Corrections to entropy and thermodynamics of charged black hole using generalized uncertainty principle

0.21 MB·English

by  Abdel Nasser Tawfik

ECTP-2013-04
.
Corrections to entropy and thermodynamics of charged black hole using
generalized uncertainty principle
Abdel Nasser Tawfik
∗
Egyptian Center for Theoretical Physics (ECTP),
Modern University for Technology and Information (MTI), 11571 Cairo, Egypt and
World Laboratory for Cosmology And Particle Physics (WLCAPP), Cairo, Egypt
5
1
Eiman Abou El Dahab
0
Faculty of Computer Sciences, Modern University for Technology and Information (MTI), 11571 Cairo, Egypt and
2
World Laboratory for Cosmology And Particle Physics (WLCAPP), Cairo, Egypt

b (Dated: February 23, 2015)
e
F Recently, there has been much attention devoted to resolving the quantum corrections to the
Bekenstein-Hawking (black hole) entropy, which relates the entropy to the cross-sectional area of
9
theblackholehorizon. Usinggeneralized uncertaintyprinciple(GUP),correctionstothegeometric
1
entropy and thermodynamics of black hole will be introduced. The impact of GUP on the entropy

near the horizon of three types of black holes; Schwarzschild, Garfinkle-Horowitz-Strominger and
]
c Reissner-Nordstr¨om is determined. It is found that the logarithmic divergence in the entropy-area
q relation turns to be positive. The entropy S, which is assumed to be related to horizon’s two-
- dimensionalarea,getsanadditionalterms,forinstance2√πα√S,whereαistheGUPparameter.
r
g
[ PACSnumbers: 04.70.Dy,04.60.Kz,05.70.-a
Keywords: Generalizeduncertaintyrelation,blackhole,entropythermodynamics

2
v
6
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2 I. INTRODUCTION
1
0
The finding that the black holes should have well-defined entropy and temperature represented one of the
.
1
greatest achievements in recent astrophysics [1–3]. In statistical physics and thermodynamics, the entropy
0
relatesthenumberofmacrostatestothatofmicrostatesofthesystemofinterest. Furthermore,theentropycan
5
be defined as the amount of additional information needed to specify the exact physical state of a system. In
1
: general relativity, the entropy of black hole got a noveldefinition. It is a pure geometric quantity so that when
v
comparing black hole with a thermodynamic system, we find an important difference. Whether the black hole
i
X hasinteriordegreesoffreedomcorrespondingtoitsentropy,theBekenstein-Hawkingentropydeliveredananswer
to this and characterizedthe statisticalmeaning [1–3]. The horizon’stwo-dimensionalareadefines the entropy.
r
a Forcompleteness,wementionthisentropyisdefinedbyBoltzmann-Gibbsstatistics. TheTsallisentropy,which
is a generalization of Boltzmann-Gibbs entropy, would have another different dependence [4]. Identification
and/or counting of the microstates was highlighted in Ref. [5]. A non-vanishing entropy of extremal charged
blackholeswasintroduced[7]. Itwassuggestedthatthispresumablylieswithinthe frameworkofthequantum
gravity. For example, the string theory [8] and loop quantum gravity [9] succeeded in presenting an statistical
explanation formulated in an entropy-area law. The proportionality relating black hole entropy with area was
derived from classicalthermodynamics, as well [10]. In this regard, we recall that a black hole is nothing but a
classical solution in general theory of relativity with some special properties.
As discussed in [5, 6], the quantum correction of geometric entropy of charged black hole would avoid being
biased in favour of a certain theory of quantum gravity. Furthermore, the tree-level quantum correction [11]
has been verified in different studies [12]. The correctionto the Bekenstein-Hawkingentropy, which relates the
entropy to the cross-sectional area of the black hole horizon, has a series of terms [5, 6]. The coefficient of
the leading-order correction, the logarithmic term, is suggested as a discriminator of prospective fundamental
theories for quantum gravity [13]. It is essential to suggest a method that fixes it, but it should not depend on
the utilized models for the quantum gravity. For instance, this might be the holographic principle [14].
