论文标题
分数统计的代数 - 从费米斯到玻色子的插值
Algebra for Fractional Statistics -- interpolating from fermions to bosons
论文作者
论文摘要
本文构建了代数$αβ-e^{iθ}βα= 1 $的希尔伯特空间，该空间在克利福德和海森伯格代数之间提供了连续的插值。这种特殊形式的灵感来自Anyons的属性。我们研究了广义数字运算符（$ {\ cal n} =βα$）的特征值，并构建由复杂坐标值（$λ_0$）的值分类的希尔伯特空间：特征值位于圆上。对于$θ$，是$2π$的不合理倍数，我们获得了无限维代表，但是对于$2π$的合理倍数（$ \ frac {m} {n} $），它是有限尺寸的，由复杂坐标$λ_0$参数化。 $ n = 2 \：; \：θ=π$是Fermions通常的Clifford代数，而对于$ n = \ infty \：; \：θ= 0 $是玻色子的Heisenberg代数，尽管有两个副本，用于正和负特征值。我们发现从费米昂（Fermion）到玻色子（Boson）情况的平稳过渡为$ n \ rightarrow \ infty $ n = 2 $。从代数构建希尔伯特空间后，可以将$ n = 2,3 $的案例映射到$ su（2）$。然后，我们通常会激励对连贯状态的研究。连贯的状态是$α$的本征态，nihilation oterator，并由非零$λ_0$的复数标记。
This article constructs the Hilbert space for the algebra $αβ- e^{i θ} βα= 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 (${\cal N} = βα$) and construct the Hilbert space, classified by values of a complex coordinate ($λ_0$): the eigenvalues lie on a circle. For $θ$ being an irrational multiple of $2 π$, we get an infinite-dimensional representation, however for a rational multiple ($\frac{M}{N}$) of $2 π$, it is finite-dimensional, parametrized by the complex coordinate $λ_0$. The case for $N=2 \: ; \: θ=π$ is the usual Clifford algebra for fermions, while the case for $N=\infty \: ; \: θ=0$ 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 $N \rightarrow \infty$ 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 $α$, the annihilation operator and are labeled by complex numbers for non-zero $λ_0$.
