Eur. Phys. J. C (2018) 78:728
https://doi.org/10.1140/epjc/s10052-018-6209-y
Regular Article - Theoretical Physics
Modified Unruh effect from generalized uncertainty principle
Fabio Scardigli
1,2,a
, Massimo Blasone
3,4,b
, Gaetano Luciano
3,4,c
, Roberto Casadio
5,6,d
1
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy
2
Institute-Lorentz for Theoretical Physics, Leiden University, P.O. Box 9506, Leiden, The Netherlands
3
Dipartimento di Fisica, Università di Salerno, 84084 Fisciano, SA, Italy
4
I.N.F.N., Gruppo Collegato di Salerno, Salerno, Italy
5
Dipartimento di Fisica e Astronomia, Alma Mater Università di Bologna, via Irnerio 46, 40126 Bologna, Italy
6
I.N.F.N., Sezione di Bologna, viale Bert Pichat 6/2, 40127 Bologna, Italy
Received: 25 April 2018 / Accepted: 1 September 2018 / Published online: 10 September 2018
© The Author(s) 2018
Abstract We consider a generalized uncertainty principle
(GUP) corresponding to a deformation of the fundamen-
tal commutator obtained by adding a term quadratic in the
momentum. From this GUP, we compute corrections to the
Unruh effect and related Unruh temperature, by first follow-
ing a heuristic derivation, and then a more standard field
theoretic calculation. In the limit of small deformations, we
recover the thermal character of the Unruh radiation. Cor-
rections to the temperature at first order in the deforming
parameter are compared for the two approaches, and found
to be in agreement as for the dependence on the cubic power
of the acceleration of the reference frame. The dependence
of the shifted temperature on the frequency is also pointed
out and discussed.
1 Introduction
In the last 30 years, many studies have converged on the idea
that the Heisenberg uncertainty principle (HUP) [1] should be
modified when gravitation is taken into account. In micro-
physics, gravity is usually neglected on the ground of its
weakness, when compared with the other fundamental inter-
actions. However, this argument should not apply when one
wants to address fundamental questions in Nature. In this per-
spective, gravity should be included, especially when we dis-
cuss the formulation of fundamental principles like Heisen-
berg’s principle. And in fact, gravitation has always played a
pivotal role in the generalization of the HUP, from the early
attempts [2–5], to the more recent proposals, like those in
a
e-mail: fabio@phys.ntu.edu.tw
b
e-mail: blasone@sa.infn.it
c
e-mail: gluciano@sa.infn.it
d
e-mail: casadio@bo.infn.it
string theory, loop quantum gravity, deformed special rela-
tivity, non-commutative geometry, and studies of black hole
physics [6–18].
A possible way for this generalization is to reconsider
the well-known classical argument of the Heisenberg micro-
scope [1]. The size δx of the smallest detail of an object,
theoretically detectable with a beam of photons of energy
E, is roughly given by (if we assume the dispersion relation
E = p)
1
δx
¯
h
2 E
, (1)
so that increasingly large energies are required to explore
decreasingly small details. In its original formulation, Heisen-
berg’s gedanken experiment ignores gravity. Nevertheless,
gedanken experiments involving formation of gravitational
instabilities in high energy scatterings of strings [6–9], or
gedanken experiments taking into account the possible for-
mation, in high energy scatterings, of micro black holes with a
gravitational radius R
S
= R
S
(E) proportional to the (centre-
of-mass) scattering energy E (see Ref. [15]), suggest that the
usual uncertainty relation should be modified as
δx
¯
h
2 E
+ β R
S
(E), (2)
where β is a dimensionless parameter. Recalling that R
S
2 G
N
E = 2
2
p
E/
¯
h, we can write
1
We shall always work with c = 1, but explicitly show the Newton
constant G
N
and the Planck constant
¯
h. The Planck length is defined
as
p
=
G
N
¯
h/c
3
10
−35
m, the Planck energy as E
p
p
=
¯
hc/2,
and the Planck mass as m
p
= E
p
/c
2
10
−8
kg, so that
p
= 2G
N
m
p
and 2
p
m
p
=
¯
h. The Boltzmann constant k
B
will be shown explicitly,
unless otherwise stated.
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