Take another look at Figure 2.6.3.
As said, what we see in these images is neither the nucleus nor the electron but the fuzzy relative position between the electron and the nucleus in various stationary states of atomic hydrogen. Nor do we see this fuzzy position “as it is.” What we see is the plot of a position probability distribution, which defines the fuzzy relative position between the electron and the nucleus. It defines it by assigning probabilities counterfactually, to the possible outcomes of unperformed measurements.
To see what I mean by this, imagine a small region V in the imaginary space of sharp positions relative to the proton, well inside the probability distribution associated with the electron’s fuzzy position relative to the proton. As long as this distribution is the correct probabilistic description of the electron’s fuzzy position, the electron is neither inside V nor outside V. For if it were inside, the probability of finding it there would be 1, and if it were outside, the probability of finding it there would be 0, neither of which is the case.
If, on the other hand, we were to ascertain (by way of a measurement) whether the electron was inside V or outside V, we would find that it was either inside V or outside V. We would change the electron’s fuzzy position relative to the proton. Hence if we want to quantitatively describe a fuzzy position, we must assume that measurements are made (inasmuch as we describe it in terms of the respective probabilities of finding the electron in regions such as V). And if we do not want to change it in the process of describing it, we must assume that no measurement is made. In other words, we must describe it by assigning probabilities to the possible outcomes of unperformed measurements.
The fact that fuzzy observables are both quantified and defined by assigning probabilities to the possible outcomes of (unperformed) measurements, goes a long way towards explaining why quantum mechanics is a probability calculus, and why measurements are the events to which its probabilities are assigned. It also shows that Bell’s criticism was beside the point. “To restrict quantum mechanics to be exclusively about piddling laboratory operations is to betray the great enterprise,” he wrote. Neither can the unperformed measurements that are to quantify and define fuzzy observables be called “piddling laboratory operations,” nor is the occurrence of measurements restricted to laboratories. Any event or state of affairs from which either the truth or the falsity of a proposition of the form “system S has the property p” (or “observable O has the value v”) can be inferred, qualifies as a measurement.
What kind of relation exists between an electron and a region V if the electron is neither inside V nor outside V? If being inside and being outside are the only relations that can hold between an object’s position and a region of space, then no kind of relation exists between the electron and V. In this case V simply does not exist as far as the electron is concerned. And since conceiving of a region V is tantamount to making the distinction between “inside V” and “outside V,” we are once again led to conclude that the distinction we make between “inside V” and “outside V” has no reality for the electron. The distinction we make between “the electron is inside V” and “the electron is outside V” corresponds to nothing in the actual world.
The reality of a spatial distinction is therefore contingent: whether the distinction we make between “inside V” and “outside V” is real for a given object O at a given time t depends on whether the proposition “O is in V at t” has a truth value (“true” or “false”), and this in turn depends on whether either O’s presence in V or O’s absence from V at the time t is indicated by an actual event or state of affairs.
But if the reality of spatial distinctions is contingent, physical space cannot be something that by itself has parts. For if the regions defined by any conceivable partition were intrinsic to space, and therefore distinct by themselves, the distinctions we make between them would be real for every object in space.
It follows that a detector is needed not only to indicate the presence of an object in its sensitive region R but also, and in the first place, to realize (make real) a region R, by realizing the distinction between being inside R and being outside R. It thereby makes the predicates “inside R” and “outside R” available for attribution.
And this bears generalization, not least because “in physics the only observations we must consider are position observations, if only the positions of instrument pointers”. The measurement apparatus that is presupposed by every quantum-mechanical probability assignment is needed not only for the purpose of indicating the possession of a particular property or value but also, and in the first place, for the purpose of realizing a set of attributable properties or values. (When measuring a spin component, for example, the apparatus is needed not only to indicate the component’s value but also to realize the axis with respect to which the component is defined.)
A possible objection: Suppose that W is a region contained in the spatial complement V of V, and that the presence of O in V is indicated by a measurement. Is not O’s absence from W indicated as well? Are we not entitled to infer that the proposition “O is in W” has a truth value — namely, “false”?
Because regions of space do not exist by themselves, the answer is negative. If W is not realized by being the sensitive region of a detector in the broadest sense of the word — anything capable of indicating the presence of something somewhere — then W does not exist, and if it does not exist, then the proposition “O is in W” cannot be in possession of a truth value. Neither the property of being inside W nor the property of being outside W is available for attribution to O. All we can infer from O’s indicated presence in V is the truth of a counterfactual: if W were the sensitive region of a detector D, then O would not be detected by D.
1. [↑] Bell, J.S. (1990). Against “measurement.” In 62 Years of Uncertainty, Plenum, pp. 17–31.
2. [↑] Bell, J.S. (1987). Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, p. 166.