Physical states in the canonical tensor model from the perspective of random tensor networks

Abstract

Tensor models, generalization of matrix models, are studied aiming for quantum gravity in dimensions larger than two. Among them, the canonical tensor model is formulated as a totally constrained system with first-class constraints, the algebra of which resembles the Dirac algebra of general relativity. When quantized, the physical states are defined to be vanished by the quantized constraints. In explicit representations, the constraint equations are a set of partial differential equations for the physical wave-functions, which do not seem straightforward to be solved due to their non-linear character. In this paper, after providing some explicit solutions for $N=2,3$, we show that certain scale-free integration of partition functions of statistical systems on random networks (or random tensor networks more generally) provides a series of solutions for general $N$. Then, by generalizing this form, we also obtain various solutions for general $N$. Moreover, we show that the solutions for the cases with a cosmological constant can be obtained from those with no cosmological constant for increased $N$. This would imply the interesting possibility that a cosmological constant can always be absorbed into the dynamics and is not an input parameter in the canonical tensor model. We also observe the possibility of symmetry enhancement in $N=3$, and comment on an extension of Airy function related to the solutions.

arXiv comment41 pages, 1 figure; typos corrected

References in corpus (39)

• The Dynamics of General Relativity
• Critical phenomena in complex networks
• 2D Gravity and Random Matrices
• Emergence of a 4D World from Causal Quantum Gravity
• Critical behavior of colored tensor models in the large N limit
• Topological Lattice Models in Four Dimensions
• Colored Tensor Models - a Review
• A Model of Three-Dimensional Lattice Gravity
• Group Field Theory: An overview
• Barrett-Crane model from a Boulatov-Ooguri field theory over a homogeneous space
• Colored Group Field Theory
• Melons are branched polymers
• Double Scaling in Tensor Models with a Quartic Interaction
• Renormalization of an SU(2) Tensorial Group Field Theory in Three Dimensions
• The Tensor Track, III
• Tensorial methods and renormalization in Group Field Theories
• Feynman Diagrams of Generalized Matrix Models and the Associated Manifolds in Dimension 4
• The 1/N Expansion of Tensor Models Beyond Perturbation Theory
• Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions
• Some classes of renormalizable tensor models
• New 1/N expansions in random tensor models
• The 1/N expansion of multi-orientable random tensor models
• Group field theory as the 2nd quantization of Loop Quantum Gravity
• Towards a double-scaling limit for tensor models: probing sub-dominant orders
• On the finite amplitudes for open graphs in Abelian dynamical colored Boulatov-Ooguri models
• Beta functions of $U(1{)}^d$ gauge invariant just renormalizable tensor models
• Discrete Renormalization Group for SU(2) Tensorial Group Field Theory
• Canonical tensor models with local time
• Tensor models from the viewpoint of matrix models: the case of loop models on random surfaces
• Uniqueness of canonical tensor model with local time
• An analysis of the intermediate field theory of $T^4$ tensor model
• Interpreting canonical tensor model in minisuperspace
• Quantum canonical tensor model and an exact wave function
• 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
• A canonical rank-three tensor model with a scaling constraint
• Tensor models and 3-ary algebras
• A Givental-like Formula and Bilinear Identities for Tensor Models

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