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.
Highlights
Limits, because of the absence of 1/N expansions,1 which played essential roles in taking the continuum limits of the matrix models
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 provides a series of solutions for general N
While the above results suggest that the canonical tensor model can be an interesting model of quantum gravity, a major question on the validity of the model would be whether its quantum dynamics can produce an object like space-time
Summary
The canonical tensor model has been introduced as a theory of dynamical fuzzy spaces [39,40,41, 45], aiming to construct a quantum theory of gravity. The canonical tensor model stands on the position such that space-time would be a time evolution of the dynamical fuzzy space satisfying (2.3) and (2.4), or equivalently the generalized hermiticity condition, (2.8). The naming of “kinematical” comes from the fact that the operators, J(ab) and J[ab], form a gl(N ) Lie algebra This is a linear Lie algebra with structure constants, and such kinematical physical states would reflect only the kinematical characters rather than the dynamics of the canonical tensor model. A naive expectation is that physically interesting dynamics is caused by the non-linear features of the constraint algebra (2.16) with structure functions, as in general relativity. From this viewpoint, the dynamical states would be of more importance. We will explicitly solve the constraint equations (2.20) to find the physical wave-function
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