Abstract
Tensor rank decomposition is a useful tool for geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able to estimate how many tensor rank decompositions can approximate a given tensor. More precisely, finding an approximate symmetric tensor rank decomposition of a symmetric tensor Q with an error allowance Δ is to find vectors ϕi satisfying ∥Q−∑i=1Rϕi⊗ϕi⋯⊗ϕi∥2≤Δ. The volume of all such possible ϕi is an interesting quantity which measures the amount of possible decompositions for a tensor Q within an allowance. While it would be difficult to evaluate this quantity for each Q, we find an explicit formula for a similar quantity by integrating over all Q of unit norm. The expression as a function of Δ is given by the product of a hypergeometric function and a power function. By combining new numerical analysis and previous results, we conjecture a formula for the critical rank, yielding an estimate for the spacetime degrees of freedom of the CTM. We also extend the formula to generic decompositions of non-symmetric tensors in order to make our results more broadly applicable. Interestingly, the derivation depends on the existence (convergence) of the partition function of a matrix model which previously appeared in the context of the CTM.
Highlights
The canonical tensor model (CTM) is a tensor model for quantum gravity which is constructed in the canonical formalism in order to introduce time into a tensor model [1] with, as its fundamental variables, the canonically conjugate pair of real symmetric tensors of degree three, Qabc and Pabc
Several remarkable connections have been found between the CTM and general relativity [3,4,5] which, combined with the fact that defining the quantised model is mathematically very simple and straightforward [6], makes this a very attractive model to study in the context of quantum gravity
Motivated by recent progress in the study of the canonical tensor model, in this work we turned our attention to the space of tensor rank decompositions
Summary
The canonical tensor model (CTM) is a tensor model for quantum gravity which is constructed in the canonical formalism in order to introduce time into a tensor model [1] with, as its fundamental variables, the canonically conjugate pair of real symmetric tensors of degree three, Qabc and Pabc. We study a related quantity ZR(∆), which we arrive at by integrating (1) over normalised tensors Q Analysing this quantity will give us information about the average amount of different decompositions, potentially representing different spaces, close to tensors, and analysing its divergent properties will lead to insights in the expected size, in terms of the amount of fuzzy points, of spaces in the CTM. Another motivation coming from the CTM to study the configuration space of tensor rank decompositions comes from the quantum CTM.
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