We consider the quantization of identical particles. We suggest an a p
riori argument for identification of the classical configuration space
. In two spatial dimensions, for two particles, this yields the (by no
w) familiar cone with deficit angle of pi, with the vertex removed. We
find two fundamental parameters which characterize the quantum theory
. The first, theta, is associated to the multiple connectedness of the
cone, while the other, alpha, is associated to the question of unitar
ity. Theta describes the statistics of the particles and gives rise to
anyons. Alpha specifies the boundary conditions to be imposed on the
wave functions at the vertex of the cone. We show by explicit example
that alpha can be regarded as a vestige of short distance interactions
between the particles, leaving theta as the truly, obligatory, appurt
enance of the quantum mechanics of identical particles in two spatial
dimensions. We also analyze the symmetries of the quantum Hamiltonian
and find a dynamical SO(2, 1) symmetry, acting on the space of Hilbert
spaces with different boundary conditions.