Spin and statistics in nonrelativistic quantum mechanics, I

A necessary and sufficient condition for Pauli's spin-statistics relation is given for nonrelativistic anyons, bosons, and fermions in two and three spatial dimensions. For any point particle species in two spatial dimensions, denote by J the total (i.e., spin plus orbital) angular momentum of a single particle, and let j be the total angular momentum of the corresponding two-particle system with respect to its center of mass. In three spatial dimensions, write Jz and jz for the z-components of these vector operators. In two spatial dimensions, the spin and statistics connection holds if and only if there exists a unitary operator U such that j=2UJU*. In three dimensions, the analogous relation cannot hold as it stands, but restricting it to an appropriately chosen subspace of the state space yields a sufficient and necessary condition for the spin-statistics connection