∗Electronicaddress: a.tawfi[email protected]
2
ItisfoundthatthecovariantentropyboundintheFriedmann-Lemaitre-Robertson-Walker(FLRW)Universe
gives an indication to the holographic nature in terms of temperature and entropy [15]. The cosmological
boundary can be chosen as the cosmological apparent horizon instead of the event horizon of a black hole. In
light of this, it was argued that the statistical (informational) entropy of a black hole can be calculated using
the brick wallmethod (BWM) [16]. Inorderto avoidthe divergencenearthe eventhorizon,a cutoff parameter
would be utilized. Since the degrees of freedom would be dominant near the horizon, the brick wall method
could be replaced by a thin-layer model making the calculation of entropy possible [17–24]. The entropy of
FLRW Universe is given by time-dependent metric.
In brick wall model, the usual position-momentum uncertainty relation is given by
~
∆x∆p . (1)
≥ 2
Then, the entropy can be calculated as follows.
1/2
S0 = β2 ∂∂Fβ0(cid:12)(cid:12)(cid:12)β=βH = βπ2 Zr+L+ǫdr √1f Zm∞√fdω ωeβω((cid:16)eβωfω2−−1m)22(cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)β=βH, (2)
where β is the inverse temperat(cid:12)ure, F0 is the free energy and L and ǫ are infrared an(cid:12)(cid:12)d ultraviolet regulators,
respectively. β is the inverse Hawking temperature. In massless limit (m=0), the entropy reads
H
1 1
S0non−ext ≈ 12 ln 2Λǫ , (3)
(cid:18) (cid:19)
where Λ stands for the cosmological constant. In deriving this expression the infrared divergent L-term is
entirely cancelled.
For the external case, we can assume that β .
→∞
Sext = β2 ∂F0 =0. (4)
0
∂β
(cid:12)β
(cid:12) →∞
Opposite to the extremal case, we assume a zero-tempera(cid:12)ture quantum mechanical system around the black
(cid:12)
hole. Then, the entropy reads
1
Sext ln , (5)
0
≈ 2Λǫ
(cid:18) (cid:19)
which can be interpreted as the physical limit that Λ should be less than 1/(2ǫ).
Althoughtheseresultsincludealogarithmicdivergence,theywillbechangedduetotheGUPeffectsthatwill
be outlined in the section that follows [25]. Section II introduces correctionsto the black hole thermodynamics
using GUP approach. The corrections to black hole entropy due by GUP are studied in section III. The GUP
approach is applied to three types of black holes; Schwarzschild, Garfinkle-Horowitz-Strominger and Reissner-
Nordstro¨m. The conclusions are outlined in section IV.
II. THERMODYNAMICS NEAR EVENT HORIZON AND GUP APPROACH
The GUP approach introduced in [26] claims to make predictions for maximum observable momentum and
minimal measurable length. Accordingly, [x ,x ] = [p ,p ] = 0 can be produced via the Jacobi identity, and
i j i j
therefore
p p
[x ,p ]=i~ δ α pδ + i j +α2 p2δ +3p p , (6)
i j ij ij ij i j
− p
(cid:20) (cid:18) (cid:19) (cid:21)
(cid:0) (cid:1)
where the parameter α=α0/Mpc=α0ℓp/~ and Mpc2 stands for Planck’s energy. Mp and ℓp is Planck’s mass
and length, respectively. Apparently, Eq. (6) implies the existence of a minimum measurable length and a
maximum measurable momentum
∆xmin α0ℓp, (7)
≈
M c
p
∆p , (8)
max
≈ α0
3
where the uncertainties ∆x ∆x and ∆p ∆p . Accordingly, for a particle having a distant originand
min max
≥ ≤
