Król Jerzy, Asselmeyer-Maluga Torsten
Chair of Cognitive Science and Mathematical Modeling, University of Information Technology and Management, ul. Sucharskiego 2, 35-225 Rzeszw, Poland.
German Aerospace Center (DLR), Rosa-Luxemburg-Str. 2, D-10178 Berlin, Germany.
Entropy (Basel). 2022 Mar 11;24(3):391. doi: 10.3390/e24030391.
In this paper, we focus on some aspects of the relation of spacetime and quantum mechanics and the study counterparts (in Set) of the categorical local symmetries of smooth 4-manifolds. In the set-theoretic limit, there emerge some exotic smoothness structures on R4 (hence the Riemannian nonvanishing curvature), which fit well with the quantum mechanical lattice of projections on infinite-dimensional Hilbert spaces. The method we follow is formalization localized on the open covers of the spacetime manifold. We discuss our findings in the context of the information paradox assigned to evaporating black holes. A black hole can evaporate entirely, but the smoothness structure of spacetime will be altered and, in this way, the missing information about the initial states of matter forming the black hole will be encoded. Thus, the possible global geometric remnant of black holes in spacetime is recognized as exotic 4-smoothness. The full-fledged verification of this proposal will presumably be possible within the scope of future quantum gravity theory research.
在本文中，我们关注时空与量子力学关系的某些方面，以及光滑4维流形的范畴局部对称性在集合范畴中的对应研究。在集合论极限下，(R^4)上出现了一些奇异的光滑结构（因此黎曼曲率非零），这与无限维希尔伯特空间上投影的量子力学格很好地契合。我们采用的方法是在时空流形的开覆盖上进行局部形式化。我们在与蒸发黑洞相关的信息悖论背景下讨论我们的发现。黑洞可以完全蒸发，但时空的光滑结构会被改变，这样，关于形成黑洞的物质初始状态的缺失信息将被编码。因此，时空中黑洞可能的全局几何遗迹被认为是奇异的4维光滑性。这一建议的全面验证大概将在未来量子引力理论研究范围内实现。
