Physics Letters B 784 (2018) 206–211
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Unruh effect of nonlocal field theories with a minimal length
Yongwan Gim
a,b
, Hwajin Um
a
, Wontae Kim
a,∗
a
Department of Physics, Sogang University, Seoul, 04107, Republic of Korea
b
Research Institute for Basic Science, Sogang University, Seoul, 04107, Republic of Korea
a r t i c l e i n f o a b s t r a c t
Article history:
Received
28 March 2018
Received
in revised form 22 June 2018
Accepted
31 July 2018
Available
online 3 August 2018
Editor: M.
Cveti
ˇ
c
The nonlocal field theory commonly requires a minimal length, and so it appears to formulate the
nonlocal theory in terms of the doubly special relativity which makes the speed of light and the minimal
length invariant simultaneously. We set up a generic nonlocal model having the same set of solutions as
the local theory but allowing Lorentz violations due to the minimal length. It is exactly corresponding
to the model with the modified dispersion relation in the doubly special relativity. For this model, we
calculate the modified Wightman function and the rate of response function by using the Unruh–DeWitt
detector method. It turns out that the Unruh effect should be corrected by the minimal length related to
the nonlocality in the regime of the doubly special relativity. However, for the Lorentz-invariant limit, it
is shown that the Wightman function and the Unruh effect remain the same as those of the local theory.
© 2018 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
There has been much attention to nonlocal theories in light
of a low energy description of fundamental nonlocal interactions.
Initially, nonlocality was mostly considered in the context of ax-
iomatic
quantum field theory [1–5]. And then many efforts have
been devoted to studying various aspects of nonlocality in connec-
tion
with gravity [6–12] and cosmology [13–17]as well as the role
of nonlocality in the framework of string theory [18–25]. In the
nonlocal field theories, the nonlocality is commonly accompanied
by a length scale because of the presence of higher derivative
terms in the equations of motion.
There
has been an intriguing issue on the Unruh effect in a spe-
cific
nonlocal field theory. In the particular nonlocal theory obey-
ing
the field equation of e
−
2
/2
φ =0with the minimal length ,
the rate of response function was calculated by using the Unruh–
DeWitt
detector method, and then it was claimed that there are
significant modifications in the Unruh effect due to the modifica-
tion
of the Feynman propagator originated from the nonlocality
[26]. However, it was proved that, for a wide class of nonlocal the-
ory
obeying the field equation given by f (
2
)φ = 0with the
everywhere nonzero and analytic function f , the Bogoliubov coeffi-
cients
should be exactly the same as the case of a local theory and
*
Corresponding author.
E-mail
addresses: yongwan89@sogang.ac.kr (Y. Gim), um16@sogang.ac.kr
(H. Um),
wtkim@sogang.ac.kr (W. Kim).
thus the Unruh effect should remain unchanged [27]. Recently, for
the specific nonlocal model of e
−
2
/2
φ = 0, it was shown that
the modified Feynman propagator due to the nonlocality consists
of the Wightman function along with the complementary error
function instead of the conventional step function [28]. It means
that the nonlocality is responsible for the error function rather
than the modification of the Wightman function, so that the Un-
ruh
effect relying on the form of the Wightman function naturally
remains the same as that of the local theory.
On
the other hand, if the nonlocal theory should respect the
special theory of relativity, the length in the nonlocal theory
will be no longer minimal length due to the length contraction de-
pending
on the inertial frame. So, the nonlocal field theory having
the minimal length should be implemented by the doubly special
relativity of the extended version of Einstein’s special relativity
[29,30], where the minimal length as well as the speed of light
is an observer-independent scale. In the framework of the doubly
special relativity, the existence of the minimal length would nec-
essarily
lead to the modification of the dispersion relation such
as f (
2
k
2
0
,
2
k
2
i
)k
μ
k
μ
=m
2
, where k
μ
k
μ
is related to the invariant
speed of light and f (
2
k
2
0
,
2
k
2
i
) makes the minimal length an
invariant scale under any inertial frames [31–33]. If one were to
consider the nonlocal model implemented by the minimal length
allowing the Lorentz violation, then the field equation would be
f ((i∂
0
)
2
, (i∂
i
)
2
)φ =m
2
. So, it would be interesting to study the
Unruh effect for the nonlocal field theory in the regime of the dou-
ble
special relativity which makes the minimal length invariant.
https://doi.org/10.1016/j.physletb.2018.07.059
0370-2693/
© 2018 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
.
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