Research Article Discreteness of Curved Spacetime...
5
Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 124543, 4 pages http://dx.doi.org/10.1155/2013/124543 Research Article Discreteness of Curved Spacetime from GUP Ahmad Adel Abutaleb Department of Mathematics, Faculty of Science, University of Mansoura, Elmansoura 35516, Egypt Correspondence should be addressed to Ahmad Adel Abutaleb; [email protected] Received 9 October 2013; Revised 13 December 2013; Accepted 16 December 2013 Academic Editor: Elias C. Vagenas Copyright © 2013 Ahmad Adel Abutaleb. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Diverse theories of quantum gravity expect modifications of the Heisenberg’s uncertainty principle near the Planck scale to a so-called Generalized uncertainty principle (GUP). It was shown by some authors that the GUP gives rise to corrections to the Schrodinger , Klein-Gordon, and Dirac equations. By solving the GUP corrected equations, the authors arrived at quantization not only of energy but also of box length, area, and volume. In this paper, we extend the above results to the case of curved spacetime (Schwarzschild metric). We showed that we arrived at the quantization of space by solving Dirac equation with GUP in this metric. 1. Introduction Diverse approaches to quantum gravity expect a minimum measurable length and a modification of the Heisenberg uncertainty principle to a so-called generalized uncertainty principle or GUP. is implies a modification of the commu- tation relations between position coordinates and momen- tum. In [1], the following proposed GUP is consistent with Doubly special relativity or DSR theories and black hole physics which ensure that [ , ] = 0 = [ , ]: [ ⋅ ] = ℎ [ − ( + )+ 2 ( 2 + 3 )] , ΔΔ ≥ ℎ 2 [1 − 2 ⟨⟩ + 4 2 ⟨ 2 ⟩] ≥ ℎ 2 [ [ [ 1+( √ 2 + 4 2 ) Δ 2 + 4 2 ⟨⟩ 2 − 2 √ ⟨ 2 ⟩ ] ] ] , (1) where = 0 / pl = 0 pl /ℎ, pl = Planck mass, pl ≃ 10 −35 m = Planck length, and pl 2 = Planck energy ≈ 10 19 GeV. GUP-induced terms become important near the Planck scale (for earlier version of GUP motivated by string theory, black hole physics, and DSR, see [2–14], and for some phenomenological implications, see [1, 15, 16]). Equation (1) implies the following minimum measurable length and maximum measurable momentum [1, 17]: Δ ≥ (Δ) min ≈ 0 pl , Δ ≤ (Δ) max ≈ pl 0 . (2) It is natural to take 0 =1; for more details see [17]. e following definitions are proposed in [1] and used in [1, 17]: = 0 , = 0 (1 − 0 + 2 2 2 0 ) (3) with 0 and 0 satisfying the canonical commutation rela- tions [ 0 , 0 ] = ℎ , such that 0 = −ℎ(/ 0 ), 2 0 = ∑ 3 =1 0 0 . In [1], it was shown that any nonrelativistic Hamiltonian of the form = 2 /2+( ⇀ ) can be written as = 2 0 /2− (/) 3 0 +( ⇀ )+( 2 ) using (3). is corrected Hamiltonian implies not only the usual quantization of energy, but also that the box length is quantized. In [17], the above results were extended to a relativistic particle in two and three dimen- sions. In this paper we study Dirac equation in Schwarzschild metric using GUP and show that we arrive at the quantization of space.
Transcript of Research Article Discreteness of Curved Spacetime...
Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013 Article ID 124543 4 pageshttpdxdoiorg1011552013124543
Research ArticleDiscreteness of Curved Spacetime from GUP
Ahmad Adel Abutaleb
Department of Mathematics Faculty of Science University of Mansoura Elmansoura 35516 Egypt
Correspondence should be addressed to Ahmad Adel Abutaleb engahamadadelyahoocom
Received 9 October 2013 Revised 13 December 2013 Accepted 16 December 2013
Academic Editor Elias C Vagenas
Copyright copy 2013 Ahmad Adel Abutaleb This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Diverse theories of quantum gravity expect modifications of the Heisenbergrsquos uncertainty principle near the Planck scale to aso-called Generalized uncertainty principle (GUP) It was shown by some authors that the GUP gives rise to corrections to theSchrodinger Klein-Gordon and Dirac equations By solving the GUP corrected equations the authors arrived at quantization notonly of energy but also of box length area and volume In this paper we extend the above results to the case of curved spacetime(Schwarzschild metric) We showed that we arrived at the quantization of space by solving Dirac equation with GUP in this metric
1 Introduction
Diverse approaches to quantum gravity expect a minimummeasurable length and a modification of the Heisenberguncertainty principle to a so-called generalized uncertaintyprinciple or GUPThis implies a modification of the commu-tation relations between position coordinates and momen-tum In [1] the following proposed GUP is consistent withDoubly special relativity or DSR theories and black holephysics which ensure that [119909
119894 119909119895] = 0 = [119901
119894 119901119895]
[119909119894sdot 119901119895] = 119894ℎ [120575
119894119895minus 119886(119901120575
119894119895+119901119894119901119895
119901) + 1198862(1199012120575119894119895+ 3119901119894119901119895)]
Δ119909Δ119901 geℎ
2[1 minus 2119886 ⟨119901⟩ + 4119886
2⟨1199012⟩]
geℎ
2
[[
[
1 +(119886
radic1199012+ 41198862)Δ119901
2
+ 41198862⟨119901⟩2minus 2119886radic⟨1199012⟩
]]
]
(1)
where 119886 = 1198860119872pl119888 = 1198860119871plℎ 119872pl = Planck mass 119871pl ≃
10minus35m = Planck length and 119872pl119888
2 = Planck energy asymp1019 GeV GUP-induced terms become important near
the Planck scale (for earlier version of GUP motivated bystring theory black hole physics and DSR see [2ndash14] andfor some phenomenological implications see [1 15 16])
Equation (1) implies the following minimum measurablelength and maximummeasurable momentum [1 17]
Δ119909 ge (Δ119909)min asymp 1198860119871pl
Δ119901 le (Δ119901)max asymp119872pl119888
1198860
(2)
It is natural to take 1198860= 1 for more details see [17]
The following definitions are proposed in [1] and used in[1 17]
119909119894= 1199090119894 119901
119894= 1199010119894(1 minus 119886119901
0+ 211988621199012
0) (3)
with 1199090119894and 119901
0119895satisfying the canonical commutation rela-
tions [1199090119894 1199010119895] = 119894ℎ120575
119894119895 such that 119901
0119894= minus119894ℎ(120597120597119909
0119894) 11990120=
sum3
119895=111990101198951199010119895
In [1] it was shown that any nonrelativistic Hamiltonianof the form119867 = 11990122119898+119881(997888119903 ) can bewritten as119867 = 1199012
02119898minus
(119886119898)1199013
0+119881(997888119903 )+119874(119886
2) using (3)This correctedHamiltonian
