论文标题
在宇宙弦背景中的保形桥
Conformal bridge in a cosmic string background
论文作者
论文摘要
使用共同桥式转换研究了非相关性$ \ mathfrak {so}（so}（so}（2,1）（2,1）（2,1）（2,1）\ cong \ mathfrak {sl}（2，{\ mathbb r}）$不变系统中的不变系统。该背景的几何特性类似于在“几何参数” $α$中编码的锥体表面的几何特性，该$α$由线的线性正/负质量密度确定。该背景上的自由粒子和谐波振荡器通过共形桥变换显示出与之相关。为了确定自由系统的积分，我们采用了将模型与平面版本相关联的本地规范转换。然后，使用保形桥变换将获得的积分映射到锥体上的谐波振荡器的积分。两种模型中定义明确的经典积分仅在$α= q/k $的情况下，$ q，k = 1,2，\ ldots，$，$ q>> 1 $是有限非线性代数的高阶生成器。对系统的任意值$α$进行量化；但是，仅当$α$是一个整数时，才存在与光谱退化性相关的明确定义的隐藏对称算子，它揭示了量子异常。
Hidden symmetries of non-relativistic $\mathfrak{so} (2,1)\cong \mathfrak{sl}(2, {\mathbb R})$ invariant systems in a cosmic string background are studied using the conformal bridge transformation. Geometric properties of this background are analogous to those of a conical surface with a deficiency/excess angle encoded in the "geometrical parameter" $α$, determined by the linear positive/negative mass density of the string. The free particle and the harmonic oscillator on this background are shown to be related by the conformal bridge transformation. To identify the integrals of the free system, we employ a local canonical transformation that relates the model with its planar version. The conformal bridge transformation is then used to map the obtained integrals to those of the harmonic oscillator on the cone. Well-defined classical integrals in both models exist only at $α=q/k$ with $q,k=1,2,\ldots,$ which for $q>1$ are higher-order generators of finite nonlinear algebras. The systems are quantized for arbitrary values of $α$; however, the well-defined hidden symmetry operators associated with spectral degeneracies only exist when $α$ is an integer, that reveals a quantum anomaly.
