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VOLUME 115 (2022) | ISSUE 10 | PAGE 610
On Lorentz invariance and the minimum length1)
Abstract
It was shown by Kirzhnits and Chechen, following an earlier paper by Mead, that the minimum length scale l is constrained by the Mössbauer effect, which leads to the result l \lesssim 10^{-20} cm, assuming the Snyder discrete space-time and conventional nuclides. Here, we note, firstly, that some recently discussed nuclides, for example 18976Os, have much narrower natural line widths, which, if excited by synchrotron radiation, could potentially decrease the limiting value to l \lesssim 10^{-24} cm in future experiments. The Snyder space-time and the superstring theory are both locally Lorentz invariant, and give rise to the same form of generalized indeterminacy principle, if we set l \approx 2\sqrt{\pi\alpha'}, where α' is the Regge slope parameter, which is thus also constrained by the Mössbauer effect. For the heterotic superstring, in particular, \sqrt{\alpha'} = 4\sqrt{2\pi G_{\rm N}} \approx 10^{-32} cm, apparently beyond experimental reach. A hadron string theory at energy \sim 250 MeV, however, would be ruled out, since then l \sim 10^{-13} cm. We emphasize that these results all presuppose a de Sitter momentum space, for the alternative anti-de Sitter momentum space implies no minimum length scale, and therefore seems unphysical.