implies not only the usual quantization of energy but also thatthe box length is quantized In [17] the above results wereextended to a relativistic particle in two and three dimen-sions In this paper we study Dirac equation in Schwarzschildmetric usingGUP and show that we arrive at the quantizationof space
2 Advances in High Energy Physics
2 GUP Dirac Equations inSchwarzschild Metric
Dirac equation in Schwarzschild metric without (GUP) canbe written as follows [18]
(119888 (997888120572 sdot997888119901) + 120573119898119888
2) 120595 =
119864
radic120577120595 (4)
where119898 is the rest mass of the particle 120595 is the Dirac spinor997888120572 and 120573 are Dirac matrices 997888119901 equiv (119901
119903 119901120579 119901120593) are momentum
operators 120577 = 1 minus 119903119904119903 119903119904is the Schwarzschild radius of
massive body related to its mass 119872 by 119903119904= 2119866119872119862
2 119866 isthe gravitational constant and 119888 is the speed of light in freespace Using 120595 = ( 1205941120594
2) (4) can be written as
119888 (997888120572 sdot997888119901)1205942+ 11989811988821205941=119864
radic1205771205941
119888 (997888120572 sdot997888119901)1205941minus 11989811988821205942=119864
radic1205771205942
(5)
Now using GUP correction (3) and (5) take the form
119888 (997888120572 sdot9978881199010) 1205942+ 11989811988821205941minus 119888119886119901
2
01205941=119864
radic1205771205941
119888 (997888120572 sdot9978881199010) 1205941minus 11989811988821205942minus 119888119886119901
2
01205942=119864
radic1205771205942
(6)
where
1199012
0= minusℎ2[radic120577
1199032120597
120597119903(1199032radic120577
120597
120597119903) +
1
1199032 sin 120579
times120597
120597120579(sin 120579 120597
120597120579) +
1
1199032sin21205791205972
1205971205932]
(7)
We studyDirac equations in (6) in Schwarzschildmetric withspherical cavity with radius 119877 defined by the potential
119880 (119903) = 0 119903 le 119877
119880 (119903) = 1198800 997888rarr infin 119903 gt 119877(8)
so we can write the corrected GUP Dirac equations withspherical cavity defined by (8) in Schwarzschild metric as
119888 (997888120572 sdot9978881199010) 1205942+ (119898119888
2+ 119880)120594
1minus 119888119886119901
2
01205941=119864
radic1205771205941
119888 (997888120572 sdot9978881199010) 1205941minus (119898119888
2+ 119880)120594
2minus 119888119886119901
2
01205942=119864
radic1205771205942
(9)
Notice that when 119886 = 0 120577 = 1 equations in (9) are usualDirac equations in flat spacetime When 119886 = 0 120577 = 1 (9) areDirac equations with GUP in flat spacetime proposed in [17]When 119886 = 0 120577 = 1 equations in (9) are Dirac equations inSchwarzschildmetric withoutGUPdefined in [18]We followthe analysis of [17 19] and related references [20 21]
We assume the form of Dirac spinor as
120595 = (1205941
1205942
) = (119892119896 (119903) 120574
1198953
119895119897(119903)
119894119891119896 (119903) 120574
1198953
119895119897minus (119903)) (10)
1205741198953
119895119897(119903) = (119897
1
21198953minus1
2
1
2| 119895 1198953)
times 1198841198953minus(12)
119897(119903) (1
0)
+ (1198971
21198953+1
2
minus1
2| 119895 1198953)
times 1198841198953+(12)
119897(119903) (0
1)
(11)
where 1198841198953minus(12)119897
(119903) and 1198841198953+(12)119897
(119903) are spherical harmonicsand (1198951 1198952 1198981 1198982 | 119895 1198953) are Clebsh-Gordon coefficients1205941 1205942are eigenstates of 1198712 (997888119871 is the angularmomentum oper-
ator) with eigenvalues ℎ2119897(119897 + 1) and ℎ2119897minus(119897minus +1) respectivelysuch that the following hold
if
119896 = 119895 +1
2gt 0 (12)
then
119897 = 119896 = 119895 +1
2 119897
minus= 119896 minus 1 = 119895 minus
1
2 (13)
and if
119896 = minus(119895 +1
2) lt 0 (14)
then
119897 = minus (119896 + 1) = 119895 minus1
2 119897
minus= minus119896 = 119895 +
1
2 (15)
We use (997888120572 sdot 997888119860)(997888120572 sdot 997888119861) = 997888119860 sdot 997888119861 + 119894997888120572 sdot (997888119860 times997888119861) and the relatedidentity (997888120572 sdot 997888119903 )(997888120572 sdot 997888119903 ) = 1199032 so we have
(997888120572 sdot9978881199010) =(997888120572 sdot997888119903 ) (997888120572 sdot997888119903 ) (997888120572 sdot9978881199010)
1199032
=
997888120572 sdot997888119903
1199032(997888119903 sdot9978881199010+ 119894997888120572 sdot (997888119903 times9978881199010))
=
997888120572 sdot997888119903
1199032(997888119903 sdot9978881199010+ 119894997888120572 sdot997888119871)
(16)
But from the definition of the momentum operators inSchwarzschild metric [18] we can write (16) as
(997888120572 sdot9978881199010) = (
997888120572 sdot 119903) (minus119894ℎradic120577
120597
120597119903+119894
119903
997888120572 sdot997888119871) (17)
Advances in High Energy Physics 3
Also we have
(997888120572 sdot997888119871 + 1) 120594
12= ∓12059412
(997888120572 sdot 119903) 120574
1198953
119895119897(119903) = minus120574
1198953
119895119897minus (119903) (
997888120572 sdot 119903) 120574
1198953
119895119897minus (119903) = minus120574
1198953
119895119897(119903)
(18)
Next from the definition of 11990120in Schwarzschild metric [18]
we can write
1199012
0119865 (119903) 119884
119898
119897= ℎ2[minusradic120577
1199032119889
119889119903(1199032radic120577
120597
120597119903) +119897 (119897 + 1)
1199032]119865 (119903) 119884
119898
119897
(19)
So using (17) (18) and (19) we can obtain from (9) the fol-lowing equations
minus 119888ℎradic120577119889119891119896
119889119903+119888 (119896 minus 1)
119903119891119896+ (119898119888
2+ 119880)119892
119896
+ 119888119886ℎ2[radic120577
1199032119889
119889119903(radic120577119903
2 119889119892119896
119889119903) minus119897 (119897 + 1)
1199032119892119896] =
119864
radic120577119892119896
119888ℎradic120577119889119892119896
119889119903+119888 (119896 + 1)
119903119892119896minus (119898119888
2+ 119880)119891
119896
+ 119888119886ℎ2[radic120577
1199032119889
119889119903(radic120577119903
2 119889119891119896
119889119903) minus119897minus(119897minus+ 1)
1199032119891119896] =
119864
radic120577119891119896
(20)
It can be shown that MIT bag boundary condition (at 119903 = 119877)is equivalent to [19 20]
120595 = 0 (21)
As in [17] we can expect new nonperturbative solutions ofthe forms 119891
119896= 119865119870(119903)119890119894120598119903119886ℎ and 119892
119896= 119866119870(119903)119890119894120598119903119886ℎ (where 120598 =
119874(1)) for which (20) simplifies to
119886ℎ1198892119892119896
