Abstract
In this paper, we demonstrate, in the context of Loop Quantum Gravity, the Quantum Holographic Principle, according to which the area of the boundary surface enclosing a region of space encodes a qubit per Planck unit. To this aim, we introduce fermion fields in the bulk, whose boundary surface is the two-dimensional sphere. The doubling of the fermionic degrees of freedom and the use of the Bogolyubov transformations lead to pairs of the spin network’s edges piercing the boundary surface with double punctures, giving rise to pixels of area encoding a qubit. The proof is also valid in the case of a fuzzy sphere.
Highlights
The holographic principle (HP) proposed by ‘t Hooft [1] and Susskind [2] states that the information entropy stored in a region of space of volume V is encoded by the area A of the boundary surface enclosing V, precisely one classical bit per unit of Planck area.HP is based on the thermodynamics of black holes and is, in a sense, a generalization of the Bekenstein bound [3]
In this paper, we demonstrate, in the context of Loop Quantum Gravity, the Quantum Holographic Principle, according to which the area of the boundary surface enclosing a region of space encodes a qubit per Planck unit
We briefly review the Holographic Principle, as well as spin networks and their application to black holes entropy in the context of Loop Quantum Gravity (LQG)
Summary
The holographic principle (HP) proposed by ‘t Hooft [1] and Susskind [2] states that the information entropy stored in a region of space of volume V is encoded by the area A of the boundary surface enclosing V, precisely one classical bit per unit of Planck area. We show that if instead of the ordinary sphere, we consider the fuzzy sphere [26] in the N = 2 representation of SU(2), the double puncture consists of two cells: one cell encodes the bit 0 and the other cell encodes the bit 1, so that the N = 2 fuzzy sphere as a whole encodes a qubit In this case, the doubling of the degrees of freedom is already inherent in the two cells and the fermion field is reduced directly to a “peak” of quantum information. We briefly review the (classical) Holographic Principle (and we will suggest a new entropy bound in the quantum case), as well as spin networks and their application to black holes entropy in the context of LQG.
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