The path integral formulation of quantum mechanics, i.e., the idea that the evolution of a quantum system is determined as a sum over all the possible trajectories that would take the system from the initial to its final state of its dynamical evolution, is perhaps the most elegant and universal framework developed in theoretical physics, second only to the standard model of particle physics. In this Tutorial, we retrace the steps that led to the creation of such a remarkable framework, discuss its foundations, and present some of the classical examples of problems that can be solved using the path integral formalism, as a way to introduce the readers to the topic and help them get familiar with the formalism. Then, we focus our attention on the use of path integrals in optics and photonics and discuss in detail how they have been used in the past to approach several problems, ranging from the propagation of light in inhomogeneous media to parametric amplification and quantum nonlinear optics in arbitrary media. To complement this, we also briefly present the path integral Monte Carlo method, as a valuable computational resource for condensed matter physics, and discuss its potential applications and advantages if used in photonics.

中文翻译：

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路径积分：从量子力学到光子学

量子力学的路径积分公式，即量子系统的演化被确定为所有可能的轨迹的总和，这些轨迹将使系统从其动力学演化的初始状态到其最终状态，这可能是最重要的。理论物理学中发展起来的优雅而通用的框架，仅次于粒子物理学的标准模型。在本教程中，我们回顾了导致创建如此出色框架的步骤，讨论了它的基础，并展示了一些可以使用路径积分形式主义解决的问题的经典示例，作为向读者介绍的一种方式并帮助他们熟悉形式主义。然后，我们将注意力集中在光学和光子学中路径积分的使用上，并详细讨论过去如何使用它们来解决几个问题，从非均匀介质中的光传播到任意介质中的参数放大和量子非线性光学. 为了补充这一点，我们还简要介绍了路径积分蒙特卡罗方法，作为凝聚态物理的宝贵计算资源，并讨论了其在光子学中的潜在应用和优势。