Equation of motion of canonical tensor model and Hamilton-Jacobi equation of general relativity

Chen, Hua; Sasakura, Naoki; Sato, Yuki

doi:10.1103/physrevd.95.066008

Cited by 14 publications

(56 citation statements)

References 34 publications

(159 reference statements)

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“…In [23] the authors discussed the correspondence between the CTM and a general relativistic system by performing a derivative expansion of P in a formal continuum limit of the CTM.…”

Section: Persistent Homologymentioning

confidence: 99%

“…These fields contain the degrees of freedom of the CTM in the formal continuum limit, which corresponds to a general relativistic system. The relation between the β fields and the scalar field φ and the metric field g µν of the relativistic system has been found by analyzing the equations of motion of the CTM and is given by [23] β ] as in (16). In particular, these weights are taken so that the weights associated to each index of P xyz are [g 1/4 ] and the integral for an index contraction d D x P xab P xcd is invariant under diffeomorphisms.…”

Section: Persistent Homologymentioning

confidence: 99%

“…15 A comment is in order. We could have used d Dω g(ω) as the volume element in (23). In this case, (27) is changed to…”

Section: Distances By a Virtual Diffusion Processmentioning

confidence: 99%

“…So in this case the φ and g µν fields completely decouple in the diffusion process. Namely, this choice gives more direct meaning to the coefficients of the diffusion equation from the point of view of the identification of the fields obtained in [23]. However, this choice is practically difficult to implement for the discrete case, since there is no natural way to know √ g while defining the kernel.…”

Section: Distances By a Virtual Diffusion Processmentioning

confidence: 99%

“…This is consistent with the rough picture that the two-sphere becomes larger in time, as discussed above in the sense of the number of points. In the following section, we perform more detailed analysis with comparison with the general relativistic system derived in [23].…”

Section: Time Evolutions Of Fuzzy Spaces In the Ctmmentioning

confidence: 99%

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Canonical tensor model through data analysis: Dimensions, topologies, and geometries

2018

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The canonical tensor model (CTM) is a tensor model in Hamilton formalism and is studied as a model for gravity in both classical and quantum frameworks. Its dynamical variables are a canonical conjugate pair of real symmetric three-index tensors, and a question in this model was how to extract spacetime pictures from the tensors. We give such an extraction procedure by using two techniques widely known in data analysis. One is the tensor-rank (or CP etc.) decomposition, which is a certain generalization of the singular value decomposition of a matrix and decomposes a tensor into a number of vectors. By regarding the vectors as points forming a space, topological properties are extracted by using the other data analysis technique called persistent homology, and geometries by virtual diffusion processes over points. Thus, time evolutions of the tensors in the CTM can be interpreted as topological and geometric evolutions of spaces. We have performed some initial investigations of the classical equation of motion of the CTM in terms of these techniques for a homogeneous fuzzy circle and homogeneous two-and three-dimensional fuzzy spheres as spaces, and have obtained agreement with the general relativistic system obtained previously in a formal continuum limit of the CTM. It is also demonstrated by some concrete examples that the procedure is general for any dimensions and topologies, showing the generality of the CTM.

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“…In [23] the authors discussed the correspondence between the CTM and a general relativistic system by performing a derivative expansion of P in a formal continuum limit of the CTM.…”

Section: Persistent Homologymentioning

confidence: 99%

Section: Persistent Homologymentioning

confidence: 99%

“…15 A comment is in order. We could have used d Dω g(ω) as the volume element in (23). In this case, (27) is changed to…”

Section: Distances By a Virtual Diffusion Processmentioning

confidence: 99%

Section: Distances By a Virtual Diffusion Processmentioning

confidence: 99%

Section: Time Evolutions Of Fuzzy Spaces In the Ctmmentioning

confidence: 99%

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Canonical tensor model through data analysis: Dimensions, topologies, and geometries

2018

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Mother canonical tensor model

Narain

Sasakura

2017

Class. Quantum Grav.

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Canonical tensor model (CTM) is a tensor model formulated in the Hamilton formalism as a totally constrained system with first class constraints, the algebraic structure of which is very similar to that of the ADM formalism of general relativity. It has recently been shown that a formal continuum limit of the classical equation of motion of CTM in a derivative expansion of the tensor up to the fourth derivatives agrees with that of a coupled system of general relativity and a scalar field in the Hamilton-Jacobi formalism. This suggests the existence of a "mother" tensor model which derives CTM through the Hamilton-Jacobi procedure, and we have successfully found such a "mother" CTM (mCTM) in this paper. The quantization of mCTM is straightforward as CTM. However, we have not been able to identify all the secondary constraints, and therefore the full structure of the model has been left for future study. Nonetheless, we have found some exact physical wave functions and classical phase spaces which can be shown to solve the primary and all the (possibly infinite) secondary constraints in the quantum and classical cases, respectively, and have thereby proven the non-triviality of the model. It has also been shown that mCTM has more interesting dynamics than CTM from the perspective of randomly connected tensor networks.

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The tensor of the exact circle: reconstructing geometry

Obster

2023

Phys. Scr.

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Developing a theory for quantum gravity is one of the big open questions in theoretical high-energy physics. Recently, a tensor model approach has been considered that treats tensors as the generators of commutative non-associative algebras, which might be an appropriate interpretation of the canonical tensor model. In this approach, the non-associative algebra is assumed to be a low-energy description of the so-called associative closure, which gives the full description of spacetime including the high-energy modes. In the previous work it has been shown how to (re)construct a topological space with a measure on it, and one of the prominent examples that was used to develop the framework was the exact circle. In this work we will further investigate this example, and show that it is possible to reconstruct the full Riemannian geometry by reconstructing the metric tensor. Furthermore, it is demonstrated how diﬀeomorphisms behave in this formalism, ﬁrstly by considering a speciﬁc class of diﬀeomorphisms of the circle, namely the ellipses, and subsequently by performing an explicit diﬀeomorphism to “smoothen” sets of points generated by the tensor rank decomposition.

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Equation of motion of canonical tensor model and Hamilton-Jacobi equation of general relativity

Cited by 14 publications

References 34 publications

Canonical tensor model through data analysis: Dimensions, topologies, and geometries

Canonical tensor model through data analysis: Dimensions, topologies, and geometries

Mother canonical tensor model

The tensor of the exact circle: reconstructing geometry

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