论文标题
具有集成性的量子机械系统中复杂的路径积分的全阶复着
All-order Resurgence from Complexified Path Integral in a Quantum Mechanical System with Integrability
论文作者
论文摘要
我们讨论了最简单的量子机械系统之一中的全阶跨系列:A U(1)对称单度系统具有一阶时间派生术语。遵循Lefschetz Thimble方法的过程,我们明确评估了Noether电荷的生成函数的路径积分,并得出其精确的跨系列表达式。使用保护定律,我们发现了动作的所有复杂鞍点,这些鞍座是导致非扰动效应和模型复兴结构的原因。每个鞍点周围的全阶功率序列贡献是在生成函数遵循的微分方程的帮助下从一环的决定因素生成的。跨系列是通过总结所有相关鞍点的贡献来构建的,我们通过确定双针和原始路径积分轮廓之间的相交数来确定。我们确认,扰动系列的Borel模棱两可是由源自相交数字不连续跳跃的非扰动歧义而取消的。在路径综合形式主义中计算出的跨系列与确切的生成函数一致,由于模型的整合性质,可以在操作员形式上获得其明确形式。该协议表明,基于Lefschetz Thimble方法,通过路径积分的半古典膨胀获得的跨系列的非扰动完整性。
We discuss all-order transseries in one of the simplest quantum mechanical systems: a U(1) symmetric single-degree-of-freedom system with a first-order time derivative term. Following the procedure of the Lefschetz thimble method, we explicitly evaluate the path integral for the generating function of the Noether charge and derive its exact transseries expression. Using the conservation law, we find all the complex saddle points of the action, which are responsible for the non-perturbative effects and the resurgence structure of the model. The all-order power-series contributions around each saddle point are generated from the one-loop determinant with the help of the differential equations obeyed by the generating function. The transseries are constructed by summing up the contributions from all the relevant saddle points, which we identify by determining the intersection numbers between the dual thimbles and the original path integration contour. We confirm that the Borel ambiguities of the perturbation series are canceled by the non-perturbative ambiguities originating from the discontinuous jumps of the intersection numbers. The transseries computed in the path-integral formalism agrees with the exact generating function, whose explicit form can be obtained in the operator formalism thanks to the integrable nature of the model. This agreement indicates the non-perturbative completeness of the transseries obtained by the semi-classical expansion of the path integral based on the Lefschetz thimble method.