an energy scale comparable to the planckian one, the momentum would be a subject of a modification
pi = p0i 1 αp0+2α2p20 , (9)
−
xi = x0i,(cid:0) (cid:1) (10)
where p20 = ip0ip0i and p0i are the components of the low energy momentum. The operators p0j and x0i
satisfy the canonical commutation relation [x0i,p0j] = i~ δij. Having the standard representation in position
space, then pP0i = i~∂/∂x0i andx0i would representthe spatialcoordinates operatorat low energy[26]. Then
−
the uncertainty relation in natural units reads
~
∆x∆p 1 2α p +4α2 p2 , (11)
≥ 2 − h i h i
(cid:2) (cid:3)
the volume of phase cell in the 1+1 dimension is changed from 2π to
2π 1 2αp+4α2 p2 , (12)
−
and the number of quantum states with ener(cid:0)gy less than ǫ [27] (cid:1)
1 1 L 1 ω2 1/2
n0(ω) = dr dpr = dr m2 , (13)
2π π √f f −
Z Zr++ǫ (cid:18) (cid:19)
where m in the mass ofthe scalarfield andω is a parameterofthe substitution ofKlein-Gordonequation. The
expression given in Eq. (13) will be changed to
1 1
n (ω) = dr dp ,
I 2π r 1 2αp+4α2p2
Z −
1/2
ω2 m2
1 1 f −
= dr , (14)
2π √f (cid:16) 1/2 (cid:17)
Z 1 2α ω2 m2 +4α2 ω2 m2
− f − f −
(cid:16) (cid:17) (cid:16) (cid:17)
where r and f are given in Eqs. (28) and (29), respectively. The WKP approximation(see Eqs. (31) and (32))
is implemented.
At Hawking temperature, Eq. (14) can be used to derive the free energy [27]
1/2
1 L 1 ∞ ωf2 −m2
F0 = −π dr√f dω(cid:16) eβω 1(cid:17) , (15)
Zr++ǫ Zm√f −
which turns to be a subject of a change to
∞ nI(ω)
F = dω ,
I − eβω 1
Zm√f −
1/2
ω2 m2
1 1 ∞ f −
= dr dω . (16)
−π √f (cid:16) (cid:17)1/2
Z Zm√f (eβω 1) 1 2α ω2 m2 +4α2 ω2 m2
− − f − f −
(cid:20) (cid:16) (cid:17) (cid:16) (cid:17)(cid:21)
III. BLACK HOLE ENTROPY AND GUP APPROACH
Near the event horizon i.e., in the range (r+,r++ǫ), f 0, the entropy can be deduced from Eq. (16)
→
1/2
β2 L 1 ∞ ωeβω ωf2 −m2
S0 = π Zr++ǫdr√f Zm√fdω ((cid:16)eβω−1)2(cid:17) (cid:12)(cid:12)(cid:12) . (17)
(cid:12)β=βH
(cid:12)
(cid:12)
(cid:12)
4
Once again, the entropy given in Eq. (17) will be changed to
1/2
β2 1 ∞ ω ωf2 −m2 eβω
S = dr dω,
I π √f (cid:16) (cid:17)1/2
Z Zm√f e2βω 2 1 2α ω2 m2 +4α2 ω2 m2
− − f − f −
(cid:20) (cid:16) (cid:17) (cid:16) (cid:17)(cid:21)
1 r++ǫ 1 ∞ f−1/2β−1x2
= dr dx, (18)
π Zr+ √f Z0 (1−e−x)(ex−1) 1−2αβ√xf +4α2βx22f
h i
where x=βω. We note that as f 0, then ω2/f is the dominant term in the bracket containing ω2/f m2.
→ −
We are interested in the thermodynamic contributions just near the horizonr+,r++ǫ, which corresponds to a
properdistanceoftheorderoftheminimallength,whichcanberelatedtoαusingnaturalunits(α=α0ℓp/~=
2πℓ ℓ ). So we have from Eq. (26)
p p
≈
r++ǫ dr
α = , (19)
f(r)
Zr+
which apparently sets a lower bound to α. Then the entroppy reads
1 r++ǫ dr ∞ a2X2
S = dX , (20)
I π αZr+ f(r)Z0 ea2X e−a2X 2 (1 2X+4X2)
− −
p (cid:16) (cid:17)
where
β
x = fX =aX. (21)
α
p
Then
1 1 ∞ a2X2
S = Σ = dX. (22)
I π I π Z0 ea2X e−a2X 2 (1 2X+4X2)
− −
(cid:16) (cid:17)
We note that as r r+, f 0, then a 0 and
→ → →
a2X2
lim =1. (23)
a 0 eaX/2 e aX/2 2
→ − −
Therefore, (cid:0) (cid:1)
∞ dX 2π
Σ = = , (24)
I Z0 1−2X+4X2 3√3
and
1 2
S = Σ = . (25)
I I
π 3√3
So far, we conclude that S is finite. It does not depend on any parameter.
