# What does the probabilistic interpretation of QM claim?

by A. Neumaier
Tags: claim, interpretation, probabilistic
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 Quote by meopemuk When a photon hits the photographic plate or a CCD detector it leaves a clear trace, whose position is well-defined (at least in our macroscopic world). So, it seems that there is no problem in measuring photon position with the precision of few micrometers, or so. We must have a theory, which would explain these kinds of measurements. Quantum mechanics is exactly this kind of theory, and the probabilistic interpretation of measurements of position (and all other obervables) is the cornerstone of quantum mechanics. If you deny that,
I deny that a photodetector measures the position of photons, or that a bubble chamber measures the position of charged quantum particles. (The trace is a trace of particle excitations as the result of the external charged field and a constant magnetic field in case of curved traces.) They measure the incident fields, nothing else.
 Quote by meopemuk then you invite a major revision of the entire foundation of quantum mechanics.
In my lecture http://www.mat.univie.ac.at/~neum/ms/optslides.pdf , I call this revision the thermal interpretation of quantum mechanics. It does not require the slightest alteration of quantum mechanics or quantum field theory. I only changed the currently accepted weird way of talking about quantum system (a long tradition introduced by many years of brainwashing) into one which matches common sense much better. So it is not a change in the foundations but only a change in the interpretation - one that is more consistent with the mathematics (such as the nonexistent of a photon position operator, and Wigner's theorem).
 Quote by meopemuk You do not deny the existence of photon's momentum eigenstates.
Indeed, photon momenta are measurable if they are large enough. (Only soft photons cannot be measured, because of the infrared problem.)
 HW Helper PF Gold P: 1,961 Do you deny that the position of an electron can be measured by letting it fall on a photographic plate?
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 Quote by dx Do you deny that the position of an electron can be measured by letting it fall on a photographic plate?
According to the quantum field theoretic view, position is only a field label, not an observable.

The plate responds to the field strength of the beam containing the electrons: Random atoms are ionized, with a rate proportional to the field strength. This effect that is subsequently magnified and becomes visible.

What is measured is therefore the field strength, although because of the randomness involved, the measurement becomes reliable only if the exposure is sufficiently long.
 P: 381 Mr Neumaier, I'm also suprised to read such a statement! (That the tracks in a bubble chamber are not a position measurement). I dont deny this immediately, i just want to understand what you're saying. Let me say first how i would define a position measurement. If the incident field/particle has a state |Ψ>, and expand this state on the basis of position eigenstates, then i would call a position measurement something that would make the wavefunction of the particle in the position representation "gather" around a point. So that, we will be able to say that it was here, in that box, and not in the andromeda galaxy. Knowing that the field/particle is located in a subregion of space, i think defines a position measurement. When charged fields/particles interact with the bubble chamber we see a trajectory. This trajectory has dimension, for example 0.5x0.5 mm^2 and that defines a subregion of space. I agree that what we see is the effect of the interaction of the particle with the atoms of the liquid, but there can be an interaction only if the paticle's wavefunction is nonzero at the point of the interaction with an atom. The fact that we see only a small trajectory -to my mind- means that the wavefunction of the particle is non-zero only in that subregion of space. It doesnt interact with the rest of the chamber, and its not in my house either. So, that fits my definition of position measurement, the wavefunction is 'gathered' in a subregion of space. Am i wrong? John
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On Wigner's no-go theorem for exact measurement:
 Quote by A. Neumaier The most accessible reference (in English translation) is the one I gave already; the book is a very useful reprint volume. The original is in German: Z. Phys. 133, 101-108.
The paper J. Math. Phys. 25 (1984), 79 -87 by Ozawa might also be of interest.
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PF Gold
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 Quote by A. Neumaier According to the quantum field theoretic view, position is only a field label, not an observable.

The position is of course a field label, since the quantum field is essentially the position version of momentum mode creation operators. But why does that imply that the position of an electron cannot be measured? As far as I know, the fact that momentum is a label of the creation operators for momentum modes does not imply that the momentum of an electron cannot be measured, so what's the difference?
 P: 381 In my opinion, we should first define what we mean by a 'position measurement' and then see if the tracks in a bubble chamber, for example, qualify.
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 Quote by dx The position is of course a field label, since the quantum field is essentially the position version of momentum mode creation operators. But why does that imply that the position of an electron cannot be measured? As far as I know, the fact that momentum is a label of the creation operators for momentum modes does not imply that the momentum of an electron cannot be measured, so what's the difference?
The difference is that momenta (like other conserved additive quantities) are asymptotic quantities, and quantum particles have meaning in an asymptotic sense only.

