Suppose that the conditions stipulated by Rule B are met: there is nothing — no event, no state of affairs, anywhere, anytime — from which the slit taken by an electron can be inferred. Can it be true, in this case,
- that each electron goes through a single slit — either L or R — and
- that the behavior of an electron that goes through one slit does not depend on whether the other slit is open or shut?
To keep the language simple, we will say that an electron leaves a mark where it is detected at the backdrop. If each electron goes through a single slit, then the observed distribution of marks when both slits are open is the sum of two distributions, one from electrons that went through L and one from electrons that went through R:
If in addition the behavior of an electron that goes through one slit does not depend on whether the other slit is open or shut, then we can observe by keeping R shut, and we can observe by keeping L shut. What we observe if R is shut is the left dashed hump, and what we observed if L is shut is the right dashed hump:
Hence if the above two conditions (as well as those stipulated by Rule B) are satisfied, we will see the sum of these two humps. In reality what we see is this:
Thus all of those conditions cannot be simultaneously satisfied. If Rule B applies, then either it is false that each electron goes through a single slit or the behavior of an electron that goes through one slit does depend on whether the other slit is open or shut.
Which is it?
According to one attempt to make physical sense of the mathematical formalism of quantum mechanics, due to Louis de Broglie and David Bohm, each electron goes through a single slit, and the behavior of an electron that goes through one slit depends on whether the other slit is open or shut.
So how does the state of, say, the right slit (open or shut) affect the behavior of an electron that goes through the left slit? In both de Broglie's pilot wave theory and Bohmian mechanics, the electron is assumed to be a well-behaved particle in the sense that it follows a precise path — its position at any moment is given by three coordinates — and in addition there exists a wave that guides the electron by exerting on it a force. If only one slit is open, this passes through one slit. If both slits are open, this passes through both slits and interferes with itself (in the "classical" sense of interference). As a result, it guides the electrons along wiggly paths that cluster at the backdrop so as to produce the observed interference pattern:
According to this story, the reason why electrons coming from the same source or slit arrive in different places, is that they start out in slightly different directions and/or with slightly different speeds. If we had exact knowledge of their initial positions and momenta, we could make an exact prediction of each electron's subsequent motion. This, however, is impossible. The uncertainty principle prevents us from making exact predictions of a particle's motion. Hence even though according to Bohm the initial positions and momenta are in possession of precise values, we can never know them.
If positions and momenta have precise values, then why can we not measure them? It used to be said that this is because a measurement exerts an uncontrollable influence on the value of the observable being measured. Yet this merely raises another question: why do measurements exert uncontrollable influences? This may be true for all practical purposes, but the uncertainty principle does not say that merely holds for all practical purposes. Moreover, it isn't the case that measurements necessarily "disturb" the systems on which they are performed.
The statistical element of quantum mechanics is an essential feature of the theory. The postulate of an underlying determinism, which in order to be consistent with the theory has to be a crypto-determinism, not only adds nothing to our understanding of the theory but also precludes any proper understanding of this essential feature of the theory. There is, in fact, a simple and obvious reason why hidden variables are hidden: the reason why they are strictly (rather than merely for all practical purposes) unobservable is that they do not exist.
At one time Einstein insisted that theories ought to be formulated without reference to unobservable quantities. When Heisenberg later mentioned to Einstein that this maxim had guided him in his discovery of the uncertainty principle, Einstein replied something to this effect: "Even if I once said so, it is nonsense." His point was that before one has a theory, one cannot know what is observable and what is not. Our situation here is different. We have a theory, and this tells in no uncertain terms what is observable and what is not.