Algebra, coherent states, generalized Hermite polynomials, and path integrals for fractional statistics—Interpolating from fermions to bosons

Abstract

This article constructs the Hilbert space for the algebra αβ − eiθβα = 1 that provides a continuous interpolation between the Clifford and Heisenberg algebras—this particular form is inspired by the properties of anyons. We study the eigenvalues of a generalized number operator (N=βα) and construct the Hilbert space, classified by values of a complex coordinate (λ0): the eigenvalues lie on a circle. For θ being an irrational multiple of 2π, we get an infinite-dimensional representation; however, for a rational multiple (MN) of 2π, it is finite-dimensional, parameterized by the complex coordinate λ0. The case for N = 2, θ = π is the usual Clifford algebra for fermions, while the case for N = ∞, θ = 0 is the Heisenberg algebra of bosons, albeit with two copies for positive and negative eigenvalues. We find a smooth transition from the fermion to the boson situation as N → ∞ from N = 2. After constructing the Hilbert space from the algebra, the cases for N = 2, 3 can be mapped to the “fuzzy sphere” of SU(2), while for general N, the “fuzzy pancake” is found to be the correct representation. Then, we motivate the study of coherent states, which are the eigenstates of α (the lowering operator), labeled by complex numbers for non-zero λ0. We specialize the study of coherent states to the very interesting case of λ0 = 0 and construct a calculus of the generalized Grassmann variables that result, applying it to compute a partition function for these particles. We then make some remarks about extending this study to that of anyons.