Abstract
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 can be 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.
Highlights
How to formulate a consistent theory for quantum gravity is one of the major problems in fundamental physics
V, we review persistent homology, a technique from topological data analysis, and demonstrate how one can apply it to the fuzzy spaces
The canonical tensor model (CTM) is a discrete model of gravity, which has a canonical conjugate pair of real symmetric three-way tensors as its dynamical variables
Summary
How to formulate a consistent theory for quantum gravity is one of the major problems in fundamental physics. The first step to answer this question would be to establish the correspondence between tensors and spacetimes To this end, we introduce two well-known techniques in data analysis to the CTM, and formulate a systematic procedure to extract topological and geometric properties of spaces held by the tensors. In Appendix, we show the algorithm of the C++ program we made and used for the tensor-rank decomposition
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
