Phenomenological approaches to quantum gravity implement a minimum resolvable length-scale but do not link it to an underlying formalism describing geometric superpositions. Here, we introduce an intuitive approach in which points in the classical spatial background are delocalised, or 'smeared', giving rise to an entangled superposition of geometries. The model uses additional degrees of freedom to parameterise the superposed classical backgrounds. Our formalism contains both minimum length and minimum momentum resolutions and we naturally identify the former with the Planck length. In addition, we argue that the minimum momentum is determined by the de Sitter scale, and may be identified with the effects of dark energy in the form of a cosmological constant. Within the new formalism, we obtain both the generalised uncertainty principle (GUP) and extended uncertainty principle (EUP), which may be combined to give an uncertainty relation that is symmetric in position and momentum. Crucially, our approach does not imply a significant modification of the positionmomentum commutator, which remains proportional to the identity matrix. It therefore yields generalised uncertainty relations without violating the equivalence principle, in contradistinction to existing models based on nonlinear dispersion relations. Implications for cosmology and the black hole uncertainty principle correspondence are briefly discussed, and prospects for future work on the smeared-space model are outlined.

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

No abstract

The existence of both a minimum mass and a minimum density in nature, in the presence of a positive cosmological constant, is one of the most intriguing results in classical general relativity. These results follow rigorously from the Buchdahl inequalities in four-dimensional de Sitter space. In this work, we obtain the generalized Buchdahl inequalities in arbitrary space-time dimensions with = 0 and consider both the de Sitter and the anti-de Sitter cases. The dependence on D, the number of space-time dimensions, of the minimum and maximum masses for stable spherical objects is explicitly obtained. The analysis is then extended to the case of dark energy satisfying an arbitrary linear barotropic equation of state. The Jeans instability of barotropic dark energy is also investigated, for arbitrary D, in the framework of a simple Newtonian model with and without viscous dissipation, and we determine the dispersion relation describing the dark energy-matter condensation process, along with estimates of the corresponding Jeans mass (and radius). Finally, the quantum mechanical implications of the mass limits are investigated, and we show that the existence of a minimum mass scale naturally leads to a model in which dark energy is composed of a 'sea' of quantum particles, each with an effective mass proportional to 1/4 .