Linear and quadratic GUP, Liouville theorem, cosmological constant, and Brick Wall entropy

Abstract

Motivated by the works on equivalence principle in the context of linear generalized uncertainty principle and, independently, in the context of quadratic generalized uncertainty principle, we expand these endeavors in the context of generalized uncertainty principle when both linear and quadratic terms in momentum are include. We demonstrate how the definitions of equations of motion change upon that expansion. We also show how to obtain an analogue of Liouville theorem in the presence of linear and quadratic generalized uncertainty principle. We employ the corresponding modified invariant unit volume of phase space to discuss the resulting density of states, the problem of cosmological constant, the black body radiation in curved spacetime, the concurrent energy and consequent no Brick Wall entropy.

Full text

Eur. Phys. J. C (2019) 79:398
https://doi.org/10.1140/epjc/s10052-019-6908-z
Regular Article - Theoretical Physics
Linear and quadratic GUP, Liouville theorem, cosmological
constant, and Brick Wall entropy
Elias C. Vagenas1, Ahmed Farag Ali2,3, Mohammed Hemeda4, Hassan Alshal5,6
1Theoretical Physics Group, Department of Physics, Kuwait University, P.O. Box 5969, 13060 Safat, Kuwait
2Department of Physics, Faculty of Science, Benha University, Benha 13518, Egypt
3Quantum Gravity Research, Los Angeles, CA 90290, USA
4Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt
5Department of Physics, Faculty of Science, Cairo University, Giza 12613, Egypt
6Department of Physics, University of Miami, Coral Gables, FL 33146, USA
Received: 28 March 2019 / Accepted: 29 April 2019
© The Author(s) 2019
Abstract Motivated by the works on equivalence principle
in the context of linear generalized uncertainty principle and,
independently, in the context of quadratic generalized uncer-
tainty principle, we expand these endeavors in the context
of generalized uncertainty principle when both linear and
quadratic terms in momentum are include. We demonstrate
how the definitions of equations of motion change upon that
expansion. We also show how to obtain an analogue of Liou-
ville theorem in the presence of linear and quadratic gen-
eralized uncertainty principle. We employ the correspond-
ing modified invariant unit volume of phase space to discuss
the resulting density of states, the problem of cosmological
constant, the black body radiation in curved spacetime, the
concurrent energy and consequent no Brick Wall entropy.
1 Introduction
As a consequence of perturbative string theory, modifying
the standard Heisenberg uncertainty principle (HUP) into
the generalized uncertainty principle (GUP), by adding an
extra quadratic term in momentum, resulted in proposing that
gravity might behave differently at the minimal length scale
compared with how it does in general relativity [1–8]. After
this proposal, in a series of papers [9–15] the first two authors
of this paper, namely ECV and AFA, together with Saurya
Das, introduced a linear and quadratic GUP (LQGUP), i.e.,
GUP with linear and quadratic terms in momentum, in such
ae-mail: elias.vagenas@ku.edu.kw
be-mails: ahmed.ali@fsc.bu.edu.eg;
ahmed@quantumgravityresearch.org
cemail: mhemeda@sci.asu.edu.eg
demail: halshal@sci.cu.edu.eg
a way that uncertainty principle becomes compatible with
doubly special relativity (DSR) theories [16–19] and consis-
tent with commutation relations of phase space coordinates
[xi,xj]=[pi,pj]=0 via Jacobi identity. In Ref. [11], the
commutation relation becomes
[xi,pj]=i¯
hδij −αδij p+pipj
p
+α2(δij p2+3pipj).(1)
In addition, this commutation relation is also associated to the
outcome of a perturbative solution, up to third order, ψ∼
eix/xmin of Schrödinger equation such that it is endowed
with a periodic nature of minimal length xmin =α0p,
suggesting that spacetime has a discrete nature [11]. Ear-
lier before that, Chang et al. [20] used the quadratic GUP
(QGUP), i.e., GUP with a quadratic term in momentum, to
study its effect on the UV/IR momentum behavior and the
implications on density of states and the cosmological con-
stant problem.1They concluded that holography in a cos-
