Canonical Tensor Models With Local Time

Abstract

It is an intriguing question how local time can be introduced in the emergent picture of spacetime. In this paper, this problem is discussed in the context of tensor models. To consistently incorporate local time into tensor models, a rankthree tensor model with first class constraints in Hamilton formalism is presented. In the limit of usual continuous spaces, the algebra of constraints reproduces that of general relativity in Hamilton formalism. While the momentum constraints can be realized rather easily by the symmetry of the tensor models, the form of the Hamiltonian constraints is strongly limited by the condition of the closure of the whole constraint algebra. Thus the Hamiltonian constraints have been determined on the assumption that they are local and at most cubic in canonical variables. The form of the Hamiltonian constraints has similarity with the Hamiltonian in the c < 1 string field theory, but it seems impossible to realize such a constraint algebras in the framework of vector or matrix models. Instead these models are rather useful as matter theories coupled with the tensor model. In this sense, a three-index tensor is the minimum-rank dynamical variable necessary to describe gravity in terms of tensor models.

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.…”

“…In this subsection, we will apply the results of the previous subsections to study the integrability of the wavefunction of the model introduced in [25]. It is a toy model closely related to a tensor model in the Hamilton formalism, called the canonical tensor model [23,24], which is studied in a quantum gravity context. Let us consider the following wave function depending on a symmetric tensor P abc , where (a, b, c = 1, 2, .…”

“…While the toy model of [25] allows any value of R, the tensor model [23,24] uniquely requires R = (N +2)(N +3)/2 [19,36] for the hermiticity of the Hamiltonian. 13 Because of the very similar form of the wave functions of the models, our interest is therefore especially in the regime R ∼ N 2 .…”

“…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.…”

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

“…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].…”