5 The importance of detectors

Take another look at Figure 2.6.3.

blank

hydrogen orbitals ray-traced

Figure 2.6.3 The posi­tion prob­a­bility dis­tri­b­u­tions asso­ci­ated with var­ious orbitals of atomic hydrogen.

blank

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.[1] 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”.[2] 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.

Next


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.