2016 International Workshop on Quantum Observers

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Note: The following text aims to elicit more discussions about the question of whether Bohm’s theory can really solve the measurement problem when considering the status of quantum observers in the theory.

Let us now consider Bohm’s approach or the hidden-variables approach to quantum mechanics. As to Bohm’s approach, an analysis of the psychophysical supervenience is also relevant and necessary (Brown, 1996). In this approach, there are two possible forms of psychophysical supervenience. One is that the mental state supervenes on both the wave function and the additional variables such as positions of Bohmian particles. The other is that the mental state supervenes only on the additional variables such as positions of Bohmian particles.

If assuming the first form of psychophysical supervenience, then our analysis will have implications for Bohm’s approach.
On the wave function part, the mental state of a quantum observer being in a superposition is also definite, and the mental content does not correspond to either branch of the superposition. Then, even although the mental state of the observer also contains the content corresponding to the branch occupied by the Bohmian particles, the whole content does not correspond to either branch of the superposition. Therefore, in this case Bohm’s approach cannot solve the measurement problem, and is not consistent with the predictions of standard quantum mechanics either.

It is usually thought that the mental state of a quantum observer being in a superposition supervenes only on the branch of the superposition occupied by the Bohmian particles. Indeed, Bohm initially assumed this form of psychophysical supervenience. He said: the packet entered by the apparatus [hidden] variable… determines the actual result of the measurement, which the observer will obtain when she looks at the apparatus.” (Bohm, 1952, p.182). In this case, the role of the Bohmian particles is merely to select the branch from amongst the other non-overlapping branches of the superposition. Brown and Wallace (2005) called this assumption Bohm’s result assumption, and they have presented convincing arguments against it (see also Stone, 1994; Brown, 1996; Zeh, 1999; Lewis, 2007). In our view, the main problem with this assumption is that the occupied branch and other empty branches have the same ontological status and ability to be supervened by the mental state. Moreover, although it is imaginable that the Bohmian particles may have influences on the occupied branch, e.g. disabling it from being supervened by the mental state, it is hardly imaginable that the Bohmian particles have influences on all other empty branches, e.g. disabling them from being supervened by the mental state.

On the other hand, if assuming the second form of psychophysical supervenience, namely assuming the mental state supervenes only on the positions of Bohmian particles, then our analysis will have no implications for Bohm’s approach, and it seems that the above inconsistency problem can also be avoided. Indeed, most Bohmians today seem to support this assumption, though they often did not state it explicitly (see, e.g. Maudlin, 1995).
However, it has been argued that this assumption is inconsistent with the functionalist approach to consciousness (Brown and Wallace, 2005; see also Bedard, 1999), and moreover, the assumption also leads to a serious problem of allowing superluminal signaling (Brown and Wallace 2005; Lewis, 2007). In our view, this problem is not as deadly as the inconsistency problem, since such superluminal signaling may exist in principle, and its existence is not inconsistent with existing experience either (Gao, 2004).

The problem with this assumption is still the inconsistency problem. Here is an argument. Consider again an observer being in the superposition:

\alpha \ket{up}_S \ket{up}_M+\beta \ket{down}_S \ket{down}_M.

In Bohm’s approach, the Bohmian particles of the observer reside in one branch of the superposition after the measurement, which indicates that the observer obtains the result corresponding to the branch.
For example, when the Bohmian particles of the observer reside in the branch $\ket{up}_M$ after the measurement, the observer obtains the x-spin up result; while when the Bohmian particles of the observer reside in the branch $\ket{down}_M$ after the measurement, the observer obtains the x-spin down result.

Now suppose these two post-measurement situations appear in two somewhat different experiments so that the Bohmian particles of the two observers in these two experiments are located in the same positions.\footnote{When each observer has only one Bohmian particle, this can always be realized by space translation. When each observer has more than one Bohmian particle, it is also possible that the relative positions of the Bohmian particles of the two observers in the two experiments are the same, since each Bohmian particle can be in any position in the region where the corresponding wave function spreads, which is the whole space in realistic situations. Then by space translation the Bohmian particles of the two observers can also be located in the same positions.} Then if assuming the second form of psychophysical supervenience, namely assuming the mental state supervenes only on the positions of Bohmian particles, then these two post-measurement situations will represent the same measurement result. But this is not the case; in the first situation the observer obtains the x-spin up result, while in the second situation she obtains the x-spin down result.\footnote{This point was also emphasized by Barrett (1999, p.123). He said:the content of a measurement record in Bohm’s theory is determined by the position of something \emph{relative} to the wave function – that is, a different wave function and the same position might produce a different record.” My analysis suggests that in order to solve this serious problem of psychophysical supervenience a wholly new hidden-variables theory is needed.}

This analysis also raises a doubt about the whole strategy of Bohm’s approach to solving the measurement problem. Why add hidden variables such as positions of Bohmian particles to quantum mechanics? It has been thought that adding these variables which have definite values at every instant is enough to ensure the definiteness of measurement results and further solve the measurement problem.
However, if the mental state cannot supervene on these additional variables, then even though these variables have definite values at every instant, they are unable to account for our definite experience and thus do not help solve the measurement problem.

