AI 中文总结
该研究分析了双原点直线上的标量与旋量量子动力学，发现标量理论对双原点不敏感，而非平凡自旋结构下的旋量理论可探测到双原点，且需附加传输条件实现自伴演化。
AI 中文摘要
我们研究配备了恒等粘合光滑结构与平坦度量的标准双原点直线 \\(\Ltwo=(\R_1\sqcup\R_2)/\\!\sim\\)（其中 \\(x\neq0\\) 时 \\((x,1)\sim(x,2)\\)）上的标量与旋量量子动力学。我们首先证明普通标量理论对双原点不敏感：从 \\(\Ltwo\\) 到豪斯多夫空间的每个连续映射都可通过商映射 \\(q:\Ltwo\to\R\\) 分解，且对于自然测度，\\(C^\infty(\Ltwo)\cong C^\infty(\R)\\)、\\(L^2(\Ltwo)\cong L^2(\R)\\)，因此自然标量自由哈密顿量与 \\(\R\\) 上的自由拉普拉斯算子是幺正等价的。然而，双原点在与非平凡自旋结构关联的旋量直线中仍可见：\\(\Ltwo\\) 上的定向平坦结构允许两种自旋结构，其区别在于图重叠的两个连通分支上的相对转移符号，非平凡结构的符号相反，这迫使每个连续整体旋量截面在两个原点处都为零，且每个光滑截面在原点处都需无穷阶消失。紧支光滑扭曲截面上的关联一阶算子 \\(-i\frac{\mathrm d}{\mathrm dx}\\) 是对称的，但非本质自伴的：其亏格指数为 \\((1,1)\\)，自伴一阶演化需要在原点处附加一个传输条件；而与平坦扭曲结构关联的自然正二次型则给出两个狄利克雷半直线拉普拉斯算子的直和，从而产生完美反射。因此，普通标量理论无法探测到双原点，而非平凡粘合的旋量直线可以，尽管仅自旋结构无法确定唯一的幺正传输定律。
英文摘要
We develop a systematic framework for quantum mechanics on a finite graph-resolved class of second-countable, T1, non-Hausdorff one-manifolds. The central problem is that local differential expressions do not by themselves determine the quantum theory: non-Hausdorff incidence, bundle transport, smooth extension, and analytic completion can impose additional global constraints on the operator domain. We formulate these constraints using transported jets, Whitney realisability, Sobolev traces, and formal return holonomy, obtaining exact closure results for finite smooth graph resolutions. At first Sobolev order, scalar dynamics reduce to a weighted quantum graph whose edge weights arise from a marked resolving presentation, leading naturally to weighted Kirchhoff Hamiltonians. Multiple-origin lines and circles, branching junctions, finite trees, and split-and-rejoin geometries then yield explicit deficiency indices, reflection and transmission laws, interference conditions, compact dark states, and embedded cavity modes. For finite-rank Hermitian bundles, unitary gluing selects a transmitting fixed subspace, so scattering becomes projection onto the invariant sector of the subgroup generated by the gluing matrices. Compact-group representations therefore turn non-Hausdorff incidence into an invariant-sector selection mechanism; connected compact semisimple groups require at most two relative gluing matrices for full invariant completion, with explicit SU(3) examples. The resulting theory separates topological non-Hausdorff data from the analytic and representation-theoretic structures that remain detectable by quantum dynamics.
Comments128 pages, 4 parts, 36 sections, 2 appendices. Develops quantum mechanics on finite graph-resolved non-Hausdorff one-manifolds: extension and Sobolev completion, formal holonomy, weighted graph dynamics, branching and interference, finite-rank unitary gluing, and compact-group invariant-sector scattering. Includes a correction to an unrestricted component-extension claim in O'Connell (2024)
