Generalised uncertainty relations from superpositions of geometries

Abstract

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.

Generalised Uncertainty Relations for Angular Momentum and Spin in Quantum Geometry

“…However, crucially, the resulting commutation relations are simply a rescaled representation of the Heisenberg algebra, with → + β, where β ∼ 3 GΛ/c 3 . The model is therefore consistent with the equivalence principle and provides a neat solution of the soccer ball problem that plagues approaches based on modified commutators [21,22].…”

“…Despite these initial promising results, more general implications of the smeared-space model have not, so far, been extensively investigated. In this paper, we extend the analysis presented in [22] to include angular momentum and spin, which represents an important contribution to the development of the theory. Although our analysis remains non-relativistic, we note the close connection between the angular momentum generators of canonical QM and the Lorentz generators [23].…”

“…An alternative approach, recently considered in [22], is to modify the canonical quantum wave function and, hence, the underlying probability distribution from which all operator uncertainties are derived. The basic idea, proposed in the so-called smeared-space model, is to associate quantum state vectors with spatial points in the classical background geometry.…”

“…Here, points in the phase space of a classical system ( r, p) become "smeared" over finite minimum volumes in the position and momentum space representations of the corresponding quantum theory. Unlike in canonical QM, absolute limits are set to each [22].…”

“…These naturally come to the fore when attempting to construct generalisations of vector relations to include the effects of quantum fluctuations of the background. In addition, we include a concise but thorough review of the smeared-space model, proposed in [22], which is intended as a self-contained introduction for readers not familiar with this work.…”