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World-Line Path Integral for the Propagator Expressed as an Ordinary Integral: Concept and Applications
Foundations of Physics ( IF 1.5 ) Pub Date : 2021-03-05 , DOI: 10.1007/s10701-021-00447-8
T. Padmanabhan

The (Feynman) propagator \(G(x_2,x_1)\) encodes the entire dynamics of a massive, free scalar field propagating in an arbitrary curved spacetime. The usual procedures for computing the propagator—either as a time ordered correlator or from a partition function defined through a path integral—requires introduction of a field \(\phi (x)\) and its action functional \(A[\phi (x)]\). An alternative, more geometrical, procedure is to define a propagator in terms of the world-line path integral which only uses curves, \(x^i(s)\), defined on the manifold. I show how the world-line path integral can be reinterpreted as an ordinary integral by introducing the concept of effective number of quantum paths of a given length. Several manipulations of the world-line path integral becomes algebraically tractable in this approach. In particular I derive an explicit expression for the propagator \(G_\mathrm{QG}(x_2,x_1)\), which incorporates the quantum structure of spacetime through a zero-point-length, in terms of the standard propagator \(G_\mathrm{std}(x_2,x_1)\), in an arbitrary curved spacetime. This approach also helps to clarify the interplay between the path integral amplitude and the path integral measure in determining the form of the propagator. This is illustrated with several explicit examples.



中文翻译:

传播者的世界线路径积分表示为普通积分:概念和应用

(Feynman)传播子\(G(x_2,x_1)\)编码在任意弯曲时空中传播的巨大自由标量场的整个动力学。用于计算传播子的常规过程(作为时间相关器或通过路径积分定义的分区函数)需要引入字段\(\ phi(x)\)及其作用函数\(A [\ phi( x)] \)。另一种更具几何意义的过程是,根据仅使用在流形上定义的曲线\(x ^ i(s)\)的世界线路径积分来定义传播器。我展示了如何将世界线路径积分重新解释为一个普通的通过引入给定长度的有效数量的量子路径的概念来积分。在这种方法中,对世界线路径积分的几种操纵在代数上变得易于处理。特别是,我导出了传播子\(G_ \ mathrm {QG}(x_2,x_1)\)显式表达式,该表达式根据标准传播子\ {G_ \ mathrm {std}(x_2,x_1)\),在任意弯曲的时空中。该方法还有助于在确定传播器的形式时阐明路径积分幅度和路径积分度量之间的相互作用。几个显式示例对此进行了说明。

更新日期:2021-03-05
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