On Lorentz Invariance and the Minimum Length

It was shown by Kirzhnits and Chechen, following an earlier paper by Mead, that the minimum length scale l is constrained by the Mossbauer effect, which leads to the result l less than or similar to 10(-20) cm, assuming the Snyder discrete space-time and conventional nuclides. Here, we note, firstly, that some recently discussed nuclides, for example (189)(76) Os, have much narrower natural line widths, which, if excited by synchrotron radiation, could potentially decrease the limiting value to l less than or similar to 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 approximate to 2 root pi alpha' where alpha' is the Regge slope parameter, which is thus also constrained by the Mossbauer effect. For the heterotic superstring, in particular, root alpha' = 4 root 2 pi G(N) approximate to 10(-32) cm, apparently beyond experimental reach. A hadron string theory at energy similar to 250 MeV, however, would be ruled out, since then l similar to 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.