Bohmian trajectories are no longer "hidden variables"
Posted Jun6-11 at 09:18 AM by Demystifier
The recent weakly measured photon trajectories
http://www.sciencemag.org/content/332/6034/1170.full [Science 3 June 2011: Vol. 332 no. 6034 pp. 1170-1173 DOI: 10.1126/science.1202218]
http://scienceblogs.com/principles/2...erfere_obs.php
coincide with the Bohmian trajectories. Hence, the Bohmian trajectories should no longer be considered "hidden variables". Now they are no longer more "hidden" than the wave function.
To avoid confusion, this experiment does not prove that the Bohmian INTERPRETATION is right. After all, this experiment does not reveal a Bohmian trajectory of a single particle. Instead, this trajectory can only be measured through a large number of measurements on an ensemble of equally prepared systems. But this is fully analogous to the wave function. A single measurement cannot measure the wave function either. Only a large number of measurements on an ensemble of equally prepared systems can determine the wave function of the system. Thus, from an experimental point of view, Bohmian trajectories are neither more nor less real than the wave function. Hence, if the wave function is not thought of as a hidden variable, then the Bohmian trajectory should not be thought of as a hidden variable either.
Of course, we still have different interpretations alive. For example, the Bohmian interpretation claims that such Bohmian trajectories exist even at the level of single systems. The Copenhagen interpretations denies it. Likewise, many-world interpretation and some variants of the "Copenhagen" interpreation claim that a wave function exists even at the level of single systems. The statistical ensemble interpretation denies it. Yet, no direct measurement is able to say which interpretation is correct. Or at least, not yet.
http://www.sciencemag.org/content/332/6034/1170.full [Science 3 June 2011: Vol. 332 no. 6034 pp. 1170-1173 DOI: 10.1126/science.1202218]
http://scienceblogs.com/principles/2...erfere_obs.php
coincide with the Bohmian trajectories. Hence, the Bohmian trajectories should no longer be considered "hidden variables". Now they are no longer more "hidden" than the wave function.
To avoid confusion, this experiment does not prove that the Bohmian INTERPRETATION is right. After all, this experiment does not reveal a Bohmian trajectory of a single particle. Instead, this trajectory can only be measured through a large number of measurements on an ensemble of equally prepared systems. But this is fully analogous to the wave function. A single measurement cannot measure the wave function either. Only a large number of measurements on an ensemble of equally prepared systems can determine the wave function of the system. Thus, from an experimental point of view, Bohmian trajectories are neither more nor less real than the wave function. Hence, if the wave function is not thought of as a hidden variable, then the Bohmian trajectory should not be thought of as a hidden variable either.
Of course, we still have different interpretations alive. For example, the Bohmian interpretation claims that such Bohmian trajectories exist even at the level of single systems. The Copenhagen interpretations denies it. Likewise, many-world interpretation and some variants of the "Copenhagen" interpreation claim that a wave function exists even at the level of single systems. The statistical ensemble interpretation denies it. Yet, no direct measurement is able to say which interpretation is correct. Or at least, not yet.
Total Comments 5
Comments
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The notion that the statistical ensemble interpretation denies definite trajectories seems a bit more subtle than implied here. If you assume a wave only model of the particle zoo the ensembles may not even refer to these particles, such as photons trajectories, but rather the substructure of them. Yet a trajectory can still be maintained in the way a soliton can have a definable trajectory. It does not even make sense to talk about an ensemble interpretation of photons that self interfere if the photons themselves define the elements of the ensemble. That is empirically ruled out by very basic double slit experiments with particles slowed to one at a time. Ensemble interpretations can only refer to a presumed substructure of photons and other particles.
As Mermin has noted it still implies a classical interpretation of probability, yet that is often the point for many who adhere to it. The bigger question is if such elements exist to define single particles as ensembles where are these elements? Perhaps it has something to do with the fact that between interactions the elements are by definition independent variables. Which by definition cannot be empirically measured directly, or else they would not be independent. Even when an interaction does occur it would only empirically appear as a point-like momentum fluctuation in space rather than revealing a pair of elements. Thus the particle and the wavefunction would be essentially one and the same thing.
This is no claim of how things are, rather it is to illustrate the over-simplicity of assuming the ensemble interpretation refers to standard particles as the singular elements of the ensemble, and the notion of the definable trajectories of such.Posted Jun8-11 at 02:16 AM by my_wan 
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To understand what weak measurement is, the following analogy from everyday life is useful.
Assume that you want to measure the weight of a sheet of paper. But the problem is that your measurement apparatus (weighing scale) is not precise enough to measure the weight of such a light object such as a sheet of paper. In this sense, the measurement of a single sheet of paper is - weak.
Now you do a trick. Instead of weighing one sheet of paper, you weigh a thousand of them, which is heavy enough to see the result of weighing. Then you divide this result by 1000, and get a number which you call - weak value. Clearly, this "weak value" is nothing but the average weight of your set of thousand sheets of papers.
But still, you want to know the weight of a SINGLE sheet of paper. So does that average value helps? Well, it depends:
1) If all sheets of papers have the same weight, then the average weight is equal to weight of the single sheet, in which case you have also measured the true weight of the sheet.
2) If the sheets have only approximately equal weights, then you can say that you have at least approximately measured the weight of a single sheet.
3) But if the weights of different sheets are not even approximately equal, then you have not done anything - you still don't have a clue what is the weight of a single sheet.
But what if you don't even know whether 1), 2) or 3) is true? Then you have different interpretations of your weak measurement. And that is precisely the case with quantum mechanics: We don't know whether particles have even approximately equal velocities at the same position (with the same wave function), so we have different interpretations. Bohmian interpretation says they have exactly equal velocities, which corresponds to the case 1), while Copenhagen interpretation corresponds to the case 3).Posted Jun8-11 at 10:39 AM by Demystifier
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Thanks, great explanation!Posted Jun13-11 at 05:02 AM by DevilsAvocado
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Today appeared the first proposal to weakly measure Bohmian particle trajectories of two entangled particles:
http://lanl.arxiv.org/abs/1207.2794Posted Jul13-12 at 04:08 AM by Demystifier
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That paper was recently published in Phys. Rev. Lett. and mentioned in ScienceDaily:Quote:Quote by Demystifier
Today appeared the first proposal to weakly measure Bohmian particle trajectories of two entangled particles:
http://lanl.arxiv.org/abs/1207.2794
Proposal to Observe the Nonlocality of Bohmian Trajectories with Entangled Photons
http://prl.aps.org/abstract/PRL/v110/i6/e060406
Researchers Explore Quantum Entanglement
http://www.sciencedaily.com/releases...0208110253.htmPosted Feb14-13 at 09:24 PM by bohm2
