Emergence of Lie group symmetric classical spacetimes in canonical tensor model

Abstract

We analyze a wave function of a tensor model in the canonical formalism, when the argument of the wave function takes Lie group invariant or nearby values. Numerical computations show that there are two phases, which we call the quantum and the classical phases, respectively. In the classical phase, fluctuations are suppressed, and there emerge configurations which are discretizations of the classical geometric spaces invariant under the Lie group symmetries. This is explicitly demonstrated for the emergence of $S^n\ (n=1,2,3)$ for $SO(n+1)$ symmetries by checking the topological and the geometric (Laplacian) properties of the emerging configurations. The transition between the two phases has the form of splitting/merging of distributions of variables, resembling a matrix model counterpart, namely, the transition between one-cut and two-cut solutions. However this resemblance is obscured by a difference of the mechanism of the distribution in our setup from that in the matrix model. We also discuss this transition as a replica symmetry breaking. We perform various preliminary studies of the properties of the phases and the transition for such values of the argument.