Effect of the generalized uncertainty principle on Galilean and Lorentz transformations

Abstract

Generalized Uncertainty Principle (GUP) was obtained in string theory and quantum gravity and suggested the existence of a fundamental minimal length which, as was established, can be obtained within the deformed Heisenberg algebra. We use the deformed commutation relations or in classical case (studied in this paper) the deformed Poisson brackets, which are invariant with respect to the translation in configurational space. We have found transformations relating coordinates and times of moving and rest frames of reference in the space with GUP in the first order over parameter of deformation. For the non-relativistic case we find the deformed Galilean transformation which is similar to the Lorentz one written for Euclidean space with signature $(+,+,+,+)$. The role of the speed of light here plays some velocity $u$ related to the parameter of deformation, which as we estimate is many order of magnitude larger than the speed of light $u\simeq 1.2 \times 10^{22} c$. The coordinates of the rest and moving frames of reference for relativistic particle in the space with GUP satisfy the Lorentz transformation with some effective speed of light. We estimate that the relative deviation of this effective speed of light $\tilde c$ from $c$ is ${(\tilde c-c)/ c}\simeq 3.5\times 10^{-45}$. The influence of GUP on the motion of particle and the Lorentz transformation in the first order over parameter of deformation is hidden in $1/c^2$ relativistic effects.