Discreteness of Space from GUP in a Weak Gravitational Field

0.16 MB·

by  Soumen Deb

Discreteness of Space from GUP in a Weak Gravitational Field
Soumen Deb1,∗ Saurya Das1,† and Elias C. Vagenas2‡
1 Theoretical Physics Group, Dept. of Physics and Astronomy, University of Lethbridge,
4401 University Drive, Lethbridge, Alberta T1K 3M4, Canada and
2 Theoretical Physics Group, Department of Physics,
Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
Quantumgravity effectsmodify theHeisenberg’suncertaintyprincipletoageneralized uncertainty
principle (GUP). Earlier work showed that the GUP-induced corrections to the Schr¨odinger equa-
tion, when applied to a non-relativistic particle in a one-dimensional box, led to the quantization
of length. Similarly, corrections to the Klein-Gordon and the Dirac equations, gave rise to length,
areaandvolumequantizations. Theseresultssuggestafundamentalgranularstructureofspace. In
6 thiswork,itisinvestigatedhowspacetimecurvatureandgravitymightinfluencethisdiscretenessof
1 space. In particular, by adding a weak gravitational background field to the above three quantum
0 equations,itisshownthatquantizationoflengths,areasandvolumescontinuetohold. However,it
2 shouldbenotedthat thenatureof thisnewquantization isquitecomplex and underproperlimits,
itreducestocaseswithoutgravity. Theseresultssuggestthatquantumgravityeffectsareuniversal.
b
e
F
I. INTRODUCTION of length, mass, and time

3
~G
c] During the last 70 years, much effort has been devoted ℓPl = r c3 ∼10−35m ,
-q towardsthe constructionofa consistenttheory ofQuan- m = ~c 10−8kg ,
r tum Gravity (QG). All approaches to QG start with an Pl G ∼
g r
[ athssautmarpetieoxntraebmoeulyt stmhealsl,trwuacytubreeyoofndsptahceetciumrreenattesxcpaelers- tPl = ~cG5 ∼10−44s .
2 imental advancement. r
v The smallnessof this scalemakes QGphenomenologists’
3 job difficult, which is to test the Planck scale effects and
However, even if not direct, experimental evidence, e.g.
9
extractusefulinformationforfurther theoreticalstudies.
8 analogue gravity experiments [1], suggests that gravity
Among the many mathematical results of String Theory
7 can show quantum effects. Therefore, since there is no
thereisonewhichisofparticularinterestandrelevantto
0 directexperimentalguidance,itisquitenaturaltotryto
1. developacorrecttheorybasedonconceptualrestrictions. QGP. This is the modification of the Heisenberg uncer-
tainty principle (HUP), which is well known as general-
0 Likeanyotheractiveresearchfield,whatQuantumGrav-
izeduncertaintyprinciple(GUP).Inthe context,mainly
6 ityPhenomenology(QGP)ideallyneedsisacombination
but not only, of String Theory, the suggested version of
1 of theory and doable experiments [2].
: GUP is [5–13]
v
i Atthemoment,QGPcanbethoughtofasacombination ~ ′∆p
X ∆x +α (1)
of all studies that might contribute to direct or indirect ≥ ∆p ~
r
observablepredictions[3,4]andanalogmodels[1]. These
a where √α′ 10−32 cm [14].
studiessupportthesmallandthelargescalestructureof
≈
Recently, the theories of Doubly Special Relativity
spacetime consistent with String Theory, or any other
(DSRs) were introduced principally to give a physical
approaches to QG.
interpretationofthePlancklength,i.e.,ℓ ,inthestruc-
Pl
ture of spacetime [15]. In particular, different values
The first step to identifying the relevant doable experi-
couldbe attributed to the Plancklength by different ob-
mentsforQGPresearchwouldbetheidentificationofthe
servers. Thus,DSRsavoidtheseviolationsofLorentzin-
workingscaleofthisnewfield. This,knownasthePlanck
variancebyconsideringthePlancklengthasanobserver-
scale,isfirstestimatedfromdimensionalarguments. The
independentscale. OneoftheconsequencesofDSRswas
Planckscaleisuniquelydefinedbythe fundamentalcon-
a similar modification of the position-momentum com-
stants,namelythespeedoflightc,thegravitationalcon-
stant G, and the Planck constant~, to provide the units mutation relation [16, 17] which leads to a modification
of the HUP as well. In this case, the suggested form of
the commutator is given by [3]
p p
∗[email protected] [xi,pj] = i~ δij α pδij + i j
− p
†[email protected] (cid:18) (cid:18) (cid:19)
‡[email protected] +α2(p2δij +3pipj) (2)
(cid:1)
2
where p can be interpreted as the magnitude of ~p since II. DISCRETENESS OF SPACE IN FLAT
3 SPACETIME
p2 = p p and α= α0 = α0ℓPl.
