论文标题
路径积分和随机演算
Path integrals and stochastic calculus
论文作者
论文摘要
路径积分是理论物理学中普遍存在的工具。但是，有时由于缺乏对各种操纵的控制而受到阻碍，例如进行整合路径的改变 - 人们想以物理学家喜欢的轻松方式进行。随机演算领域也出现了类似的问题，我们会审查以准备正确构建路径积分的基础。在路径积分级别和任意空间维度的层面上，我们不仅报告基于Riemannian几何的方法，这些方法使路径积分可与微积分的标准规则相吻合，而且还基于完全消除时间的方法来实现相同目标。我们在简单的示例上说明了路径积分的各种定义，例如粒子在球体上的扩散。
Path integrals are a ubiquitous tool in theoretical physics. However, their use is sometimes hindered by the lack of control on various manipulations -- such as performing a change of the integration path -- one would like to carry out in the light-hearted fashion that physicists enjoy. Similar issues arise in the field of stochastic calculus, which we review to prepare the ground for a proper construction of path integrals. At the level of path integration, and in arbitrary space dimension, we not only report on existing Riemannian geometry-based approaches that render path integrals amenable to the standard rules of calculus, but also bring forth new routes, based on a fully time-discretized approach, that achieve the same goal. We illustrate these various definitions of path integration on simple examples such as the diffusion of a particle on a sphere.
