Algebra for Fractional Statistics -- interpolating from fermions to bosons

Abstract

This article constructs the Hilbert space for the algebra $\alpha \beta - e^{i \theta} \beta \alpha = 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 (${\cal N} = \beta \alpha$) and construct the Hilbert space, classified by values of a complex coordinate ($\lambda_0$): the eigenvalues lie on a circle. For $\theta$ being an irrational multiple of $2 \pi$, we get an infinite-dimensional representation, however for a rational multiple ($\frac{M}{N}$) of $2 \pi$, it is finite-dimensional, parametrized by the complex coordinate $\lambda_0$. The case for $N=2 \: ; \: \theta=\pi$ is the usual Clifford algebra for fermions, while the case for $N=\infty \: ; \: \theta=0$ 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 $N \rightarrow \infty$ from $N=2$. After constructing the Hilbert space from the algebra, the cases for $N=2,3$ can be mapped to $SU(2)$. Then, we motivate the study of coherent states, rather generally. The coherent states are eigenstates of $\alpha$, the annihilation operator and are labeled by complex numbers for non-zero $\lambda_0$.