When can identical particles collide?

It is customary, when discussing configuration spaces of identical particles in two or more dimensions, to discard the configurations where two or more particles overlap, the justification being that the configuration space ceases to be a manifold at those points, and also to allow for nonbosonic statistics. We show that there is in general a loss of physical information in discarding these points by studying the simple system of two free particles moving in the plane and requiring that the Hamiltonian be self-adjoint. We find that the Hamiltonian for fermions is unique, but that in all other cases (i.e., for particles obeying properly fractional or Bose statistics) there is a one-parameter family of possible self-adjoint extensions. We show how a plausible limiting procedure selects a unique extension from each family, the favored extension being the one for which the wave function remains finite at the points of overlap. We also test our procedure by applying it to the known case of the hydrogen atom.