1198891199032= radic120577
119889119891119896
119889119903
119886ℎ1198892119891119896
1198891199032= minusradic120577
119889119892119896
119889119903
(22)
where we have dropped terms which are ignorable for small119886
When 120577 = 1 equations in (22) are identical to the (60)-(61) in reference [17] and in this case we have the followingsolutions 119891119873
119896= 119894119873119890
119894119903119886ℎ 119892119873119896= 119873119890119894119903119886ℎ 119873 is constant so we
can assume the solutions of (22) as
119891119873
119896= 119894119873119890
119894119903119886ℎ
119892119873
119896= 119873119861 (119903) 119890
119894119903119886ℎ
(23)
By applying (23) on (22) we find that
1198862ℎ2 1198892119861 (119903)
1198891199032+ 119861 (119903) + radic120577 minus
2
radic120577= 0 (24)
Consider that 119903119904is very small so we can approximate (24) to
1198862ℎ2 1198892119861 (119903)
1198891199032+ 119861 (119903) minus 1 minus
3119903119904
2119903= 0 (25)
The solution of (25) takes the form
119861 (119903) = 1198881 sin(119903
119886ℎ) + 1198882cos( 119903
119886ℎ)
+15119903119904
119886ℎ[119862119894(119903
119886ℎ) sin( 119903
119886ℎ)
minus 119878119894(119903
119886ℎ) cos ( 119903
119886ℎ)] + 1
(26)
where 1198881and 1198882are constants 119878
119894(119903119886ℎ) and 119862
119894(119903119886ℎ) are the
sine integral function and cosine integral function defined as119878119894(119910) = int
119910
0(sin 119905119905)119889119905 and 119862
119894(119910) = minus int
infin
119910(cos 119905119905)119889119905 For more
details about this functions see [22 23]Therefore the solutions of (22) take the form
119891119873
119896= 119894119873119890
119894119903119886ℎ
119892119873
119896= 119873 [1 + 120590 (119903)] 119890
119894119903119886ℎ
(27)
where
120590 (119903) = 1198881 sin(119903
119886ℎ) + 1198882cos( 119903
119886ℎ)
+15119903119904
119886ℎ[119862119894(119903
119886ℎ) sin( 119903
119886ℎ)
minus 119878119894(119903
119886ℎ) cos( 119903
119886ℎ)]
(28)
Here one must have lim119903119904rarr1199001198881= lim
119903119904rarr1199001198882= 0 and in this
case (120590(119903) = 0 119903119904= 0) the results are the same of [17]
Now the boundary condition (21) gives
|119892119896 (119877) + 119892
119873
119896(119877)|2
= |119891119896 (119877) + 119891
119873
119896(119877)|2
(29)
which to 119874(119886) translates to
(1198922
119896minus 1198912
119896) + 2119873[119892
119896 (1 + 120590 (119877)) cos(119877
119886ℎ) minus 119891119896sin( 119877
119886ℎ)]
+ 1198732[(1 + 120590 (119877))
2minus 1] = 0
(30)
From (30) we have
119891119896= 119892119896 (31)
tan( 119877119886ℎ) = 1 + 120590 (119877)
120590 (119877) = 0 or 120590 (119877) = minus2
(32)
Observing (28) we can write
120590 (119877)
cos (119877119886ℎ)= 1198881tan( 119877
119886ℎ) + 1198882
+15119903119904
119886ℎ[119862119894(119877
119886ℎ) tan( 119877
119886ℎ) minus 119878119894(119877
119886ℎ)]
(33)
4 Advances in High Energy Physics
From (32) we have
if 120590 (119877) = 0 then tan( 119877119886ℎ) = 1
if 120590 (119877) = minus2 then tan( 119877119886ℎ) = minus1
(34)
For the case of120590(119877) = 0 and 119903119904= 0wehave the same result
of discreteness of space in flat spacetime [17]For the second case 120590(119877) = minus2 we have from (33)
4119886ℎ
3119903119904cos (119877119886ℎ)
+2119862119886ℎ
3119903119904
= 119862119894(119877
119886ℎ) + 119878119894(119877
119886ℎ)
119862 is constant(35)
If we take the constant 119862 = 0 then we have
4119886ℎ
3119903119904cos (119877119886ℎ)
= 119862119894(119877
119886ℎ) + 119878119894(119877
119886ℎ) (36)
Suppose that we choose 119903119904= (43)119886ℎ (very small as
assumption) then we have
1
cos (119877119886ℎ)= 119862119894(119877
119886ℎ) + 119878119894(119877
119886ℎ) (37)
Equation (37) has infinite number of solutions we writesome numerical values of it (119877119886ℎ) = 5 565 (119877119886ℎ) = 7159(119877119886ℎ) = 11 and 755 (119877119886ℎ) = 13448 So the radius of cav-ity 119877 has been quantized in terms of 119886ℎ and we again arrivedat the quantization of space in Schwarzschild-like metric
3 Conclusion
Dirac equationswithGUP in Schwarzschildmetric have beenstudied We showed that the assumption of existence of aminimummeasurable length and a corresponding modifica-tion of uncertainty principle yields discreteness of space inthismetric But the question now arises what is the guaranteethat this result will continue to hold for more generic curvedspacetimes We expect that this discreteness will alwaysappear provided that generalized uncertainty principle entersinto the theory but in fact we have no mathematical proofof existence of such discreteness of space if we work on thegeneral metric of the general relativity theory
References
[1] A F Ali S Das and E C Vagenas ldquoDiscreteness of space fromthe generalized uncertainty principlerdquo Physics Letters B vol678 no 5 pp 497ndash499 2009
[2] D Amatti M Ciafaloni and G Veneziano ldquoCan spacetime beprobed below the string sizerdquo Physics Letters B vol 216 no 1-2pp 41ndash47 1980
[3] M Maggiore ldquoA generalized uncertainty principle in quantumgravityrdquo Physics Letters B vol 304 no 1-2 pp 65ndash69 1993
[4] M Maggiore ldquoQuantum groups gravity and the generalizeduncertainty principlerdquo Physical Review D vol 49 no 10 pp5182ndash5187 1994
[5] F Scardigli ldquoGeneralized uncertainty principle in quantumgravity from micro-black hole gedanken experimentrdquo PhysicsLetters B vol 452 no 1-2 pp 39ndash44 1999
[6] S HossenfelderM Bleicher S Hofmann J Ruppert S Schererand H Stoecker ldquoSignatures in the Planck regimerdquo PhysicsLetters B vol 575 no 1-2 pp 85ndash99 2003
[7] A Kempf G Mangano and R B Mann ldquoHilbert space repre-sentation of the minimal length uncertainty relationrdquo PhysicalReview D vol 52 no 2 pp 1108ndash1118 1995
[8] A Kempf ldquoNon-pointlike particles in harmonic oscillatorsrdquoJournal of Physics A vol 30 no 6 pp 2093ndash2101 1997
[9] F Brau ldquoMinimal length uncertainty relation and the hydrogenatomrdquo Journal of Physics A vol 32 no 44 pp 7691ndash7696 1999
[10] G A Camelia ldquorelativity in spacetimes with short-distancestructure governed by an observer-independent (Planckian)length scalerdquo International Journal of Modern Physics D vol 11no 1 p 35 2002
[11] J Magueijo and L Smolin ldquoGeneralized Lorentz invariancewith an invariant energy scalerdquo Physical Review D vol 67 no4 Article ID 044017 12 pages 2003
[12] J L Cortes and J Gamboa ldquoQuantum uncertainty in doublyspecial relativityrdquo Physical Review D vol 71 no 6 Article ID065015 4 pages 2005
