Emergence of classical spacetimes in canonical tensor model

Abstract

We study the wave function of a tensor model in the canonical formalism by Hamiltonian Monte Carlo method for Lie group symmetric or nearby values for the argument of the wave function, and show that there emerge Lie-group symmetric semi-classical spacetimes. More precisely, we consider some 𝑆𝑂 ( 𝑛 + 1 ) ( 𝑛 = 1 , 2 , 3 ) symmetric values for the tensor argument of the wave function, and show that there emerge discrete 𝑛 -dimensional spheres. A key fact is that there exist two phases, the classical phase and the quantum phase, depending on the values of the argument of the wave function, and emergence of classical spaces above occurs in the former phase, while fluctuations of configurations are too large for such emergence in the latter phase. The transition between the two phases has similarity with the Gross-Witten-Wadia transition, or that between the one-cut and the two-cut solutions in the matrix model. Based on the results, we give some speculations on how spacetimes evolve in the tensor model.