  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.
