Quantum Subterfuge

An account of the double-slit experiment by a former professor of mathematical physics is supposed to show the necessity of a paradoxical conclusion, but under examination it shows only the logical confusions of its author.

A few years ago, wanting to gain some knowledge of quantum mechanics, I started reading Quantum Theory: A Very Short Introduction by John Polkinghorne, formerly Professor of Mathematical Physics in the University of Cambridge (Oxford University Press, 2002). I recall being exasperated by the book on my first attempt to read it, but I did not recall the reason until more recently, when I reread Polkinghorne’s commentary on the double-slit experiment.

Polkinghorne opens his exposition by quoting a comment of Richard Feynman on the experiment:

In reality it contains the only mystery. We cannot make the mystery go away by ‘explaining’ how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics. [Polkinghorne, p. 22; no citation of source provided]

I recall thinking, when I first read this: “Excellent! Now all I have to do is pay very close attention to the account of the experiment, and I shall understand the basic peculiarities of all quantum mechanics.” This expectation was disappointed.

Describing the physical setup and the results of the experiment is not difficult. Electrons or other quantum particles are fired at a barrier in which there is a pair of adjacent slits. On the far side of the slits is a screen that detects the impacts of the particles. Classical physics predicts that the particles will impinge on the screen in a scattering pattern, with two areas of greatest intensity located directly across from the two slits. But in fact what emerges is an interference pattern, with a single area of greatest intensity located across from the midpoint of the two slits, exactly as if one were sending waves of some sort through the slits.

One might try to explain this on classical-physical lines by supposing that the electrons, though individually particulate in nature, behave in a wave-like fashion when they are shot together in a stream: it is the stream of electrons, not the individual electron, that behaves like a wave. It is true that one cannot observe wave behavior in an individual electron but only in a collection of electrons. But that does not mean that the wave behavior can be explained on classical lines as an effect of physical interaction among electrons in a stream, for the fact is that the electrons form an interference pattern even when they are fired at the slits one at a time. So either one must suppose that the behavior of each electron is influenced by the path taken by its predecessors, or one must attribute wave properties to each electron.

Things get even weirder when the experiment is set up so as to allow the option of detecting which slit the particles pass through. In so-called “quantum eraser” experiments (q.v., Wikipedia), photons passing through circular polarizers form a scattering pattern or an interference pattern according to whether the polarization that distinguishes which slit they went through is preserved or “erased” by a second, diagonal polarizer. And with so-called the “delayed choice” quantum eraser experiment (q.v., Wikipedia again), things get even weirder. But those are other stories, not covered in Polkinghorne’s book. Polkinghorne finds weirdness enough in the original plain double-slit experiment. He writes:

Electrons arriving one by one is particlelike behaviour; the resulting collective interference pattern is wavelike behaviour. But there is something much more interesting than that to be said. We can probe a little deeper into what is going on by asking the question, When an indivisible single electron is traversing the apparatus, through which slit does it pass in order to get to the detector screen? Let us suppose that it went through the top slit, A. If that were the case, the lower slit, B, was really irrelevant and it might just as well have been temporarily closed up. But, with only A open, the electron would not be most likely to arrive at the midpoint of the far screen, but instead it would be most likely to end up at the point opposite A. Since this is not the case [emphasis mine], we conclude that the electron could not have gone through A. Standing the argument on its head, we conclude that the electron could not have gone through B either. What then was happening? That great and good man, Sherlock Holmes, was fond of saying that when you have eliminated the impossible, whatever remains must have been the case, however improbable it may seem to be. Applying this Holmesian principle leads us to the conclusion that the indivisible electron went through both slits (pp. 23–24, latter emphasis in original).

Pay attention to the words “this is not the case” and try to identify their antecedent: what is not the case? The only thing mentioned in the preceding sentences that can in any clear sense be said to be the case or not the case is that the electron in question struck the screen at the point opposite slit A. But it would be insane to say casually that this is not the case, since there is nothing in the preceding stipulations that would justify such a conclusion. No, what Polkinghorne seems to mean by “this is not the case” is that it is not the case that the electron would be “most likely” to arrive at the point opposite slit A.

(Note: The remainder of this post has been extensively revised since I first posted it. The paragraphs that immediately follow analyze and criticize Polkinghorne’s argument in a very detailed fashion. Readers whose patience or interest is tried by such a treatment may profit by skipping down to the paragraph just before the first graph, in which I restate my criticisms by means of an analogy with an argument whose defects are much easier to recognize.)

Let “E” designate a randomly selected electron that is fired at the slits and that strikes the screen on the far side of them. The first part of Polkinghorne’s reasoning can then be summed up as follows:

1. If E passes through slit A, then E is most likely to strike the screen at a point opposite A.
2. E is not most likely to strike the screen at a point opposite A.
Therefore,
3. E does not pass through A.
   

The argument appears to be impeccable as far as its logical form is concerned. If that is so then the only question to be raised about its cogency is whether both premises are true. But in fact, reflection on the premises reveals an ambiguity that defeats the argument.

Consider premise 2 first. Given the setup of the experiment, the only evidence that we have for attributing to a randomly selected electron a probability of hitting one or another part of the screen is the interference pattern that emerges on the screen.  That pattern shows the highest incidence of impacts at the midpoint between the two slits. From this fact we can conclude that a randomly selected electron that strikes the screen is most likely to do so at the midpoint and not opposite either slit. This reasoning justifies Polkinghorne’s second premise.

