The purpose of this paper is to challenge the existing paradigm on which contemporary models of generalised uncertainty relations (GURs) are based, that is, the assumption of modified commutation relations. We review an array of theoretical problems that arise in modified commutator models, including those that have been discussed in depth and others that have received comparatively little attention, or have not been considered at all in the existing literature, with the aim of stimulating discussion on these topics. We then show how an apparently simple assumption can solve, or, more precisely, evade these issues, by generating GURs without modifying the basic form of the canonical Heisenberg algebra. This simplicity is deceptive, however, as the necessary assumption is found to have huge implications for the quantisation of space-time and, therefore, gravity. These include the view that quantum space-time should be considered as a quantum reference frame and, crucially, that the action scale characterising the quantum effects of gravity, β, must be many orders of magnitude smaller than Planck’s constant, β ∼ 10–61 × ℏ, in order to recover the present day dark energy density. We argue that these proposals should be taken seriously, as a potential solution to the pathologies that plague minimum length models based on modified commutators, and that their implications should be explored as thoroughly as those of the existing paradigm, which has dominated research in this area for almost three decades.

(Ba0.85Ca0.15) Zr0.1Ti0.9O3 (BCZT) ceramics with donor (La3+ and Bi3+ ions) and acceptor (Na+ ions) doping at the A-site were fabricated in this study. The effects of crystal structure, microstructure, and chemical defects on all samples’ dielectric properties and piezoelectric coeffi cient (d33) were observed. Bi3+ ions decreased grain size, decreased phase transition temperature, and caused the diffuse dielectric spectrum, resulting in a low d33 value. Na+ ions induced abnormal grain growth, leading to enhancement in maximum dielectric constant; meanwhile, the phase transition temperature increased. The d33 value of Na-doped BCZT was comparable with BCZT ceramic. The addition of La3+ ions increased the phase transition temperature and decreased grain size. The d33 value of La-doped BCZT was lower than BCZT ceramic. The result suggested that BCZT, BCZT-Na, and BCZT-La ceramics could be alternative materials for piezoelectric energy harvesting devices.

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We present a simple gedanken experiment in which a compact object traverses a spacetime with three macroscopic spatial dimensions and n compact dimensions. The compactification radius is allowed to vary, as a function of the object’s position in the four-dimensional space, and we show that the conservation of gravitational self-energy implies the dimensional dependence of the mass-radius relation. In spacetimes with extra dimensions that are compactified at the Planck scale, no deviation from the four-dimensional result is found, but, in spacetimes with extra dimensions that are much larger than the Planck length, energy conservation implies a deviation from the normal Compton wavelength formula. The new relation restores the symmetry between the Compton wavelength and Schwarzschild radius lines on the mass-radius diagram and precludes the formation of black holes at TeV scales, even if large extra dimensions exist. We show how this follows, intuitively, as a direct consequence of the increased gravitational field strength at distances below the compactification scale. Combining these results with the heuristic identification between the Compton wavelength and the minimum value of the position uncertainty, due to the Heisenberg uncertainty principle, suggests the existence of generalised, higher-dimensional uncertainty relations. These relations may be expected to hold for self-gravitating quantum wave packets, in higher-dimensional spacetimes, with interesting implications for particle physics and cosmology in extra-dimensional scenarios.

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