mological background might introduce another scale other
than 1
α0pdue to the suppressed density of states in UV case.
Therefore, the number of degrees of freedom contributing
to the vacuum energy density would be very small. Follow-
ing this line of research, one of the authors, namely AFA,
did the same calculations [22] upon considering only the
linear GUP (LGUP), i.e., GUP with a linear term in momen-
tum. The linear term in momentum of LGUP changes the
power of the unit volume of phase space from D,asinRef.
[20]to D+1, but it does not suppress the density of states.
1The cosmological constant problem has also been discussed in the
context of the LQGUP-deformed Wheeler-DeWitt equation [21].
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398 Page 2 of 9 Eur. Phys. J. C (2019) 79:398
Therefore, the effect of LGUP on holographic entropy of the
cutoff phase space disagrees with ’t Hooft’s standard result,
that forces disagreement between the micro-canonical and
canonical ensembles for such system with large number of
degrees of freedom.
The rest of this work is structured as follows. In Sect. 2
we reconsider the effect of LQGUP on the equivalence prin-
ciple and the equations of motions. In Sect. 3we examine
the effect of LQGUP on the unit volume of phase space,
and whether we should consider the correction factor to be
raised to power Dor D+1. Then, in Sect. 4we see the conse-
quences on the cosmological constant problem. Moreover, in
Sect. 5we investigate the outcome of introducing LQGUP to
energy distribution of massless black body radiation. In addi-
tion, in Sect. 6, we compare the effect of LQGUP with the
effect of LGUP and QGUP on massless particles in general
static spherically symmetric curved spacetime. Furthermore,
in Sect. 7we introduce LQGUP to the Brick Wall entropy of
black holes. Finally, we discuss the contrasts and similarities
among the different orders of GUPs and, therefore, conclude
We take the units G=c=¯
h=kB=1.
2 LQGUP equivalence principle and equations of
motion
For the classical limit of Eq. (1) of any two canonical conju-
gates ˆ
Pand ˆ
Q, the correspondence principle states that
1
i¯
h[ˆ
P,ˆ
Q]→{P,Q},(2)
where the square brackets stand for the Lie brackets while the
curly ones stand for Poisson brackets. Meanwhile the relation
between the expectation value of any QM observable and the
expectation value of the commutator of that observable with
the Hamiltonian of the system is given by
d
dt A= 1
i¯
h[A,H] + ∂
∂tA.(3)
Upon employing the correspondence principle, as stated in
Eq. (2), on Eq. (3) for position, we obtain
˙xi={xi,H}=δij
∂H
∂pj={xi,pj}∂H
∂pj
,(4)
and for the momentum we get
˙pj=−{xi,pj}∂V
∂xj
.(5)
Then, we utilize Eq. (1) in the above two expressions to get
˙x=(1−2αp+4α2p2)p
m
˙p=−(1−2αp+4α2p2)∂V
∂x.(6)
Consequently, the definition of the force reads
F=m¨x=m{˙x,H}
=(1−4αp+12α2p2){p,H}
=−(1−4αp+12α2p2)(1−2αp+4α2p2)∂V
∂x
=−[1−6αp+24α2p2+O(α3)]∂V
∂x.(7)
It is noteworthy that pand Fare no longer equal to m˙xand
−∂V/∂x, respectively. The αterm matches with the results
obtained in Ref. [22]. In addition, we have an α2term, as
expected, and this α2term does not contradict the conclu-
sion about the dynamical violation of equivalence principle
obtained in Ref. [22]. LQGUP controls the UV divergences
such that it shows similar cosmological implications of the
dark sector where the associated long-range force acts only
between nonbaryonic particles [23]. It should be stressed
that the violation of equivalence principle obtained here also
agrees with that obtained from tidal forces in the domains of
string theory [6,24].
3 LQGUP and Liouville theorem
In the light of Eq. (1), it is evident the momentum depen-
dence of the unit volume of each quantum state in the phase
space. This would contradict that laws of physics should not
change their form with respect to any change in space and
time, i.e., the unit volume of the space has to be invariant
upon the change in the momentum for every state. Therefore,
we look for an analogue to Liouville theorem by assum-
ing the change in position and momentum in a time δt
as
x
i=xi+δxi=xi+˙xiδt+O(δt2)
p
i=pi+δpi=pi+˙piδt+O(δt2). (8)
We demand the Jacobian – which relates the states of phase
space before and after a time δt–tobe

