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Physics Letters B
www.elsevier.com/locate/physletb
Dispersion relations in finite-boost DSR
Nosratollah Jafari
∗, Michael R.R. Good
DepartmentofPhysics,NazarbayevUniversity,KabanbayBatyrAve53,Nur-Sultan,010000,Kazakhstan
a r t i c l e i n f o a b s t ra c t
Articlehistory:
Received26February2020 Receivedinrevisedform4June2020 Accepted24August2020
Availableonline31August2020 Editor:B.Grinstein
We find finite-boost transformations DSR theories in first order of the Planck length lp, by solving differentialequationsforthemodifiedgenerators.Weobtaincorrespondingdispersionrelationsforthese transformations,whichhelpusclassifytheDSRtheoriesviafourtypes.Thefinaltypeofourclassification has the same special relativistic dispersion relation butthe transformations are not Lorentz. In DSR theories,thevelocityofphotonsisgenerallydifferentfromtheordinaryspeedcandpossesstimedelay, however inthis new DSRlight has thesame special relativistic speed withno delay.A special case demonstratesthatanysearchforquantumgravityeffectsinobservationswhichgivesaspecialrelativistic dispersionrelationisconsistentwithDSR.
©2020TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
Contents
1. Introduction . . . 2
2. Lorentztransformationsfromthedifferentialequations . . . 2
3. DSRtheoriesinfirstorderofPlancklength . . . 3
3.1. DSRboosts . . . 3
3.2. Generalizationofthefirstorderκ-Poincarésymmetry . . . 4
3.3. Differentialequationsandfinite-boosttransformations . . . 4
3.4. Dispersionrelationanditsclassifications . . . 5
3.5. CompositionofboostsandWignerrotation . . . 5
4. Specialexamplesofthefinite-boostDSRtransformations . . . 6
4.1. MS-DSR . . . 6
4.2. AC-DSR . . . 7
4.3. Dispersionrelationwiththecubicterm . . . 7
4.4. Specialrelativisticdispersionrelation . . . 7
4.5. Dispersionrelationfamily . . . 7
5. ObservationalconsequencesofthefirstorderDSR . . . 8
5.1. SpecialrelativisticdispersionrelationandnullresultsinobservationsforQGeffects . . . 8
6. Conclusion . . . 9
Declarationofcompetinginterest . . . 9
Acknowledgements . . . 9
Appendix A. Appendixes . . . 9
A.1. Findingsolutionsofthedifferentialequations . . . 9
A.2. MStransformationsinthefirstorderofthePlancklength . . . 10
A.3. ACtransformationsinthefirstorderofthePlancklength . . . 10
A.4. TransformationsfortheDSRwithcubicterm . . . 10
A.5. Transformationsforspecialrelativisticdispersionrelationfamily . . . 11
References . . . 11
*
Correspondingauthor.E-mailaddress:[email protected](N. Jafari).
https://doi.org/10.1016/j.physletb.2020.135735
0370-2693/©2020TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3.
1. Introduction
DSRtheorieshavebeenproposedforquantumgravity(QG)modificationstoEinstein’sspecialrelativity[1–5].Thetransformationsof the energyandmomentum underfinite boosts havebeen obtainedin [6]. In thispaper, we continue thisstudy by finding the finite- boostDSR transformations toleading orderofthe Plancklength fromthesolutions ofthedifferential equations,andby looking atthe observational consequencesof thesetransformations.We obtain thecorresponding dispersion relations, whichallow usto classifyDSR theoriestofourtypeswitheachtyperesultinginadifferentspecifickindofDSRtofirstorderofthePlancklength.
Infact,thefirstdifferentialequationforDSRwasgivenbyAmelino-Camelia[2].However,despitethispaperbeingoneofthepioneering worksinDSR,theproposalanddifferentialequationswere specificanddidnot containall DSRtheoriesto firstorder.Also, bystarting from
κ
-Poincaré boost generators [7], Amelino-Camelia andcolleagues obtaineddifferential equations fora specific DSR theory. They solved thedifferentialequationsandfound finitetransformations forthegenerators [3]. Thesefinitetransformations areimportantbut theyarenotthemostgeneral.Wegeneralizetheirmethodandresults.Maguijio and Smolin obtained a different realization of DSR theories by non-linear action of the Lorentz group on the energy- momentumspace[4].Also, they gaveaprocedure forfindingcorresponding transformationsforanygivenmodified dispersion relation [5].ThismodifieddispersionshouldleavethePlanckscaleinvariant.
TheAmelino-Camelia(AC)[1],andtheMaguejo-Smolin(MS)[4] DSRtheoriesarecanonicalandprototypical,buttheyarenotgeneral DSRtheoriesatfirstorderPlancklength.TheyarespecificexamplesofDSRproposals.
