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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¯

hδij −αδij 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 =α0p,

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

α0pdue 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].

0123456789().: V,-vol 123

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

dDxdDp=

∂(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=− ∂

∂piδij −αδij 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−αδij 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+1pi.(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+1pj

∂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

α22

D+1+1

2p2∼α22

D+1+1

2(p2+2piδpi)

∼α22

D+1+1

2p2−2pi{xi,pj}∂H

∂xj

δt

∼α2(2

D+1+1

2)p2−2piδij

∂H

∂xj

δt+O(α3)

∼α22

D+1+1

2p2−2pj

∂H

∂xj

δt.(16)

Then, we combine Eqs. (15) and (16) to get

1−αp+α22

D+1+1

2p2

∼1−αp+α22

D+1+1

2p2

+α

p(1−2αp)−α2−4α2

D+1pj

∂H

∂xj

δt.(17)

We factor out 1−αp+α22

D+1+1

2p2in the RHS

such that Eq. (17) becomes

1−αp+α22

D+1+1

2p2

∼1−αp+α22

D+1+1

2p2

×1+(1−2αp)(α/p)

1−αp+α22

D+1+1

2p2

−α2+(4α2/(D+1))

1−αp+α22

D+1+1

2p2pj

∂H

∂xj

δt

∼1−αp+α22

D+1+1

2p2

×1+α

p(1−2αp)(1+αp)−α2−4α2

D+1

+O(α3)pj

∂H

∂xj

δt.(18)

Or,

1−αp+α22

D+1+1

2p2

∼1−αp+α22

D+1+1

2p2

×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+α22

D+1+1

2p2−(D+1)

∼1−αp+α22

D+1+1

2p2−(D+1)

×1−(D+1)α

p−2α2−4α2

D+1pj

∂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)D1−αp+2

D+1+1

2α2p2(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)4p2+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)=2π4

34√3

2α

π/2

−π/6

cos6θ√3tanθ+1

2α2

×⎡

⎣√3tanθ+1

2α2

+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

2α4

3427√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∞

01

(2π)3

s3

es±1+4a

(2π)4

s4

es±1

+6a2

(2π)5

s5

es±1ds.(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

β24(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)3dr dθdϕdp

rdpθdpϕ

(1−αω/f1/2+α2ω2/f)4

=1

(2π)3dr dθdϕ

(1−αω/f1/2+α2ω2/f)4

×2

f1/2ω2

f−1

r2p2

θ−1

r2sin2θp2

ϕ1/2

×dpθdpϕ

=4πω3

3(2π)3r2dr

f2(1−αω/f1/2+α2ω2/f)4

×sin θdθdϕ

=2ω3

3πr2dr

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

βdg(ω) ln(1−e−βω)

=−∞

0

g(ω)dω

eβω −1

=−2

3πr0

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

3πr0

r2dr

f2∞

0

eβωω4dω

(eβω −1)2(1−αω/f1/2+α2ω2/f)4

=2β−3

3πr0

r2dr

f2

×∞

0

x4dx

(1−e−x)(ex−1)1−αx

βf1/2+α2x2

β2f4.(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

3πr0

r2dr

f2∞

0

(x3+x2)dx

1−αx

βf1/2+α2x2

β2f4

=2β−3

3πr0

r2dr

f228π√3+81

243(α/β)4f2+32π√3+81

243(α/β)3f3/2

=2

3π

28π√3+81

243α4βr0

r2dr +2

3π

32π√3+81

243α3r0

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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