arXiv 2012.046132020-12hep-thgr-qc

宇宙弦背景中的共形桥

Conformal bridge in a cosmic string background

Luis Inzunza, Mikhail S. Plyushchay

备注：: 44 pages
发表信息：: JHEP 2021, 165 (2021)
DOI：: 10.1007/JHEP05(2021)165

摘要：

利用保角桥变换研究了宇宙弦背景下非相对论性$\mathfrak{so} (2,1)\cong \mathfrak{sl}(2, {\mathbb R})$不变系统的隐对称性。这种背景的几何性质类似于圆锥表面的几何性质，其缺角/余角编码在“几何参数”$α$中，由弦的线性正/负质量密度确定。通过保角桥变换，证明了自由粒子和谐振子在此背景下的联系。为了确定自由系统的积分，我们采用了一个本地的正则变换，涉及到它的平面版本的模型。然后利用保角桥变换将得到的积分映射到锥上谐振子的积分。两个模型中定义良好的经典积分仅存在于$α=q/k$和$q,k=1,2,\ldots,$中，对于$q>1$来说，$q,k=1,2,\ldots,$是有限非线性代数的高阶生成元。该系统被量化为任意值的$α$，然而，定义良好的隐藏对称运营商与频谱简并只存在时，$α$是一个整数，揭示了量子异常。

英文摘要：

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.

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