Physical states in the canonical tensor model from the perspective of random tensor networks

Abstract

Tensor models, generalization of matrix models, are studied aiming for quantum gravity in dimensions larger than two. Among them, the canonical tensor model is formulated as a totally constrained system with first-class constraints, the algebra of which resembles the Dirac algebra of general relativity. When quantized, the physical states are defined to be vanished by the quantized constraints. In explicit representations, the constraint equations are a set of partial differential equations for the physical wave-functions, which do not seem straightforward to be solved due to their non-linear character. In this paper, after providing some explicit solutions for N = 2, 3, we show that certain scale-free integration of partition functions of statistical systems on random networks (or random tensor networks more generally) provides a series of solutions for general N . Then, by generalizing this form, we also obtain various solutions for general N . Moreover, we show that the solutions for the cases with a cosmological constant can be obtained from those with no cosmological constant for increased N . This would imply the interesting possibility that a cosmological constant can always be absorbed into the dynamics and is not an input parameter in the canonical tensor model. We also observe the possibility of symmetry enhancement in N = 3, and comment on an extension of Airy function related to the solutions.

Numerical and analytical analyses of a matrix model with non-pairwise contracted indices

“…• The expectation values of some observables are computed by the numerical simulations, and it is observed that there exists a transition region around R ∼ N 2 /2. Intriguingly, the location is in good coincidence with R = (N + 2)(N + 3)/2 that is required by the consistency of the tensor model (the hermiticity of its Hamiltonian constraint) [18,15,29]. Presently, this coincidence is mysterious, since there are no apparent connections between the transition and the consistency.…”

“…where we have determined the overall factor by requiring f (2) N,R,Λ ij (0) = 1, the product is over all the eigenvalues of the matrix Λ ij , and h N,R (x) is defined in (15).…”

“…If the wave function has a peak at a configuration that can well be described by a macroscopic spacetime picture, then the model can be considered to be potentially successful. An indirect motivation for the present paper is to understand the properties of the wave function [15] that is an exact solution to a tensor model in the Hamilton formalism [16,17]. It has been argued and shown for some simple cases that the wave function has peaks at the tensor values that are invariant under Lie groups [18].…”

A random matrix model with non-pairwise contracted indices

“…In [25], it was argued and explicitly shown for some simple cases that the wave function (76) has coherent peaks for some specific loci of P abc where P abc is invariant under Lie-group transformations (namely, P abc = h a a h b b h c c P a b c for ∀ h ∈ H with a Lie-group representation H). In fact, a tensor model [20,21,22] in the Hamilton formalism [23,24] has a similar wave functionψ(P ) R with a power R andψ(P ) very similar to ψ(P ) [18], and it was shown in [19] that the wave function of this tensor model has similar coherent peaks. To consistently interpret this phenomenon as the preference for Lie-group symmetric configurations in the tensor model, we first have to show that we can apply the quantum mechanical probabilistic interpretation to the wave function, namely, the wave function must be absolute square integrable.…”

“…One of our motivations to initiate the study of the model (1) is to investigate the properties of the wave function [18,19] of a tensor model [20,21,22] in the Hamilton formalism [23,24], which is studied in a quantum gravity context. The expression (1) can be obtained after integrating over the tensor argument of the wave function of the toy model introduced in [25], which is closely related to this tensor model.…”