∂(x
1,...,x
D;p
1,..., p
D)
∂(x1,...,xD;p1,..., pD)=1+∂δxi
∂xi+∂δpi
∂pi
+···,(9)
such that the phase space volume element after δtbecomes
dDxdDp=
∂(x
1,...,x
D;p
1,..., p
D)
∂(x1,...,xD;p1,..., pD)
dDxdDp.(10)
Upon combining Eqs. (1), (4), (5), and (8), we express the
variation term in the RHS of Eq. (9)as
∂δxi
∂xi+∂δpi
∂pi=− ∂
∂piδij −αδij p+pipj
p
+α2(δij p2+3pipj)∂H
∂xj
δt,(11)
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Eur. Phys. J. C (2019) 79:398 Page 3 of 9 398
where again the αterm matches with the one in Ref. [22] and
is evaluated there to be
−∂
∂pi−αδij p+pipj
p=α(D+1)pi
p,(12)
meanwhile the α2term is evaluated as
−∂
∂pi[α2δij p2+3pipj]
=−2α2(D+1)1+2
D+1pi.(13)
Now we substitute Eqs. (12) and (13)intheRHSofEq.(9)
to get
1+∂δxi
∂xi+∂δpi
∂pi
=1+(D+1)α
p−2α2−4α2
D+1pj
∂H
∂xj
δt.(14)
To obtain the correct scale factor that makes LQGUP com-
patible with Liouville theorem, we consider the infinitesimal
time evolution in the linear term to the first order in αand δt
from Ref. [22]as
(1−αp)∼(1−αp)1+αpi
p
∂H
∂xj
δt,(15)
and the infinitesimal time evolution in the quadratic term to
the second order in αand first order in δtas
α22
D+1+1
2p2∼α22
D+1+1
2(p2+2piδpi)
∼α22
D+1+1
2p2−2pi{xi,pj}∂H
∂xj
δt
∼α2(2
D+1+1
2)p2−2piδij
∂H
∂xj
δt+O(α3)
∼α22
D+1+1
2p2−2pj
∂H
∂xj
δt.(16)
Then, we combine Eqs. (15) and (16) to get
1−αp+α22
D+1+1
2p2
∼1−αp+α22
D+1+1
2p2
+α
p(1−2αp)−α2−4α2
D+1pj
∂H
∂xj
δt.(17)
We factor out 1−αp+α22
D+1+1
2p2in the RHS
such that Eq. (17) becomes
1−αp+α22
D+1+1
2p2
∼1−αp+α22
D+1+1
2p2
×1+(1−2αp)(α/p)
1−αp+α22
D+1+1
2p2
−α2+(4α2/(D+1))
1−αp+α22
D+1+1
2p2pj
∂H
∂xj
δt
∼1−αp+α22
D+1+1
2p2
×1+α
p(1−2αp)(1+αp)−α2−4α2
D+1
+O(α3)pj
∂H
∂xj
δt.(18)
Or,
1−αp+α22
D+1+1
2p2
∼1−αp+α22
D+1+1
2p2
×1+α
p−2α2−4α2
D+1+O(α3)pj
∂H
∂xj
δt.
(19)
Finally, we raise the last result to power −(D+1)then expand
it to the first order of binomial coefficient such that the weight
factor of LQGUP, that corrects the definition of unit volume
phase space, is defined as
1−αp+α22
D+1+1
2p2−(D+1)
∼1−αp+α22
D+1+1
2p2−(D+1)
×1−(D+1)α
p−2α2−4α2
D+1pj
∂H
∂xj
δt.
(20)
By comparing Eq. (14) with Eq. (20), the corrected LQGUP
invariant-under-time unit volume of phase space is given by
dDxdDp
(2π)D1−αp+2
D+1+1
2α2p2(D+1),(21)
which, technically, will later define the number of quan-
tum states per momentum space volume upon integrating
over dDx. Consequently, this would affect the calculations
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398 Page 4 of 9 Eur. Phys. J. C (2019) 79:398
Fig. 1 The behavior of weight factor (1−αp+α2p2)−4of LQGUP
compared to (1+βp2)−3of Ref. [20] when D = 3. The horizontal axis
is the logarithm of every weight factor. We have set α2=β=1
of energy, holographic entropy, and cosmological constant.
Before we discuss these, we want to emphasize on the differ-
ent results obtained in Refs. [20,22]. In Ref. [20], the power
that appears in the corresponding equation to Eq. (21) is not
(D+1)but Dand, in addition, there is no αterm. In Ref.
[22], it has the same power as we have even even if it does
not have the α2term. Since α2∼β, where βis the minimal
length factor in Ref. [20], we expect the behavior of LQGUP
weight factor to be close to that of Ref. [20], as shown in
Fig. 1. However, the computational results are quite differ-
ent, due to the divergent behavior of the linear term we have,
as we will see in next sections. This numerical difference
between QGUP and LQGUP is crucial when we consider
the quantum gravity effects within the vicinity of the mini-
mal length.
4 LQGUP effect on cosmological constant
Based on the LQGUP analogue of Liouville theorem derived
in the previous section, the sum over all harmonic oscillator
momentum states per unit volume will now read2
(m)=2π
∞

