Minimum Length Uncertainty Relations in the Presence of Dark Energy

Abstract

We introduce a dark energy-modified minimum length uncertainty relation (DE-MLUR) or dark energy uncertainty principle (DE-UP) for short. The new relation is structurally similar to the MLUR introduced by Károlyházy (1968), and reproduced by Ng and van Dam (1994) using alternative arguments, but with a number of important differences. These include a dependence on the de Sitter horizon, which may be expressed in terms of the cosmological constant as l dS ∼ 1/ √ Λ. Applying the DE-UP to both charged and neutral particles, we obtain estimates of two limiting mass scales, expressed in terms of the fundamental constants {G, c, , Λ, e}. Evaluated numerically, the charged particle limit corresponds to the order of magnitude value of the electron mass (me), while the neutral particle limit is consistent with current experimental bounds on the mass of the electron neutrino (mν e ). Possible cosmological consequences of the DE-UP are considered and we note that these lead naturally to a holographic relation between the bulk and the boundary of the Universe. Low and high energy regimes in which dark energy effects may dominate canonical quantum behaviour are identified and the possibility of testing the model using near-future experiments is briefly discussed. Acknowledgments 24Appendix: Conceptual issues -is r < (∆x total ) min physical? 24References 25 arXiv:1712.00271v3 [gr-qc]

Generalised Uncertainty Relations for Angular Momentum and Spin in Quantum Geometry

“…Hence, it is clear that, if the fundamental quanta of space-time are fermionic, universal expansion can also be viewed as a result of their continuous pair-production. Such a view is consistent with the model of particulate dark energy proposed in [48][49][50][51][52].…”

“…both hold, independently of Eqs. (48) and (51). We denote the position and momentum uncertainties by ∆ g x ′i = σ i g and ∆ g p ′ j =σ gj , respectively, when |g| 2 is chosen to be a Gaussian function.…”

Generalised uncertainty relations from superpositions of geometries

“…as required. Here, modes with k > 2π/l Λ immediately stimulate the pair-production of dark energy particles [62,63], triggering universal expansion in place of increased energy density, as described above.…”

“…IV A, we show that an object in smeared-space, described by a wave function Ψ that optimises the lower bound on the product of uncertainties, ∆ Ψ X∆ Ψ P , has an energy density of order ∼ ρ Λ . We then discuss possible observational consequences of the smeared-space uncertainty relations (62)- (63) and their implications for the nature of dark energy. In Sec.…”

“…Second, m Λ is the unique mass-scale for which the Compton wavelength of a particle is equal to its gravitational turn-around radius, i.e., the radius at which dark energy repulsion overcomes canonical (Newtonian) gravitational attraction [60]. This gives a neat interpretation of the stability condition ρ part ≥ ρ Λ and suggests that m Λ is the unique scale for which the (positive) rest mass of a body is counter-balanced by its (negative) gravitational energy [61][62][63]. In this way, particles with rest mass m Λ can be pair-produced ad infinitum, leading to eternal universal expansion and the existence of an asymptotic de Sitter phase, a(τ ) ∝ e √ Λ/3cτ , where τ is the cosmic time and a(τ ) is the scale factor of the universe.…”