Things start to get semiclassical (where the particle concept begins to be applicable) only when field concentrations are so large that their density peaks at reasonably well-defined locations in phase space. At this point, these peaks behave like particles, and position and momentum of the peak behaves approximately classically.
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 Quote by JK423 In my opinion, we should first define what we mean by a 'position measurement' and then see if the tracks in a bubble chamber, for example, qualify.
Why don't you start with a proposal for a definition what _you_ mean by a 'position measurement'?
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 Quote by A. Neumaier Why don't you start with a proposal for a definition what _you_ mean by a 'position measurement'?
I've done it at post #22.
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PF Gold
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 Quote by A. Neumaier The difference is that momenta (like other conserved additive quantities) are asymptotic quantities, and quantum particles have meaning in an asymptotic sense only.
Could you expand on this part a bit. What's an asymptotic quantity?
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 Quote by dx Could you expand on this part a bit. What's an asymptotic quantity?
An observable still visible at times t-->inf or t-->-inf, so that scattering theory says something interesting about it. This is relevant since quantum dynamics is very fast but measurements take time. Measuring times are already very well approximated by infinity, on the time scale of typical quantum processes. Thus only asymptotic quantities have a reasonably well-defined response.

That's why information about microsystems is always collected via scattering experiments described by the S-matrix, which connects asymptotic preparation at time t=-inf with asymptotic measurement at time t=+inf.
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 Quote by JK423 I've done it at post #22.
Ah, I missed the details in that post.
 Quote by JK423 Let me say first how i would define a position measurement. If the incident field/particle has a state |Ψ>, and expand this state on the basis of position eigenstates, then i would call a position measurement something that would make the wavefunction of the particle in the position representation "gather" around a point. So that, we will be able to say that it was here, in that box, and not in the andromeda galaxy. Knowing that the field/particle is located in a subregion of space, I think defines a position measurement.
This recipe cannot cover a photon position measurement since the photon disappears upon exciting an electron. Do you want to improve upon your definition of a position measurement, or do you want to treat photons and electrons on a different footing?
 Quote by JK423 When charged fields/particles interact with the bubble chamber we see a trajectory. This trajectory has dimension, for example 0.5x0.5 mm^2 and that defines a subregion of space. I agree that what we see is the effect of the interaction of the particle with the atoms of the liquid, but there can be an interaction only if the particle's wavefunction is nonzero at the point of the interaction with an atom.
In the quantum field view, one would say that there can be a response only if the field intensity is nonzero at the point of interaction. This works independent of the number of particles present.
 Quote by JK423 The fact that we see only a small trajectory -to my mind- means that the wavefunction of the particle is non-zero only in that subregion of space. It doesnt interact with the rest of the chamber, and its not in my house either. So, that fits my definition of position measurement, the wavefunction is 'gathered' in a subregion of space. Am I wrong?
If you assume the collapse postulate, your view is consistent, as long as you don't claim that position can be measured arbitrarily well. This is just the Copenhagen interpretation.

The problem with this is that there is no known mechanism for causing the collapse. (Decoherence reduces the pure state to a mixture, but we don't observe a mixture of tracks - only a single one. This accounts correctly for the long-term average, but not of the collapse at each single instance.)

The quantum field picture doesn't need to assume a collapse; ordinary randomness is enough.
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 Quote by A. Neumaier Yes. In quantum field theory and hence in multiparticle quantum mechanics where particles are indistinguishable, position is a mere parameter, like time, that cannot be measured. Only for a single massive particle it seems to be different - but even here it causes the typical qauntum weirdness of propertyless particles suddenly materializing when measured.
I agree with you that parameter x in quantum field $$\psi(x,t)$$ has absolutely no relationship to physically measurable position. However, this does not mean that the observable of position cannot be defined in quantum field theory. We've discussed this point with you already. In any n-particle sector of the Fock space I can define a state in which one particle has position x_1, second particle has position x_2, third particle ... etc. You were correct to point out that in the case of indistinguishable particles this does not allow to form a Hermitian "particle position" operator. But the above construction of n-particle localized states is sufficient to describe position measurements in the Fock space.

You would possibly object that the Fock space is not valid for interacting particles. But this has no relevance, because we've been discussing the measurements of position of a single electron, which is not interacting with anything.

Another point is that refusing the measurability of positions you are are not saving yourself from the "weird" quantum collapse. You've mentioned elsewhere that the momentum-space wavefunction $$\psi(p)$$ does have a measurable probabilistic interpretation. So, it does require a collapse. This time in the momentum space.