Motivatedby theDSR proposal,we starting fromtheboost formulainthe
κ
-Poincaré andDSRtheories[7,8], take itsleading order in Planck length, andfind a generalizedboost by adding some free parameters βi to this first order boost (βi are real numbers and i=0,...,4.) The generalized boost parametrizes the DSR theories to first order. By using this boost, we obtain differential equations which governthe evolution ofthe energy and momentums in momentum space. Thesedifferential equations are some extensions of the differential equations like dd2ξp20 −p0=0, and dpdξ2 =0 in specialrelativity. With perturbative methods, we obtain solutions of the aforementioneddifferentials,whichleadustothefinite-boostDSRtransformationsin(3+1)dimensionalmomentumspace.WeshowthattheAmelino-Camelia(AC)[1],andtheMaguejo-Smolin(MS)[4] DSRtheoriesinfirstorderofthePlancklengthlp,are specialcasesofourDSRfinite-boosttransformations.Ourstudyisinthesamedirectionas[11–14],i.e.‘beyondspecialrelativitytheories’
andcanberegardedasalogicalcontinuationandextension.ItmayalsoproveusefulforinvestigatingthegeometricalnatureofDSR[15].
Otherapproaches,suchasthegeneralizeduncertaintyprinciple(GUP)isrelatedtoDSR[18].Theseisaresultofrecastingthecommu- tatorsbetweenpandxbyaddingmodificationsinvolvingthePlanckenergy[16,17].Asiswell known,theDSRtheoriesareobtainedby addingthePlanckscaleasaninvarianttotheboostsectorsofthePoincarégroup.Thus,ageneralexpressionforthefirstorderDSRfinite transformationscanassistinfindingaconsistentGUPtheorytofirstorderinthePlanckenergy.
Observationsofquantumgravityeffectsforastrophysicalandcosmologicalobservationsarechallenging,andfindingthesetiny effects are notoriously difficultin practice [25,26]. Onthe theoretical side, there many approachesfor including quantum gravity effects (see e.g.[19,20] fora review).Finite-boost transformationDSRtheoriesandclassificationsoffirstorderDSR theories(byuseofa dispersion relation)providemorepossibilitiesforinvestigatingquantumgravityeffectsinobservations.Thefreedomcomesfromtheuseofadjustable parametersinthefinite-boosttransformations.
2. Lorentztransformationsfromthedifferentialequations TheLorentztransformationsinmomentumspaceare
pμ
=
μνpν,
(1)whereμν arethecomponentsoftheLorentzmatrices. The pμ aretheenergyandmomentacomponentsfora particleintheprimed inertial systemwhich moveswith velocity v withrespect to the unprimed inertial system. For the infinitesimal Lorentz matrices, we have
μν
δ
μν+ ω
μν,
(2)where
ω
μν arethecomponentsoftheanti-symmetric[ω
μν]matrices.Inthe(1+1)-dimensionalcase,wehaveω
μν= (δ
μ1δ
0ν+ δ
μ0δ
1ν)ξ.
(3)Here,ξistherapidityparameter.Thus,theexplicitformoftheinfinitesimalLorentztransformationsare:
⎧ ⎪
⎪ ⎪
⎨
⎪ ⎪
⎪ ⎩
p0
=
p0+
p1ξ,
p1=
p1+
p0ξ,
p2=
p2,
p3=
p3.
(4)
Fromwhich,wecanwritethefirstorderdifferentialequationsforthe p0 andp1 componentsas
⎧ ⎨
⎩
dp0 dξ
=
p1,
dp1 dξ
=
p0.
(5)
Thesecondorderdifferentialequationswhichcontainsonlyp0andp1are:
⎧ ⎪
⎨
⎪ ⎩
d2p0
dξ2
−
p0=
0,
d2p1
dξ2
−
p1=
0.
(6)
ThefirstequationofthedifferentialequationsinEq. (6) intheprimedsystemhasthesolution
p0
(ξ ) =
C1coshξ +
C2sinhξ,
(7)whereC1 andC2areconstants,andcanbedeterminedfromtheinitialconditionsp0(ξ=0)=p0and dpdξ0(ξ=0)=p1. Forthep2 andp3 components,theinfinitesimaltransformationsEq. (4) givethefirstorderdifferentialequations
⎧ ⎨
⎩
dp2 dξ
=
0,
dp3 dξ
=
0.
(8)
Intheprimedsystem,thesolutionofthefirstdifferentialequationsinEq. (8) willbe
p2
(ξ ) =
p2+
C1,
(9)whereC1 isaconstant.Byusingtheinitialcondition p2(ξ=0)=p2,onefindsthatthisconstantshouldbezero.
Therefore,fromEq. (7) andEq. (9),wefindthewell-knownLorentztransformationsintheenergymomentumspaceas
⎧ ⎪
⎪ ⎪
⎨
⎪ ⎪
⎪ ⎩
p0
=
p0coshξ +
p1sinhξ,
p1=
p1coshξ +
p0sinhξ,
p2=
p2,
p3
=
p3(10)
Inthenextsection,wewillextendthismethodforfindingsolutionsofthefinite-boostDSRinfirstorderofPlancklengthtransformations whicharenonlinearextensionsofthesystemofdifferentialequationsinEq. (6).