i=1 i i mPlc ~
The sPuggested form of the commutator given in Eq. (2) In this section, we briefly review the non-relativistic sit-
is satisfied by the modified operators uation where a particle is trapped in a one-dimensional
box and one finds the GUP-corrected Schr¨odinger equa-
x =x ,
i 0i (3) tion [18]. In particular, we consider one of the standard
p =p (1 αp +2α2p2), i=1,2,3 .
i 0i − 0 0 examples in quantum mechanics, namely the problem of
a particle moving in a one-dimensional infinite potential
Here, x , p satisfy the canonical commutation rela-
0i 0i well. The well or the one dimensional box of length L
tions [x ,p ] = i~δ , implying that p = i~ ∂ is
0i 0i ij 0i − ∂x0i is defined by the potential V(x) = 0 for 0 x L and
the standard momentum (operator) at low energies and ≤ ≤
outside this box. The quantum mechanical equation
p the modifiedmomentumathigherenergies. Notethat ∞
i governing such a particle is the Schr¨odinger equation
3
p2 = p p [18].
0 0i 0i
i=1 Hψ =Eψ
P
The specific modification of the commutator (see Eq.
except for the fact that the position and momentum op-
(2)), with the modified operators as given in Eq. (3),
erators are now modified due to the GUP-effects.
leads to a version of GUP which reads [19–21]
Incorporating the GUP corrections, one can write the
~ modified Schr¨odinger equation as
∆x∆p 1 2α<p>+4α2 <p2 >
≥ 2 −
d2 d3
~(cid:2) α (cid:3) ψ+k2ψ+2iα~ ψ =0 (5)
1+ +4α2 ∆p2 dx2 0 dx3
≥ 2 " <p2 > !
where k = 2mE/~2.
+4α2 <pp>2 2α <p>2 (4) 0
− Atthispoint, itshouldbe stressedthattheα-dependent
p
p i term in the above equation is only important when en-
withthedimensionlessparameterα generallyconsidered
0 ergies are comparable to Planck energy and lengths are
to be of order of unity.
comparable to Planck length. The general solution of
It is evident that QGP indicates an irremovable uncer-
this equation is
tainty in distance measurements [2]. In the framework
of String Theory, the modified commutation relations of ψ =Aeik0′x+Be−ik0′′x+Ceix/2α~ .
position and momentum operators result in a version of
GUP. A similar, but subtler, consequence of this version
The first two terms along with the boundary conditions
isthattheapparentlycontinuous-lookingspaceonavery
V(x = 0) = 0 = V(x = L) lead to the standard energy
fine scale is actually grainy. One can ask whether this is
quantization. It is the new third α-dependent term that
a sole influence of gravity or a fundamental structure of
gives rise to a new condition [18]
the spacetime. Now, if one admits the fact that classi-
cal gravity is a derived effect of curvature of spacetime
L
caused by mass, then one can expect to find this dis- cos θ =cos(k L+θ )=cos(nπ+θ +δ )
2α~ − C 0 C C 0
continuity even in the regions of the universe far from a (cid:18) (cid:19)
massive object.
The nature of this discreteness may or may not change which in turn implies that1
whenthespacetimeisnomoreflat,namelyitisacurved
spacetime due to the presence ofa gravitationalfield. In L L
= = nπ+2qπ pπ (6)
orderto investigatethis,we trapaparticleinaboxwith 2α~ 2α ℓ − ≡
0 Pl
agravitationalpotentialinsidethe boxandseeifgravity
wherep 2q nisanaturalnumber. Theaboveexpres-
influences the discreteness shown in [18, 22].
≡ ±
sion shows that the length L is quantized. This result
The outline of this work is as follows. In the next Sec-
canbe interpretedasthe factthat, likethe energyofthe
tion, we briefly review the problem of a particle moving
particleinside the box, the length ofthe box canassume
inaone-dimensionalpotential. Spacetimeisflatbutdue
only certainvalues. Inparticular,L has to be in units of
to GUP-effects, it effectively shows a discrete structure.
