论文标题
迈向相对论量子理论的概率基础：弯曲的时空中的一体规则
Towards a Probabilistic Foundation of Relativistic Quantum Theory: The One-Body Born Rule in Curved Spacetime
论文作者
论文摘要
在这项工作中，我们为相对论量子理论的基础建立了一种新颖的方法，该方法基于概括量子力学诞生规则，用于确定粒子位置概率的弯曲时空。这项研究的主要动机是克服量子场理论（QFT）的内部数学问题，例如“无限问题”（重生）（重生），而公理的QFT方法不仅显示出数学上的数学，而且是概念性的。此处介绍的方法是通过构造概率的，可以容纳各种动力学模型，不依赖Minkowski时空的对称性，并尊重相对性的一般原理。 在这项工作的分析部分中，我们考虑了涉及数量的数学数量平滑度的$ 1 $体案例。这被确定为一般偏见的连续性方程理论的特殊情况。尽管相关的方法对诞生规则的相对论概括假设感兴趣的超级表面是空间般的，并且时空是全球夸张的，但我们采用了C. Eckart和J. Ehlers的事先贡献，以表明前者自然地被横向条件替代，而后者的横向状态则是过时的。我们讨论了$ 1 $ body案的两个不同的表述，这些案例是从非相关主义类似物中借用术语的，我们称拉格朗日和欧拉利亚图片。我们提供两者的全面处理。这项工作对数学物理学文献的主要贡献是拉格朗日图片的发展。 Langrangian图片显示了如何在这种方法中解决“时间问题”，因此可以作为对许多身体的概括的蓝图，并且没有保留物体的数量（后者给出的示例）。
In this work we establish a novel approach to the foundations of relativistic quantum theory, which is based on generalizing the quantum-mechanical Born rule for determining particle position probabilities to curved spacetime. A principal motivator for this research has been to overcome internal mathematical problems of quantum field theory (QFT) such as the `problem of infinities' (renormalization), which axiomatic approaches to QFT have shown to be not only of mathematical but also of conceptual nature. The approach presented here is probabilistic by construction, can accommodate a wide array of dynamical models, does not rely on the symmetries of Minkowski spacetime, and respects the general principle of relativity. In the analytical part of this work we consider the $1$-body case under the assumption of smoothness of the mathematical quantities involved. This is identified as a special case of the theory of the general-relativistic continuity equation. While related approaches to the relativistic generalization of the Born rule assume the hypersurfaces of interest to be spacelike and the spacetime to be globally hyperbolic, we employ prior contributions by C. Eckart and J. Ehlers to show that the former condition is naturally replaced by a transversality condition and that the latter one is obsolete. We discuss two distinct formulations of the $1$-body case, which, borrowing terminology from the non-relativistic analog, we term the Lagrangian and Eulerian pictures. We provide a comprehensive treatment of both. The main contribution of this work to the mathematical physics literature is the development of the Lagrangian picture. The Langrangian picture shows how one can address the `problem of time' in this approach and therefore serves as a blueprint for the generalization to many bodies and the case that the number of bodies is not conserved (example given for the latter).