3. DSRtheoriesinfirstorderofPlancklength 3.1.DSRboosts
ThegeneratoroftheDSRtheoriesin(3+1)dimensionsisgivenby
N1
=
p1∂
∂
p0+
lp2p2
+
1−
e−2lpp0 2lp∂
∂
p1−
lpp1 pj∂
∂
pj.
(11)Itoriginatesfromthe
κ
-Poincarégroup[7–9].Inleadingorderoflp,thisgeneratorwillbe N1=
p1∂
∂
p0+
p0
−
lpp20+
lp 2p2∂
∂
p1−
lpp1 pj∂
∂
pj.
(12)In fact, thisgenerator representsonly one class of DSRs. In order to includea wider range of theories, we modify this generator by generalizingtopreserveparityandtime-reversalsymmetry:
N
˜
1= (
p1+
lpβ
0p0p1) ∂
∂
p0+
p0
+
lpβ
1p20+
lpβ
2p2∂
∂
p1+
lpβ
3p1 pj∂
∂
pj+
lpβ
41jkpj
∂
∂
pk,
(13)whereβ0,..., β4 are arbitraryrealnumbers.Weparameterizethe extended
κ
-Poincaré symmetryinthefirstorderofthePlancklength bythesereal numbers.One cannot obtainMS-DSRandother DSR theoriesfromEq. (12),butone can obtainthem fromEq. (13). This generalization gives usDSR theories to the first order in the Planck length. We also have extended freedom in finding more general dispersionrelations tomatchobservationalresults.Inother words,Eq. (13) generalizes firstorderκ
-Poincarésymmetry. Thisextension obeysthegroupstructureanddoesnotviolatetherelativityprinciple[11].ByusingthegeneralgeneratorEq. (13),wecanwritetheinfinitesimaltransformationinthefirstorderofthePlancklengthlp [10,11], as
⎧ ⎪
⎪ ⎪
⎪ ⎪
⎨
⎪ ⎪
⎪ ⎪
⎪ ⎩
p0
=
p0+
p1ξ +
lpβ
0p0p1ξ,
p1
=
p1+
p0ξ +
lpβ
1p20ξ +
lpβ
2p2ξ +
lpβ
3p21ξ,
p2=
p2+
lpβ
3p1p2ξ −
lpβ
4p0p3ξ,
p3
=
p3+
lpβ
3p1p3ξ +
lpβ
4p0p2ξ.
(14)
3.2. Generalizationofthefirstorder
κ
-Poincaré symmetryMajidandRuegg[7],demonstratedthatthe
κ
-Poincaréalgebraisasemi-directproductoftheclassicalLorentz group S O(1,3)acting inadeformedwayonthemomentumsector.Thereisaback-reactionofthemomentumsectorontheLorentzrotations,whiletherotation partoftheLorentz sectorisnotdeformed.Amelino-Camelia andMaguijioandSmolinusedsimilarprocedurestoobtaintheAC-DSRand MS-DSRtheories[1,2,4].Theymodifiedonlytheboosts ofthePoincaré groupandkepttherotationsunmodified.Thesemodifiedboosts actonthemomentumspaceinanonlinearway.We continue this procedure for modifying the boost generator by adding the parameters βi, but we keep the other generators {p0,pi,Mi} unmodified.We haveadded only some parameters to Eq. (12) to obtain more generalmodifications inthe resulting new boostinEq. (13).Thenewboosts (arbitrarydirections)androtationsshouldbesatisfiedinanextendedgrouplikethe
κ
-Poincarégroup tofirstorder.WewillcheckthegrouppropertiesforthemodifiedboostEq. (13) insubsection3.5.Some comments are warranted with respect to the motivation of the new modified boost in Eq. (13) from the boost in Eq. (12).
Addingfreeparameters enlargesthesymmetryofthe
κ
-Poincarégroup.There ismathematicalmotivationtostudy theextensionoftheκ
-Poincarégroupanditsinterestingnonlinearactions.Moreover,therelevantdifferentialequationsinvolvenonlinearsolutionsofsecond orderwhichareimportantformoregeneraldynamicalsystemsandtheirsymmetries.Toobtainthecorrespondingalgebraforthisextendedgroup,werewritethegeneralgeneratorinEq. (13) foranyarbitrarydirection N
˜
i= (
pi+
lpβ
0p0pi) ∂
∂
p0+
p0
+
lpβ
1p20+
lpβ
2p2∂
∂
pi+
lpβ
3pipj
∂
∂
pj+
lpβ
4i jkpj
∂
∂
pk.
(15)Wefindthecommutatorsofthisgeneratorwith pμandthecommutatorsofthisgeneratorwithMi(thegeneratorsoftherotations).We alsofindthecommutatorsofthisgeneratorwithitself.Thismodifiedalgebraisgiveninthefollowing,
[ ˜
Ni,
pj] = (
p0+
lpβ
1p20+
lpβ
2p2)δ
i j+
lpβ
3pipj−
lpβ
4i jkpk
,
(16)[ ˜
Ni,
p0] =
pi+
lpβ
0p0pi.