In Sec. III, we investigate the discreteness of spacetime
in the problem of a non-relativistic particle moving in a
one-dimensional potential when gravity is present. Fur-
thermore, we explore the discreteness of spacetime for 1 Asalreadymentioned,forbrevitythemathematicaldetailshave
beenomittedhere. However,fortheinterestedreader,thederiva-
the caseofrelativistic0-spinand1/2-spinparticlesmov-
tionof thequantization condition, i.e. Eq. (6), canbefoundin
ing again in a one-dimensional potential when gravity is
[18]. The whole analysis goes from Eq.(11) to Eq.(21) of refer-
present. Finally,inSec. V,webrieflypresentourresults. ence[18].
3
α ℓ . This indicates that the space, at least in a con- Without considering the GUP effects, the Schr¨odinger
0 Pl
finedregionandwithouttheinfluenceofgravity,islikely equation governing the motion of a particle of mass m
to be discrete. inside this box is given by [23]
Further work has shown that this consequence of the
GUP effects can be extended to relativistic scenarios d2ψ(x) 2m
in one, two, and three dimensions [22]. There are sev- 0 (kx E)ψ (x)=0 . (8)
dx2 − ~2 − 0
eral reasons why one needs to investigate the relativis-
tic cases. High energy particles are much more likely with ψ(x)=0 when x 0 or x L, since the potential
0 ≤ ≥
to probe the fabric of spacetime near the Planck scale, outside the box becomes .
∞
which means that they are necessarily relativistic or The above equation, namely Eq.(8), is an Airy equation
ultra-relativistic particles. In addition, the fact, that whose general solution reads [24]
most elementary particles are fermions, leads us to in-
2m(kx E) 2m(kx E)
vestigateDiracequationinsteadoftheSchr¨odingerequa- ψ (x)=C Ai ~2 − +C Bi ~2 − (9)
tion. 0 1 " (2~m2 k)32 # 2 " (2~m2k)32 #
where Ai[u] andBi[u] are Airy functions ofthe firstand
III. DISCRETENESS IN CURVED SPACETIME second kind, respectively.
We now use this wavefunction, i.e., ψ, for solving
0
the GUP-corrected Schr¨odinger equation. Utilizing the
Ithasbeenproventhatthe GUPcorrectionsimposedon
GUP-modified operators given in Eq. (3) in order to
afreeparticleleadtothediscretenessofspace. Although
modify the Hamiltonian of the system under study, the
the moving particle was kept in a box, no force field in-
GUP-correctedone-dimensionalSchr¨odingerequationfor
side the box was assumed, i.e., the particle was free to
a non-relativistic particle moving in a box of length L
move in a flat spacetime. If we wish to claim that the
with a linear potential reads, cf. Eq. (5),
quantum gravityeffects areuniversalthen we should ex-
pect that the length quantizationwill also emerge in the d2ψ 2m d3ψ
+ (E kx)ψ+2iα~ =0 . (10)
presence of external forces. In other words, discreteness dx2 ~2 − dx3
of space must hold whether or not there is an external
field present. It is seenthat the additionalthird term, 2iα~dd3xψ3, which
depends on the GUP parameter, i.e., α, becomes sig-
nificant at high energies (comparable to Planck energy),
A. Non-relativistic case or, equivalently, at small lengths (comparable to Planck
length). Therefore, we can consider a perturbative ap-
proach in order to solve Eq. (10). A suitable trial solu-
The first step towards this generalization would be to
tion can be of the form
considergravityastheexternalforcefieldinsidethebox,
sinceitistheweakestamongthefourfundamentalforces ψ = ψ(E+ǫα,k,x)
1 0
as well as being universal. Additionally, as we have dis- d
cussed before, our goalis to find how gravity determines = ψ0(E,k,x)+ǫαdEψ0(E,k,x) (11)
thenatureofdiscreteness. Withagravitationalpotential
where the form of ψ is given by Eq. (9), and ǫ is a
present inside the box, we ignore all but the first term 0
coefficient that will be determined later.