Now consider the first premise. E was defined as an electron randomly selected from among all the electrons that reach the screen. But premise 1 concerns an electron that is randomly selected from among those that have passed through slit A. The pattern on the screen provides no evidence whatever relevant to a conclusion about the most likely point of arrival of such an electron as that. The only way to get evidence relevant to a conclusion about a randomly selected electron that has passed through slit A is either to block off slit B or to use a device that distinguishes the impacts of electrons that have passed through A from the impacts of electrons that have passed through B, as in the quantum eraser experiments. It is established that if we do either of these things then no interference pattern emerges. If we block off slit B, the highest incidence of impacts is opposite slit A, and if a device is used that distinguishes the electrons passing through A from those passing through B, then there will be an area of highest incidence opposite each slit. Under such conditions, Polkinghorne’s first premise is true. But his second premise is either false or irrelevant to the conclusion—false if it concerns an electron that has passed through slit A; irrelevant if it concerns an electron whose place of passage is undetermined.

Of course, this is not Polkinghorne’s entire argument, but only one half of its preliminary part. The second half of the preliminary part is the repetition of this argument with “slit B” taking the place of “slit A” in the premises and the conclusion.

4. If E passes through slit B, then E is most likely to strike the screen at a point opposite B.
5. E is not most likely to strike the screen at a point opposite B.
Therefore,
6. The electron did not go through B.
   

That part of the argument is, obviously, just as futile with “B” in it as it was with “A,” but let it stand for the moment, so that we can consider the would-be clinching part of the argument: the supposedly Holmesian conclusion that the electron passed through both slits. Polkinghorne’s stated reasoning is:

3. E does not pass through slit A.
6. E does not pass through slit B.
Therefore,
7. E passes through both slits.

There is nothing Holmesian about such a conclusion at all: Sherlock Holmes would not mistake such a bald non sequitur for a deduction. Nor does it take the intellect of Sherlock Holmes to see that what follows from lines 3 and 6 is:

E does not pass through either slit.

Of course, this is not the desired conclusion at all. Not only does Polkinghorne fail to establish his desired intermediate conclusions (3 and 6); he has set up his entire argument to establish the wrong final conclusion. The undesired conclusion can be avoided if we go back to re-write the previous two subordinate arguments by inserting the qualifier “solely” before the phrases “through slit A” and “through slit B,” and adding a further premise, “E passes through at least one slit.” But the fact remains that the intermediate conclusions 3 and 6 are not established or even made in the slightest degree credible by any of Polkinghorne’s reasoning.

The defects in Polkinghorne’s argument can be brought out by means of an analogy. A graph of the distribution of heights among adults in the United States looks like this (this graph and the two that follow are taken from this Web page by John D. Cook Consulting):

Graph 1: height distribution of all adults in US

The numbers along the bottom represent height in inches. The midpoint of the peak is around 67 inches. Let R be a randomly selected adult resident of the United States. According to this graph, R is most likely to be 67 inches tall. So R is not most likely to be, say, 64 inches tall, or 70 inches tall.

But R may, and indeed (setting aside rare cases of indeterminate sex) must, be either female or male. Suppose that R is female. The distribution of heights for adult females has a peak around 64 inches:

Graph 2: height distribution of adult females in the US

So if R is female, R is most likely to be about 64 inches tall. By contrast, the distribution for adult males has a peak around 70 inches.

Graph 3: height distribution of adult males in the US

So if R is male, R is most likely to be about 70 inches tall. Now imagine that, with these facts in hand, statistician John Jokinghorne presents us with the following argument:

1. If R is female, then R is most likely to be about 64 inches tall (by graph 2).
2. R is not most likely to be about 64 inches tall (by graph 1).
Therefore,
3. R is not female (from 1 and 2).
4. If R is male, then R is most likely to be about 70 inches tall (by graph 3).
5. R is not most likely to be about 70 inches tall (by graph 1).
Therefore,
6. R is not male (from 4 and 5).
Therefore,
7. R is both female and male (supposedly Holmesian conclusion from 3 and 6).

Clearly, all three of the conclusions in this argument are non sequiturs. Conclusion 3 does not follow from (1) and (2), because adding the supposition that R is female, as in (1), makes (2) false or irrelevant to (3). The same applies to the relation of premises 4 and 5 to conclusion 6. And the would-be Holmesian conclusion is of stupefying inconsequence. One may think that Polkinghorne’s argument cannot be as bad as Jokinghorne’s, because it is not so obviously bogus; but logically considered, it is every bit as bad. Its logical defects are exactly analogous. They just happen to be less conspicuous because of the more recondite subject matter.

One last observation: Presumably, Polkinghorne intends his argument to establish something not just about some randomly selected electron in the experiment but about every electron in the experiment, namely that it passes through both slits. The analogous conclusion of Jokinghorne’s argument would be:

8. Every American adult is both female and male.

If Jokinghorne’s argument does not incline you to accept this conclusion (and it shouldn’t), then neither should Polkinghorne’s argument incline you to accept his conclusion about the double-slit experiment.

There may be compelling reasons in quantum mechanics to say that each electron goes through both slits, but whatever those reasons may be, Polkinghorne fails to state them. Making a popular exposition of quantum mechanics requires making the reasoning that leads to its paradoxical conclusions clear. Instead of this, Polkinghorne’s book offers the kind of confused thinking that can at best produce only incomprehension and that at worst produces the false belief that one has understood something when in fact one has merely participated in the author’s own confusions.