(17)Theothercommutatorsare
[
Mi,
N˜
j] =
i jkN˜
k−
lpβ
4p0i jkMk
,
(18)[ ˜
Ni,
N˜
j] = −
i jkMk−
lp(β
0+
2β
1+
3β
2− β
3)
i jkMk−
2lpβ
4p0i jkN
˜
K,
(19) andtherotationsofmomentumswhichremainunmodifiedare[
Mi,
Mj] =
i jkMk,
(20)[
Mi,
pj] =
i jkpk, [
Mi,
p0] =
0.
(21) Thealgebraicstructureofourextendedκ
-Poincarégroupisimportantforstudyingsymmetriesofthisgroup.3.3. Differentialequationsandfinite-boosttransformations
From Eq. (14), we find differentialequations which governthe evolution of energyand momentums in momentum space, andwe solvethedifferentialequationsforthe p0 and p1 componentstogetherandforthe p2 and p3 componentstogether.Forthe p0 and p1 componentswesolvethesecondorderequationswhicharemoresimple,butforthep2andp3componentsitisbesttofindthesolutions ofthefirstorderdifferentialequations.Forthe p0 andp1componentsthefirstorderdifferentialequationsare
⎧ ⎨
⎩
dp0
dξ
=
p1+
lpβ
0p0p1,
dp1
dξ
=
p0+
lpβ
1p20+
lpβ
2p2+
lpβ
3p21.
(22)
Byintroducingtheconstants
a0
= β
0+ β
2,
a1= β
0+ β
2+ β
3,
and a3=
2β
0+
2β
1+
2β
2+ β
3,
(23) weobtainthesecondorderdifferentialequations,⎧ ⎪
⎨
⎪ ⎩
d2p0
dξ2
=
p0+
lpa0p20+
lpa1(
dpdξ0)
2+ β
2(
p22+
p23),
d2p1
dξ2
=
p1+
lpa3p1dp1 dξ.
(24)
Forthep2andp3componentsthedifferentialequationsare
⎧ ⎨
⎩
dp2
dξ
=
lpβ
3p1p2−
lpβ
4p0p3,
dp3
dξ
=
lpβ
3p1p3+
lpβ
4p0p2.
(25)
SolutionsofthedifferentialequationsinEq. (24) andEq. (25) giveusthefinite-boostDSRtransformationsforallordersoftherapidity parameterξ,butonlyinthefirstorderofthePlancklength.SolutionsofthesedifferentialequationsaregiveninAppendixA.1.
Thefinite-boosttransformationstothefirstorderinlp are
⎧ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎨
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎩
p0
=
p0coshξ +
p1sinhξ
+
lp( α
1p20+ α
2p21)
sinh2ξ +
lp( α
2p20+ α
1p21)
cosh2ξ
+
2lp( α
1+ α
2)
p0p1sinhξ
coshξ −
lp( α
2p20+ α
1p21)
coshξ −
lpα
3p0p1sinhξ +
lpβ
2(
p22+
p23)(
coshξ −
1),
p1
=
p1coshξ +
p0sinhξ
+
lpα
3p0p1sinh2ξ +
lpα
3p0p1cosh2ξ
+
lpα
3(
p20+
p21)
sinhξ
coshξ −
lpα
3p0p1coshξ −
lp( α
2p20+ α
1p21)
sinhξ +
lpβ
2(
p22+
p23)
sinhξ,
p2
=
p2+
lp(β
3p1p2− β
4p0p3)
sinhξ +
lp(β
3p0p2− β
4p1p3)(
coshξ −
1),
p3=
p3+
lp(β
3p1p3+ β
4p0p2)
sinhξ +
lp(β
3p0p3+ β
4p1p2)(
coshξ −
1),
(26)
where
α
1≡ β
0+
2β
1− β
2− β
33
, α
2≡ β
0− β
1+
2β
2+
2β
33
,
andα
3≡ β
0+
2β
1+
2β
2+
2β
33
,
(27)forconvenience.Thesetransformationsarefinite-boostDSRtransformationstofirstorderinlpbutforallordersofrapidityξ.Weconfirm theyarethesameasthetransformationsin[6],whichhavebeenobtainedusingcommutators.
3.4.Dispersionrelationanditsclassifications
Taking pμ=(p0,p)=(m,0) fortheinitial pμ inthe transformationsin Eq. (26), we obtain generalexpressions forthe coshξ and sinhξ as
cosh
ξ =
p0−
lpα
1p2−
lpα
2p0m
,
sinhξ = |
p| −
lpα
3p0|
p|
m
.
(28)Here,mistherestmassoftheparticle.Usingcosh2ξ−sinh2ξ=1,wefindthecorrespondingdispersionrelationforthetransformations ofEq. (26) tofirstorderinlp as
p20
−
p2−
2lpα
2p30+
2lp( α
3− α
1)
p0p2=
m2.
