Physics Letters B 767 (2017) 242–246
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Physics Letters B
www.elsevier.com/locate/physletb
GUP parameter from quantum corrections to the Newtonian potential
Fabio Scardigli
a,b,c
, Gaetano Lambiase
d,e
, Elias C. Vagenas
f,∗
a
Dipartimento di Matematica, Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milano, Italy
b
Department of Applied Mathematics, University of Waterloo, Ontario N2L 3G1, Canada
c
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
d
Dipartimento di Fisica “E.R. Caianiello”, Universita’ di Salerno, I-84084 Fisciano (SA), Italy
e
INFN – Gruppo Collegato di Salerno, Italy
f
Theoretical Physics Group, Department of Physics, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
a r t i c l e i n f o a b s t r a c t
Article history:
Received
9 November 2016
Received
in revised form 24 January 2017
Accepted
24 January 2017
Available
online 27 January 2017
Editor:
M. Cveti
ˇ
c
We propose a technique to compute the deformation parameter of the generalized uncertainty principle
by using the leading quantum corrections to the Newtonian potential. We just assume General Relativity
as theory of Gravitation, and the thermal nature of the GUP corrections to the Hawking spectrum. With
these minimal assumptions our calculation gives, to first order, a specific numerical result. The physical
meaning of this value is discussed, and compared with the previously obtained bounds on the generalized
uncertainty principle deformation parameter.
© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Research on generalizations of the uncertainty principle (GUP)
of quantum mechanics has nowadays a long history [1]. One of
the main lines of investigation focuses on understanding how the
Heisenberg Uncertainty Principle (HUP) should be modified once
gravity is taken into account. Given the pivotal role played by
gravitation in these arguments, it is not surprising that the most
relevant modifications to the HUP have been proposed in string
theory, loop quantum gravity, deformed special relativity, and stud-
ies
of black hole physics [2–7].
As
it is well know, the dimensionless deforming parameter of
the GUP, henceforth denoted by β, is not (in principle) fixed by
the theory, although it is generally assumed to be of order one
(this happens, in particular, in some models of string theory, see
for instance Ref. [2]).
There
have been many studies that aim at setting bounds on β,
for instance Refs. [8–10]. In these works, a specific (in general,
non-linear) representation of the operators in the deformed fun-
damental
commutator is utilized
1
*
Corresponding author.
E-mail
addresses: fabio@phys.ntu.edu.tw (F. Scardigli), lambiase@sa.infn.it
(G. Lambiase),
elias.vagenas@ku.edu.kw (E.C. Vagenas).
1
We shall work with c =k
B
=1, but explicitly show the Newton constant G
N
and
Planck constant
¯
h. We also recall that the Planck length is defined as
2
p
= G
N
¯
h/c
3
,
the Planck energy as E
p
p
=
¯
hc/2, and the Planck mass as m
p
= E
p
/c
2
, so that
G
N
=
p
/2 m
p
and
¯
h = 2
p
m
p
.
ˆ
X,
ˆ
P
=
i
¯
h
1 +β
ˆ
P
2
m
2
p
(1)
in order to compute corrections to quantum mechanical quantities,
such as energy shifts in the spectrum of the hydrogen atom, or to
the Lamb shift, the Landau levels, Scanning Tunneling Microscope,
charmonium levels, etc. The bounds so obtained on β are quite
stringent, ranging from β<10
21
to β<10
50
.
A
further group of bounds can be found in Refs. [11] and [12],
where a deformation of classical Newtonian mechanics is intro-
duced
by modifying the standard Poisson brackets in a way that
resembles the quantum commutator
ˆ
x,
ˆ
p
=
i
¯
h
1 +β
0
ˆ
p
2
⇒{
X, P }=
1 +β
0
P
2
(2)
where β
0
= β/m
2
p
. However, in the limit β → 0, Ref. [11] recov-
ers
only the Newtonian mechanics but not GR, and GR corrections
must be added as an extra structure. Clearly, the physical relevance
of this approach and the bound that follows for β remains there-
fore
questionable.
Finally,
in Refs. [13] and [14], the authors consider the gravi-
tational
interaction when evaluating bounds on β. They use a co-
variant
formalism which firstly is defined in Minkowski space, with
the metric η
μν
=diag(1, −1, −1, −1), which can be easily general-
ized
to curved space–times via the standard procedure η
μν
→ g
μν
.
In addition, these papers, as the previous ones, start from a defor-
mation
of classical Poisson brackets, although posited in covariant
form. From the deformed covariant Poisson brackets, they obtain
http://dx.doi.org/10.1016/j.physletb.2017.01.054
0370-2693/
© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.