that the observed effect was real. The original experiment used the fact that two bosons tend to arrive at two separate detectors at the same time. Morgan and Mandel used a thermal photon source to create a dim beam of photons and observed the tendency of the photons to arrive at the same time on a single detector. Both of these effects used the wave nature of light to create a correlation in arrival time – if a single photon beam is split into two beams, then the particle nature of light requires that each photon is only observed at a single detector, and so an anti-correlation was observed in 1977 by H. Jeff Kimble. Finally, bosons have a tendency to clump together, giving rise to Bose–Einstein correlations, while fermions due to the Pauli exclusion principle, tend to spread apart, leading to Fermi–Dirac (anti)correlations. Bose–Einstein correlations have been observed between pions, kaons and photons, and Fermi–Dirac (anti)correlations between protons, neutrons and electrons. For a general introduction in this field, see the textbook on Bose–Einstein correlations by Richard M. Weiner A difference in repulsion of Bose–Einstein condensate in the "trap-and-free fall" analogy of the HBT effect affects comparison. Also, in the field of particle physics, Goldhaber et al. performed an experiment in 1959 in Berkeley and found an unexpected angular correlation among identical pions, discovering the ρ0 resonance, by means of decay. From then on, the HBT technique started to be used by the heavy-ion community to determine the space–time dimensions of the particle emission source for heavy-ion collisions. For recent developments in this field, see for example the review article by Lisa. Wave mechanics The HBT effect can, in fact, be predicted solely by treating the incident electromagnetic radiation as a classical wave. Suppose we have a monochromatic wave with frequency on two detectors, with an amplitude that varies on timescales slower than the wave period . (Such a wave might be produced from a very distant point source with a fluctuating intensity.) Since the detectors are separated, say the second detector gets the signal delayed by a time , or equivalently, a phase ; that is, The intensity recorded by each detector is the square of the wave amplitude, averaged over a timescale that is long compared to the wave period but short compared to the fluctuations in : where the overline indicates this time averaging. For wave frequencies