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 hiddenvariables 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 nonoverlapping 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 xspin up result; while when the Bohmian particles of the observer reside in the branch $\ket{down}_M$ after the measurement, the observer obtains the xspin down result.
Now suppose these two postmeasurement 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 postmeasurement situations will represent the same measurement result. But this is not the case; in the first situation the observer obtains the xspin up result, while in the second situation she obtains the xspin 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 hiddenvariables 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.

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