Is the Born Rule Justifiable in Particle Detection?

[Jilang]

i have been trying to obtain a schema for the Born rule and have got to this result.
The incidence of detection at a point (x,t) = the incidence of the particle being at point (x,t) multiplied by the incidence of the detector being at point (x,t) on detection. The incidence of the second term is proportional to the probability of the the particle being at (x,t). Is this justifiable?

 

[houlahound]

Diagram?

 

[Jilang]

Yes please, if you can it would be helpful.

 

[bhobba]

It's got nothing to do with diagrams.

The justification for the Born Rule is non contextuality then you can use Gleason:
http://www.kiko.fysik.su.se/en/thesis/helena-master.pdf

Thanks
Bill

 

[vanhees71]

To put it less complicated. The justification for Born's rule simply is that it works. Today, there's seems to be no way to derive Born's rule from the other postulates of QT, and that's why it's taken as an independent postulate. To a certain extent, Gleason's theorem, mentioned by bhobba above shows that it's the only way to define probabilities in terms of a (rigged) Hilbert space as used in QT as the mathematical foundation of quantum theory. The meaning of states is given by Born's probabilistic rule and in my opinion nothing else and also nothing else is needed to do physics. You can browse this forum to find gigantic threads on "interpretation", leading to no conclusion. The good thing is that these debates are unnecessary for physics. It's about philosophy and epistemology, not about the hard observable facts described by QT as a physical theory, and that's just the minimal statistical interpretation. So far there's no observational fact hinting at something missing in QT.

The only fundamental physical problem concerning QT is that we are still lacking a consistent description of gravity within QT, but I don't think that the solution will be an issue about interpretation. One needs a new ingenious physical idea and/or new observations hinting in the right direction of model/theory building, not philosophical pseculations, which never have lead to any results in the history of physics.

 

[ShayanJ]

not philosophical speculations, which never have lead to any results in the history of physics.

I appreciate your good post but this sentence doesn't seem right to me! If there was no EPR, there was no Bell's theorem either! At their time, EPR were doing philosophical speculations and years later, with Bell's theorem, it was revealed that there was a little physics there. How can you be sure that there is no physics in things that you now call philosophy?(Of course, people can go too far in that direction!)
Sure, you see no problem with QM, but it was the same with EPR. People were ok with QM but still EPR led to new insights!

 

[Jilang]

This is not quite what I am asking. Would it not be correct to interpret the probability of detection being In part due to the location of the particle and in part due to the location of the detector?

 

[ShayanJ]

This is not quite what I am asking. Would it not be correct to interpret the probability of detection being In part due to the location of the particle and in part due to the location of the detector?

What does "being in part due to" even mean?
Quantum mechanics is contextual which means what you measure depends on your process of measurement but I don't think simply moving your detector non-trivially changes anything!

 

[Jilang]

Well if you move the detector to a region where the amplitude of the particle is low there is less chance of detection.

 

[ShayanJ]

Well if you move the detector to a region where the amplitude of the particle is low there is less chance of detection.

Yeah, but I said "non-trivially", and that's what I call a trivial change. Because of course that happens, whether we're talking about classical or quantum physics.

You haven't answered my question!

What does "being in part due to" even mean?

 

[Jilang]

Sure, but doesn't the Born rule also apply to macroscopic entities?

 

[ShayanJ]

Sure, but doesn't the Born rule also apply to macroscopic entities?

If you don't use quantum mechanics to describe them, no!
What's your point anyway?

 

[Jilang]

But can't they also be described by QM?

 

[ShayanJ]

But can't they also be described by QM?

Yes, but then you have to consider the interaction of your measurement device with its environment which induces decoherence and makes things complicated.
I'm not sure where you're going with this but I'm sure you need to learn more before you can continue.

 

[bhobba]

This is not quite what I am asking. Would it not be correct to interpret the probability of detection being In part due to the location of the particle and in part due to the location of the detector?

What location of the particle? Why do you think in QM a particle has the property of location independent of measurement?

Thanks
Bill

 

[Jilang]

What location of the particle? Why do you think in QM a particle has the property of location independent of measurement?

Thanks
Bill

The unmeasured location as a probability distribution as specified by the wave function. The measured location is a function of both.

 

[bhobba]

The unmeasured location as a probability distribution as specified by the wave function. The measured location is a function of both.