of the Taylor expansion of this potential, which is a lin-
Skipping intermediate mathematical steps, the general
ear term. This is reasonable because we are interested
solution of the GUP-corrected Schr¨odinger equation is
in the behavior of spacetime fabric near Planck scale
given by
and the gravitational potential changes very little over
suuscehthsmeaglrladviisttaatniocneas.l Fpuorttehnetiramloernee,rignyparpacptriocxei,mwaeteodfteans ψ(x) = A ξ−1/4sin 2ξ32 + π +
√π 3 4
V(h) = mgh over a small vertical distance h and, thus, (cid:20) (cid:18) (cid:19)
tdheentfitehldatsttrheinsgatlhsorjeuasdtsifiEesht=he−pmr1ev∂di∂oVduh(sh)cl=aim−go.fIuttiilsizeivnig- 2~m2 1/3k−2/3ǫα −41ξ−5/4sin 23ξ3/2+ π4
(cid:18) (cid:19) (cid:18) (cid:18) (cid:19)
a linearized potential term. 2 π
+ξ1/4cos ξ3/2+ +
Let us now consider a one-dimensional box of length L
3 4
(0<x<L)with a linear potentialinside, whichhas the (cid:18) (cid:19)(cid:19)(cid:21)
form B ξ−1/4cos 2ξ32 + π +
√π 3 4
(cid:20) (cid:18) (cid:19)
kx, if 0<x<L 1/3
V(x)= (7) 2m 2 π
, otherwise k−2/3ǫα ξ1/4sin ξ3/2+
(cid:26) ∞ ~2 − 3 4
(cid:18) (cid:19) (cid:18) (cid:18) (cid:19)
with k to be a parameter of unit J/m and the smallness 1ξ−5/4cos 2ξ3/2+ π +Ceix/2~α (12)
of k is assumed. −4 3 4
(cid:18) (cid:19)(cid:19)(cid:21)
4
with
1
ξ = 2m 3 k−32 (E kx)
~2 −
(cid:18) (cid:19)
11
ǫ = (2i~)3 2m 12 k67E−14
4 ~2
"
(cid:18) (cid:19)
π π
C sin ξ + C cos ξ +
1 0 2 0
× 4 − 4
(cid:16) (cid:16) 17(cid:17) (cid:16) (cid:17)(cid:17)
+ α(2i~) 2m 12 k61E45
~2
(cid:18) (cid:19)
π π
× C2sin ξ0+ 4 −C1cos ξ0+ 4 FIG. 1: Comparison between L0 (solid lines) which is the
(cid:16) (cid:16)11 (cid:17) (cid:16) (cid:17)(cid:17)i quantizedlength with GUPcorrections in flat spacetimeand
2m 12 k61E−14 L(dottedlines)whichisthequantizedlengthwithGUPcor-
÷ "(cid:18) ~2 (cid:19) rections in curved spacetime.
π π
C sin ξ + C cos ξ +
1 0 2 0
× 4 − 4 with
(cid:16) (cid:16) (cid:17) 3 (cid:16) (cid:17)(cid:17)i
1 2
ξ0 = 32 (cid:18)2~m2 (cid:19)3 k−32E! H1 = √1π "(cid:18)2~m2 (cid:19)31 (Ek−23kx)#−14 (17)
and A, B, C are constants. In particular, we can absorb 1
trheaelpchoanssetaonftAwhinileψB, scuacnh btheawtrAittceannabseBtr=eateBdeaisθBa. H2 = 23(cid:18)2~m2 (cid:19)2 E23 . (18)
| |
Furthermore, C is such a constant that its magnitude
Without loss of generality, we let A = sinθ and B =
C becomes zero in the limit α 0, since the last term 1 1
| | → cosθ for an arbitrary θ; thus, Eq. (13) becomes
must vanish in this limit.
Next,byimposingtheboundaryconditionsψ(x=0)=0 cos(L /2~α) = cosθcos(κL ) sinθsin(κL )(19)
0 0 0
and ψ(x = L) = 0, we arrive at the following condition −
= cos(κL +θ) . (20)
on the length of the box 0
Accordingto the analysisin[18], the aboveequationim-
−1/4
cos(L/2~α) = 1 kL plies that 2L~0α = pπ, p ∈ N. Since L is the perturbation
− E × of L , Eq. (13) yields
(cid:18) (cid:19) 0
2 2m(E kL)3/2 π L
"A∗sin 3r ~2 −k + 4!+ 2~α =f(k)p1π+pπ (21)
2 2m(E kL)3/2 π wherep1 N andforeachp there is afinite setofvalues
B∗cos 3r ~2 −k + 4!#(13) of p1 ∈ N∈. Moreover, since the first term on the RHS of
Eq. (21) is a small perturbative term, the number of p
1
values, for each p, depends on the smallness of function
where A and B are constants that depend on A, B, k,
∗ ∗
f(k).
and E.
