Emergence of Lie group symmetric classical spacetimes in the 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 Sn (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.

Splitting-merging transitions in tensor-vectors systems in exact large-$N$ limits

“…In fact, in the examples considered in [10], the values of R c agreed with the ranks of C (or very near values in a few large system cases). Therefore the system (1) seems to have an intrinsic dynamics which automatically detects an appropriate rank for a given C. A purpose of this paper is to study this interesting property, which was found in the previous numerical simulation, by an exact method.…”

“…Especially, the errors are very large in the right region (1RSB) of the transition point, which is supposed to be glassy. 10 This rescaling is necessary, because the SO(N ) invariant expression has the form,…”

“…A motivation of considering CTM is to overcome the issue by introducing a temporal direction into tensor models, trying to follow the success of the causal dynamical triangulation over the dynamical triangulation in generating macroscopic spacetimes [9], where the former has a temporal direction, while the latter not. Indeed, in [10], it was explicitly shown by numerical simulations that a wave function of CTM seems to have a twofold phase structure, and classical spacetimes emerge in the one which we call the classical phase. 1 Here the transition between the two phases can be characterized by splitting-merging transitions of distributions of the dynamical variables, like those in the matrix counterparts, such as the Gross-Witten-Wadia transition [11,12] and the transitions among multi-cut large-N solutions [13].…”

“…While a matrix can be decomposed by straightforward procedures, a practical method for the tensor case is to optimize φ i a so that the cost function (C − φφφ) 2 be minimized or vanish [20]. Here one of the difficult issues is that we do not have prior knowledge of an appropriate value of R for a given C: If we take a too large R for the optimization, the decomposition will be overfitting, and, if a too small R is taken, we will miss some properties of C. What was found in [10] and is interesting in the system (1) is that, when β is taken large enough, the dominant configurations of φ i a are such that they are separated into two parts, the dominant and minor parts 5 ,…”

“…Integrating over ψ in (10) generates a new term −(N − n)/2 log detλ in the exponent, and then by assuming large-N and carrying out the λ integration by taking the saddle point, we obtain…”