Principle of Diminishing Potentialities in Large N Algebras

• URL: http://arxiv.org/abs/2508.11688v3
• Date: Wed, 24 Sep 2025 09:21:59 GMT
• Title: Principle of Diminishing Potentialities in Large N Algebras
• Authors: Bik Soon Sia,
• Abstract summary: In thermal equilibrium, we show that the Principle of Diminishing Potentialities ( PDP) holds for the large $N$ algebra of $mathcalN=4$ Super Yang-Mills (SYM) theory.<n>We extend the large $N$ algebra by performing crossed product by the maximal abelian subgroup $H $ of the compact symmetry group $G$ of the two-sided eternal black hole.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A connection between the completion of quantum mechanics featuring events and the theory of emergent spacetime in quantum gravity where von Neumann algebra plays a vital role is established. In thermal equilibrium, we show that the Principle of Diminishing Potentialities (PDP) holds for the large $N$ algebra of $\mathcal{N}=4$ Super Yang-Mills (SYM) theory with gauge group $SU(N)$ when the temperature is higher than Hawking-Page temperature. Below Hawking-Page transition and for the case of zero temperature, PDP does not hold. Since the centralizer of thermofield double state on the large $N$ algebra of $\mathcal{N}=4$ SYM theory coincides with the center of the large $N$ algebra which is trivial, we extend the large $N$ algebra by performing crossed product by the maximal abelian subgroup $H $ of the compact symmetry group $G$ of the two-sided eternal black hole. In this case, the centralizer of an extension of thermofield double state is non-trivial and it is given by the action of the maximal abelian subgroup $H$ on the Hilbert space $\mathcal{H}_{TFD} \otimes L^{2} (H)$. This centralizer is by itself commutative and it coincides with its own center. This implies that the first actual event that initiate the ``Events-Trees-Histories'' dynamical evolution in this framework is given by the spectral projectors associated to the action of the Cartan subalgebra $\mathfrak{h}$ of the Lie algebra $\mathfrak{g}$ associated to the group $G$ on the Hilbert space $\mathcal{H}_{TFD} \otimes L^{2} (H)$.

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