(29)Expressingthisdispersionrelationintermsofβi yields p20
−
p2−
2lpβ
0− β
1+
2β
2+
2β
33
p30
+
2lp(β
2+ β
3)
p0p2=
m2.
(30)Thus,ifweknowtheinfinitesimaltransformationsforeverygivenDSRinfirstorderofPlancklength asinEq. (14) wecanreadtheβ0, ...,β4 parametersfromthem.Thenwecancompute
α
1 toα
3 parametersinEq. (27).ByputtingtheminEq. (26),we canconstructthe correspondingfinite-boost transformationsforthe giveninfinitesimal transformations.Moreover, thecorresponding modified dispersion relationcanbe foundfromEq. (29).Thisapproachhighlightsthefactthatan infinitenumberofthefinite-boostDSRtransformationsin firstorderofPlancklengththeoriesarepossible[9].Wecanclassifythefinite-boostDSRtransformationsinfirstorderwithrespecttothedifferenttypesofmodifieddispersionrelations.
In first order of lp, the modified dispersion relation is given by Eq. (29) which contains the p0p21 andthe p30 additional terms. The dispersionrelationforMS-DSRinEq. (38) containsthesetwoadditionaltermsaswell,whichwe regardasafirsttype ofclassification.
ThedispersionrelationforAC-DSRinEq. (41) tothefirstorderinlpcontainsonlythecouplingp0p21additionalterm,whichisthesecond type.Thethirdtypeoftheclassificationcontainsonlythep30additionalterm.Thefourthtype,whichisourfinaltypeofclassificationhas noadditionaltermbutthetransformationsaredifferentfromLorentz.
3.5.CompositionofboostsandWignerrotation
FortestingthegrouppropertiesoftheboostsinEq. (26),wecheckthatthecompositionoftwoboostsisanewdifferentboostsimilar toLorentz.The compositionofboosts fortwo parallelboosts shouldbe anewboost,andfortwo perpendicularboosts weshouldhave anewboostwitharotation(Wignerrotation).Thesepropertieshavebeencheckedin[6] forthegeneralizedboostinEq. (13).Here,we checkandconfirmindetailsomeoftheseproperties.
Firstwe checkcombinationpropertyfortheinfinitesimal transformations Eq. (14).We assumeanother infinitesimaltransformations likeEq. (14),butfrompμto pμwithrapidityparameterξinthesamedirectionwhichisgivenby
⎧ ⎪
⎪ ⎪
⎪ ⎪
⎨
⎪ ⎪
⎪ ⎪
⎪ ⎩
p0
=
p0+
p1ξ
+
lpβ
0p0p1ξ
,
p1
=
p1+
p0ξ
+
lpβ
1p02ξ
+
lpβ
2p2ξ
+
lpβ
3p12ξ
,
p2=
p2+
lpβ
3p1p2ξ
−
lpβ
4p0p3ξ
,
p3
=
p3+
lpβ
3p1p3ξ
+
lpβ
4p0p2ξ
.
(31)
CombinationsoftheinfinitesimaltransformationsinEq. (14) andEq. (31) willgiveusothersimilarinfinitesimaltransformationsfrom pμtopμinthesamedirectionwithrapidityparameterξ,whichisthesumoftworapidityparameters,
ξ
= ξ + ξ
.
(32)In the finite-boosts transformations case, as given by Eq. (26), we have expressions that include coshξ, sinhξ, cosh2ξ, sinh2ξ, and coshξsinhξ,whichyieldthesametransformationsaftertwosuccessiveboostsinthesamedirectionandtherapidityparameterssatisfy Eq. (32) asintheinfinitesimaltransformationscase.
Fortwoperpendicular boosts,asmentioned,wehaveanewboostandarotation.Inspecialrelativitythisrotationisthewell-known WignerrotationθW asdiscussedin[21,22] andisgivenby
tanh
θ
W= − γ γ
ξ ξ
γ + γ
.
(33)InDSRtheories,therotationpartofthePoincarégroup isunmodifiedandonlytheboosts arechanged.Thereforenocorrection willbe addedtotheWignerrotation[6].
Theinteresting consequencesoftheWignerrotationinspecialrelativityhavebeenstudiedbeforein[21–23].Animportanteffectis relatedtothespinofacompositesystemsuchasaprotonwhichisacompositesystemofquarks.Thesumofthespinsforacomposite systemcan violateLorentzinvariance.Infact,spinisrelatedtothePoincarégroup, or,to the
κ
-Poincaré groupandits extensions,such asthe extension insubsection 3.2. Thespin S ofa moving particle withmassm canbe defined bytransforming its Pauli-Lubinski 4- vector wμ=(1/2)νρσ μJρσpν toitsrestframeby arotationless DSRboost D(p),where D(p)p=(m,0),and(0,S)=D(p)w/m.Under anarbitraryDSRtransformation˜,thespinandmomentumofaparticlewillbetransformedvia
S
=
Rw( , ˜
p)
S,
p= ˜
p,
(34)where
Rw
( , ˜
p) =
D(
p) ˜
D−1(
p),
(35)istheWignerrotation[23].ByusingEq. (35),wecanshowthattheWignerrotationforDSRisequaltotheLorentzcase.