[13] A Ashoorioon and R B Mann ldquoOn the tensorscalar ratio ininflation with UV cutoffrdquoNuclear Physics B vol 716 no 1-2 pp261ndash279 2005
[14] A Ashoorioon A Kempf and R B Mann ldquoMinimum lengthcutoff in inflation and uniqueness of the actionrdquoPhysical ReviewD vol 71 no 2 Article ID 023503 11 pages 2005
[15] S Das and E Vagenas ldquoUniversality of quantum gravity correc-tionsrdquo Physical Review Letters vol 101 no 22 Article ID 2213014 pages 2008
[16] S Das and E Vagenas ldquoPhenomenological implications of thegeneralized uncertainty principlerdquo Canadian Journal of Physicsvol 87 no 3 pp 233ndash240 2009
[17] S Das E C Vagenas and A F Ali ldquoDiscreteness of space fromGUP II relativistic wave equationsrdquo Physics Letters B vol 690no 4 pp 407ndash412 2010
[18] C C Barros Jr ldquoQuantum mechanics in curved space-timerdquoTheEuropean Physical Journal C vol 42 no 1 pp 119ndash126 2005
[19] R K Bhadri Models of the Nucleon from Quarks to Solitionschapter 2 Addison-Wesley 1988
[20] A W Thomas ldquoChiral symmetry and the BAG model a newstarting point for nuclear physicsrdquo in Advances in NuclearPhysics vol 13 pp 1ndash137 1984
[21] A J Hey Lecture Notes in Physics vol 77 Springer Berlin Ger-many 1978
[22] J Havil Gamma Exploring Eulers Constant Princeton Univer-sity Press Princeton NJ USA 2003
[23] M Abramowits and I A Stegun Handbook of MathematicalFunctions with Formulas Graphes and Mathematical TablesDover New York NY USA 1972
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2 Advances in High Energy Physics
2 GUP Dirac Equations inSchwarzschild Metric
Dirac equation in Schwarzschild metric without (GUP) canbe written as follows [18]
(119888 (997888120572 sdot997888119901) + 120573119898119888
2) 120595 =
119864
radic120577120595 (4)
where119898 is the rest mass of the particle 120595 is the Dirac spinor997888120572 and 120573 are Dirac matrices 997888119901 equiv (119901
119903 119901120579 119901120593) are momentum
operators 120577 = 1 minus 119903119904119903 119903119904is the Schwarzschild radius of
massive body related to its mass 119872 by 119903119904= 2119866119872119862
2 119866 isthe gravitational constant and 119888 is the speed of light in freespace Using 120595 = ( 1205941120594
2) (4) can be written as
119888 (997888120572 sdot997888119901)1205942+ 11989811988821205941=119864
radic1205771205941
119888 (997888120572 sdot997888119901)1205941minus 11989811988821205942=119864
radic1205771205942
(5)
Now using GUP correction (3) and (5) take the form
119888 (997888120572 sdot9978881199010) 1205942+ 11989811988821205941minus 119888119886119901
2
01205941=119864
radic1205771205941
119888 (997888120572 sdot9978881199010) 1205941minus 11989811988821205942minus 119888119886119901
2
01205942=119864
radic1205771205942
(6)
where
1199012
0= minusℎ2[radic120577
1199032120597
120597119903(1199032radic120577
120597
120597119903) +
1
1199032 sin 120579
times120597
120597120579(sin 120579 120597
120597120579) +
1
1199032sin21205791205972
1205971205932]
(7)
We studyDirac equations in (6) in Schwarzschildmetric withspherical cavity with radius 119877 defined by the potential
119880 (119903) = 0 119903 le 119877
119880 (119903) = 1198800 997888rarr infin 119903 gt 119877(8)
so we can write the corrected GUP Dirac equations withspherical cavity defined by (8) in Schwarzschild metric as
119888 (997888120572 sdot9978881199010) 1205942+ (119898119888
2+ 119880)120594
1minus 119888119886119901
2
01205941=119864
radic1205771205941
119888 (997888120572 sdot9978881199010) 1205941minus (119898119888
2+ 119880)120594
2minus 119888119886119901
2
01205942=119864
radic1205771205942
(9)
Notice that when 119886 = 0 120577 = 1 equations in (9) are usualDirac equations in flat spacetime When 119886 = 0 120577 = 1 (9) areDirac equations with GUP in flat spacetime proposed in [17]When 119886 = 0 120577 = 1 equations in (9) are Dirac equations inSchwarzschildmetric withoutGUPdefined in [18]We followthe analysis of [17 19] and related references [20 21]
We assume the form of Dirac spinor as
120595 = (1205941
1205942
) = (119892119896 (119903) 120574
1198953
119895119897(119903)
119894119891119896 (119903) 120574
1198953
119895119897minus (119903)) (10)
1205741198953
119895119897(119903) = (119897
1
21198953minus1
2
1
2| 119895 1198953)
times 1198841198953minus(12)
119897(119903) (1
0)
+ (1198971
21198953+1
2
minus1
2| 119895 1198953)
times 1198841198953+(12)
119897(119903) (0
1)
(11)
where 1198841198953minus(12)119897
(119903) and 1198841198953+(12)119897
(119903) are spherical harmonicsand (1198951 1198952 1198981 1198982 | 119895 1198953) are Clebsh-Gordon coefficients1205941 1205942are eigenstates of 1198712 (997888119871 is the angularmomentum oper-
ator) with eigenvalues ℎ2119897(119897 + 1) and ℎ2119897minus(119897minus +1) respectivelysuch that the following hold
if
119896 = 119895 +1
2gt 0 (12)
then
119897 = 119896 = 119895 +1
2 119897
minus= 119896 minus 1 = 119895 minus
1
2 (13)
and if
119896 = minus(119895 +1
2) lt 0 (14)
then
119897 = minus (119896 + 1) = 119895 minus1
2 119897
minus= minus119896 = 119895 +
1
2 (15)
We use (997888120572 sdot 997888119860)(997888120572 sdot 997888119861) = 997888119860 sdot 997888119861 + 119894997888120572 sdot (997888119860 times997888119861) and the relatedidentity (997888120572 sdot 997888119903 )(997888120572 sdot 997888119903 ) = 1199032 so we have
(997888120572 sdot9978881199010) =(997888120572 sdot997888119903 ) (997888120572 sdot997888119903 ) (997888120572 sdot9978881199010)
1199032
=
997888120572 sdot997888119903
1199032(997888119903 sdot9978881199010+ 119894997888120572 sdot (997888119903 times9978881199010))
=
997888120572 sdot997888119903
1199032(997888119903 sdot9978881199010+ 119894997888120572 sdot997888119871)
(16)
But from the definition of the momentum operators inSchwarzschild metric [18] we can write (16) as
(997888120572 sdot9978881199010) = (
997888120572 sdot 119903) (minus119894ℎradic120577
120597
120597119903+119894
119903
997888120572 sdot997888119871) (17)
Advances in High Energy Physics 3
Also we have
(997888120572 sdot997888119871 + 1) 120594
12= ∓12059412
(997888120572 sdot 119903) 120574
1198953
119895119897(119903) = minus120574
1198953
119895119897minus (119903) (
997888120572 sdot 119903) 120574
1198953
119895119897minus (119903) = minus120574
1198953
119895119897(119903)
(18)
Next from the definition of 11990120in Schwarzschild metric [18]
we can write
1199012
0119865 (119903) 119884
119898
119897= ℎ2[minusradic120577
1199032119889
