Quantum Mechanics of Paraparticles

We show that it is possible to formulate a consistent first-quantized theory of paraparticles, i.e., particles which are neither bosons nor fermions. We examine a number of properties of the theory and show that the formulation of Messiah and Greenberg in terms of generalized rays can be replaced by an equivalent formulation in which states are represented by rays in the usual way. We use this alternative formulation to establish some results of ordinary quantum mechanics. We examine in detail the consistency of the theory with the cluster law and show that paraparticles must have states associated with whole families of different permutation symmetries, according to the following rule: If a given particle has $N$-particle states associated with a given Young diagram, then it must have ($N-1\mathrm{}$)-, ($N-2\mathrm{}$)-,..., two-particle states associated with all Young diagrams which can be obtained from the first by successively removing squares. This gives rise to infinitely many different kinds of paraparticle, all with rather complicated properties.