As in the case of flat spacetime, we have arrived at a
Itcanbeshownthatinthelimitk 0thewavefunction
→ length quantization condition. Moreover, we have a fine
given by Eq. (12) becomes the solution of Schr¨odinger
structure(splitting)ofthelengthquantizationduetothe
equation for an infinite potential well. Thus, taking the
presence of gravity (see Fig. 1). This is similar to the
limit k 0 the RHS of Eq. (13) reads
→ energyquantizationofthehydrogenatom,inpresenceof
B cos(κL ) A sin(κL ) (14) an external electromagnetic field.
1 0 1 0
−
where L is the length of the box in flat spacetime, κ =
0
B. Relativistic Case
2mE, and
~2
q
Thesmall-scalestructureofspacetimeshouldnotdepend
H π H π
A =H A cos 2 + B sin 2 + (15) onthe useofrelativisticornon-relativistictestparticles.
1 1 ∗ ∗
k 4 − k 4
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) However, particles with speeds comparable to the speed
H2 π H2 π of light should be treated relativistically, and the fun-
B =H A sin + +B cos + (16)
1 1 ∗ k 4 ∗ k 4 damental spacetime structure should be reexamined. In
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
5
this subsection, we take a closer look at the relativis- We can deal with this term which is a non-local one us-
tic equivalent of the Schr¨odinger equation, i.e., Klein- ingfractionalcalculus[26],butamuchsimplerapproach
Gordon equation and, in particular, the modification in- would be to employ the Dirac equation.
duced by GUP. First, we will derive the GUP-version of The free-particle Dirac equation is given by [27]
the Klein-Gordon equation with a linear potential and
then we will try to solve it in order to obtain possible ∂Ψ
i = βmc2+cα~ P~ Ψ (24)
length quantization. Notwithstanding its relative sim- ∂t ·
plicity,Klein-Gordonequationhasmathematicaldifficul- (cid:16) (cid:17)
ties, especially when it comes to dimensions higher than with
one. For this reason, it is much easier to implement the
I 0
more versatile Dirac equation. Therefore, we will also β γ0 = 2 (25)
solve the Dirac equationin order to find a similar length ≡ (cid:18) 0 −I2 (cid:19)
quantization as in [22].
and
I 0 0 σ 0 σ
1. Klein-Gordon Equation αi ≡γ0γi = 02 I2 σi 0i = σi 0i
(cid:18) − (cid:19)(cid:18)− (cid:19) (cid:18) (cid:19)
The Klein-Gordon equation with no force field is given where σ , with i=x, y, z for the 3 spatial dimensions, are
i
by [25] thePaulispinmatrices. Thesematricesaregivenby[28]
(~2(cid:3)+m2c2)ψ =0 (22) 0 1 0 i
σx = 1 0 , σy = i −0 ,
where (cid:3)= 1 ∂2 2 and ~ = ∂ ˆi+ ∂ ˆj+ ∂ kˆ. (cid:18) (cid:19) (cid:18) (cid:19)
c2∂t2 −∇ ∇ ∂x ∂y ∂z 1 0
Next, we take into account a gravitational force field by σz = 0 1 (26)
utilizing a linearized potential. In this case, the GUP- (cid:18) − (cid:19)
correctedKlein-Gordonequationin one dimension reads
Hereβmc2+cα~ P~ istheDiracHamiltonianwithnoforce
·
d2ψ 1 d3ψ field to be present. It should be noted that α~ is distinct
dx2+~2c2 E2−m2c4−2Ekx ψ+2iα~dx3 =0. (23) from the GUP parameter, i.e., α.
At this point, we take into account a gravitational force
(cid:0) (cid:1)
Comparing Eq. (23) with Eq. (10), i.e., field by utilizing a potential term in the form V(~r). In
this case, the GUP-corrected Dirac equation reads
d2ψ 2m d3ψ
+ (E kx)ψ+2iα~ =0 ,
dx2 ~2 − dx3 i∂Ψ = βmc2+cα~ P~ +V(~r)I4 Ψ . (27)
∂t ·
and by making the following “transformations” (cid:16) (cid:17)
Specifically, for the case of one spatial dimension, say z,
2mE 1 (E2 m2c4) the GUP-corrected Dirac equation reads
~2 → ~2c2 −
2Ek 2mk d d2
ic~α +cα~2 +βmc2+kzI ψ(z)=Eψ(z) .
~2c2 → ~2 − zdz dz2 4
(cid:18) (cid:19)
wearriveatalengthquantizationsimilartotheonegiven
This equation represents a relativistic particle in a one-
by Eq. (21).
dimensional box with a potential of the form kz inside.