That's an error in your reasoning. Without being measured the formalism says nothing - it may not even have the property of location. Think of a spinning coin when you toss it - it doesn't even have the property of heads up or down until it is 'measured' ie hits the ground. BTW is just an analogy - one side is still heads and the other tales - it isn't like that in QM - it literally may not have the property of head's or tales until measured in QM. Unmeasured location is an interpretive assumption you are making. Once you understand in QM what's going on when not measured is the area of interpretations and not the QM formalism things will be a lot clearer - weirder - yes - but clearer.

To understand QM better studying actual interpretation's IMHO will help a lot.

Consistent histories is a good place to start - but its just a start:
http://quantum.phys.cmu.edu/CHS/histories.html

Thanks
Bill

 

[Jilang]

Thanks Bill, it looks like in CH the Born rule is also a postulate.

 

[vanhees71]

I appreciate your good post but this sentence doesn't seem right to me! If there was no EPR, there was no Bell's theorem either! At their time, EPR were doing philosophical speculations and years later, with Bell's theorem, it was revealed that there was a little physics there. How can you be sure that there is no physics in things that you now call philosophy?(Of course, people can go too far in that direction!)
Sure, you see no problem with QM, but it was the same with EPR. People were ok with QM but still EPR led to new insights!

The EPR paper is clearly a physics paper, not philosophy. However, even Einstein himself didn't like it too much, and he wrote a much better one with him as single author to make his point of view clear. His issue was the "inseparability" of far-distant parts of a quantum system described by entanglement. Bell's great achievement was to make the philosophical assumption that the world should be separable in Einstein's sense a physically decidable question by showing that any deterministic local hidden-variable theory implies his famous inequality, which is violated by QT, and all corresponding "Bell tests" show that the inequality is violated and that the predictions of QT prevail. This rules out the validity of any deterministic local hidden-variable theory.

Einstein's paper is unfortunately in German, and I don't know, whether there is an English translation:
A. Einstein, Quantenmechanik und Realität, Dialectica 2, 320 (1948)

 

[Jilang]

Well, that is kind of pertinent to my original question. Would not the correlations between the observations have as much to do with the correlations between the detectors as between the states of the particles?

 

[bhobba]

Well, that is kind of pertinent to my original question. Would not the correlations between the observations have as much to do with the correlations between the detectors as between the states of the particles?

The detectors are not correlated.

Thanks
Bill

 

[Jilang]

The detectors are not correlated.

Thanks
Bill

Are they not aligned at certain relative angles in some experiments?

 

[bhobba]

Are they not aligned at certain relative angles in some experiments?

Of course you have to have them 'aimed' at what you are detecting - in that sense they are of course 'correlated', but that is not the usual context in such experiments. I am sure some experimenter can devise a setup where the detector is spherical - its just a daft thing to do and would, obviously, have zero effect on its outcome.

Thanks
Bill

 

[vanhees71]

This is the famous question about loop holes, i.e., indeed it could be that the detectors are causing the correlations rather than the state of the measured objects itself. As far as I know, all loop holes are fixed now, but not all in one single experiment. However, perhaps the experts about this issue in this forum should review the experimental status. Maybe I'm not up to date :-).

 

[Jilang]

To understand QM better studying actual interpretation's IMHO will help a lot.

Consistent histories is a good place to start - but its just a start:
http://quantum.phys.cmu.edu/CHS/histories.html

Thanks
Bill

Following your tip, I have been studying consistent histories and the trade off appears to be that although it respects realism, there is a stochastic process at work. Under this formalism I would regard the probability of finding a particle at a point X for a short period of time as being proportional to the amplitude at point X and also the amplitude of finding the particle at points X + dx adjacent to it. In the limit as dx goes to zero this amplitude becomes proportional to the amplitude of the particle at point X. I think I am happy that this would result in the Born rule.

 

[rkastner]

To put it less complicated. The justification for Born's rule simply is that it works. Today, there's seems to be no way to derive Born's rule from the other postulates of QT, and that's why it's taken as an independent postulate..

One can indeed derive the Born Rule as a natural consequence of the direct-action picture of fields (in which both the emission of a quantum and the absorption of that quantum are required for what counts as 'measurement'). This is shown explicitly here: https://arxiv.org/abs/1711.04501
In a nutshell: you must multiply the amplitudes for emission and absorption to get the amplitude for the entire process (which is the only way that radiative processes occur in the direct action theory), and of course that total amplitude turns out to be the Born probability. If one only considers part of the process (either emission OR absorption), then that gives you the probability for either one. The reason this is not derivable in the 'standard' QT approach is because it tacitly assumes that emission OR absorption occur unilaterally, and thus neglects the additional complex conjugate factor, so one has to put it in as an ad hoc postulate.

 

[Jilang]

Thanks Ruth! ; )