On the quantization of identical particles

R. MacKenzie ; P. K. Panigrahi ; M. V. Paranjape ; S. Sakhi

Teoretičeskaâ i matematičeskaâ fizika, Tome 98 (1994) no. 1, pp. 80-89

Résumé

We consider the quantization of identical particles. We suggest an apriori arguement for identification of the classical configuration space. In two spatial dimensions, for two particles, this yields the (by now) 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 unitarity. $\theta$ 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, appurtenance 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.