Abstract
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 (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.
Highlights
The proper description of the regime whereby classical spacetime itself overlaps with the quantum regime of reality and becomes subject to quantum rules is currently one of the biggest unsolved puzzles in theoretical physics
This is why certain, sometimes subtle, indications may come from the mathematics of very successful physical theories, such as quantum field theory or quantum mechanics (QM), which are embedded in the field of operator algebras, and general relativity (GR), which is from the field of differential geometry and topology
When black holes (BH) is formed in spacetime, this method can be extended such that the descrepancies of the singular quantum region and smooth cosmological 4-domain are expressible in terms of random forcings and B -local modifications of spacetime
Summary
The proper description of the regime whereby classical spacetime H and the forcing extensions of ZFC models M of local data has been analyzed in some papers (e.g., [6,10,12]) This is why we report on this in the key terminologies section below, which aims to provide a brief overview of the state of the art in this subject and the methods applied. The main result is stated in Theorem 1, where it is shown that the smoothness resulting from the QM lattice L is the locally B modified 4-smoothness of spacetime In the Appendix A, we illustrate the phenomenon in the case of Schwarzschild BH
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