frame in which the electron is at rest in the sense that repeated position measurements would the same result. There is a frame in which the expectation value of the second position measurement does not change with time and is equal to the originally measured position, and we could reasonably agree to call that frame "the electron's own frame of reference". However, in that frame there is uncertainty about the electron's position; the expectation value is constant, but a position measurement doesn't in general yield that result.

 

[mikeyork]

"The frame of its observer" and "its own frame of reference" are common but sloppy ways of saying "a frame in which the electron is at rest" and "a frame in which the observer is at rest", respectively.

Or "The electron is at the origin" and "the observer is at the origin". You can't specify which until you specify whether you are measuring position or momentum. In the absence of measurement, what is the distinction?

There is no frame in which the electron is at rest in the sense that repeated position measurements would the same result. There is a frame in which the expectation value of the second position measurement does not change with time and is equal to the originally measured position, and we could reasonably agree to call that frame "the electron's own frame of reference". However, in that frame there is uncertainty about the electron's position; the expectation value is constant, but a position measurement doesn't in general yield that result.

Again this argument is one about measurement. It seems to me you are invoking option (3) and/or (4) but interpreting it as option (1) when they are not the same thing.

I have no disagreement with option (3). I do however have doubts about option (1).

 

[Khashishi]

Aharonov and Kaufherr (1984) covers this very topic
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.30.368

They define quantum reference frames. They write,
"The reference frames...can be thought of as laboratories containing rulers, clocks, etc., all of which are rigidly attached to the walls of the laboratory."
The laboratories are finite mass, and therefore the laboratory obeys the uncertainty principle when viewed from another external reference frame.

I haven't perused the paper yet, but they present certain paradoxes and claim to resolve them.

 

[mikeyork]

Aharonov and Kaufherr (1984) covers this very topic
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.30.368

They define quantum reference frames. They write,
"The reference frames...can be thought of as laboratories containing rulers, clocks, etc., all of which are rigidly attached to the walls of the laboratory."
The laboratories are finite mass, and therefore the laboratory obeys the uncertainty principle when viewed from another external reference frame.

I haven't perused the paper yet, but they present certain paradoxes and claim to resolve them.

Thanks very much for this. I was aware that Aharonov had written extensively on quantum frames but have always had difficulty accessing his papers. Do you have an electronic version of the paper you recommend? It's always difficult justifying paying for a paper when you don't really know very much about the relevance of its contents.

 

[mikeyork]

vanhees71,

Have you been watching this thread? I strongly suspect you would have something interesting to say.

 

[Ben Wilson]

Thanks very much for this. I was aware that Aharonov had written extensively on quantum frames but have always had difficulty accessing his papers. Do you have an electronic version of the paper you recommend? It's always difficult justifying paying for a paper when you don't really know very much about the relevance of its contents.

What they show is the interesting properties that arise from assigning such a coordinate system to a quantum system, resulting from uncertainty of momentum p with a fully determined position x=0.

here are some freebies: https://arxiv.org/abs/quant-ph/0610030 , https://arxiv.org/abs/quant-ph/0612126

reference frames are quantized and [x,p]≠0.

 

[Ben Wilson]

"The frame of its observer" and "its own frame of reference" are common but sloppy ways of saying "a frame in which the electron is at rest" and "a frame in which the observer is at rest", respectively. It's easier to speak this way and the sloppiness generally doesn't matter, which is why we often use these not quite precise terms, but sometimes it can mislead - and this is one of those times.

Why i like the term co-moving frame which lends well to the implausibility of co-moving with the electron. But I'm sure this is also an inadequate piece of language for qm.

 

[fresh_42]

I have my own resolution of this paradox, centered on (3), but the implications are many and complex and would probably be considered speculative (though I think they are obvious and focused on the distinction between translations and boosts and their unitary representations*) so I won't relate it here, but I'd love to hear what others think.

We don't discuss personal theories on PF.

*If anyone wants to know my resolution, pm me.

Neither do we support hidden parallel platforms.

Thread closed.