0
p2
(1−αp+α2p2)4p2+m2dp.(22)
Upon considering tan θ=2αp−1
√3, the above integral
becomes
2The numerical factor 2πin front of the integral of LQGUP in Eq. (22)
should have been 1/2π2as it is in next section (see Eq. (25)). However,
for the sake of comparison between our result given here and the result
for QGUP obtained in Ref. [20], we keep it 2π.
Fig. 2 The effect of different weight factors on the calculations of
cosmological constant upon considering LGUP, QGUP and LQGUP
when D = 3. We have set α=1andm=0
(m)=2π4
34√3
2α
π/2

−π/6
cos6θ√3tanθ+1
2α2
×⎡
⎣√3tanθ+1
2α2
+m2⎤
⎦
1/2
dθ. (23)
This integral is not easy to be exactly solved. However, we
still can compare our result here with those obtained in Refs.
[20,22]. This is done in Fig. 2, after setting m=0.3The
massless cosmological constant corresponding to LQGUP
reads
LQGUP(0)=2π√3
2α4
3427√3+28π
384α3∼2π
α4,(24)
which is much larger than the QGUP(0)obtained in
Ref. [20].4It is easily seen that the cosmological con-
stant of LQGUP is still finite with αand α2to be the
UV cutoff. However, we agree with Chang et al. in Ref.
[20] that this does not resolve the cosmological constant
problem since α2∼MPwith MPto be the Planck
mass.
5 LQGUP effect on energy distribution of black body
massless radiation
In this section, we calculate the energy distribution of black-
body massless radiation in the framework of LQGUP. First,
we set m=0 so that for the single massless particle we get
E=p2+m2=p. Then, the total number of quantized
3In the context of Gravity’s Rainbow, a similar plot was obtained in
Ref. [25].
4Remember that, in Ref. [20], β∼α2.
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Eur. Phys. J. C (2019) 79:398 Page 5 of 9 398
Fig. 3 The effect of different weight factors on the calculation of the
number of states upon considering LGUP, QGUP and LQGUP in D =
3. We have set α=1
modes for massless bosonic field in a cubic box (with D=3)
of size Lreads
N=L3
2π2
∞

