Quantum Canonical Tensor Model and an Exact Wave Function

Takeuchi

2020

Eur. Phys. J. C

Self Cite

We study a matrix model that has φ i a (a = 1, 2, . . . , N, i = 1, 2, . . . , R) as its dynamical variable, whose lower indices are pairwise contracted, but upper ones are not always done so. This matrix model has a motivation from a tensor model for quantum gravity, and is also related to the physics of glasses, because it has the same form as what appears in the replica trick of the spherical p-spin model for spin glasses, though the parameter range of our interest is different. To study the dynamics, which in general depends on N and R, we perform Monte Carlo simulations and compare with some analytical computations in the leading and the next-leading orders. A transition region has been found around R ∼ N 2 /2, which matches a relation required by the consistency of the tensor model. The simulation and the analytical computations agree well outside the transition region, but not in this region, implying that some relevant configurations are not properly included by the analytical computations. With a motivation coming from the tensor model, we also study the persistent homology of the configurations generated in the simulations, and have observed its gradual change from S 1 to higher dimensional cycles with the increase of R around the transition region.

“…A caution is that each leg on one vertex brings independent lower indices, but a common upper index. (28) with (29). We find the two diagrams shown in the right figure of Figure 1.…”

Section: Let Us Definementioning

confidence: 83%

Section: Introductionmentioning

confidence: 83%

“…B Derivation of (29) In this section, we derive (29). This was previously derived in [21], but this is repeated here to make the present paper self-contained.…”

Section: A R = 2 Casementioning

confidence: 98%

“…In this appendix, we will compute the fourth-order term (Pφ 3 ) 4 c φ in (27). From the definition (25) of cumulants and the formula (29), we obtain…”

Section: A R = 2 Casementioning

confidence: 99%

See 2 more Smart Citations

Numerical and analytical analyses of a matrix model with non-pairwise contracted indices

Takeuchi

2020

Eur. Phys. J. C

Self Cite

“…While the toy model of [25] allows any value of R, the tensor model [23,24] uniquely requires R = (N +2)(N +3)/2 [19,36] for the hermiticity of the Hamiltonian. 13 Because of the very similar form of the wave functions of the models, our interest is therefore especially in the regime R ∼ N 2 .…”

Section: Application To a Tensor Modelmentioning

confidence: 99%

A random matrix model with non-pairwise contracted indices

Lionni

Progress of Theoretical and Experimental Physics

2019

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We consider a random matrix model with both pairwise and non-pairwise contracted indices. The partition function of the matrix model is similar to that appearing in some replicated systems with random tensor couplings, such as the p-spin spherical model for the spin glass. We analyze the model using Feynman diagrammatic expansions, and provide an exhaustive characterization of the graphs which dominate when the dimensions of the pairwise and (or) nonpairwise contracted indices are large. We apply this to investigate the properties of the wave function of a toy model closely related to a tensor model in the Hamilton formalism, which is studied in a quantum gravity context, and obtain a result in favor of the consistency of the quantum probabilistic interpretation of this tensor model. *

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

Narain

Sato

2015

J. High Energ. Phys.

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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.