Equation of motion of canonical tensor model and Hamilton-Jacobi equation of general relativity

TL;DR

This paper demonstrates that the classical equations of motion of the canonical tensor model in a continuum limit align with those of a gravity-scalar field system derived from the Hamilton-Jacobi equation, revealing critical dimensional behavior.

Contribution

It shows the classical continuum limit of the canonical tensor model corresponds to a gravity-scalar field system from Hamilton-Jacobi theory, identifying a critical dimension and emergent conformal symmetry.

Findings

Equations of motion match those of a gravity-scalar system in continuum limit.

Critical dimension identified as six, above which instability occurs.

De Sitter spacetime emerges as a solution at the critical dimension.

Abstract

The canonical tensor model (CTM) is a rank-three tensor model formulated as a totally constrained system in the canonical formalism. The constraint algebra of CTM has a similar structure as that of the ADM formalism of general relativity, and is studied as a discretized model for quantum gravity. In this paper, we analyze the classical equation of motion (EOM) of CTM in a formal continuum limit through a derivative expansion of the tensor up to the forth order, and show that it is the same as the EOM of a coupled system of gravity and a scalar field derived from the Hamilton-Jacobi equation with an appropriate choice of an action. The action contains a scalar field potential of an exponential form, and the system classically respects a dilatational symmetry. We find that the system has a critical dimension, given by six, over which it becomes unstable due to the wrong sign of the scalar kinetic term. In six dimensions, de Sitter spacetime becomes a solution to the EOM, signaling the emergence of a conformal symmetry, while the time evolution of the scale factor is power-law in dimensions below six.