[Peter J. Lewis]

Thanks for the interesting post, Shan. Let me attempt to respond to your main argument. It is true that simply specifying the position of a Bohmian particle doesn’t pick out a measurement outcome. But neither does specifying the position of a classical particle. You need to specify the position of the particle relative to the apparatus — if it hits the top half of the detection screen it is spin up, and if it hits the bottom half it is spin down. But a Bohmian can interpret the screen as a set of particle positions too. That is, the outcome supervenes on a pattern among a number of particles . (And probably on the behavior of the particles over time, too.)

[Quantum Speculations]

Thanks for your insightful comments, Peter. I should have made my argument more accurate. I agree that if assuming the measurement outcome supervenes on a pattern among a number of Bohmian particles, then Bohm’s theory can solve the measurement problem. But I think the assumption seems inconsistent with SQM. Let me restate my argument more clearly.

Consider a simple example in which a measuring device is composed of two Bohmian particles. In this case, the relative position of the two particles may indicate the measurement outcome. Now consider two post-measurement situations in two experiments: in one situation the two Bohmian particles reside in the branch $\ket{up}_M$ after the measurement, and the experiment obtains the x-spin up result; while in the other situation the two Bohmian particles reside in the branch $\ket{down}_M$ after the measurement, and the experiment obtains the x-spin down result. Since each Bohmian particle can be in any position in the region where the corresponding wave function spreads, which is the whole space in realistic situations, it is always possible that the relative positions of the two Bohmian particles in the two post-measurement situations are the same. This is irrelevant to the overlap of the two branches $\ket{up}_M$ and $\ket{down}_M$. Then if assuming the relative position of the two Bohmian particles indicates the measurement outcome, then these two experiments will obtain the same result. But this is not the case; the first experiment obtains the x-spin up result, while the second experiment obtains the x-spin down result.

In my view, in order to solve this problem a new hidden-variables theory is needed, which replaces the wave function with some other variables and thus will be quite different from Bohm’s theory. However, since there are already several proofs of the reality of the wave function, such as my recent argument in terms of protective measurements, this seems to be not a promising solution either.

[Peter J. Lewis]

OK, good. But I guess I’m inclined to stick to my guns here — the relative positions of the particles determine the measurement outcome. Suppose the first particle is the one whose spin is being measured, and the second particle marks the top of the detection screen. Then if the two particles end up close together, the incoming particle is spin-up, and if they end up far apart it is spin-down.

Your challenge is that since the spin-up and spin-down wave packets extend over the whole of space, the two particles might be associated with the spin-down wave packet even if they are close together. But I don’t know how to associate a Bohmian particle with a wave packet. A Bohmian particle is just a location in (configuration) space, and as such is influenced by all wave packets that are non-zero at that point. The particles aren’t associated with either packet, and neither packet picks out the “actual” measurement result (the particle configuration performs that job).

I think maybe the rhetoric of “empty” and “occupied” wave packets in the context of Bohm’s theory (which I’m as guilty as anyone for using) is misleading, and should be retired.

[Aurelien Drezet]

Dear Shan dear all, I gave a kind of reply to Shan’s work concerning psycho physical parallelism in the topic ‘The End of the Many-Worlds? (or Could we save Everett’s interpretation)’

with best regards Aurélien

[Quantum Speculations]

Thanks, Aurélien. I will have a look at your reply. Best, Shan

[Quantum Speculations]

Thanks for your further clarification, Peter. I still think that your assumption that the relative positions of the particles determine the measurement outcome requires a wholly new hidden-variables theory which is different from Bohm’s theory.

Consider your example. If after being measured by the Stern-Gerlach magnet, the two separated wave packets of the first particle are reflected by two mirrors so that their vertical positions are exchanged (during which the first particle in one packet will also be reflected along with the packet, and the position of the second particle is not changed), then according to your assumption, the measurement result will be changed. But this is not the case. The key point, in my view, is that the spin-magnetic field interaction is described by the wave function, and thus the measurement result being spin-up or spin-down is essentially encoded in the wave function too. (I think this also means that your assumption requires a new theory, in which, for example, the spin-magnetic field interaction is directly described by the particles.)

In addition, I think the concepts of “empty” and “occupied” wave packets are still valid at least in the context of measurement. For one, the Bohmian particle residing in one of many separated wave packets will ensure that the measurement record can be stable. If all wave packets greatly overlap and interfere with each other, then the motion of the Bohmian particles in the overlap region will in general be very chaotic and not stable, and thus their relative position can hardly represent a stable measurement record.

Finally, even if your assumption is true, you still need a detailed theory to give the corresponding relationship between different relative positions of the Bohmian particles and different measurement outcomes. In my view, this theory will be very different from SQM and Bohm’s theory, since the wave function will play minor or even no role in determining the appearance of a measurement outcome in the theory.

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