I
We note thatincontrastto the case ofbrickwallmethod, there is no divergencewithin the just vicinity near
the horizon due to the effect of the generalized uncertainty relation on the quantum states.
Inthesectionsthatfollows,weestimatethecorrectionsinthegeometricentropyofthreetypesofblackholes;
Schwarzschild, Garfinkle-Horowitz-Stromingerand Reissner-Nordstro¨mdue to the GUP approach.
5
A. Schwarzschild black hole
In the Schwarzschild gauge, the metric and field tensors, respectively, are conjectured to be expressed as
1
ds2 = f(r)dt2+ dr2, (26)
− f(r)
F = F (r). (27)
rt rt
The function f(r) in the static solution is defined as
M Q2
f(r) = 1 e 2Λr + e 4Λr, (28)
− Λ − 4Λ2 −
where M is the mass of black hole and Q gives its charge. The outer event horizon has the radius
2 2
1 M M Q
r+ = ln + . (29)
2Λ 2Λ s 2Λ − 2Λ 
(cid:18) (cid:19) (cid:18) (cid:19)
 
In light of this, its derivative vanishes and the Klein-Gordon equation is reduced to
d2R 1 df dR 1 ω2
+ + m2 R = 0, (30)
dr2 f dr dr f f −
(cid:18) (cid:19)
where φ(r)=exp( iωt)R(r). Using WKB approximation,then R exp(iS(r)),
− ∼
1 ω2
p2 = m2 , (31)
r f f −
(cid:18) (cid:19)
and p =dS/dr and
r
ω2
p2 = m2. (32)
f −
In natural units, ~=c=G=k =1, the line element in Schwarzschildblack hole reads
B
1
ds2 = 1 2M dt2+ 1 2M − dr2+r2dΩ2. (33)
2
− − r − r
(cid:18) (cid:19) (cid:18) (cid:19)
Then, Hawking radiation temperature T, horizon area A and entropy S, respectively, read
1 1
T = = , (34)
4πr 8πM
H
A = 4πr2 =16πM2, (35)
H
S = πr2 =4πM2, (36)
H
where r = 2M is the location of the black hole horizon. The increase (decrease) in the horizon area due to
H
absorbing (radiating) a particle of energy dM can be expressed as
dA = 8πr dr =32πMdM. (37)
H H
This particle is conjectured to satisfy Heisenberg’s uncertainty relation ∆x ∆p δ .
i j ij
≥
There are different approaches for GUP. For review, reader are referred to [25, 28].
6
1. Quadratic QUP approach
When Heisenberg’s uncertainty relation is replaced by quadratic GUP [29]
1
∆x +α2∆p , (38)
i i
≥ ∆p
i
where Plank length l2 =((~G )/c3)1/2 equals unity in natural units. From Eq. (38), we have
pl d
1 α2 α2 2
∆p 1+ +2 + . (39)
i ≥ ∆xi " (∆xi)2 (cid:18)(∆xi)2(cid:19) ···#
Then,
1
dA = 32πM , (40)
∆x
α2 α2 2
dA = dA 1+ +2 + . (41)
GUP " (∆x)2 (cid:18)(∆x)2(cid:19) ···#
If we utilize the notations of Ref. [5], then
A
∆x = 2r = . (42)
H
π
r
After straightforwardintegration, we derive
A = A+α2π lnA 2 α2π 2 1 + . (43)
GUP
− A ···
(cid:0) (cid:1)
The Bekenstein-HawkingarealawstatesthatS =A/4. Then, the blackhole’s entropycanbe derivedfromEq.
