Canonical tensor models with local time

TL;DR

This paper introduces a rank-three tensor model with local time and first class constraints, reproducing the algebra of general relativity in the continuum limit, and discusses its implications for emergent spacetime and gravity.

Contribution

It presents a novel tensor model incorporating local time with a consistent constraint algebra, advancing the understanding of emergent spacetime in tensor models.

Findings

Reproduces the algebra of general relativity in the continuum limit

Determines Hamiltonian constraints that are local and cubic in variables

Suggests three-index tensors as minimal variables for gravity in tensor models

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 rank- three 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.