0
p2dp
(1−αp+α2p2)4
=L3
2π2×32π√3+81
243α3.(25)
In Fig. 3, we plot the number of states, i.e., N, as a function of
the momentum, i.e., p, of the single massless particle, when
computed in different versions of GUP. The effect of the dif-
ferent weight factors on Nis easily seen. It is also noteworthy
that the number of states of the LQGUP is higher than the
suppressed one of the QGUP [21], due to the contribution of
the linear term, i.e., αterm. However, the number of states of
LQGUP remains convergent compared to that of the LGUP.
Now, we compute the corresponding energy of the black body
massless radiation
E=L3
2π2
∞

0
p3dp
(1−αp+α2p2)4
=L3
2π2×28π√3+81
243α4.(26)
It should be pointed out that the energy of the black body
massless radiation has similar behavior with (m=0)with
respect to the GUP parameter α. This is easily seen since both
Eqs. (24) and (26) are inversely proportional to α4. Thus, it
is expected that the plot of energy of the black body massless
radiation given by Eq. (26) as a function of a function of the
momentum, i.e., p, of the single massless particle will be
very similar to Fig. 2.
At this point, it is very important to introduce the following
functions
g0(w, T)≡(w/wα)3
e(w/wα)(Tα/T)−1
gα(w, T)≡1
[1−w/wα+(w/wα)2]4g0(w, T)(27)
with wto be the frequency of the spectral function, and the
constants wα∼1
α, and Tα∼1
kBα. These functions will help
to compute the energy of the black body massless radiation
in a curved spacetime when the LQGUP is taken into con-
sideration.
6 LQGUP and massless particles in curved spacetime
In this section, we expand the analysis of Ref. [26]. In partic-
ular, in Ref. [26] the total energy density of massless particles
was computed in the context of QGUP and using the unit vol-
ume of phase space obtained in Ref. [20]. Now, we employ
Eq. (21) in such a way that at the WKB level, the norm of
3-momentum vector of a massless particle reads
p2=pipi=w2
f(r),(28)
where f(r)≡−gtt is the metric element of any static spher-
ically symmetric metric like the Schwarzschild, Reissner–
Nordström, Bardeen, Hayward, and (anti-)de Sitter space-
time background, or any combination of them. If we set
D=3, then the total energy density for all frequencies will
be
ρ( f,β) =γ
∞

0
f2w3
2π2(f−α√fw+α2w2)4
×1
eβw ±1dw(29)
where f=f(r),γis the spin degeneracy, the negative
sign in the denominator stands for the massless bosons,
while the positive sign stands for the massless fermions.
Upon considering the change of variables x=βw/2πand
T(r)=1/(β√f), with T(r)to be the local temperature in
a curved spacetime and βis the reciprocal temperature,5Eq.
(29) becomes
ρ(x,T)=8π2γT4∞

0
x3
(1−ax +a2x2)4×1
e2πx±1dx,
(30)
5Henceforth, the βwill be the reciprocal temperature, and not the GUP
parameter βthat appears in Ref. [20].
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398 Page 6 of 9 Eur. Phys. J. C (2019) 79:398
Fig. 4 The total energy density ρ(x)versus the variable x for massless
bosons in a general static spherically symmetric spacetime, where a=1
with a=2παT. This integral is not easy to be exactly solved,
but it is indeed a convergent integral. So upon expanding the
denominator up to O(a3(α)) and setting x=s/2π, we get
ρ(s,T)∼4πγ T4∞

01
(2π)3
s3
es±1+4a
(2π)4
s4
es±1
+6a2
(2π)5
s5
es±1ds.(31)
For the case of massless bosons, we use the Riemann zeta
function
ζ(s)=1
(s)
∞

0
xs−1
ex−1dx where s∈{4,5,6}(32)
and, for the case of massless fermions, we use Dirichlet eta
function
η(s)=1
(s)
∞