Inthe primedmoving frame,the protonisboostedwitha rotationless DSRtransformationalong its spin direction.Eachquark spin willbeaffectedbyaWignerrotation,andtheserotationscanchangethevectorspinsumofthequarksandantiquarks.Thischangingof thespinsumhasconsequencesforthepartonmodeloftheproton.However,theprotonspininthemovingframewillbethesameasin therestframe[23].SincetheWignerrotationinDSRisthesameasthespecialrelativisticcase,therearenonewconsequences(i.e.no newkindsofspin)oftheWignerrotationsbesidestheusualspecialrelativisticeffects.
ThegroupoftheDSRtheoriesisthe
κ
-Poincarégroup[7–9],andourextensionofthisgroup inEq. (13) infirstorderPlancklength, is an extension of theκ
-Poincaré symmetry. The commutators betweengenerators for every two boosts in one direction, or in two differentdirectionssatisfyEq. (19),whichshowsthatcombinationsoftwoboosts willgive anotherboost,oraboostandarotation.We donotcheckthesepropertiesinfulldetail;however,theycanbetestedwiththetransformationsinEq. (26) bylongbutstraightforward calculations.We have testedthe group property and findthe generator in Eq. (13) satisfies an extension ofthe
κ
-Poincaré algebra asgiven in subsection 3.2. This is assurance that the finite-boost transformations inEq. (26) arenot just reparameterizations of the compact Lie groupsasdiscussedin[30].4. Specialexamplesofthefinite-boostDSRtransformations
Inthissection,weprovidesomeexamplesofthefinite-boostDSRtransformationsinfirstorderofPlancklengthbyfindingfinite-boost transformationsandtheir correspondingdispersion relationstofirstorderinlp fromthegeneralformalism.The firsttwoexamples are thewell-knownMSandAC-DSRtheories,whichhelpconfirmthevalidityoftheformalism.Thelastfinaltwoexamplesareofparticular importance,becausetheyhavespecialformdispersionrelations.
4.1. MS-DSR
FortheMS-DSRwehave
β
0= −
1, β
1=
0, β
2=
0, β
3= −
1,
andβ
4=
0,
(36)andwecancomputetheconstants
α
1 toα
3 fromtherelationsinEq. (27) asα
1=
0, α
2= −
1,
andα
3= −
1.
(37)ByputtingthesevaluesinEq. (29) wefindthedispersionrelationoftheMS-DSRtothefirstorderinlp,
p20
−
p2+
2lpp30−
2lpp0p2=
m2.
(38)TransformationsforthisDSRinfirstorderisgiveninAppendixA.2.
4.2.AC-DSR
FortheAC-DSRwehave
β
0=
0, β
1= −
1, β
2=
12
, β
3= −
1,
andβ
4=
0,
(39)andwecancomputetheconstants
α
1 toα
3byuseoftherelationsinEq. (27) asα
1= −
12
, α
2=
0,
andα
3= −
1.
(40)Byusingthesevalues,wefinddispersionrelationoftheAC-DSRinthefirstorderoflp as
p20
−
p2−
lpp0p2=
m2,
(41)andtransformationsforthisDSRisgiveninAppendixA.3.
4.3.Dispersionrelationwiththecubicterm
Usingtheconditions
α
1=α
3,andα
2=0 inEq. (29) onefindsamodifieddispersionrelationwithacubicterm,p20
−
p2−
2lpα
2p30=
m2.
(42)TransformationsforthistypeofDSRisgiveninAppendixA.4.
4.4.Specialrelativisticdispersionrelation
Thefinalinteresting exampleutilizes
α
1=α
3,andα
2=0 inEq. (29). Theseconditionsleadusto theunmodifiedspecialrelativistic dispersionrelationp20
−
p2=
m2.
(43)Intermsoftheβiparameterstheseconditionswillbe
β
2= −β
3and, β
0= β
1,
(44)whichisvalidformanygeneralDSRtheoriesinfirstorderofPlancklength.ByputtingtheconditionsofEq. (44) intoEq. (27) wefind
α
1=
B, α
2=
0, α
3=
B.
(45)Here,wehavetakenB≡β0,whichisthefreeparameter.Therealsotwootherfreeparametersβ2 andβ4,whicharetakentobezerofor thefollowingtransformations.Byusingthevaluesof
α
1 toα
3parametersinEq. (26),wefindthefollowingtransformations⎧ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎨
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎩
p0
=
p0coshξ +
p1sinhξ +
lpB(
p0sinhξ +
p1coshξ )
2−
p21coshξ −
p0p1sinhξ ,
p1
=
p1coshξ +
p0sinhξ +
lpBp0p1
(
sinh2ξ +
cosh2ξ )
+(
p20+
p21)
sinhξ
coshξ −
p0p1coshξ −
p21sinhξ ,
p2=
p2,
p3
=
p3.