The four linearly independent solutions to this equation
are given by
2. Dirac Equation in one dimension
4ikακz
ψ = N 1
The three-dimensionalversionof Klein-Gordonequation 1 1 − c/z+2iακ(c(1 2ακ~2) 2E) ×
suffersfromthe non-localityofthedifferentialoperators. (cid:18) − − (cid:19)
χ
In particular, the term p2, when GUP is considered, be- eiκz
rσ χ
comes (cid:18) z (cid:19)
p2 =p2 2αp3 = ~2 2+2iα~3 3 ψ2 = N2eiz/α~ σχχ (28)
0− 0 − ∇ ∇ (cid:18) z (cid:19)
and, thus, the second term reads withχtobeanormalizedspinorthatsatisfiestherelation
χ†χ=I.
∂2 ∂2 ∂2 3/2 Imposing the boundary conditions directly here, we end
2iα~3 + + .
∂x2 ∂y2 ∂z2 up having the so-calledKlein paradox. In order to avoid
(cid:18) (cid:19)
6
this, we will resort to the MIT bag model of quark con- where δ is the usual Kronecker delta, qˆis an arbitrary
ij
finement [29]. Imposing the MIT bag boundary condi- unit vector, and δ is given by
l
tions and omitting some straightforward steps, the con-
2rκˆ
dition on the length of the box is given by δ =κ L =tan−1 l + (lnα)
l l l r2κˆ2 1 O
(cid:18) l − (cid:19)
ρ1(ir 1) ei(δ−κL)−ei(cid:18)κL−tan−1(cid:18)r22−r1(cid:19)(cid:19) with κˆl to be the l component of the unit vector of the
L  − ! wave vector~κ with components κl.
= arg
α~ F′ Moreover, ρ1 and ρ2 are defined as
 
 
  4ikακ1x
π+2nπ, n N, (29) ρ1 = (cid:18)1− c/x+2iακ1(c(1−2ακ1~2)−2E)(cid:19)
−4 ∈ 4ikακ x
1
ρ = 1+ .
whereκ=κ0+α~κ20 withκ0 tobe thewavenumberthat 2 (cid:18) c/x−2iακ1(c(1+2ακ1~2)−2E)(cid:19)
satisfiestherelationE2 =(~κ0)2+(mc2)2. Additionally, The number of terms in the first row is 2d+1 and that
r, δ, ρ1, and F′ are defined as in the second row is (2d+1) d.
×
~κ c Using the MIT bag model again, we obtain conditions
r = 0 on the dimensions of the box. In this case, these condi-
E+mc2
tions are not symmetrical unlike the case in flat space-
δ = tan−1 2r time. Along x-direction, the length quantization has the
r2 1 following form
(cid:18) − (cid:19)
4ikακz
ρ1 = (cid:18)1− c/z+2iακ(c(1−2ακ~2)−2E)(cid:19) qˆα1L~1 = αqˆ01ℓLP1l
F′ = √2F ρ (irκˆ 1) ρ (irκˆ +1)eiδ1
1 1 2 1
= −θ1+arg − −F′ f¯1
with F αs and s>0. (cid:18) (cid:19)
∼ +2n π, n N (30)
1 1
∈
d
3. Dirac Equation in Three Dimensions with f¯l(xi,κi,δi)= eiκixi +e−i(κixi−δi) . Along
i=1(i6=l)
y and z directions, theQquan(cid:0)tization conditions a(cid:1)re iden-
In the most general case, let us consider a box defined
tical
by 0 x L , i=1...d, d being the dimension of the
i i
≤ ≤
box,i.e.,d=1,2,or3. Thatis,thisboxcanbeone,two, qˆlLl qˆlLl
= = 2θ +2n π (31)
or three-dimensional. The box has a linearized potential α~ α ℓ − l l
0 Pl
inside, as before. Without loss of generality, we orient
with n N and θ =tan−1(qˆ).
the box such that the direction in which the potential l l l
∈
This is also consistent with the fact that the potential
changes is our x-direction. The Dirac Hamiltonian with
inside the box increases linearly along x-direction and
the linear potential term can now be written as
remains zero along y and z directions.
