  In 2005, Couder, Protiere, Fort and Badouad showed that oil droplets bouncing
on a vibrating tray of oil can display nonlocal interactions reminiscent of the
particle-wave associations in quantum mechanics; in particular they can move,
attract, repel and orbit each other. Subsequent experimental work by Couder,
Fort, Protiere, Eddi, Sultan, Moukhtar, Rossi, Molacek, Bush and Sbitnev has
established that bouncing drops exhibit single-slit and double-slit
diffraction, tunnelling, quantised energy levels, Anderson localisation and the
creation/annihilation of droplet/bubble pairs.
  In this paper we explain why. We show first that the surface waves guiding
the droplets are Lorentz covariant with the characteristic speed c of the
surface waves; second, that pairs of bouncing droplets experience an
inverse-square force of attraction or repulsion according to their relative
phase, and an analogue of the magnetic force; third, that bouncing droplets are
governed by an analogue of Schroedinger's equation where Planck's constant is
replaced by an appropriate constant of the motion; and fourth, that orbiting
droplet pairs exhibit spin-half symmetry and align antisymmetrically as in the
Pauli exclusion principle. Our analysis explains the similarities between
bouncing-droplet experiments and the behaviour of quantum-mechanical particles.
It also enables us to highlight some differences, and to predict some
surprising phenomena that can be tested in feasible experiments.
