κ-Deformed Exponentials, Minimal Length and Renormalization Group Flow

The κ -deformed exponential exp κ ( x )=exp( 1 κ arcsinh( κx ) ) studied by Kaniadakis [1]-[3] allows to construct “deformed” Lorentz transformations associated with the ordinary velocity boost rapidity parameter ξ and which can be recast in terms of ordinary Lorentz transformations (involving the ordinary exponential) but associated with a κ -deformed (modified) rapidity parameter ξ κ =ξf( κξ ) given by a ξ -dependent scaling of the original ξ rapidity parameter. It is shown that when both the κ parameter and ξ→∞ , and the double scaling limit ξ ln( 2κξ ) κξ =∞×0 is finite and nonzero, it leads to a finite value for the κ -deformed boost rapidity parameter ξ κ=∞ = ξ ∞ ≠∞ , such that the κ -deformed velocity (in units of c=1 ) tanh( ξ κ )= v κ 1 is less than the speed of light, and in turn, the Lorentz dilation factor γ( v κ )≠∞ no longer blows up. Consequently, there is a lower bound in the length L ′ = L γ( v k ) ≠0 due to a finite Lorentz length contraction. After imposing that the lower bound L ′ should not be smaller than the postulated minimum Planck scale one arrives at the length-scale-dependent relation arccosh( L L P )= ξ ∞ >0 that admits a physical interpretation analogous to the running of the physical couplings and masses with the energy scale in the Renormalization Group program in Quantum Field Theory.