H = cα~ p~+βmc2+V(~r)I To obtain the area and volume quantizations, we simply
· multiply the above conditions
= c(α p +α p +α p )+βmc2+kxI
x x y y z z
= cα~ ·p~0−cα(α~ ·p~0)(α~ ·p~0)+βmc2+kxI . A = N qˆlLl = N (2n π 2θ )(2n π θ
N l l 1 1
α ℓ − −
NotethatweemployedtheGUP-correctedmomenta,i.e., l=1 0 Pl l=2
Y Y
p = p (1 αp ), i = 1,..,3, where p = i~ d , and ρ (irκˆ 1) ρ (irκˆ +1)eiδ1
foillowe0di Di−rac p0rescription,i.e., we repl0aiced−p0 bdyxiα~ p~0. +arg 1 1− −F′2 1 f¯1 (32)
· (cid:18) (cid:19)(cid:19)
The wavefunction inside the box turns out to be
with n N, and where N =2 and N =3 represent the
l
∈
d area and volume quantization, respectively.
ρδi1eiκixi +ρδi1e−i(κixi−δi)
1 2
 (cid:20)iQ=1(cid:16) +Feiqαˆ.~~r χ (cid:17) 
 
ψ = d d ρδi1eiiκixi+  IV. CONCLUSIONS
 1 
 (−1jP)=δ1ij(cid:20)ρiQδ2=i11e(cid:16)−i(κixi−δi) rκˆj  Ionnet-hdisimweonrskio,nwaelhbaovxeosfhsoiwzenLth,aintcilfuwdeeatrgarpaaviptaatritoicnlaelipnoa-
 +Feiqαˆ.~~rqˆj σjχ (cid:17)  tentialinsidetheboxandthentrytomeasurethelength
 
 i 
7
ofthebox,thelengthLwillappearasaquantizedquan- complete theory of quantum gravity, once formulated,
tityinunitsofα ℓ whereℓ isthePlancklength. This should be able to address the issues discussed here, with
0 Pl Pl
resultcanbeinterpretedasthediscretenessofspacenear backgroundspacetime which may be fluctuating. In this
the Planckscaleholding for curvedspacetime as it holds case,wehopethatthe resultsderivedinthis workwould
for flat spacetime, as shown in previous works [18, 22]. continue to hold, at least approximately, and almost ex-
For the gravitational potential, we have used the first actly in the limit when such fluctuations can be ignored.
term of a Taylor series to describe it as a linearized po- Finally,onemaybeinterestedindelvingintothepossible
tential. This is reasonable because we are interested in connection between the non-relativistic particle moving
the behavior of spacetime fabric near Planck scale and inaboxinsidewhichalinearpotentialispresent,andthe
the gravitational potential changes at a very slow rate hydrogen atom. In both systems, a fine structure (split-
over such small distances. ting)showsup. Inparticular,forthefirstsystemitisthe
We have implemented our method for a non-relativistic finestructureofthelengthquantization,whileforsecond
particleincurvedspacetimeandforarelativisticone. In systemitisthe finestructureoftheenergyquantization.
the latter case, the GUP-corrected Klein-Gordon equa- This apparent coincidence suggests further investigation
tion in one dimension has been solved as well as the of the discreteness of spacetime. In addition, although
GUP-corrected Dirac equation in one, two and three di- the original HUP is restricted to position-momentum
mensions. As already mentioned, in all cases the length commutation while the time-energy uncertainty princi-
of the box appears as a quantized quantity in units of ple has been merely thought of as a statistical measure
α0ℓPl. The presence of lengths that are proportional to of variance, a more generalized idea of GUP-corrected
the Planck length in all cases strengthens the claim of commutationrelationinvolving 4-momentummight give
the existence of a minimum measurable length. Further- rise to discontinuity of time.
more, in two and three dimensions, the area and volume
quantizations were also obtained.
Extension of the method employed in this work for ar-
bitrary curved spacetime would be quite interesting. In
V. ACKNOWLEDGMENTS
particular, it is expected that subsequent terms in the
Taylor series would give rise to a more general curved
spacetime. Hence, an arbitrary form of the gravitational We would like to thank the referee for constructive com-
potentialcouldbeanalyzedfollowingthesameapproach. ments. This work is supported in part by the Natural
This would still assume a fixed classical background. A Sciences and Engineering Research Council of Canada.
[1] C.Barcelo,S.LiberatiandM.Visser,LivingRev.Rel.8, [13] C. Bambi and F. R. Urban, Class. Quant. Grav.
12(2005)[LivingRev.Rel.14,3(2011)][gr-qc/0505065]. 25, 095006 (2008) doi:10.1088/0264-9381/25/9/095006
[2] G. Amelino-Camelia, Living Rev. Rel. 16, 5 (2013) [arXiv:0709.1965 [gr-qc]].
doi:10.12942/lrr-2013-5 [arXiv:0806.0339 [gr-qc]]. [14] E. Witten, Phys. Today 49N4, 24 (1996).
