Integrability for Relativistic Spin Networks
The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lorentz group appearing in the direct integral decom...
| Main Authors: | , |
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| Format: | Article |
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2001
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| Online Access: | https://eprints.nottingham.ac.uk/7/ |
| _version_ | 1848790365193109504 |
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| author | Barrett, John W. Baez, John C. |
| author_facet | Barrett, John W. Baez, John C. |
| author_sort | Barrett, John W. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lorentz group appearing in the direct integral decomposition of the space of L^2 functions on three-dimensional hyperbolic space. To `evaluate' such a spin network we must do an integral; if this integral converges we say the spin network is `integrable'. Here we show that a large class of relativistic spin networks are integrable, including any whose underlying graph is the 4-simplex (the complete graph on 5 vertices). This proves a conjecture of Barrett and Crane, whose validity is required for the convergence of their state sum model. |
| first_indexed | 2025-11-14T18:11:27Z |
| format | Article |
| id | nottingham-7 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T18:11:27Z |
| publishDate | 2001 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-72020-05-04T20:32:38Z https://eprints.nottingham.ac.uk/7/ Integrability for Relativistic Spin Networks Barrett, John W. Baez, John C. The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lorentz group appearing in the direct integral decomposition of the space of L^2 functions on three-dimensional hyperbolic space. To `evaluate' such a spin network we must do an integral; if this integral converges we say the spin network is `integrable'. Here we show that a large class of relativistic spin networks are integrable, including any whose underlying graph is the 4-simplex (the complete graph on 5 vertices). This proves a conjecture of Barrett and Crane, whose validity is required for the convergence of their state sum model. 2001 Article NonPeerReviewed Barrett, John W. and Baez, John C. (2001) Integrability for Relativistic Spin Networks. Classical and Quantum Gravity, 18 (4683-4). |
| spellingShingle | Barrett, John W. Baez, John C. Integrability for Relativistic Spin Networks |
| title | Integrability for Relativistic Spin Networks |
| title_full | Integrability for Relativistic Spin Networks |
| title_fullStr | Integrability for Relativistic Spin Networks |
| title_full_unstemmed | Integrability for Relativistic Spin Networks |
| title_short | Integrability for Relativistic Spin Networks |
| title_sort | integrability for relativistic spin networks |
| url | https://eprints.nottingham.ac.uk/7/ |