(46)
Thesetransformationsarethefinite-boostDSRtransformationsinthefirstorderofPlancklengthlp,withanunmodifiedspecialrelativistic dispersion relation.Despite this, they are not the usual Lorentz transformations. It is surprising that the special relativisticdispersion relationaccompaniesthesenon-Lorentztransformations.
4.5.Dispersionrelationfamily
ThetransformationsEq. (46) whichcorrespondstothedispersionrelationforthespecialrelativityarenotunique.InconditionEq. (44), therearethree freeparameters:β0,β2,andβ4.Asecond exampleofthetransformationswhich havethespecialrelativisticdispersion relation in the first order of the Planck length is found by taking β4=1 instead while keeping β0=B and β2=0 as before. These transformationsaregiveninAppendixA.5.
Werefertothesetransformationsasthesecondfinite-boostDSRtransformationswithspecialrelativisticdispersionrelation.Ofcourse, forotherchoicesofthefreeparametersonecanfindothertransformations.Thus,onehasafamilyoftransformationsoftheDSRtheories infirstorder.Forthedispersionrelationwithacubicterm,onehasanothersimilarfamily.
5. ObservationalconsequencesofthefirstorderDSR
TheproposalforobservationofquantumgravityeffectsingammarayburstswasgiveninitiallybyAmelino-Cameliaandhiscolleagues in[24].Theyassumedadeformeddispersionrelationforthephotonsintheformof
p21
=
E21
+
f EEQ G
,
(47)where EQ G isaneffectivequantum gravityscale,and f isamodeldependent functionof E/EQ G.One canfindan upperlimit onthe scalewithobservations anditmaybe differentfromthe Planckenergy M1.22×1019GeV.This dispersionrelation tofirst orderin E/EQ G is
p21
=
E21
+ η
EEQ G
,
(48)where
η
issomenumber.Theproposalisthatthevelocityofthegammarayburstsshouldbedifferentfromc,whichisgivenbyv
= ∂
E∂
p1=
c1
− η
EEQ G
.
(49)Moreover,asignalwhichiscomingfromdistanceD withenergy E willbereceivedwithtimedelay
tL I V
= η
EEQ G D
c
,
(50)different than the ordinary casewith the speed c. A linear correction to the velocity ofhigh energy photons like Eq. (49) has been suggestedinmanypapersfortestingquantumgravityeffects[25–32].
Wefindthe velocityofphotonsandtime delayforthe photonsin ourformalismforfinite-boost DSRtransformations.Byusing the dispersionrelationinEq. (29),thevelocityofthephotonswillbe
v
c1
+ ( α
1+ α
2− α
3)
lpE,
(51)andthetimedelaywillbe
tL I V
= ( α
1+ α
2− α
3)
lpE Dc
.
(52)Ascanbeseen,the
η
parameterinEq. (48) hasbeenreplacedby(α
1+α
2−α
3)inthefinite-boostDSRtransformations.Tobeclear,the distanceD isacosmologicaldistancethatdependsontheredshift z,matterdensitym,cosmologicalconstantdensity,andHubble constant H0.ThetimedelayistL I V
= ( α
1+ α
2− α
3)
lpE H0 z 0(
1+
z)
dz 1+
m(
1+
z)
3+
.
(53)Here,wetakem=0.3,=0.7, H0=72 km s−1Mpc−1 forthecosmologicalparameters[31].
The time delay in Eq. (52) and Eq. (53) is the only Lorentz invariance violating (LIV) term fortime delay betweena high energy astrophysicalphotonsourceandreceiver.Thegeneralexpressionfortimedelayis
t
=
tL I V+
tint+
tspe+
tD M+
tgrav,
(54)wheretint isthetimedelay duetothefact thatphotonswithhighandlowenergiesdonot leavethesourcesimultaneously. tspe is aresult ofthespecialrelativistic effectsifthephotonshave non-zeromassandtD M isdueto dispersionby the line-of-sightof free electroncontent,whichisnon-negligibleespeciallyforlowenergyphotons.Finally,tgrav isduetothegravitationalpotentialcontribution alongthephotonspropagationpathsforpossibleviolationofEinstein’sequivalenceprinciple[32].