(43)
S = S+ α2 π ln S α2π 2 1 +C, (44)
GUP
4 − 8S −···
(cid:0) (cid:1)
whereC isanarbitraryconstant. Wenoticethatthecoefficientofthelogarithmiccorrectiontermispositive. It
is a leading-order correction known as logarithmic ”prefactor”. This result obviously contradicts the one given
in [5].
2. Linear GUP approach
If we use GUP introduced in [26],
∆x∆p [1 2α∆p], (45)
≥ −
then
1 1
∆p . (46)
≥ ∆x 1+ 2α
(cid:18) ∆x(cid:19)
Accordingly, the area and entropy, respectively, can be re-written as
A
A = A 4α√π√S+8πα2 ln +2α , (47)
GUP
− rπ !
S = S 2α√π√S+α2π lnS+C, (48)
GUP
−
where α A/π. We notice that the coefficient of lnS is also positive, but the entropy gets an additional
≪
term, 2α√π√S. C is an arbitrary constant.
p
7
B. Garfinkle-Horowitz-Strominger black hole
The line element in Garfinkle-Horowitz-Stromingerspace-time [30] reads
1
ds2 = 1 2M dt2+ 1 2M − +r(r 2a)dΩ2, (49)
2
− − r − r −
(cid:18) (cid:19) (cid:18) (cid:19)
where a = Q2/2M and Q is the electric charge of the black hole. The horizon area and entropy, respectively,
are
A = 4πr (r 2a)=16πM(M a), (50)
H H
− −
S = 4πM(M a). (51)
−
Assuming that a is constant, then the change in A reads
dA = 16πM(2M a) dM. (52)
−
When assuming, as before, that ∆x=2r and a r , then Eq. (41) leads to corrected area and entropy
H H
≪
π2α2 π3/2α2 α2π 2 α2π 2 π
A = A+α2π lnA+8a2 +8a 2 +16a2 , (53)
GUP A A1/2 − A A2
(cid:0) (cid:1) (cid:0) (cid:1)
α2π π2α2 π3/2α2 α2π 2 α2π 2 π
S = S+ lnS+a2 +a +a2 +C. (54)
GUP 4 2S S1/2 − 8S 4S2
(cid:0) (cid:1) (cid:0) (cid:1)
If we use GUP that introduced in [26],
2a
A = 4πr2 1 , (55)
H − r
(cid:18) H(cid:19)
which is valid for a r . Expression(55) can be used to estimate r ,
H H
≪
1 A
r = +a. (56)
H
2 π
r
Then, the change in area of black hole is
1
2α −
dA = 1+ dA. (57)
GUP
∆x
(cid:18) (cid:19)
After integration, we get expressions for corrected area and entropy
A
A = A 4α√π√A+8απ(a+α) ln +2(a+α) , (58)
GUP
− rπ !
S = S 2α√π√S+απ(a+α) lnS+C, (59)
GUP
−
where C is an arbitrary constant. It is assumed that a+α (A/π)1/2. Again, we notice that the logarithmic
≪
term is positive and an additional term 2α√π√S appears.
