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

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.

38 pages, one figure. Some explanations, a figure, and an appendix added, a misprinted equation corrected, no changes of physical contents

References in corpus (4)

• Dynamics of dark energy
• Critical phenomena in complex networks
• TASI Lectures on the Higher Spin - CFT duality
• Renormalization procedure for random tensor networks and the canonical tensor model

Cited by in corpus (4)

• Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models
• Phase profile of the wave function of canonical tensor model and emergence of large spacetimes
• Emergence of Lie group symmetric classical spacetimes in canonical tensor model
• Counting tensor rank decompositions