Statistics in Space Dimension Two

E. C. G. Sudarshan

Quantum Field Theory formulated in terms of hermitian fields automatically leads to a spin-statistics connection when invariance under rotations is required. In three (or more) dimensions of space this implies Bose statistics for integer spin fields and Fermi statistics for half-integer spin fields. One should recall that spin-1/2 fields in three dimensions have two nonhermitian or four hermitian components. This automatic doubling of the number of components enables one to define a pseudoscalar matrix, and this in turn allows one to prove the TCP theorem for rotationally invariant field theories. In two space dimensions one obtains anyon statistics independent of the “spin”. For the quantum mechanics of identical particles we obtain only the possibility of either statistics for either spin as long as the spatial dimension is three (or higher). For two space dimensions we get anyon statistics. This difference is due to the contractibility of closed loops in three or more dimensions. The relation to the arguments of Broyles, of Bacry, and of Berry and Robbins is discussed.

Alexios P. Polychronakos

Ingemar Bengtsson, Karol Życzkowski

Nathan L. Harshman, Adam Knapp

The standard topological approach to indistinguishable particles formulates exchange statistics by using the fundamental group to analyze the connectedness of the configuration space. Although successful in two and more dimensions, this approach gives only trivial or near-trivial exchange statistics in one dimension because two-body coincidences are excluded from configuration space. Instead, we include these path-ambiguous singular points and consider configuration space as an orbifold. This orbifold topological approach allows unified analysis of exchange statistics in any dimension and predicts possibilities for anyons in one-dimensional systems, including non-Abelian anyons obeying alternate strand groups. These results clarify the nontopological origin of fractional statistics in one-dimensional anyon models.

Yong-Shi Wu

Because of complicated topology of the configuration space for indistinguishable particles in two dimensions, Feynman's path-integral formulation allows exotic statistics. All possible quantum statistics in two-space are characterized by an angle parameter $\ensuremath{\theta}$ which interpolates between bosons and fermions. The current formalisms in terms of topological action of multivalued wave functions can be derived in a model-independent way.