Discrete $Z_4$ symmetry in quantum gravity
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914926469054464 |
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| author | Volovik, G. E. |
| author_facet | Volovik, G. E. |
| contents | We consider the discrete $Z_4$ symmetry ${\hat {\bf i}}$, which takes place in the scenario of quantum gravity where the gravitational tetrads emerge as the order parameter. Under this symmetry operation $\hat {\bf i}$, the emerging tetrads are multiplied by the imaginary unit, $\hat {\bf i}\,e^a_μ= - i e^a_μ$. The existence of such symmetry and the spontaneous breaking of this symmetry are also supported by the consideration of the symmetry breaking scheme in the topological superfluid $^3$He-B. The order parameter $^3$He-B is the analogue of the tetrad field, but it has complex values. The $\hat {\bf i}$-symmetry operation changes the phase of the complex order parameter by $π/2$, which corresponds to the $Z_4$ discrete symmetry in quantum gravity. We also considered the alternative scenario of the breaking of this $Z_4$ symmetry, in which the $\hat {\bf i}$-operation changes sign of the scalar curvature, $\hat {\bf i} \,{\cal R} = - {\cal R}$, and thus the Einstein-Hilbert action violates the $\hat {\bf i}$-symmetry. In the alternative scenario of symmetry breaking, the gravitational coupling $K=1/16πG$ plays the role of the order parameter, which changes sign under $\hat {\bf i}$-transformation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00718 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Discrete $Z_4$ symmetry in quantum gravity Volovik, G. E. Other Condensed Matter General Relativity and Quantum Cosmology High Energy Physics - Phenomenology We consider the discrete $Z_4$ symmetry ${\hat {\bf i}}$, which takes place in the scenario of quantum gravity where the gravitational tetrads emerge as the order parameter. Under this symmetry operation $\hat {\bf i}$, the emerging tetrads are multiplied by the imaginary unit, $\hat {\bf i}\,e^a_μ= - i e^a_μ$. The existence of such symmetry and the spontaneous breaking of this symmetry are also supported by the consideration of the symmetry breaking scheme in the topological superfluid $^3$He-B. The order parameter $^3$He-B is the analogue of the tetrad field, but it has complex values. The $\hat {\bf i}$-symmetry operation changes the phase of the complex order parameter by $π/2$, which corresponds to the $Z_4$ discrete symmetry in quantum gravity. We also considered the alternative scenario of the breaking of this $Z_4$ symmetry, in which the $\hat {\bf i}$-operation changes sign of the scalar curvature, $\hat {\bf i} \,{\cal R} = - {\cal R}$, and thus the Einstein-Hilbert action violates the $\hat {\bf i}$-symmetry. In the alternative scenario of symmetry breaking, the gravitational coupling $K=1/16πG$ plays the role of the order parameter, which changes sign under $\hat {\bf i}$-transformation. |
| title | Discrete $Z_4$ symmetry in quantum gravity |
| topic | Other Condensed Matter General Relativity and Quantum Cosmology High Energy Physics - Phenomenology |
| url | https://arxiv.org/abs/2406.00718 |