5.1. SpecialrelativisticdispersionrelationandnullresultsinobservationsforQGeffects
For
α
1=α
3,andα
2=0,whichgivesthespecialrelativisticdispersioninEq. (43),thevelocityofthephotonswillbetheunchanged specialrelativisticc,andthetimedelay inEq. (52) willbe zero.Therefore,thetransformationsinEq. (46) (andalsoall transformations inthespecialrelativisticdispersionrelationfamilysuchasEq. (A.10))aredifferentfromtheLorentztransformationsinfirstorderofthe Plancklength,butthevelocityofthephotonsarethesameasspecialrelativity.Thesephotonswillbereceivedinthesametimeasinthe usualspecialrelativisticcase. Simplyput,ashasbeendonee.g. in[25,26],measuringthevelocity andtimedelay ofGRBphotonsisnot sufficientforinvestigatingquantumgravityeffectstofirstorder.6. Conclusion
FindinggeneralexpressionsforDSR tofirstorderis aninteresting andimportantmatterboth mathematicallyandphysically.Inthis paperwe startedwiththe boostfromthe
κ
-Poincarégroup, addingfree parameters βi,usefulforgeneralization. The newgeneralized boostgivesfinite-boosttransformationswhichallowustoobtainthegeneralexpressionsforDSRtheoriesinthefirstorderofthePlanck length.Afterfinding thedispersionrelationforthesegeneralfinite-boosttransformations,we classifytheresultingDSRtheoriesto four types.Thewell-knownMSandACtheoriesarethefirsttwotypesofthisclassification.ThesecondtwotypesaretheDSRtheorieswitha cubicterminthedispersionrelationandtheDSRtheorieswithspecialrelativisticdispersion.ThereareimportantobservationalconsequencesofthesedifferentDSRtypes.Perhapsthemostnotableconsequenceistheexpectation thatthe speed oflight should bedifferent fromtheusual speed c. Infact, thespeed ofhighenergyphotons andlow energyphotons shouldbedifferentanddependsontheirenergycontent.Inthisstudy,wehavefoundaninterestingpossibilitywhereDSRtheorieswith transformationsdifferentfromLorentztransformations,canalsocorrespondtoanunmodifiedspecialrelativisticdispersionrelation.Thus, anysearch forquantumgravity effectsinastrophysicalandcosmologicalobservationswhichgivea specialrelativisticresultcanalsobe interpretedasconsistentwithDSR. Anullresultisnotenough forinvestigatingcorresponding quantumgravity effectstofirst orderin Plancklength.
Therearetwopossibleextensionsandapplicationsofthesefinite-boostDSRtransformations.First,thesolutionsoftheextendedsecond orderdifferentialequationscanbe investigatedforabetter understandingoftheir dynamicsandbehavior[33,34].Second,the possible extensionsofthePoincaréor
κ
-Poincarégroup maybefruitfulforinvestigations(e.g.dualDSR theory[35,36])ofquantumgravitational effects.Declarationofcompetinginterest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influencetheworkreportedinthispaper.
Acknowledgements
Fundingby theFY2018-SGP-1-STMMCompetitiveResearchGrantNo.090118FD5350atNazarbayevUniversity isappreciated,aswell asthestate-targetedprogram“CenterofExcellenceforFundamentalandAppliedPhysics”(BR05236454)bytheMinistryofEducationand ScienceoftheRepublicofKazakhstan.
Appendix A. Appendixes
A.1.Findingsolutionsofthedifferentialequations
Forobtaining thefinite-boosttransformations,firstwe solvethedifferentialequationsforthe p0 and p1 components.Intheprimed inertialsystem,theformofEqs. (24) arethesame.Weintroducetheperturbativesolutionsofthesedifferentialequationsinfirstorderof thePlancklengthlp as
p0=
p0coshξ +
p1sinhξ +
lpA(ξ ),
p1
=
p1coshξ +
p0sinhξ +
lpD(ξ ).
(A.1)Theunknownfunction A(ξ )shouldalsosatisfythefollowingadditionaldifferentialequation
d2A
d
ξ
2−
A= (
a0p21+
a1p20)
sinh2ξ + (
a0p20+
a1p21)
cosh2ξ +
2(
a0+
a1)
p0p1sinhξ
coshξ.
(A.2) ThegeneralsolutionforA(ξ )isA
(ξ ) = λ
1sinh2ξ + λ
2cosh2ξ + λ
3sinhξ
coshξ + λ
4coshξ + λ
5sinhξ − β
2(
p22+
p23),
(A.3)where λ1=2a03−a1p20+ 2a13−a0p21, λ2= 2a13−a0p20+ 2a03−a1p21, and λ3= 2(a03+a1)p0p1. To find λ4, and λ5 multipliers we use the initial conditions p0(0)=p0, and dpdξ0(0)=p1+lpβ0p0p1, which yield λ4= −λ2+β2(p22+p23), andλ5=β0p0p1−λ3. The other unknown function D(ξ )can befound inasimilarway.Byputting A(ξ )and D(ξ ) inEq. (A.1) wecanfindthetransformationsforthe p0 andp1 components.
Forthep2 andp3 components,wetake
p2=
p2+
lpA1sinhξ +
lpA2coshξ +
A3,
p3
=
p3+
lpA4sinhξ +
lpA5coshξ +
A6,
(A.4)where A1,..., A6,areunknown functionsof p0,...,p3.Byputtingthesesolutionsinthedifferentialequationsforp2 andp3 inEq. (25), wecanfindtheunknownfunctionsA1,..., A6,whichleadustothetransformationsforthep2 andp3 components.