arXiv 2104.083512021-04hep-thquant-ph

共形桥变换与PT对称性

Conformal bridge transformation and PT symmetry

Luis Inzunza, Mikhail S. Plyushchay

备注：: 42 pages, review article
发表信息：: J. Phys.: Conf. Ser. 2038 (2021) 012014
DOI：: 10.1088/1742-6596/2038/1/012014

摘要：

从$\mathcal{PT}$对称性的角度对保角桥变换（CBT）进行了评述。最初，CBT是作为一个非幺正变换（在经典情况下是一个复杂的正则变换）提出的，它在狄拉克意义上联系了两种不同形式的动力学。也就是说，它将渐近自由形式映射到与$\mathfrak{so}(2,1)\cong \mathfrak{sl}(2,{\mathbb R})$共形对称性相关的动力学的谐波约束形式。然而，由于变换将非厄米算子$i\hat{D}$（其中$\hat{D}$是膨胀的生成元）与$\mathfrak{sl}(2,{\mathbb R})$代数的紧致厄米生成元$\hat{\mathcal{J}}_0$相关联，因此CBT生成元可以与$\mathcal{PT}$对称度量相关联。在这项工作中，我们审查的应用程序，这种转换的一维和二维系统，以及系统的宇宙弦背景下，并为共形扩展带电粒子的领域中的狄拉克粒子。我们还比较和统一了CBT和达布变换。后者用于构造具有极端波性质的KdV族方程的$\mathcal{PT}$ -对称解。作为一个新的结果，我们利用一个修正的CBT将一维$\mathcal{PT}$ -正则化的渐近自由共形力学模型与de Alfaro，Fubini和Furlan系统的$\mathcal{PT}$ -正则化版本联系起来.

英文摘要：

The conformal bridge transformation (CBT) is reviewed in the light of the $\mathcal{PT}$ symmetry. Originally, the CBT was presented as a non-unitary transformation (a complex canonical transformation in the classical case) that relates two different forms of dynamics in the sense of Dirac. Namely, it maps the asymptotically free form into the harmonically confined form of dynamics associated with the $\mathfrak{so}(2,1)\cong \mathfrak{sl}(2,{\mathbb R})$ conformal symmetry. However, as the transformation relates the non-Hermitian operator $i\hat{D}$, where $\hat{D}$ is the generator of dilations, with the compact Hermitian generator $\hat{\mathcal{J}}_0$ of the $\mathfrak{sl}(2,{\mathbb R})$ algebra, the CBT generator can be associated with a $\mathcal{PT}$-symmetric metric. In this work we review the applications of this transformation for one- and two-dimensional systems, as well as for systems on a cosmic string background, and for a conformally extended charged particle in the field of Dirac monopole. We also compare and unify the CBT with the Darboux transformation. The latter is used to construct $\mathcal{PT}$-symmetric solutions of the equations of the KdV hierarchy with the properties of extreme waves. As a new result, by using a modified CBT we relate the one-dimensional $\mathcal{PT}$-regularized asymptotically free conformal mechanics model with the $\mathcal{PT}$-regularized version of the de Alfaro, Fubini and Furlan system.

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