Principle of diminishing potentialities in large N algebras

A bstract 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, it can be shown that the Principle of Diminishing Potentialities (PDP) holds for the large N algebra of 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 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 ℋ TFD ⊗ 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 𝔥 of the Lie algebra 𝔤 associated to the group G on the Hilbert space ℋ TFD ⊗ L 2 ( H ).