$1+1$ 维相对论性狄拉克系统中真实三体相互作用的相位编码

中文摘要 AI 辅助

我们展示了在 $(1+1)$ 维相对论性三粒子狄拉克系统中，真实的三体相位信息如何进入不变质量。作为一个可解的参考系统，我们考虑了具有成对接触相互作用 $g_{ij}(1-\alpha_i\alpha_j)\delta(x_i-x_j)$ 的 Sakamoto--Munakata--Ino 模型。这些奇异相互作用可以通过一个不连续幺正变换转化为依赖于扇区的相位和匹配条件。尽管显式接触项因此被移除，非零组分质量算符被旋转并保留了非平凡的谱信息。我们引入了一个由 $Q_3=\alpha_1\alpha_2\alpha_3$ 生成的真实三体和乐（holonomy）。动能和成对相互作用部分与 $Q_3$ 对易，而组分质量算符与其反对易。因此，无质量系统分离为 $Q_3=\pm1$ 扇区，这些扇区获得相反的相位 $e^{\pm i\theta_3}$，而非零组分质量混合这两个扇区。这种相位-扇区混合机制使得相对三体相位在动力学上可被束缚态谱访问，并建立了一个算符级机制，通过该机制三体和乐产生物理三体不变质量对 $\theta_3$ 的依赖。我们进一步强调，拓扑三体和乐并不自动等价于裸三接触势；这种等价性需要一个正则化的自伴实现和一个满足庞加莱代数的、与相互作用相容的相容升压。因此，所得到的框架将真实三体相位信息与相对论性复合系统的质量谱联系起来，同时将受控的和乐构造与未解决的短距离三接触实现明确区分开来。

英文摘要

We show how genuine three-body phase information can enter the invariant mass of a relativistic three-particle Dirac system in $(1+1)$ dimensions. As a solvable reference system, we consider the Sakamoto--Munakata--Ino model with pairwise contact interactions $g_{ij}(1-α_iα_j)δ(x_i-x_j)$. These singular interactions can be transferred into sector-dependent phases and matching conditions by a discontinuous unitary transformation. Although the explicit contact terms are thereby removed, the nonzero constituent-mass operator is rotated and retains nontrivial spectral information. We introduce a genuine three-body holonomy generated by $Q_3=α_1α_2α_3$. The kinetic and pair-interaction parts commute with $Q_3$, while the constituent-mass operator anticommutes with it. Consequently, the massless system separates into the $Q_3=\pm1$ sectors, which acquire opposite holonomy phases $e^{\pm iθ_3}$, whereas nonzero constituent masses mix the two sectors. This phase-sector-mixing mechanism makes the relative three-body phase dynamically accessible to the bound-state spectrum and establishes an operator-level mechanism through which the three-body holonomy generates a $θ_3$ dependence of the physical three-body invariant mass. We further emphasize that the topological three-body holonomy is not automatically equivalent to a bare triple-contact potential; such an equivalence requires a regulated self-adjoint realization and a compatible interaction-dependent boost satisfying the Poincaré algebra. The resulting framework therefore connects genuine three-body phase information to the mass spectrum of a relativistic composite system while clearly separating the controlled holonomy construction from the unresolved short-distance triple-contact realization.

发表机构

• Nagoya Aoi University（名古屋青大学）

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