0
xs−1
ex+1dx where s∈{4,5,6}.(33)
Finally, we provide the Figs. 4,5,6, and 7to demonstrate
and compare the effect of HUP, QGUP, and LQGUP on the
total energy density of massless particles. For fixed αand
T(r), we assume ato be small compared with x. When
αand T(r)conspire to render a very diminutive values of
afor general static spherically symmetric spacetime, as in
Figs. 6and 7, we notice that GUP correction tends to be HUP,
as expected, for both massless bosons and fermions. Since
HUP dies slower than LQGUP, we agree with Chang et al. that
the distortion to the black body radiation is undetectable, and
the spectrum of the cosmic microwave background (CMB)
stays unaffected too. As an example for the effect of LQGUP
on the radiation distribution of massless particles, we discuss
in Ref. [27] the case of an ultracold RNdS-like spacetime and
its corresponding massless charged particles.
Fig. 5 The total energy density ρ(x)versus the variable x for massless
fermions in a general static spherically symmetric spacetime, where
a=1
Fig. 6 The total energy density ρ(x)versus the variable x for massless
bosons in a general static spherically symmetric spacetime, where a=
0.01
Fig. 7 The total energy density ρ(x)versus the variable x for massless
fermions in a general static spherically symmetric spacetime, where
a=0.01
7 LQGUP effect on Brick Wall entropy
Motivated by Ref. [28], Li calculated, in the context of QGUP,
the energy density of the black body radiation as follows [29]
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Eur. Phys. J. C (2019) 79:398 Page 7 of 9 398
u=∞
0
ω3dω
(eβω −1)(1+α2ω2)3(34)
=β−4∞
0
x3dx
(ex−1)(1+ax2)3(35)
where a=(α/β)2and x=βω. The above integral was
solved asymptotically first by setting the HUP condition,
namely α→0 which means the temperature is much less
than Planck temperature, and then by setting the upper bound
condition of the energy density. Thus, we adopt the same
analysis here except that we introduce our new weight factor.
Similar to the result we obtained from Eq. (24), the numerical
correction that comes from LQGUP is expected to be much
larger than that of QGUP. However, it will not substantially
change the convergent behavior of the function as we have
seen before. The upper bound of energy density is given by
u<β
−4∞
0
x2dx
1−αx
β+α2x2
β24(36)
=32π√3+81
243α3β−1(37)
where the inequality comes from the fact that (ex−1)>x
which means that when the temperature is higher than the
Planck temperature, the state equation of the thermal radia-
tion is different from that of HUP, i.e., u∼β−4[29] . From
Eqs. (21) and (27), the number of quantum states with energy
less than ωis given by
g(ω) =1
(2π)3dr dθdϕdp
rdpθdpϕ
(1−αω/f1/2+α2ω2/f)4
=1
(2π)3dr dθdϕ
(1−αω/f1/2+α2ω2/f)4
×2
f1/2ω2
f−1
r2p2
θ−1
r2sin2θp2
ϕ1/2
×dpθdpϕ
=4πω3
3(2π)3r2dr
f2(1−αω/f1/2+α2ω2/f)4
×sin θdθdϕ
=2ω3
3πr2dr
f2(1−αω/f1/2+α2ω2/f)4,(38)
and when α→0, Eq. (38) goes back to the standard expres-
sion in the HUP limit. Furthermore, the free energy reads
F(β) =1
βdg(ω) ln(1−e−βω)
=−∞
0
g(ω)dω
eβω −1
=−2
3πr0
r2dr
f2∞
0
ω3dω
(eβω −1)(1−αω/f1/2+α2ω2/f)4.
(39)
Therefore, the entropy is written in the from
S=β2∂F
∂β
=2β2
3πr0
r2dr
f2∞
0
eβωω4dω
(eβω −1)2(1−αω/f1/2+α2ω2/f)4
=2β−3
3πr0
r2dr
f2
×∞
0
x4dx
(1−e−x)(ex−1)1−αx
βf1/2+α2x2
β2f4.(40)
In the light of the following inequalities
1−e−x>x
1+x
ex−1>x(41)
the entropy satisfies the inequality
S<2β−3
3πr0
r2dr
f2∞
0
(x3+x2)dx
1−αx
βf1/2+α2x2
β2f4
=2β−3
3πr0
r2dr
f228π√3+81
243(α/β)4f2+32π√3+81
243(α/β)3f3/2
=2
3π
28π√3+81
243α4βr0
r2dr +2
3π
32π√3+81
243α3r0
r2dr
f1/2.
(42)
Since we consider the upper bound, we only want to get
contribution from the domain close to the horizon, [r0,r0+
], that corresponds to the minimal length ∼α, i.e., it is
just the neglected vicinity in the Brick Wall model [30,31].
Therefore, we have
2α=r0+
r0
dr
√f
∼r0+
r0
dr
√2κ(r−r0)
∼2
κ(43)
where κ=2πβ−1is the surface gravity at the horizon of
black hole. Finally, the entropy is written as
123

