Quantum gravity in three dimensions, Witten spinors and the quantisation of length
In this paper, I investigate the quantisation of length in euclidean quantum gravity in three dimensions. The starting point is the classical hamiltonian formalism in a cylinder of finite radius. At this finite boundary, a counter term is introduced that couples the gravitational field in the interi...
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| Format: | Article |
| Language: | English |
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Elsevier
2018-05-01
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| Series: | Nuclear Physics B |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S0550321318300671 |
| _version_ | 1818180614584532992 |
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| author | Wolfgang Wieland |
| author_facet | Wolfgang Wieland |
| author_sort | Wolfgang Wieland |
| collection | DOAJ |
| description | In this paper, I investigate the quantisation of length in euclidean quantum gravity in three dimensions. The starting point is the classical hamiltonian formalism in a cylinder of finite radius. At this finite boundary, a counter term is introduced that couples the gravitational field in the interior to a two-dimensional conformal field theory for an SU(2) boundary spinor, whose norm determines the conformal factor between the fiducial boundary metric and the physical metric in the bulk. The equations of motion for this boundary spinor are derived from the boundary action and turn out to be the two-dimensional analogue of the Witten equations appearing in Witten's proof of the positive mass theorem. The paper concludes with some comments on the resulting quantum theory. It is shown, in particular, that the length of a one-dimensional cross section of the boundary turns into a number operator on the Fock space of the theory. The spectrum of this operator is discrete and matches the results from loop quantum gravity in the spin network representation. |
| first_indexed | 2024-12-11T21:22:34Z |
| format | Article |
| id | doaj.art-5bcbce284b024f889f95c7580cbd5a2c |
| institution | Directory Open Access Journal |
| issn | 0550-3213 |
| language | English |
| last_indexed | 2024-12-11T21:22:34Z |
| publishDate | 2018-05-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Nuclear Physics B |
| spelling | doaj.art-5bcbce284b024f889f95c7580cbd5a2c2022-12-22T00:50:25ZengElsevierNuclear Physics B0550-32132018-05-01930219234Quantum gravity in three dimensions, Witten spinors and the quantisation of lengthWolfgang Wieland0Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, CanadaIn this paper, I investigate the quantisation of length in euclidean quantum gravity in three dimensions. The starting point is the classical hamiltonian formalism in a cylinder of finite radius. At this finite boundary, a counter term is introduced that couples the gravitational field in the interior to a two-dimensional conformal field theory for an SU(2) boundary spinor, whose norm determines the conformal factor between the fiducial boundary metric and the physical metric in the bulk. The equations of motion for this boundary spinor are derived from the boundary action and turn out to be the two-dimensional analogue of the Witten equations appearing in Witten's proof of the positive mass theorem. The paper concludes with some comments on the resulting quantum theory. It is shown, in particular, that the length of a one-dimensional cross section of the boundary turns into a number operator on the Fock space of the theory. The spectrum of this operator is discrete and matches the results from loop quantum gravity in the spin network representation.http://www.sciencedirect.com/science/article/pii/S0550321318300671 |
| spellingShingle | Wolfgang Wieland Quantum gravity in three dimensions, Witten spinors and the quantisation of length Nuclear Physics B |
| title | Quantum gravity in three dimensions, Witten spinors and the quantisation of length |
| title_full | Quantum gravity in three dimensions, Witten spinors and the quantisation of length |
| title_fullStr | Quantum gravity in three dimensions, Witten spinors and the quantisation of length |
| title_full_unstemmed | Quantum gravity in three dimensions, Witten spinors and the quantisation of length |
| title_short | Quantum gravity in three dimensions, Witten spinors and the quantisation of length |
| title_sort | quantum gravity in three dimensions witten spinors and the quantisation of length |
| url | http://www.sciencedirect.com/science/article/pii/S0550321318300671 |
| work_keys_str_mv | AT wolfgangwieland quantumgravityinthreedimensionswittenspinorsandthequantisationoflength |