119889119903(1199032radic120577
120597
120597119903) +119897 (119897 + 1)
1199032]119865 (119903) 119884
119898
119897
(19)
So using (17) (18) and (19) we can obtain from (9) the fol-lowing equations
minus 119888ℎradic120577119889119891119896
119889119903+119888 (119896 minus 1)
119903119891119896+ (119898119888
2+ 119880)119892
119896
+ 119888119886ℎ2[radic120577
1199032119889
119889119903(radic120577119903
2 119889119892119896
119889119903) minus119897 (119897 + 1)
1199032119892119896] =
119864
radic120577119892119896
119888ℎradic120577119889119892119896
119889119903+119888 (119896 + 1)
119903119892119896minus (119898119888
2+ 119880)119891
119896
+ 119888119886ℎ2[radic120577
1199032119889
119889119903(radic120577119903
2 119889119891119896
119889119903) minus119897minus(119897minus+ 1)
1199032119891119896] =
119864
radic120577119891119896
(20)
It can be shown that MIT bag boundary condition (at 119903 = 119877)is equivalent to [19 20]
120595 = 0 (21)
As in [17] we can expect new nonperturbative solutions ofthe forms 119891
119896= 119865119870(119903)119890119894120598119903119886ℎ and 119892
119896= 119866119870(119903)119890119894120598119903119886ℎ (where 120598 =
119874(1)) for which (20) simplifies to
119886ℎ1198892119892119896
1198891199032= radic120577
119889119891119896
119889119903
119886ℎ1198892119891119896
1198891199032= minusradic120577
119889119892119896
119889119903
(22)
where we have dropped terms which are ignorable for small119886
When 120577 = 1 equations in (22) are identical to the (60)-(61) in reference [17] and in this case we have the followingsolutions 119891119873
119896= 119894119873119890
119894119903119886ℎ 119892119873119896= 119873119890119894119903119886ℎ 119873 is constant so we
can assume the solutions of (22) as
119891119873
119896= 119894119873119890
119894119903119886ℎ
119892119873
119896= 119873119861 (119903) 119890
119894119903119886ℎ
(23)
By applying (23) on (22) we find that
1198862ℎ2 1198892119861 (119903)
1198891199032+ 119861 (119903) + radic120577 minus
2
radic120577= 0 (24)
Consider that 119903119904is very small so we can approximate (24) to
1198862ℎ2 1198892119861 (119903)
1198891199032+ 119861 (119903) minus 1 minus
3119903119904
2119903= 0 (25)
The solution of (25) takes the form
119861 (119903) = 1198881 sin(119903
119886ℎ) + 1198882cos( 119903
119886ℎ)
+15119903119904
119886ℎ[119862119894(119903
119886ℎ) sin( 119903
119886ℎ)
minus 119878119894(119903
119886ℎ) cos ( 119903
119886ℎ)] + 1
(26)
where 1198881and 1198882are constants 119878
119894(119903119886ℎ) and 119862
119894(119903119886ℎ) are the
sine integral function and cosine integral function defined as119878119894(119910) = int
119910
0(sin 119905119905)119889119905 and 119862
119894(119910) = minus int
infin
119910(cos 119905119905)119889119905 For more
details about this functions see [22 23]Therefore the solutions of (22) take the form
119891119873
119896= 119894119873119890
119894119903119886ℎ
119892119873
119896= 119873 [1 + 120590 (119903)] 119890
119894119903119886ℎ
(27)
where
120590 (119903) = 1198881 sin(119903
119886ℎ) + 1198882cos( 119903
119886ℎ)
+15119903119904
119886ℎ[119862119894(119903
119886ℎ) sin( 119903
119886ℎ)
minus 119878119894(119903
119886ℎ) cos( 119903
119886ℎ)]
(28)
Here one must have lim119903119904rarr1199001198881= lim
119903119904rarr1199001198882= 0 and in this
case (120590(119903) = 0 119903119904= 0) the results are the same of [17]
Now the boundary condition (21) gives
|119892119896 (119877) + 119892
119873
119896(119877)|2
= |119891119896 (119877) + 119891
119873
119896(119877)|2
(29)
which to 119874(119886) translates to
(1198922
119896minus 1198912
119896) + 2119873[119892
119896 (1 + 120590 (119877)) cos(119877
119886ℎ) minus 119891119896sin( 119877
119886ℎ)]
+ 1198732[(1 + 120590 (119877))
2minus 1] = 0
(30)
From (30) we have
119891119896= 119892119896 (31)
tan( 119877119886ℎ) = 1 + 120590 (119877)
120590 (119877) = 0 or 120590 (119877) = minus2
(32)
Observing (28) we can write
120590 (119877)
cos (119877119886ℎ)= 1198881tan( 119877
119886ℎ) + 1198882
+15119903119904
119886ℎ[119862119894(119877
119886ℎ) tan( 119877
119886ℎ) minus 119878119894(119877
119886ℎ)]
(33)
4 Advances in High Energy Physics
From (32) we have
if 120590 (119877) = 0 then tan( 119877119886ℎ) = 1
if 120590 (119877) = minus2 then tan( 119877119886ℎ) = minus1
(34)
For the case of120590(119877) = 0 and 119903119904= 0wehave the same result
of discreteness of space in flat spacetime [17]For the second case 120590(119877) = minus2 we have from (33)
4119886ℎ
3119903119904cos (119877119886ℎ)
+2119862119886ℎ
3119903119904
= 119862119894(119877
119886ℎ) + 119878119894(119877
119886ℎ)
119862 is constant(35)
If we take the constant 119862 = 0 then we have
4119886ℎ
3119903119904cos (119877119886ℎ)
= 119862119894(119877
119886ℎ) + 119878119894(119877
119886ℎ) (36)
Suppose that we choose 119903119904= (43)119886ℎ (very small as
assumption) then we have
1
cos (119877119886ℎ)= 119862119894(119877
119886ℎ) + 119878119894(119877
119886ℎ) (37)
Equation (37) has infinite number of solutions we writesome numerical values of it (119877119886ℎ) = 5 565 (119877119886ℎ) = 7159(119877119886ℎ) = 11 and 755 (119877119886ℎ) = 13448 So the radius of cav-ity 119877 has been quantized in terms of 119886ℎ and we again arrivedat the quantization of space in Schwarzschild-like metric
3 Conclusion
Dirac equationswithGUP in Schwarzschildmetric have beenstudied We showed that the assumption of existence of aminimummeasurable length and a corresponding modifica-tion of uncertainty principle yields discreteness of space inthismetric But the question now arises what is the guaranteethat this result will continue to hold for more generic curvedspacetimes We expect that this discreteness will alwaysappear provided that generalized uncertainty principle entersinto the theory but in fact we have no mathematical proofof existence of such discreteness of space if we work on thegeneral metric of the general relativity theory
References
[1] A F Ali S Das and E C Vagenas ldquoDiscreteness of space fromthe generalized uncertainty principlerdquo Physics Letters B vol678 no 5 pp 497ndash499 2009
[2] D Amatti M Ciafaloni and G Veneziano ldquoCan spacetime beprobed below the string sizerdquo Physics Letters B vol 216 no 1-2pp 41ndash47 1980
[3] M Maggiore ldquoA generalized uncertainty principle in quantumgravityrdquo Physics Letters B vol 304 no 1-2 pp 65ndash69 1993
[4] M Maggiore ldquoQuantum groups gravity and the generalizeduncertainty principlerdquo Physical Review D vol 49 no 10 pp5182ndash5187 1994