398 Page 8 of 9 Eur. Phys. J. C (2019) 79:398
S∼2
3π
(28π√3+81)
243α4βr2
0+2
3π
(32π√3+81)
243α32r2
0α
∼0.239 A
α2.(44)
It is evident that the entropy, S, is proportional to the black
hole horizon area A=4πr2
0and, in addition, the entropy is
less than A/4α2, as expected. Furthermore, it is also antici-
pated from the previous sections that by introducing the lin-
ear term to the QGUP it will cause the convergent QGUP
energy distribution, and, consequently, the entropy to signif-
icantly increase.6However, the convergent behavior remains
the same. So, in contrary to LGUP effect on entropy [22], we
agree with QGUP of Ref. [29] that LQGUP does not need
any cutoff near the horizon. The last result emphasizes that
minimal length contributes to black holes such that it may
provide simpler interpretation without introducing a diver-
gent assumption like the Brick Wall model.7
8 Conclusion
Motivated by the unexpected ramification upon employing
LQGUP to no-cloning theorem [34], we discuss the conse-
quences of applying the LQGUP on the characteristics of
the momenta distribution in phase space, particularly IR/UV
behaviors. It is shown that in QGUP of Ref. [20] that the
UV behavior is convergent, while it is divergent in LGUP
of Ref. [22]. So we reconcile them through LQGUP. Upon
employing LQGUP on equations of motion, we agree with
Ref. [22] that the acceleration is no longer mass-independent,
and hence, the equivalence principle is dynamically violated.
Then, we modify the Liouville theorem in the presence of
LQGUP and show that the weight factor has power (D+1)
as in Ref. [22] and a quadratic term as in Ref. [20], but with
a numerical factor that depends on D. Next, we encounter
the cosmological constant problem. We deduce that LQGUP
has similar convergent form of that in Ref. [20] rather than
the divergent behavior of that in Ref. [22]. However, it still
can not resolve the cosmological constant problem as in Ref.
[20] due to the (0)∼1/α4together with the fact that
α2∼Mp. After that, we compare the different consequences
of each corresponding weight factor of LGUP, QGUP, and
LQGUP on the number of massless bosonic states of black
body radiation. The LQGUP shows a convergent behavior
similar to that of QGUP despite it is much larger in the num-
ber of states. That larger number is due to the linear term,
which by its own has divergent behavior as in Ref. [22]. It is
6In Ref. [29], λ=α2.
7The claim that there is no need for the introduction of the Brick Wall
model in order to keep under control the divergences appearing when
one approaches the horizon, was also supported in the framework of
Gravity’s Rainbow [32,33].
obvious that the linear and quadratic terms together conspire
to give such behavior. Moreover, we show how that reflects
on the calculation of the energy distribution and gives the
same behavior. Later, we introduce the gravitational effects
on the energy of massless bosons and fermions. We notice the
agreement with Ref. [20] on the unaffected CMB and undis-
torted radiation of black body, and that is guaranteed by the
faster decay of LQGUP compared with HUP. Furthermore,
we get the bosons’ behavior to be with slightly higher energy
density than that of fermions, as expected. HUP, QGUP, and
LQGUP get very close to each other for very small values
of α, as expected too. Finally through LQGUP and QGUP
of Ref. [29] but not the LGUP of Ref. [22], we agree that
minimal length would “guard” the entropy of black holes so
that there is no need for any Brick Wall model.
Data Availability Statement This manuscript has no associated data
or the data will not be deposited. [Authors’ comment: No datasets were
generated and/or analyzed during the current study.]
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
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