 Quote by meopemuk photographic plates, bubble chambers and other not-so-sophisticated devices to do so.
 Quote by A. Neumaier These never measure particles, but macroscopic distributions of silver atoms or bubbles.
Our difference is that I believe that the blackening of silver atoms or the formation of bubbles are direct local effects of incident particles. So, by measuring positions of exposed grains of photoemulsion or bubbles we measure (albeit indirectly) positions of particles, which created these effects.

If I understand correctly, your position is that the blackened grain of photoemulsion or the formed bubble is not a proof that the particle really hit that spot. You invoke a (rather strange, in my opinion) detection theory from Mandel & Wolf, where they represent the particle by an extended continuous field. Then creation of the local photographic image or a small bubble is "explained" by a sequence of non-trivial condensation events happening in the bulk of the detector. These events require migration of charge to macroscopic distances, entanglement, and other complicated and not fully explained things.

If I understand correctly, your motivation for applying these non-trivial models of particle detection is to avoid using the quantum-mechanical wave function collapse. So, you replace the collapse with some chaotic and yet mysteriously choreographed (condensation of the originally distributed particle energy at one fixed but random point) processes inside the macroscopic detector.

Eugene.
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 Quote by A. Neumaier [...] the analysis in the book by Mandel & Wold shows that the clicks in the photodetector are produced by the photodetector already for a classical external e/m radiation field, showing that photodetection is a random measurement of the intensity of the incident radiation field, and nothing else. [...] For alpha particles, the corresponding analysi analysis is given in Mott's 1929 paper (reprinted in pp.129-134 in: Wheeler & Zurek, Quantum theory and measurement, Princeton 1983). He shows that the tracks formed in a cloud chamber are already produced by the cloud chamber in a classical external radial charged field - in which case the quantum system considered does not contain an alpha-particle at all. Thus the tracks cannot be said to measure a particle position. Instead they form a random measurement of the intensity and direction of the incident charged field, and nothing else.
To others who may be interested in studying the latter (formation of
tracks by charged particles) in more detail...

There's an extended treatment in Schiff's textbook, pp335-339. He uses
2nd-order perturbation theory to consider the probability of a fast
electron participating in an ionizing interaction with the electrons in
two separate atoms. The result is that the probability is very small
unless the atoms are on a line parallel to the momentum of the incident
electron (approximated as an incident plane wave field).

Thus, Mandel & Wolf are not the only ones who treat the subject in
this more careful way.
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 Quote by strangerep To others who may be interested in studying the latter (formation of tracks by charged particles) in more detail... There's an extended treatment in Schiff's textbook, pp335-339. He uses 2nd-order perturbation theory to consider the probability of a fast electron participating in an ionizing interaction with the electrons in two separate atoms. The result is that the probability is very small unless the atoms are on a line parallel to the momentum of the incident electron (approximated as an incident plane wave field). Thus, Mandel & Wolf are not the only ones who treat the subject in this more careful way.
strangerep,

I agree that some aspects of particle detection can be explained by Mandel & Wolf type arguments. However, there are situations, where these arguments fail completely. I think the most spectacular failure is related to electrons registered by a photographic plate. If you describe the incident electron by a plane wave or other continuous charge density field, you will have a hard time to explain how this distributed charge density condenses to a single location of one emulsion grain. I think it is well established that after "observation" the entire electron charge is located in the neighborhood of the blackened emulsion grain. Apparently, there should be a mechanism by which the distributed charge density condenses to a point and overcomes a strong Coulomb repulsion in the process. This doesn't look plausible even from the point of view of energy conservation.

Eugene.
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 Quote by A. Neumaier Yes. particle tracks _are_ important; but because they allow one to measure the momentum of a particle. But particle position is irrelevant, and doesn't exist on the quantum field level.
I'm sorry, but this statement is simply false. I just gave an example where particle position is relevant. One can mention quantum fields all one wants, but that doesn't change the fact that as a practical matter particle positions can be meaningful and useful approximations. Even in quantum field theory.
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 Quote by Physics Monkey I'm sorry, but this statement is simply false. I just gave an example where particle position is relevant. One can mention quantum fields all one wants, but that doesn't change the fact that as a practical matter particle positions can be meaningful and useful approximations. Even in quantum field theory.
I have nothing against particle positions as meaningful and useful semiclassical _approximations_, as is appropriate for particles assumed to have collapsed already, and hence described by an effective particle picture along a track. This is a change of the representation, simplifying the picture and the analysis in cases where the physics allows this.

Nevertheless, even in a track, one only has a measurement of the projection of the position on the plane transversal to the momentum.

But before the detector is reached, there is just a radially expanding quantum field for each particle kind involved in the decay (before and after), and Mott's analysis applies. The secondary bubble traces start at random positions along the track, for the same reason that the primary trace start at a random position anywhere at the surface of the detector where the field density is large enough and continues inside the detector.

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