We analyze a dynamics of ultracold neutrons (UCNs) caused by interactions violating Lorentz invariance within the Standard Model Extension (SME) (Colladay and Kostelecký (1997) [3] and Kostelecký (2004) [1]). We use the effective non–relativistic potential for interactions violating Lorentz invariance derived by Kostelecký and Lane (1999) [47] and calculate contributions of these interactions to the transition frequencies of transitions between quantum gravitational states of UCNs bouncing in the gravitational field of the Earth. Using the experimental sensitivity of qBounce experiments we make some estimates of upper bounds of parameters of Lorentz invariance violation in the neutron sector of the SME which can serve as a theoretical basis for an experimental analysis. We show that an experimental analysis of transition frequencies of transitions between quantum gravitational states of unpolarized and polarized UCNs should allow to place some new constraints in comparison to the results adduced by Kostelecký and Russell (2011) [16].
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