Quantization of causal diamonds in (2+1)-dimensional gravity -- Part II: Group-theoretic quantization
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912506023247872 |
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| author | Silva, Rodrigo Andrade e |
| author_facet | Silva, Rodrigo Andrade e |
| contents | We develop the non-perturbative reduced phase space quantization of causal diamonds in (2+1)-dimensional gravity with a nonpositive cosmological constant. In Part I we described the classical reduction process and the reduced phase space, $\widetilde{\mathcal P} = T^*(\text{Diff}^+\!(S^1)/\text{PSL}(2, \mathbb R))$, while in Part II we discuss the quantization of the phase space and quantum aspects of the causal diamonds. Because the phase space does not have a natural linear structure, a generalization of the standard canonical (coordinate) quantization is required. In particular, as the configuration space is a homogeneous space for the $\text{Diff}^+\!(S^1)$ group, we apply Isham's group-theoretic quantization scheme. We propose a quantization based on (projective) unitary irreducible representations of the $\text{BMS}_3$ group, which is obtained from a natural prescription for extending $\text{Diff}^+\!(S^1)$ into a transitive group of symplectic symmetries of the phase space. We find a class of suitable quantum theories labelled by a choice of a coadjoint orbit of the Virasoro group and an irreducible unitary representation of the corresponding little group. The most natural choice, justified by a Casimir matching principle, corresponds to a Hilbert space realized by wavefunctions on $\text{Diff}^+\!(S^1)/\text{PSL}(2, \mathbb R)$ valued in some unitary irreducible representation of $\text{SL}(2, \mathbb R)$. A surprising result is that the twist of the diamond corner loop is quantized in terms of the ratio of the Planck length to the corner perimeter. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_03100 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantization of causal diamonds in (2+1)-dimensional gravity -- Part II: Group-theoretic quantization Silva, Rodrigo Andrade e High Energy Physics - Theory General Relativity and Quantum Cosmology Mathematical Physics We develop the non-perturbative reduced phase space quantization of causal diamonds in (2+1)-dimensional gravity with a nonpositive cosmological constant. In Part I we described the classical reduction process and the reduced phase space, $\widetilde{\mathcal P} = T^*(\text{Diff}^+\!(S^1)/\text{PSL}(2, \mathbb R))$, while in Part II we discuss the quantization of the phase space and quantum aspects of the causal diamonds. Because the phase space does not have a natural linear structure, a generalization of the standard canonical (coordinate) quantization is required. In particular, as the configuration space is a homogeneous space for the $\text{Diff}^+\!(S^1)$ group, we apply Isham's group-theoretic quantization scheme. We propose a quantization based on (projective) unitary irreducible representations of the $\text{BMS}_3$ group, which is obtained from a natural prescription for extending $\text{Diff}^+\!(S^1)$ into a transitive group of symplectic symmetries of the phase space. We find a class of suitable quantum theories labelled by a choice of a coadjoint orbit of the Virasoro group and an irreducible unitary representation of the corresponding little group. The most natural choice, justified by a Casimir matching principle, corresponds to a Hilbert space realized by wavefunctions on $\text{Diff}^+\!(S^1)/\text{PSL}(2, \mathbb R)$ valued in some unitary irreducible representation of $\text{SL}(2, \mathbb R)$. A surprising result is that the twist of the diamond corner loop is quantized in terms of the ratio of the Planck length to the corner perimeter. |
| title | Quantization of causal diamonds in (2+1)-dimensional gravity -- Part II: Group-theoretic quantization |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology Mathematical Physics |
| url | https://arxiv.org/abs/2310.03100 |