This article develops the algebraic structure that results from the $\theta$-commutator $\alpha \beta - e^{i \theta} \beta \alpha = 1 $ that provides a continuous interpolation between the Clifford and Heisenberg algebras. We first demonstrate the most general geometrical picture, applicable to all values of $N$. After listing the properties of this Hilbert space, we study the generalized coherent states that result when $\xi^N=0$, for $N \ge 2$. We also solve the generalized harmonic oscillator problem and derive generalized versions of the Hermite polynomials for general $N$. Some remarks are made to connect this study to the case of anyons. This study represents the first steps towards developing an anyonic field theory.

中文翻译：

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代数、相干态、广义 Hermite 多项式和分数统计的路径积分 - 从费米子到玻色子的插值

本文开发了由 $\theta$-commutator $\alpha \beta - e^{i \theta} \beta \alpha = 1 $ 产生的代数结构，它提供了 Clifford 和 Heisenberg 代数之间的连续插值。我们首先演示最通用的几何图形，适用于 $N$ 的所有值。在列出这个 Hilbert 空间的性质之后，我们研究了当 $\xi^N=0$ 时产生的广义相干状态，对于 $N\ge 2$。我们还解决了广义谐振子问题，并推导出了通用 $N$ 的 Hermite 多项式的广义版本。一些评论是为了将这项研究与任何人的情况联系起来。这项研究代表了发展任意场理论的第一步。