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- W3099001985 abstract "We derive generalised uncertainty relations (GURs) for angular momentum and spin in the smeared-space model of quantum geometry. The model implements a minimum length and a minimum linear momentum, and recovers both the generalised uncertainty principle (GUP) and the extended uncertainty principle (EUP) within a single formalism. In this paper, we investigate the consequences of these results for particles with extrinsic and intrinsic angular momentum, and obtain generalisations of the canonical ${rm so(3)}$ and ${rm su(2)}$ algebras. We find that, although ${rm SO(3)}$ symmetry is preserved on three-dimensional slices of an enlarged phase space, individual subcomponents of the generalised generators obey nontrivial subalgebras. These give rise to GURs for angular momentum while leaving the canonical commutation relations intact except for a simple rescaling, $hbar rightarrow hbar + beta$. The value of the new parameter, $beta simeq hbar times 10^{-61}$, is determined by the ratio of the dark energy density to the Planck density. Here, we assume the former to be of the order of the Planck length and the latter to be of the order of the de Sitter momentum $sim hbarsqrt{Lambda}$, where $Lambda$ is the cosmological constant. In the smeared-space model, $hbar$ and $beta$ are interpreted as the quantisation scales for matter and geometry, respectively, and a quantum state vector is associated with the spatial background. We show that this also gives rise to a rescaled Lie algebra for generalised spin operators, together with associated subalgebras that are analogous to those for orbital angular momentum. Remarkably, consistency of the algebraic structure requires the quantum state associated with a flat background to be fermionic, with spin eigenvalues $pm beta/2$. Finally, the modified spin algebra leads to GURs for spin measurements." @default.
- W3099001985 created "2020-11-23" @default.
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- W3099001985 date "2020-04-19" @default.
- W3099001985 modified "2023-10-18" @default.
- W3099001985 title "Generalised Uncertainty Relations for Angular Momentum and Spin in Quantum Geometry" @default.
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- W3099001985 doi "https://doi.org/10.3390/universe6040056" @default.
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