Some Notes on Quantum Information in Spacetime

The results obtained since the 70s with the study of Hawking radiation and the Unruh effect have highlighted a new domain of authority of relativistic principles [...].

It is known that there are well-defined whormhole solutions in General Relativity and Yang Mills Theory, and the recent ER = EPR conjecture proposes the question of the emergence of metric space-time from a non-local background [34][35][36][37][38]. A suggestion in the direction of the laboratory comes from the Bose-Marletto-Vedral conjecture on the possible coalescence of two quantum systems in a non-local phase, which would reveal the limits of the local metric description and the non-classical aspects of space-time [39,40]. A covariant analysis of this situation shows that discrete effects could prove to be an overlap of geometries measurable through entanglement entropy [41,42].
Furthermore, localization appears as the production of a new degree of freedom. We assume, in accordance with a recent proposal [30,43], that the localization R of a process is associated with the genesis of a micro-horizon of de Sitter of center O and radius cθ0 ≈ 10-13 cm (chronon, corresponding to the classical radius of the electron), with O generally delocalized according to the wave function entering/leaving the process. The constant θ0 is independent of cosmic time, so the ratio t0/θ0 ≈ 10 41 is also independent of cosmic time, with ct0 ≈ 1028 cm. This ratio expresses the number of totally distinct temporal locations accessible by the R process within the horizon of cosmological de Sitter. In practice, the time line segment on which an observer at the center of the horizon places the process R has length t0, while the duration of the process R is in the order of θ0; the segment is therefore divided into separate t0/θ0 ≈ 10 41 "cells". Each cell can be in two states: "on" or "off". The temporal localization of a single process R corresponds to the situation in which all the cells are switched off minus one. Configurations with multiple cells on will correspond to the location of multiple distinct R processes on the same time line. If you accept the idea that each cell is independent, you have 2 10 41 distinct configurations in all. The positional information associated with the location of 0, 1, 2, . . . , 10 41 R processes then amounts to 1041 bits, the binary logarithm of the number of configurations. This is a kind of coded information on the time axis contained within the observer's de Sitter horizon.
The R processes are in fact real interactions between real particles, during which an amount of action is exchanged in the order of the Planck quantum h. Therefore, in terms of phase space, the manifestation of one of these processes is equivalent to the ignition of an elementary cell of volume h3. The number of "switched on" cells in the phase space of a given macroscopic physical system is an estimator of the volume it occupies in this space, and therefore of its entropy. It is therefore conceivable that the location information of the R processes is connected to entropy through the uncertainty principle. This possibility presupposes the "objective" nature of the R processes.
It is therefore natural to ask whether some form of Bekenstein's limit on entropy applies in some way to the two horizons mentioned. If we assume that the information on the temporal location of the processes R, I = 10 41 bits, is connected to the area of the micro-horizon, A = (cθ0) 2 ≈ 10 −26 cm 2 from the holographic relationship: A Then, the spatial extension l of the "cells" associated with an information bit is ≈10 −33 cm, the Planck scale! It is necessary to underline that the Planck scale presents itself in this way as a consequence of the holographic conjecture (1), combined with the "two horizons" hypothesis, and therefore of the finiteness of the information I. It in no way represents a limit to the continuity of spacetime, nor to the spatial or temporal distance between two events (which remains a continuous variable). Furthermore, since I = t0/θ0 and t0 is related to the cosmological constant λ by the relation λ = 4/3t02, the (1) is essentially a definition of the Planck scale as a function of the cosmological constant. A global-local relationship is exactly what we expect from a holographic vacuum theory.
Funding: This research received no external funding.

Conflicts of Interest:
The author declare no conflict of interest.