Probability and interference

According to Feynman, single-particle interference is ‘the only mystery’ of quantum theory (Feynman, Leighton and Sands 1965, ch. 1, p. 1). Feynman considered an experiment in which particles are fired, one at a time, towards a screen with two holes labelled 1 and 2. With both holes open, the distribution P12 of particles at the backstop displays an oscillatory pattern of bright and dark fringes. If P1 is the distribution with only hole 1 open, and P2 is the distribution with only hole 2 open, then experimentally it is found that P12 ≠ P1 + P2. According to the argument given by Feynman (as well as by many other authors), this result is inexplicable by ‘classical’ reasoning.

By his presentation of the two-slit experiment (as well as by his development of the path-integral formulation of quantum theory), Feynman popularised the idea that the usual probability calculus breaks down in the presence of quantum interference, where it is probability amplitudes (and not probabilities themselves) that are to be added. As pointed out by Koopman (1955), and by Ballentine (1986), this argument is mistaken: the probability distributions at the backstop – P12, P1 and P2 – are conditional probabilities with three distinct conditions (both slits open, one or other slit closed), and probability calculus does not imply any relationship between these. Feynman's argument notwithstanding, standard probability calculus is perfectly consistent with the two-slit experiment.