[5] F Scardigli ldquoGeneralized uncertainty principle in quantumgravity from micro-black hole gedanken experimentrdquo PhysicsLetters B vol 452 no 1-2 pp 39ndash44 1999
[6] S HossenfelderM Bleicher S Hofmann J Ruppert S Schererand H Stoecker ldquoSignatures in the Planck regimerdquo PhysicsLetters B vol 575 no 1-2 pp 85ndash99 2003
[7] A Kempf G Mangano and R B Mann ldquoHilbert space repre-sentation of the minimal length uncertainty relationrdquo PhysicalReview D vol 52 no 2 pp 1108ndash1118 1995
[8] A Kempf ldquoNon-pointlike particles in harmonic oscillatorsrdquoJournal of Physics A vol 30 no 6 pp 2093ndash2101 1997
[9] F Brau ldquoMinimal length uncertainty relation and the hydrogenatomrdquo Journal of Physics A vol 32 no 44 pp 7691ndash7696 1999
[10] G A Camelia ldquorelativity in spacetimes with short-distancestructure governed by an observer-independent (Planckian)length scalerdquo International Journal of Modern Physics D vol 11no 1 p 35 2002
[11] J Magueijo and L Smolin ldquoGeneralized Lorentz invariancewith an invariant energy scalerdquo Physical Review D vol 67 no4 Article ID 044017 12 pages 2003
[12] J L Cortes and J Gamboa ldquoQuantum uncertainty in doublyspecial relativityrdquo Physical Review D vol 71 no 6 Article ID065015 4 pages 2005
[13] A Ashoorioon and R B Mann ldquoOn the tensorscalar ratio ininflation with UV cutoffrdquoNuclear Physics B vol 716 no 1-2 pp261ndash279 2005
[14] A Ashoorioon A Kempf and R B Mann ldquoMinimum lengthcutoff in inflation and uniqueness of the actionrdquoPhysical ReviewD vol 71 no 2 Article ID 023503 11 pages 2005
[15] S Das and E Vagenas ldquoUniversality of quantum gravity correc-tionsrdquo Physical Review Letters vol 101 no 22 Article ID 2213014 pages 2008
[16] S Das and E Vagenas ldquoPhenomenological implications of thegeneralized uncertainty principlerdquo Canadian Journal of Physicsvol 87 no 3 pp 233ndash240 2009
[17] S Das E C Vagenas and A F Ali ldquoDiscreteness of space fromGUP II relativistic wave equationsrdquo Physics Letters B vol 690no 4 pp 407ndash412 2010
[18] C C Barros Jr ldquoQuantum mechanics in curved space-timerdquoTheEuropean Physical Journal C vol 42 no 1 pp 119ndash126 2005
[19] R K Bhadri Models of the Nucleon from Quarks to Solitionschapter 2 Addison-Wesley 1988
[20] A W Thomas ldquoChiral symmetry and the BAG model a newstarting point for nuclear physicsrdquo in Advances in NuclearPhysics vol 13 pp 1ndash137 1984
[21] A J Hey Lecture Notes in Physics vol 77 Springer Berlin Ger-many 1978
[22] J Havil Gamma Exploring Eulers Constant Princeton Univer-sity Press Princeton NJ USA 2003
[23] M Abramowits and I A Stegun Handbook of MathematicalFunctions with Formulas Graphes and Mathematical TablesDover New York NY USA 1972
Submit your manuscripts athttpwwwhindawicom
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FluidsJournal of
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Advances in Condensed Matter Physics
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Superconductivity
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Statistical MechanicsInternational Journal of
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GravityJournal of
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Physics Research International
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Journal of
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Volume 2014
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PhotonicsJournal of
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Journal of
Biophysics
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ThermodynamicsJournal of
Advances in High Energy Physics 3
Also we have
(997888120572 sdot997888119871 + 1) 120594
12= ∓12059412
(997888120572 sdot 119903) 120574
1198953
119895119897(119903) = minus120574
1198953
119895119897minus (119903) (
997888120572 sdot 119903) 120574
1198953
119895119897minus (119903) = minus120574
1198953
119895119897(119903)
(18)
Next from the definition of 11990120in Schwarzschild metric [18]
we can write
1199012
0119865 (119903) 119884
119898
119897= ℎ2[minusradic120577
1199032119889
119889119903(1199032radic120577
120597
120597119903) +119897 (119897 + 1)
1199032]119865 (119903) 119884
119898
119897
(19)
So using (17) (18) and (19) we can obtain from (9) the fol-lowing equations
minus 119888ℎradic120577119889119891119896
119889119903+119888 (119896 minus 1)
119903119891119896+ (119898119888
2+ 119880)119892
119896
+ 119888119886ℎ2[radic120577
1199032119889
119889119903(radic120577119903
2 119889119892119896
119889119903) minus119897 (119897 + 1)
1199032119892119896] =
119864
radic120577119892119896
119888ℎradic120577119889119892119896
119889119903+119888 (119896 + 1)
119903119892119896minus (119898119888
2+ 119880)119891
119896
+ 119888119886ℎ2[radic120577
1199032119889
119889119903(radic120577119903
2 119889119891119896
119889119903) minus119897minus(119897minus+ 1)
1199032119891119896] =
119864
radic120577119891119896
(20)
It can be shown that MIT bag boundary condition (at 119903 = 119877)is equivalent to [19 20]
120595 = 0 (21)
As in [17] we can expect new nonperturbative solutions ofthe forms 119891
119896= 119865119870(119903)119890119894120598119903119886ℎ and 119892
119896= 119866119870(119903)119890119894120598119903119886ℎ (where 120598 =
119874(1)) for which (20) simplifies to
119886ℎ1198892119892119896
1198891199032= radic120577
119889119891119896
119889119903
119886ℎ1198892119891119896
1198891199032= minusradic120577
119889119892119896
119889119903
(22)
where we have dropped terms which are ignorable for small119886
When 120577 = 1 equations in (22) are identical to the (60)-(61) in reference [17] and in this case we have the followingsolutions 119891119873
119896= 119894119873119890
119894119903119886ℎ 119892119873119896= 119873119890119894119903119886ℎ 119873 is constant so we
can assume the solutions of (22) as
119891119873
119896= 119894119873119890
119894119903119886ℎ
119892119873
119896= 119873119861 (119903) 119890
119894119903119886ℎ
(23)
By applying (23) on (22) we find that
1198862ℎ2 1198892119861 (119903)
1198891199032+ 119861 (119903) + radic120577 minus
2
radic120577= 0 (24)
Consider that 119903119904is very small so we can approximate (24) to
1198862ℎ2 1198892119861 (119903)
1198891199032+ 119861 (119903) minus 1 minus
3119903119904
2119903= 0 (25)
The solution of (25) takes the form
119861 (119903) = 1198881 sin(119903
119886ℎ) + 1198882cos( 119903
119886ℎ)
+15119903119904
119886ℎ[119862119894(119903
119886ℎ) sin( 119903
119886ℎ)