C. Reissner-Nordstr¨om Black Hole
In space-time of Reissner-Nordstro¨mblack hole, the line element is given by [31, 32]
ds2 = 1 2M + Q2 dt2+ 1 2M + Q2 −1dr2+r2dΩ2, (60)
− − r r2 − r r2 2
(cid:18) (cid:19) (cid:18) (cid:19)
8
where r = M (M2 Q2)1/2 are locations of outer and inner horizon, respectively. The area and entropy
of outer±horizon±, are g−iven as A = 4πr2 and S = πr2, respectively. When the electric charge Q is taken
+ +
invariable and Q r [33], then the corrected area and entropy are
H
≪
A = A+α2π lnA 16α2π2 Q2 +8Q4 2 α2π 2 1 +16Q2 + 32Q4 , (61)
GUP − A A2 − A A2 3A3
(cid:18) (cid:19) (cid:18) (cid:19)
S = S+ α2π lnS α2π2 Q2 +2Q4 α2π(cid:0)2 (cid:1)1 + Q2 + Q4 +C+ . (62)
GUP 4 − S S2 − 8S 2S2 12S3 ···
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
If we use for GUP the approachwhich was introduced in [26], then the corrected area is
A 4πQ2
dA = − dA, (63)
GUP A 4πQ2+2α√πA
−
which can be integrated
A = A 4α√π√A+4α2πln √A √A+2α√π 4πQ2
GUP
− −
(cid:16) (cid:16) (cid:17) (cid:17)
1 √A+2α√π κ
8απ√π α2+2πQ2 ln− − , (64)
− "κ √A+2α√π+κ#
−
(cid:0) (cid:1)
where κ= 4α2π+16πQ2. For finite α and Q √A, the entropy reads
≪
p S = S 2α√π√S+α2πln S πQ2
GUP
− −
α2+2Q2 √(cid:0)S+α√π(cid:1) α2π+4πQ2
πα ln − − +C. (65)
− α2+4Q2 √S+α√π+pα2π+4πQ2!
−
p p
Again C is an arbitrary constant.
IV. CONCLUSIONS
SGUP
1.000
0.500
0.100
0.050
0.010
0.005
S
0.02 0.05 0.10 0.20 0.50 1.00 2.00
Fig. 1: Comparison between GUP-modifications of entropy calculated for Schwarzschild (solid curve), Garfinkle-
Horowitz-Strominger (dashed curve) and Reissner-Nordstr¨om (dash-dotted curve) black holes in log-log scale. The
unmodified entropy is given bysolid straight line. At very large entropy,the modifications entirely disappear.
As discussed above, the quantum correction of the geometric entropy of charged black hole has one great
advantage. Doing this one can avoid being biased in favour of a certain theory of the quantum gravity. For
9
SGUP
2.00
1.00
0.50
0.20
0.10
0.05
0.02
S
0.02 0.05 0.10 0.20 0.50 1.00 2.00
Fig.2: The same as in Fig. 1 butat vanishing α.
example, the correction to the Bekenstein-Hawking entropy, which relates the entropy to the cross-sectional
area of the black hole horizon, includes a series of terms where the coefficient of the leading-order correction,
the logarithmic term, is suggested as a discriminator of prospective fundamental theories for quantum gravity.
Itisessentialtosuggestamethodthatfixesit,butitshouldnotdependontheutilizedmodelsforthequantum
gravity. For instance, this might be the holographic principle.
Brick wall method is used to calculate the statistical (informational) entropy of black hole. In doing this, a
cutoff parameter is assuming in order to avoid the divergence near the event horizon. Because the degrees of
freedomare likely dominantnearthe horizon,itis assumedthat the brickwallmethod shouldbe replacedby a
thin-layer model making the calculation of entropy possible. For instance, the entropy of FLRW Universe can
be given by time-dependent metric. It is found that the black hole entropy is logarithmically related to the
ultraviolet regulator ǫ, so that the physical entropy is limited to Λ<2ǫ.
The statistical entropy of a system likely has a clear microscopic interpretation. The question whether the
blackholeentropywouldhaveasimilarstatisticalinterpretationisrelevant. Whencomparingblackholeentropy
with the one that counts for the microstates Ω, we can simply relate A/4 to lnΩ. This is valid as long as the
gravity is sufficiently strong so that the horizon radius is much larger than the Compton wavelength. In order
toapplyGUPapproach,westartwiththemodifiedmomentumandstatisticallyderiveexpressionsforareaand
entropy. Then, we apply the holographic principle.
Using GUP approach, the black hole thermodynamics and entropy get substantial corrections. The cor-
rections are studied for three types of black holes; Schwarzschild,Garfinkle-Horowitz-Stromingerand Reissner-
Nordstro¨m. Wefoundthatthelogarithmicdivergenceintheentropy-arearelationturnstobepositive. Further-
more we find that the entropy S gets an additional terms, such as 2α√π√S, where α is the GUP parameter.
