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.

arXiv comment38 pages, one figure. Some explanations, a figure, and an appendix added, a misprinted equation corrected, no changes of physical contents

References in corpus (23)

• Dynamics of dark energy
• The Dynamics of General Relativity
• Critical phenomena in complex networks
• The dS/CFT Correspondence
• Emergence of a 4D World from Causal Quantum Gravity
• Critical behavior of colored tensor models in the large N limit
• Colored Tensor Models - a Review
• Inflation and the dS/CFT Correspondence
• Colored Group Field Theory
• Melons are branched polymers
• Holographic Renormalization Group
• TASI Lectures on the Higher Spin - CFT duality
• Canonical tensor models with local time
• Uniqueness of canonical tensor model with local time
• Interpreting canonical tensor model in minisuperspace
• Abelian gauge fields coupled to simplicial quantum gravity
• Phase Structure of Four-dimensional Simplicial Quantum Gravity with a U(1) Gauge Field
• Exact Free Energies of Statistical Systems on Random Networks
• Ising model on random networks and the canonical tensor model
• Random volumes from matrices
• Renormalization procedure for random tensor networks and the canonical tensor model
• Matter fields in triangle-hinge models
• Emergent classical geometries on boundaries of randomly connected tensor networks

Cited by in corpus (9)

• Consistent Scalar and Tensor Perturbation Power Spectra in Single Fluid Matter Bounce with Dark Energy Era
• Canonical tensor model through data analysis -- Dimensions, topologies, and geometries --
• Emergent symmetries in the canonical tensor model
• Numerical and analytical analyses of a matrix model with non-pairwise contracted indices
• 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
• Mother canonical tensor model
• Emergence of Lie group symmetric classical spacetimes in canonical tensor model
• Counting tensor rank decompositions