TITLE:
κ-Deformed Exponentials, Minimal Length and Renormalization Group Flow
AUTHORS:
Carlos Castro Perelman
KEYWORDS:
Kaniadakis Entropy, -Exponentials, Deformed Lorentz Transformations, Minimal Length, Renormalization Group
JOURNAL NAME:
Journal of High Energy Physics, Gravitation and Cosmology,
Vol.12 No.1,
January
13,
2026
ABSTRACT: 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.