三阱到双阱同伦中的实时瞬子动力学
Real-Time Instanton Dynamics in a Triple-Well to Double-Well Homotopy
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- Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles University（查尔斯大学数学与物理学院粒子与核物理研究所）
- Department of Low-Temperature Physics, Faculty of Mathematics and Physics, Charles University（查尔斯大学数学与物理学院低温物理系）
- Department of Physics, Faculty of Applied Sciences, University of West Bohemia（西波希米亚大学应用科学学院物理系）
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中文总结 AI 辅助
本文通过三阱到双阱的同伦构造，证明实时瞬子奇点是双阱极限的退化现象，并给出奇点出现条件的完整代数与几何刻画。
中文摘要 AI 辅助
我们证明，在解析延拓的双阱瞬子中发现的实时奇点并非闵可夫斯基隧穿的固有特征，而是一种退化的极限现象。利用三阱势与双阱势之间的线性同伦，我们构造了一个由$p\in(0,1)$参数化的势函数族，其实时（$\alpha=0$）瞬子解处处正则且有界。对于任意威克旋转角$\alpha\in(0,\pi/2)$，我们证明正则性是普遍的但非普适的。存在一个可数无穷的、闭式形式的势参数族$\{p_{s,k}(\alpha)\}_{k\geq0}$，在这些参数下，瞬子恰好发展出两个实时奇点，且绝不会更多。在$\alpha=0$处取严格双阱极限（$p\to1$）时，这些奇点坍缩为Cherman和Ünsal发现的无穷奇点梳。通过用Weierstrass椭圆函数求解复化运动方程，我们将该奇点梳追溯到退向$\pm\mathrm{i}\infty$的经典虚转向点，从而对实时瞬子何时以及恰好多少次变得奇异给出了统一的几何与代数解释。
英文摘要
We show that the real-time singularities found in the analytically continued double-well instanton are not an inherent feature of Minkowski tunneling, but a degenerate limiting phenomenon. Using a linear homotopy between a triple-well and a double-well potential, we construct a family of potentials, parametrized by $p\in(0,1)$, whose real-time ($α=0$) instanton solutions are everywhere regular and bounded. For any Wick rotation angle $α\in(0,π/2)$, we prove that regularity is generic but not universal. There exists a countably infinite, closed-form family of potential parameters $\{p_{s,k}(α)\}_{k\geq0}$ at which the instanton develops exactly two real-time singularities, never more. In the strict double-well limit ($p\to1$) taken at $α=0$, these collapse into the infinite singular comb found by Cherman and Ünsal. By solving the complexified equations of motion in terms of Weierstrass elliptic functions, we trace this comb to classical imaginary turning points receding to $\pm\mathrm{i}\infty$, giving a unified geometric and algebraic account of when, and exactly how often, real-time instantons become singular.