Figure 1 illustrates an intensive comparison between GUP-modifications of the entropies of Schwarzschild
(solidcurve),Garfinkle-Horowitz-Strominger(dashedcurve)andReissner-Nordstro¨m(dash-dottedcurve)black
holes in a log-log scale. We also present the unmodified entropy by the solid straight line. We note that the
modifications entirely disappear, at large values of S. In these calculations, the parameters, α, a, Q and C are
fixedandkeptconstant. ItisobviousthattheentropyofSchwarzschildblackholegetsapositivecontribution. A
considrableamountis subtractedfromtheentropiesofGarfinkle-Horowitz-StromingerandReissner-Nordstro¨m
black holes. The results at vanishing α are illustrated in Fig. 2. It is obvious that the entropies of the three
types of black holes are dominated by the parameter C which apparently becomes negligible at large S.
10
Appendix A: Modifications of uncertainty principle
1. Generalized (gravitational) uncertainty principle (GUP)
The commutatorrelation[26,34,35], whichareconsistentwiththe stringtheory,theblackholesphysicsand
DSR leads to
p p
[xˆ,pˆ]=i~ δ α pδ + i j +α2 p2δ +3p p , (A1)
i j ij ij ij i j
− p
(cid:20) (cid:18) (cid:19) (cid:21)
(cid:0) (cid:1)
implying a minimal length uncertainty anda maximum measurablemomentum when implementing convenient
representation of the commutation relations of the momentum space wave-functions [28, 36]. The constant
coefficient α = α0 1/(Mpc2) = α0 lp2/~2 is referring to the quantum-gravitational effects on the Heisenberg
uncertainty principple. The momentuqm pˆj and the position xˆi operators are given as
xˆiΨ(p) = x0i(1 αp0+2α2p20)Ψ(p),
−
pˆjΨ(p) = p0jΨ(p). (A2)
We notice that p20 = 3jp0jp0j satisfies the canonical commutation relations [x0i, p0j] = i~δij. Then, the
minimal length uncertainty [26, 34, 35] and maximum measurable momentum [28, 36] read
P
∆x (∆x) ~α,
min
≥ ≈
1
p , (A3)
max
≈ 4α
where the maximum measurable momentum agrees with the value which was obtained in the doubly special
relativity (DSR) theory [28, 45]. By using natural units, the one-dimensional uncertainty reads [26, 34, 35]
~
∆x∆p 1 2α∆p+4α2∆p2 . (A4)
≥ 2 −
(cid:0) (cid:1)
This representation of the operators product satisfies the non-commutative geometry of the spacetime [36]
[pˆ,pˆ] = 0,
i j
1
[xˆi, xˆj] = i~α 4α 1 αp0+2α2p~02 Lˆij. (A5)
− − P −
(cid:18) (cid:19) (cid:16) (cid:17)
The rotational symmetry does not break by the commutation relations [36]. In fact, the rotation generators
can still be expressed in terms of position and momentum operators as [28, 36]
Xˆ Pˆ Xˆ Pˆ
i j j i
L = − . (A6)
ij 1 αp0+2α2p~02
−
2. Modified Dispersion Relation (MDR)
Various observations support the conjectured that the Lorentz invariance might be violated. The velocity
of light should differ from c. Any tiny adjustment leads to modification of the energy-momentum relation
and modifies he dispersion relation in vacuum state by δv [70–74]. In particular, at the Planck scale, the
modifications of energy-momentum dispersion relation have been considered in Refs. [62–64]. Two functions
p(E) as expansions with leading Planck-scale correction of order L E3 and L2E4 respectively, reads [54],
p p
p~2 E2 m2+α1LpE3, (A7)
≃ −
p~2 ≃ E2−m2+α2L2pE4. (A8)
ThesearevalidaparticleofmassM atrest,whosepositionisbeingmeasuredbyaprocedureinvolvingacollision
with a photon of energy E and momentum p. Since the relations are originated from Heisenberg uncertainty