minus 119878119894(119903
119886ℎ) cos ( 119903
119886ℎ)] + 1
(26)
where 1198881and 1198882are constants 119878
119894(119903119886ℎ) and 119862
119894(119903119886ℎ) are the
sine integral function and cosine integral function defined as119878119894(119910) = int
119910
0(sin 119905119905)119889119905 and 119862
119894(119910) = minus int
infin
119910(cos 119905119905)119889119905 For more
details about this functions see [22 23]Therefore the solutions of (22) take the form
119891119873
119896= 119894119873119890
119894119903119886ℎ
119892119873
119896= 119873 [1 + 120590 (119903)] 119890
119894119903119886ℎ
(27)
where
120590 (119903) = 1198881 sin(119903
119886ℎ) + 1198882cos( 119903
119886ℎ)
+15119903119904
119886ℎ[119862119894(119903
119886ℎ) sin( 119903
119886ℎ)
minus 119878119894(119903
119886ℎ) cos( 119903
119886ℎ)]
(28)
Here one must have lim119903119904rarr1199001198881= lim
119903119904rarr1199001198882= 0 and in this
case (120590(119903) = 0 119903119904= 0) the results are the same of [17]
Now the boundary condition (21) gives
|119892119896 (119877) + 119892
119873
119896(119877)|2
= |119891119896 (119877) + 119891
119873
119896(119877)|2
(29)
which to 119874(119886) translates to
(1198922
119896minus 1198912
119896) + 2119873[119892
119896 (1 + 120590 (119877)) cos(119877
119886ℎ) minus 119891119896sin( 119877
119886ℎ)]
+ 1198732[(1 + 120590 (119877))
2minus 1] = 0
(30)
From (30) we have
119891119896= 119892119896 (31)
tan( 119877119886ℎ) = 1 + 120590 (119877)
120590 (119877) = 0 or 120590 (119877) = minus2
(32)
Observing (28) we can write
120590 (119877)
cos (119877119886ℎ)= 1198881tan( 119877
119886ℎ) + 1198882
+15119903119904
119886ℎ[119862119894(119877
119886ℎ) tan( 119877
119886ℎ) minus 119878119894(119877
119886ℎ)]
(33)
4 Advances in High Energy Physics
From (32) we have
if 120590 (119877) = 0 then tan( 119877119886ℎ) = 1
if 120590 (119877) = minus2 then tan( 119877119886ℎ) = minus1
(34)
For the case of120590(119877) = 0 and 119903119904= 0wehave the same result
of discreteness of space in flat spacetime [17]For the second case 120590(119877) = minus2 we have from (33)
4119886ℎ
3119903119904cos (119877119886ℎ)
+2119862119886ℎ
3119903119904
= 119862119894(119877
119886ℎ) + 119878119894(119877
119886ℎ)
119862 is constant(35)
If we take the constant 119862 = 0 then we have
4119886ℎ
3119903119904cos (119877119886ℎ)
= 119862119894(119877
119886ℎ) + 119878119894(119877
119886ℎ) (36)
Suppose that we choose 119903119904= (43)119886ℎ (very small as
assumption) then we have
1
cos (119877119886ℎ)= 119862119894(119877
119886ℎ) + 119878119894(119877
119886ℎ) (37)
Equation (37) has infinite number of solutions we writesome numerical values of it (119877119886ℎ) = 5 565 (119877119886ℎ) = 7159(119877119886ℎ) = 11 and 755 (119877119886ℎ) = 13448 So the radius of cav-ity 119877 has been quantized in terms of 119886ℎ and we again arrivedat the quantization of space in Schwarzschild-like metric
3 Conclusion
Dirac equationswithGUP in Schwarzschildmetric have beenstudied We showed that the assumption of existence of aminimummeasurable length and a corresponding modifica-tion of uncertainty principle yields discreteness of space inthismetric But the question now arises what is the guaranteethat this result will continue to hold for more generic curvedspacetimes We expect that this discreteness will alwaysappear provided that generalized uncertainty principle entersinto the theory but in fact we have no mathematical proofof existence of such discreteness of space if we work on thegeneral metric of the general relativity theory
References
[1] A F Ali S Das and E C Vagenas ldquoDiscreteness of space fromthe generalized uncertainty principlerdquo Physics Letters B vol678 no 5 pp 497ndash499 2009
[2] D Amatti M Ciafaloni and G Veneziano ldquoCan spacetime beprobed below the string sizerdquo Physics Letters B vol 216 no 1-2pp 41ndash47 1980
[3] M Maggiore ldquoA generalized uncertainty principle in quantumgravityrdquo Physics Letters B vol 304 no 1-2 pp 65ndash69 1993
[4] M Maggiore ldquoQuantum groups gravity and the generalizeduncertainty principlerdquo Physical Review D vol 49 no 10 pp5182ndash5187 1994
[5] F Scardigli ldquoGeneralized uncertainty principle in quantumgravity from micro-black hole gedanken experimentrdquo PhysicsLetters B vol 452 no 1-2 pp 39ndash44 1999
[6] S HossenfelderM Bleicher S Hofmann J Ruppert S Schererand H Stoecker ldquoSignatures in the Planck regimerdquo PhysicsLetters B vol 575 no 1-2 pp 85ndash99 2003
[7] A Kempf G Mangano and R B Mann ldquoHilbert space repre-sentation of the minimal length uncertainty relationrdquo PhysicalReview D vol 52 no 2 pp 1108ndash1118 1995
[8] A Kempf ldquoNon-pointlike particles in harmonic oscillatorsrdquoJournal of Physics A vol 30 no 6 pp 2093ndash2101 1997
[9] F Brau ldquoMinimal length uncertainty relation and the hydrogenatomrdquo Journal of Physics A vol 32 no 44 pp 7691ndash7696 1999
[10] G A Camelia ldquorelativity in spacetimes with short-distancestructure governed by an observer-independent (Planckian)length scalerdquo International Journal of Modern Physics D vol 11no 1 p 35 2002
[11] J Magueijo and L Smolin ldquoGeneralized Lorentz invariancewith an invariant energy scalerdquo Physical Review D vol 67 no4 Article ID 044017 12 pages 2003
[12] J L Cortes and J Gamboa ldquoQuantum uncertainty in doublyspecial relativityrdquo Physical Review D vol 71 no 6 Article ID065015 4 pages 2005
[13] A Ashoorioon and R B Mann ldquoOn the tensorscalar ratio ininflation with UV cutoffrdquoNuclear Physics B vol 716 no 1-2 pp261ndash279 2005
[14] A Ashoorioon A Kempf and R B Mann ldquoMinimum lengthcutoff in inflation and uniqueness of the actionrdquoPhysical ReviewD vol 71 no 2 Article ID 023503 11 pages 2005
[15] S Das and E Vagenas ldquoUniversality of quantum gravity correc-tionsrdquo Physical Review Letters vol 101 no 22 Article ID 2213014 pages 2008
[16] S Das and E Vagenas ldquoPhenomenological implications of thegeneralized uncertainty principlerdquo Canadian Journal of Physicsvol 87 no 3 pp 233ndash240 2009
[17] S Das E C Vagenas and A F Ali ldquoDiscreteness of space fromGUP II relativistic wave equationsrdquo Physics Letters B vol 690no 4 pp 407ndash412 2010
[18] C C Barros Jr ldquoQuantum mechanics in curved space-timerdquoTheEuropean Physical Journal C vol 42 no 1 pp 119ndash126 2005
[19] R K Bhadri Models of the Nucleon from Quarks to Solitionschapter 2 Addison-Wesley 1988
[20] A W Thomas ldquoChiral symmetry and the BAG model a newstarting point for nuclear physicsrdquo in Advances in NuclearPhysics vol 13 pp 1ndash137 1984
