We investigate the dynamics of a composite system (𝒞𝒮) consisting of an interacting fermion–antifermion pair in the three-dimensional space–time background generated by a static point source. By considering the interaction between the particles as Dirac oscillator coupling, we analyze the effects of space–time topology on the energy of such a 𝒞𝒮. To achieve this, we solve the corresponding form of a two-body Dirac equation (fully-covariant) by assuming the center-of-mass of the particles is at rest and locates at the origin of the spatial geometry. Under this assumption, we arrive at a nonperturbative energy spectrum for the system in question. This spectrum includes spin coupling and depends on the angular deficit parameter (𝒜𝒟𝒫) of the geometric background. This provides a suitable basis to determine the effects of the geometric background on the energy of the 𝒞𝒮 under consideration. Our results show that such a 𝒞𝒮 behaves like a single quantum oscillator. Then, we analyze the alterations in the energy levels and discuss the limits of the obtained results. We show that the effects of the geometric background on each energy level are not same and there can be degeneracy in the energy levels for small values of the 𝒜𝒟𝒫.

中文翻译：

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点源诱导时空复合系统的动力学

我们研究了复合系统的动力学（𝒞𝒮) 由静态点源生成的三维时空背景中的相互作用的费米子-反费米子对组成。通过将粒子之间的相互作用视为狄拉克振子耦合，我们分析了时空拓扑对这种粒子能量的影响。𝒞𝒮. 为了实现这一点，我们通过假设粒子的质心处于静止状态并位于空间几何的原点来求解相应形式的二体狄拉克方程（完全协变）。在这个假设下，我们得出了所讨论系统的非微扰能谱。该谱包括自旋耦合并取决于角赤字参数(𝒜𝒟𝒫)的几何背景。这为确定几何背景对能量的影响提供了合适的基础。𝒞𝒮在考虑中。我们的结果表明，这样一个𝒞𝒮表现得像一个单一的量子振荡器。然后，我们分析了能级的变化并讨论了所得结果的局限性。我们表明几何背景对每个能级的影响是不一样的，并且对于较小的值，能级可能存在退化。𝒜𝒟𝒫.