Canonical tensor models with local time

Abstract

It is an intriguing question how local time can be introduced in the emergent picture of spacetime. In this paper, this problem is discussed in the context of tensor models. To consistently incorporate local time into tensor models, a rank- three tensor model with first class constraints in Hamilton formalism is presented. In the limit of usual continuous spaces, the algebra of constraints reproduces that of general relativity in Hamilton formalism. While the momentum constraints can be realized rather easily by the symmetry of the tensor models, the form of the Hamiltonian constraints is strongly limited by the condition of the closure of the whole constraint algebra. Thus the Hamiltonian constraints have been determined on the assumption that they are local and at most cubic in canonical variables. The form of the Hamiltonian constraints has similarity with the Hamiltonian in the c < 1 string field theory, but it seems impossible to realize such a constraint algebras in the framework of vector or matrix models. Instead these models are rather useful as matter theories coupled with the tensor model. In this sense, a three-index tensor is the minimum-rank dynamical variable necessary to describe gravity in terms of tensor models.

25 pages, v3: misleading sentences concerning time reversal symmetry corrected, v2:minor corrections

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• A Renormalizable 4-Dimensional Tensor Field Theory
• Physical states in the canonical tensor model from the perspective of random tensor networks
• Canonical tensor model through data analysis -- Dimensions, topologies, and geometries --
• Symmetric configurations highlighted by collective quantum coherence
• Emergent symmetries in the canonical tensor model
• Equation of motion of canonical tensor model and Hamilton-Jacobi equation of general relativity
• A random matrix model with non-pairwise contracted indices
• A canonical rank-three tensor model with a scaling constraint
• Ising model on random networks and the canonical tensor model
• Numerical and analytical analyses of a matrix model with non-pairwise contracted indices
• Renormalization procedure for random tensor networks and the canonical tensor model
• Matter fields in triangle-hinge models
• Phases of a matrix model with non-pairwise index contractions
• Emergent classical geometries on boundaries of randomly connected tensor networks
• An $OSp$ extension of Canonical Tensor Model
• Splitting-merging transitions in tensor-vectors systems in exact large-$N$ limits
• Phase profile of the wave function of canonical tensor model and emergence of large spacetimes
• Symmetry enhancement in a two-logarithm matrix model and the canonical tensor model
• Constraint algebra of general relativity from a formal continuum limit of canonical tensor model
• Emergence of Lie group symmetric classical spacetimes in canonical tensor model
• Mother canonical tensor model
• Counting tensor rank decompositions
• Triangle-hinge models for unoriented membranes