Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models
The quest for a consistent theory for quantum gravity is one of the most challenging problems in theoretical high-energy physics. An often-used approach is to describe the gravitational degrees of freedom by the metric tensor or related variables, and to find a way to quantise this. In the canonical...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2023 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2023
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212008 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models. Dennis Obster. SIGMA 19 (2023), 076, 43 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862740148476706816 |
|---|---|
| author | Obster, Dennis |
| author_facet | Obster, Dennis |
| citation_txt | Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models. Dennis Obster. SIGMA 19 (2023), 076, 43 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The quest for a consistent theory for quantum gravity is one of the most challenging problems in theoretical high-energy physics. An often-used approach is to describe the gravitational degrees of freedom by the metric tensor or related variables, and to find a way to quantise this. In the canonical tensor model, the gravitational degrees of freedom are encoded in a tensorial quantity ₐbc, and this quantity is subsequently quantised. This makes the quantisation much more straightforward mathematically, but the interpretation of this tensor as a spacetime is less evident. In this work, we take a first step towards fully understanding the relationship to spacetime. By considering ₐbc as the generator of an algebra of functions, we first describe how we can recover the topology and the measure of a compact Riemannian manifold. Using the tensor rank decomposition, we then generalise this principle to have a well-defined notion of the topology and geometry for a large class of tensors ₐbc. We provide some examples of the emergence of a topology and measure of both exact and perturbed Riemannian manifolds, and of a purely algebraically-defined space called the semi-local circle.
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| first_indexed | 2026-04-17T17:37:17Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212008 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T17:37:17Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Obster, Dennis 2026-01-22T09:16:35Z 2023 Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models. Dennis Obster. SIGMA 19 (2023), 076, 43 pages 1815-0659 2020 Mathematics Subject Classification: 83C45; 46C05; 16S15 arXiv:2203.03633 https://nasplib.isofts.kiev.ua/handle/123456789/212008 https://doi.org/10.3842/SIGMA.2023.076 The quest for a consistent theory for quantum gravity is one of the most challenging problems in theoretical high-energy physics. An often-used approach is to describe the gravitational degrees of freedom by the metric tensor or related variables, and to find a way to quantise this. In the canonical tensor model, the gravitational degrees of freedom are encoded in a tensorial quantity ₐbc, and this quantity is subsequently quantised. This makes the quantisation much more straightforward mathematically, but the interpretation of this tensor as a spacetime is less evident. In this work, we take a first step towards fully understanding the relationship to spacetime. By considering ₐbc as the generator of an algebra of functions, we first describe how we can recover the topology and the measure of a compact Riemannian manifold. Using the tensor rank decomposition, we then generalise this principle to have a well-defined notion of the topology and geometry for a large class of tensors ₐbc. We provide some examples of the emergence of a topology and measure of both exact and perturbed Riemannian manifolds, and of a purely algebraically-defined space called the semi-local circle. The author would like to thank N. Sasakura for all the fruitful discussions, advice, and encouragement that made this work possible. Furthermore, the author would like to thank the referees who gave valuable input to improve the work. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models Article published earlier |
| spellingShingle | Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models Obster, Dennis |
| title | Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models |
| title_full | Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models |
| title_fullStr | Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models |
| title_full_unstemmed | Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models |
| title_short | Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models |
| title_sort | tensors and algebras: an algebraic spacetime interpretation for tensor models |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212008 |
| work_keys_str_mv | AT obsterdennis tensorsandalgebrasanalgebraicspacetimeinterpretationfortensormodels |