迈向相对论量子理论的概率基础：弯曲时空中的单体玻恩定则
Towards a Probabilistic Foundation of Relativistic Quantum Theory: The One-Body Born Rule in Curved Spacetime
AI总结：
本文针对量子场论的固有数学与概念问题，将玻恩定则推广到弯曲时空，提出无需依赖闵氏时空对称性与整体双曲假设的相对论量子理论概率基础框架，重点发展了可解决时间问题的单体拉格朗日图像。
摘要：
本研究建立了一种研究相对论量子理论基础的新方法，其核心是将用于确定粒子位置概率的量子力学玻恩定则推广至弯曲时空场景。开展本研究的主要动因是解决量子场论（QFT）的内部数学问题，例如“无穷大问题”（即重整化问题），QFT的公理化研究路径已表明，这类问题不仅是数学层面的，更属于概念层面的固有问题。本文提出的方法从构造上就具备概率属性，可适配多种动力学模型，不依赖闵可夫斯基时空的对称性，且符合广义相对性原理。在本研究的分析部分，我们在所涉及数学量满足光滑性的假设下考察了单体场景，该场景被认定为广义相对论连续性方程理论的一个特例。现有针对玻恩定则的相对论推广相关研究，通常假设所考察的超曲面为类空超曲面，且时空是整体双曲的；我们借助C. Eckart和J. Ehlers的前期成果证明，类空超曲面条件可自然替换为横截性条件，而整体双曲时空的假设则完全不必要。我们讨论了单体场景的两种不同表述，借鉴非相对论对应场景的术语，将其分别命名为拉格朗日图像和欧拉图像，并对两种表述进行了全面分析。本研究对数学物理领域的主要贡献是发展了拉格朗日图像，该图像展示了如何在这一研究路径下解决“时间问题”，因此可作为推广到多体场景、粒子数不守恒场景的通用蓝图，文中也给出了粒子数不守恒场景的示例。
英文摘要：
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J. Ehlers to show that the former condition is naturally replaced by a transversality condition and that the latter one is obsolete.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).
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