统一的精确圆柱狄拉克模式与对称分辨量子几何
Unified Exact Cylindrical Dirac Modes and Symmetry-Resolved Quantum Geometry
- University of Science and Technology Beijing（北京科技大学）
- Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究推导了固定参数下的一般正则正能圆柱狄拉克解，将多种常用模式嵌入二维解空间，通过守恒横向算子分解为对称分支，结合量子几何与对称性约束，建立了跨场景的统一框架。
AI中文摘要：
自由狄拉克方程的精确圆柱解为一大类轴对称相对论费米子系统提供了天然的单粒子模式，包括电子涡旋、扭曲粒子散射、旋转物质和圆柱场量子化，但这些模式通常在不同的内部基矢中表述。我们推导了固定能量、横向和纵向动量以及总角动量下的一般正则正能圆柱解，并展示了常用的自旋极化、分离、螺旋度和横向螺旋度模式如何嵌入所得的二维解空间。一个守恒的横向算子将剩余的二重态分解为两个对称定义的分支。它们依赖于参数的本征空间具有规范不变的量子几何特征，具有相反的贝里曲率、共同的量子度量，且满足两能级度量-曲率关系。我们进一步推导了分支转换的对称性约束及相应的简化两态动力学。所得框架将不同圆柱狄拉克场景下的模式分类、对称分辨、量子几何与动力学联系起来。
英文摘要：
Exact cylindrical solutions of the free Dirac equation provide natural single-particle modes for a broad class of axially symmetric relativistic fermion systems, including electron vortices, twisted-particle scattering, rotating matter, and cylindrical field quantization, but are commonly formulated in different internal bases. We derive the general regular positive-energy cylindrical solution at fixed energy, transverse and longitudinal momenta, and total angular momentum, and show how the commonly used spin-polarized, separation, helicity, and transverse-helicity modes are embedded in the resulting two-dimensional solution space. A conserved transverse operator resolves the residual doublet into two symmetry-defined branches. Their parameter-dependent eigenspaces admit a gauge-invariant quantum-geometric characterization, with opposite Berry curvatures, a common quantum metric, and saturation of the two-level metric--curvature relation. We further derive symmetry constraints on branch conversion and the corresponding reduced two-state dynamics. The resulting framework connects mode classification, symmetry resolution, quantum geometry, and dynamics across distinct cylindrical Dirac settings.
