Constraint algebra of general relativity from a formal continuum limit of canonical tensor model

TL;DR

This paper demonstrates that the constraint algebra of the canonical tensor model (CTM) in a formal continuum limit reproduces the algebra of general relativity's ADM formalism, linking a discrete tensor model to continuum gravity.

Contribution

It shows the exact correspondence between CTM constraints and ADM constraints in a continuum limit, providing a potential discrete approach to quantum gravity.

Findings

CTM reproduces ADM algebra without cosmological constant

Metric tensor expressed in terms of CTM tensor

Cosmological constant induces non-local dynamics

Abstract

Canonical tensor model (CTM for short below) is a rank-three tensor model formulated as a totally constrained system in the canonical formalism. In the classical case, the constraints form a first-class constraint Poisson algebra with structures similar to that of the ADM formalism of general relativity, qualifying CTM as a possible discrete formalism for quantum gravity. In this paper, we show that, in a formal continuum limit, the constraint Poisson algebra of CTM with no cosmological constant exactly reproduces that of the ADM formalism. To this end, we obtain the expression of the metric tensor field in general relativity in terms of one of the dynamical rank-three tensors in CTM, and determine the correspondence between the constraints of CTM and those of the ADM formalism. On the other hand, the cosmological constant term of CTM seems to induce non-local dynamics, and is inconsistent with an assumption about locality of the continuum limit.