[3] A. F. Ali, S. Das and E. C. Vagenas, Phys. Rev. doi:10.1063/1.881493
D 84, 044013 (2011) doi:10.1103/PhysRevD.84.044013 [15] G.Amelino-Camelia, Int.J.Mod.Phys.D11,35(2002)
[arXiv:1107.3164 [hep-th]]. doi:10.1142/S0218271802001330 [gr-qc/0012051].
[4] I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim [16] J. Magueijo and L. Smolin, Phys. Rev. D 71,
and C. Brukner, Nature Phys. 8, 393 (2012) 026010 (2005) doi:10.1103/PhysRevD.71.026010 [hep-
doi:10.1038/nphys2262 [arXiv:1111.1979 [quant-ph]]. th/0401087].
[5] G. Veneziano, Europhys. Lett. 2, 199 (1986). [17] J. L. Cortes and J. Gamboa, Phys. Rev. D 71,
doi:10.1209/0295-5075/2/3/006 065015 (2005) doi:10.1103/PhysRevD.71.065015 [hep-
[6] D. J. Gross and P. F. Mende, Nucl. Phys. B 303, 407 th/0405285].
(1988). doi:10.1016/0550-3213(88)90390-2 [18] A. F. Ali, S. Das and E. C. Vagenas, Phys. Lett.
[7] D.Amati,M.Ciafaloni andG.Veneziano,Phys.Lett.B B 678, 497 (2009) doi:10.1016/j.physletb.2009.06.061
216, 41 (1989). doi:10.1016/0370-2693(89)91366-X [arXiv:0906.5396 [hep-th]].
[8] K.Konishi,G.PaffutiandP.Provero,Phys.Lett.B234, [19] A.Kempf,J.Phys.A30,2093 (1997) doi:10.1088/0305-
276 (1990). doi:10.1016/0370-2693(90)91927-4 4470/30/6/030 [hep-th/9604045].
[9] M. Maggiore, Phys. Lett. B 304, 65 (1993) [20] A. Kempf, G. Mangano and R. B. Mann, Phys. Rev.
doi:10.1016/0370-2693(93)91401-8 [hep-th/9301067]. D 52, 1108 (1995) doi:10.1103/PhysRevD.52.1108 [hep-
[10] L. J. Garay, Int. J. Mod. Phys. A 10, 145 (1995) th/9412167].
doi:10.1142/S0217751X95000085 [gr-qc/9403008]. [21] F. Brau, J. Phys. A 32, 7691 (1999) doi:10.1088/0305-
[11] F. Scardigli, Phys. Lett. B 452, 39 (1999) 4470/32/44/308 [quant-ph/9905033].
doi:10.1016/S0370-2693(99)00167-7 [hep-th/9904025]. [22] S. Das, E. C. Vagenas and A. F. Ali, Phys.
[12] S. Hossenfelder, M. Bleicher, S. Hofmann, J. Ruppert, Lett. B 690, 407 (2010) [Phys. Lett. 692,
S.SchererandH.Stoecker,Phys.Lett.B575,85(2003) 342 (2010)] doi:10.1016/j.physletb.2010.05.052,
doi:10.1016/j.physletb.2003.09.040 [hep-th/0305262]. 10.1016/j.physletb.2010.07.025 [arXiv:1005.3368 [hep-
8
th]]. Universidad de Tarapaca, Arica, Chile, Lecture Notes,
[23] L. D. Landau, E. M. Lifschitz, Quantum Mechanics 41-54 (2005).
(Non-Relativistic Theory),3rdedition,Oxford,England: [27] P.A.M.Dirac,Proc.Roy.Soc.Lond.A117,610(1928).
Pergamon Press (1991). doi:10.1098/rspa.1928.0023
[24] A. D. Polyanin, V. F Zeitsev, Handbook of Exact Solu- [28] C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Me-
tions for Ordinary Differential Equations, Chapman and chanics, vol. 1. Wiley-VC(1991).
Hall (2003). [29] A.ChodosandC.B.Thorn,Phys.Lett.B53,359(1974).
[25] A. Das, Lectures in Quantum Mechanics, Singapore: doi:10.1016/0370-2693(74)90402-X
World Scientific Publishing (2008).
[26] M. Bologna, Short Introduction to Fractional Calculus,