Fractional Statistics in Quantum Mechanics

Abstract

One usually does not think of quantum statistics in terms of a continuous parameter, such as a coupling constant. We are accustomed to the notion that many-particle wave functions are either symmetric or antisymmetric:

$$ \psi ( \cdots j \cdots i \cdots ) = {e^{i\theta }}\psi ( \cdots i \cdots j \cdots ), $$

(1)

where θ = 2nπ for bosons and θ = (2n + 1)π for fermions. Interpolating in θ, e.g., θ = π/2, seems to make no sense because iterating Eq. (1) twice gives

$$ \psi ( \cdots j \cdots i \cdots ) = {e^{2i\theta }}\psi ( \cdots j \cdots i \cdots ) $$

(2)

and the single-valuedness of Ψ demands that e^{2i\emptyset }= 1, so one concludes that Bose and Fermi statistics exhaust all the allowed values of θ.