[21] A J Hey Lecture Notes in Physics vol 77 Springer Berlin Ger-many 1978
[22] J Havil Gamma Exploring Eulers Constant Princeton Univer-sity Press Princeton NJ USA 2003
[23] M Abramowits and I A Stegun Handbook of MathematicalFunctions with Formulas Graphes and Mathematical TablesDover New York NY USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
4 Advances in High Energy Physics
From (32) we have
if 120590 (119877) = 0 then tan( 119877119886ℎ) = 1
if 120590 (119877) = minus2 then tan( 119877119886ℎ) = minus1
(34)
For the case of120590(119877) = 0 and 119903119904= 0wehave the same result
of discreteness of space in flat spacetime [17]For the second case 120590(119877) = minus2 we have from (33)
4119886ℎ
3119903119904cos (119877119886ℎ)
+2119862119886ℎ
3119903119904
= 119862119894(119877
119886ℎ) + 119878119894(119877
119886ℎ)
119862 is constant(35)
If we take the constant 119862 = 0 then we have
4119886ℎ
3119903119904cos (119877119886ℎ)
= 119862119894(119877
119886ℎ) + 119878119894(119877
119886ℎ) (36)
Suppose that we choose 119903119904= (43)119886ℎ (very small as
assumption) then we have
1
cos (119877119886ℎ)= 119862119894(119877
119886ℎ) + 119878119894(119877
119886ℎ) (37)
Equation (37) has infinite number of solutions we writesome numerical values of it (119877119886ℎ) = 5 565 (119877119886ℎ) = 7159(119877119886ℎ) = 11 and 755 (119877119886ℎ) = 13448 So the radius of cav-ity 119877 has been quantized in terms of 119886ℎ and we again arrivedat the quantization of space in Schwarzschild-like metric
3 Conclusion
Dirac equationswithGUP in Schwarzschildmetric have beenstudied We showed that the assumption of existence of aminimummeasurable length and a corresponding modifica-tion of uncertainty principle yields discreteness of space inthismetric But the question now arises what is the guaranteethat this result will continue to hold for more generic curvedspacetimes We expect that this discreteness will alwaysappear provided that generalized uncertainty principle entersinto the theory but in fact we have no mathematical proofof existence of such discreteness of space if we work on thegeneral metric of the general relativity theory
References
[1] A F Ali S Das and E C Vagenas ldquoDiscreteness of space fromthe generalized uncertainty principlerdquo Physics Letters B vol678 no 5 pp 497ndash499 2009
[2] D Amatti M Ciafaloni and G Veneziano ldquoCan spacetime beprobed below the string sizerdquo Physics Letters B vol 216 no 1-2pp 41ndash47 1980
[3] M Maggiore ldquoA generalized uncertainty principle in quantumgravityrdquo Physics Letters B vol 304 no 1-2 pp 65ndash69 1993
[4] M Maggiore ldquoQuantum groups gravity and the generalizeduncertainty principlerdquo Physical Review D vol 49 no 10 pp5182ndash5187 1994
[5] F Scardigli ldquoGeneralized uncertainty principle in quantumgravity from micro-black hole gedanken experimentrdquo PhysicsLetters B vol 452 no 1-2 pp 39ndash44 1999
[6] S HossenfelderM Bleicher S Hofmann J Ruppert S Schererand H Stoecker ldquoSignatures in the Planck regimerdquo PhysicsLetters B vol 575 no 1-2 pp 85ndash99 2003
[7] A Kempf G Mangano and R B Mann ldquoHilbert space repre-sentation of the minimal length uncertainty relationrdquo PhysicalReview D vol 52 no 2 pp 1108ndash1118 1995
[8] A Kempf ldquoNon-pointlike particles in harmonic oscillatorsrdquoJournal of Physics A vol 30 no 6 pp 2093ndash2101 1997
[9] F Brau ldquoMinimal length uncertainty relation and the hydrogenatomrdquo Journal of Physics A vol 32 no 44 pp 7691ndash7696 1999
[10] G A Camelia ldquorelativity in spacetimes with short-distancestructure governed by an observer-independent (Planckian)length scalerdquo International Journal of Modern Physics D vol 11no 1 p 35 2002
[11] J Magueijo and L Smolin ldquoGeneralized Lorentz invariancewith an invariant energy scalerdquo Physical Review D vol 67 no4 Article ID 044017 12 pages 2003
[12] J L Cortes and J Gamboa ldquoQuantum uncertainty in doublyspecial relativityrdquo Physical Review D vol 71 no 6 Article ID065015 4 pages 2005
[13] A Ashoorioon and R B Mann ldquoOn the tensorscalar ratio ininflation with UV cutoffrdquoNuclear Physics B vol 716 no 1-2 pp261ndash279 2005
[14] A Ashoorioon A Kempf and R B Mann ldquoMinimum lengthcutoff in inflation and uniqueness of the actionrdquoPhysical ReviewD vol 71 no 2 Article ID 023503 11 pages 2005
[15] S Das and E Vagenas ldquoUniversality of quantum gravity correc-tionsrdquo Physical Review Letters vol 101 no 22 Article ID 2213014 pages 2008
[16] S Das and E Vagenas ldquoPhenomenological implications of thegeneralized uncertainty principlerdquo Canadian Journal of Physicsvol 87 no 3 pp 233ndash240 2009
[17] S Das E C Vagenas and A F Ali ldquoDiscreteness of space fromGUP II relativistic wave equationsrdquo Physics Letters B vol 690no 4 pp 407ndash412 2010
[18] C C Barros Jr ldquoQuantum mechanics in curved space-timerdquoTheEuropean Physical Journal C vol 42 no 1 pp 119ndash126 2005
[19] R K Bhadri Models of the Nucleon from Quarks to Solitionschapter 2 Addison-Wesley 1988
[20] A W Thomas ldquoChiral symmetry and the BAG model a newstarting point for nuclear physicsrdquo in Advances in NuclearPhysics vol 13 pp 1ndash137 1984
[21] A J Hey Lecture Notes in Physics vol 77 Springer Berlin Ger-many 1978
[22] J Havil Gamma Exploring Eulers Constant Princeton Univer-sity Press Princeton NJ USA 2003
[23] M Abramowits and I A Stegun Handbook of MathematicalFunctions with Formulas Graphes and Mathematical TablesDover New York NY USA 1972
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FluidsJournal of
Atomic and Molecular Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in Condensed Matter Physics
OpticsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstronomyAdvances in
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Superconductivity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Statistical MechanicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GravityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
AstrophysicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Physics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solid State PhysicsJournal of
Computational Methods in Physics
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Soft MatterJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
AerodynamicsJournal of
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PhotonicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Biophysics
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ThermodynamicsJournal of
