Non-relativistic bosonization and fractional statistics

Stefano Forte

These lectures discuss the formulation of quantum mechanics with fractional spin and statistics in 2+1 dimensions in a relativistic setting, emphasizing the path-integral approach. The non-relativistic theory is reviewed from a path-integral viewpoint. The group-theoretical underpinnings of relativistic fractional spin are discussed, then the path-integral quantization of spin and of massive fermions without using spinors is reviewed. The path integral for a system of n relativistic particles with fractional spin is constructed, the spin-statistics relation and the Lorentz and Poincare' representation content of physical states are discussed. Some problems in formulating a field theory with fractional statistics are presented, and their resolution in the operator cocycle approach is reviewed.

Avinash Khare

Avinash Khare

Масуд Чайчиан, R. González Felipe, Claus Montonen

The statistics of q-oscillators, quons, and, to some extent, of anyons are studied and the basic differences among these objects are pointed out In particular, the statistical distributions for different bosonic and fermionic q-oscillators are found for their corresponding Fock space representations in the case when the Hamiltonian is identified with the number operator. In this case and for non-relativistic particles, the single-particle temperature Green function is defined with q-deformed periodicity conditions. The equations of state for nonrelativistic and ultrarelativistic bosonic q-gases in an arbitrary space dimension are found near Bose statistics, as well as that for an anyonic gas near Bose and Fermi statistics. The first corrections to the second virial coefficients are also evaluated.

Joan Soto, José Goity

A 2+1 dimensional model is presented which consists of two commuting non-relativistic fields with opposite abelian charges minimally coupled to an abelian Chern-Simons theory, which implements the fractional statistics. The model enjoys a discrete symmetry which prevents the ficticious magnetic field to develop a non-zero expectation value in the finite density ground state, thus suppressing parity violating effects. The Hugenholz-Pines relation which guarantees the stability of standard superfluids holds. For statistics close to the bosonic limit we find a solution which exibits a superfluid component.

Daniel P. Arovas