Fundamental Klein-Gordon Equation from Stochastic Mechanics in Curved Spacetime

IF 1 3区物理与天体物理 Q3 PHYSICS, MULTIDISCIPLINARY

Foundations of Physics Pub Date : 2025-07-09 DOI:10.1007/s10701-025-00873-y

Eric S. Escobar-Aguilar, Tonatiuh Matos, J. I. Jiménez-Aquino

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Abstract

This work presents an alternative approach to obtain the quantum field equations in curved spacetime, considering that sufficiently small particles follow stochastic trajectories around geodesic. Our proposal is based on a stochastic differential equation in which the noise term experienced by the quantum particles is a consequence of the stochastic background in spacetime. This fact allows the particles to describe erratic movements and locally the universe exhibits characteristics akin to a lake with gentle ripples rather than a flat unyielding surface. Building upon this foundational understanding, we investigate the influence of this background on quantum-scale particles without considering the metric to be stochastic, rather we let test particles move randomly around the geodesic of macroscopic particles. Their behavior aligns with solutions to the Klein-Gordon (KG) equation specific to this curved spacetime. As the KG equation, in its non-relativistic limit within a flat spacetime, reduces to the Schrödinger equation, consequently, we propose a compelling connection: the Schrödinger equation may emerge directly from a spacetime lacking local smoothness.

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弯曲时空中随机力学中的基本Klein-Gordon方程

考虑到足够小的粒子沿着测地线的随机轨迹运动，本文提出了一种在弯曲时空中获得量子场方程的替代方法。我们的建议是基于一个随机微分方程，其中量子粒子所经历的噪声项是时空随机背景的结果。这一事实使粒子能够描述不稳定的运动，而宇宙局部表现出类似于湖泊的特征，有柔和的涟漪，而不是平坦的坚硬表面。在此基础上，我们研究了这种背景对量子尺度粒子的影响，而不考虑度量是随机的，而是让测试粒子在宏观粒子的测地线上随机移动。它们的行为与克莱因-戈登（KG）方程的解是一致的，这个方程是专门针对弯曲时空的。由于KG方程在平坦时空内的非相对论性极限下约化为Schrödinger方程，因此，我们提出了一个令人信服的联系：Schrödinger方程可能直接来自缺乏局部光滑性的时空。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Foundations of Physics 物理-物理：综合

CiteScore

2.70

自引率

6.70%

发文量

104

审稿时长

6-12 weeks

期刊介绍： The conceptual foundations of physics have been under constant revision from the outset, and remain so today. Discussion of foundational issues has always been a major source of progress in science, on a par with empirical knowledge and mathematics. Examples include the debates on the nature of space and time involving Newton and later Einstein; on the nature of heat and of energy; on irreversibility and probability due to Boltzmann; on the nature of matter and observation measurement during the early days of quantum theory; on the meaning of renormalisation, and many others. Today, insightful reflection on the conceptual structure utilised in our efforts to understand the physical world is of particular value, given the serious unsolved problems that are likely to demand, once again, modifications of the grammar of our scientific description of the physical world. The quantum properties of gravity, the nature of measurement in quantum mechanics, the primary source of irreversibility, the role of information in physics – all these are examples of questions about which science is still confused and whose solution may well demand more than skilled mathematics and new experiments. Foundations of Physics is a privileged forum for discussing such foundational issues, open to physicists, cosmologists, philosophers and mathematicians. It is devoted to the conceptual bases of the fundamental theories of physics and cosmology, to their logical, methodological, and philosophical premises. The journal welcomes papers on issues such as the foundations of special and general relativity, quantum theory, classical and quantum field theory, quantum gravity, unified theories, thermodynamics, statistical mechanics, cosmology, and similar.