Generalised Uncertainty Relations for Angular Momentum and Spin in Quantum Geometry

Abstract

We derive generalised uncertainty relations (GURs) for orbital angular momentum and spin in the recently proposed 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 extended uncertainty principle (EUP), previously proposed in the quantum gravity literature, 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 so(3) and su(2) algebras. We find that, although SO(3) symmetry is preserved on three-dimensional slices of an enlarged phase space, corresponding to a superposition of background geometries, individual subcomponents of the generalised generators obey nontrivial subalgebras. These give rise to GURs for orbital angular momentum while leaving the canonical commutation relations intact except for a simple rescaling, → + β. The value of the new parameter, β ≃ × 10 −61 , is determined by the ratio of the dark energy density to the Planck density and its existence is required by the presence of both minimum length and momentum uncertainties. 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 ∼ √ Λ, where Λ is the cosmological constant, which is consistent with the existence of a finite cosmological horizon. In the smeared-space model, and β 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 the background space to be fermionic. However, this is no way contradicts the standard result that the graviton is expected to be a spin-2 boson. (We explain why, at length, in the text.) Finally, the modified spin algebra leads to GURs for spin measurements. The potential implications of these results for cosmology and high-energy physics, and for the description of spin and angular momentum in relativistic theories of quantum gravity, are briefly discussed. Contents

Why space could be quantised on a different scale to matter

“…We have shown that relaxing this stringent requirement by introducing a new quantum of action for geometry, β = ħ h, leads to interesting possibilities, with the potential to open up brand new avenues in quantum gravity research [19,32]. These include the proposal that the observed vacuum energy, and the present-day accelerated expansion of the universe that it drives, are related to the quantum properties of space-time [17,18]. In this model, a measurement of the dark energy density constitutes a de facto measurement of the geometry quantisation scale, β, fixing its value to β ħ h × 10 −61 .…”

“…In this model, a measurement of the dark energy density constitutes a de facto measurement of the geometry quantisation scale, β, fixing its value to β ħ h × 10 −61 . This essay was written as a non-technical introduction to the smeared-space model, whose formalism was developed over a series of published works [16][17][18][19]32]. It is based on the material presented at the 4th International Conference on Holography, Hanoi, Vietnam (August 2020), and designed to be accessible to a wide and diverse audience.…”

“…for each i, j ∈ {1, 2, 3}, where Ψ := ψg denotes the composite wave function of a particle in smeared space [16][17][18][19]. 3 Finally, we note that the existence of a finite cosmological horizon implies a corresponding limit on the particle momentum, which may be satisfied by setting ∆ g p ħ h Λ/3.…”

Schwinger pair production and the extended uncertainty principle: can heuristic derivations be trusted?

“…In a later analysis by Lake et al, e.g. [28], the EUP term is obtained from a superposition of Euclidean geometries, which are also intrinsically flat like S 1 , but of course with different topology than S 1 , and the EUP term turns out to be positive. Note that, on the other hand, [21] gives good arguments that if the underlying spatial geometry is positively curved, then the corresponding EUP should have a negative correction term and conversely, a negatively curved spatial geometry should give rise to a positive correction term, at least when the geometry is of constant sectional curvature (This is in contrast with the discussion in the previous paragraph, in which the sign of the EUP term appears to be related to the topology of the background, rather than its geometry).…”

Modified Commutators vs Modified Operators in a Quantum Gravity minimal length scale

“…Modifying the commutator as in ( 11) comes with some challenges as reviewed in [20][21][22][23]. These challenges are: (i) violation of the equivalence principle; (ii) the "soccer" ball problem [24] (i.e.…”