7002 raM 1 1v8003070/ht-pe:hviXraBrankoDragovich∗
InstituteofPhysics
Pregrevica118,P.O.Box57,11001Belgrade,Serbia
Abstract
Weintroducenonlinearscalarfieldmodelsforopenandopen-closedstringswithspacetimederivativesencodedintheoperatorval-uedRiemannzetafunction.ThecorrespondingtwoLagrangiansarederivedinanadelicapproachstartingfromtheexactLagrangiansforeffectivefieldsofp-adictachyonstrings.Asaresulttachyonsareab-sentinthesemodels.Thesenewstringsweproposetocallzetastrings.Somebasicclassicalpropertiesofthezetastringsareobtainedandpresentedinthispaper.
1Introduction
Therearejusttwentyyearsfromthepublicationofthefirstpaperonthep-adicstring[1].Sofarp-adicstructureshavebeenobservednotonlyinstringtheorybutalsoinmanyothermodelsofmodernmathematicalphysics(forareviewoftheearlydaysdevelopments,seee.g.[2],[3]).
Oneofthegreatestachievementsinp-adicstringtheoryisaneffectivefielddescriptionofscalaropenandclosedp-adicstrings[4],[5].ThecorrespondingLagrangiansareverysimpleandexact.Theydescribenotonlyfour-pointscatteringamplitudesbutalsoallhigheronesatthetree-level.
Thisp-adicstringtheoryhasbeensignificantlypushedforwardwhenwasshown[6]thatitdescribestachyoncondensationandbranedescentrelations
simplerthanbyordinarybosonicstrings.Afterthatsuccess,manyaspectsofp-adicstringdynamicshavebeeninvestigatedandcomparedwithdynamicsofordinarystrings(see,e.g.[7]andreferencestherein).Noncommutativede-formationofp-adicstringworld-sheetwithaconstantB-fieldwasinvestigatedin[8](onp-adicnoncommutativityseealso[9]).Asystematicmathematicalstudyofspatiallyhomogeneoussolutionsofthenonlinearequationofmo-tionwasperformedin[10].Somepossiblecosmologicalimplicationsofp-adicstringtheoryhavebeenalsoinvestigated[11].Itwasrecentlyproposed[12]thatp-adicstringtheoriesprovidelatticediscretizationtotheworld-sheetofordinarystrings.Asaresultofthesedevelopmentsmanynontrivialfea-turesofordinarystringtheoryhavebeenreproducedfromthep-adiceffectiveaction.
Adelicapproachtothestringscatteringamplitudesisaveryusefulwaytoconnectp-adicandordinarycounterparts(see[2]asareview).More-over,iteliminatesunwantedprimenumberparameterpcontainedinp-adicamplitudesandalsocurestheproblemofp-adiccausalityviolation.Adelicgeneralizationofquantummechanicswasalsosuccessfullyformulated,andwasfoundaconnectionbetweenadelicvacuumstateoftheharmonicoscil-latorandtheRiemannzetafunction[13].Recently,aninterestingapproachtowardfoundationofafieldtheoryandcosmologybasedontheRiemannzetafunctionwasproposedin[14].AnadelicapproachwiththeRiemannzetafunctionisoneofthemotivationsforthispaper.
Thepresentpaperisaresultoftheattempttointegrateallp-adiceffectivefieldactionsintoonebosonicfieldtheory.Inthenexttwosectionsweexplorethecasesofopenandopen-closedbosonicstrings.
2Openscalarzetastring
Theexacttree-levelLagrangianforeffectivescalarfieldϕwhichdescribesopenp-adicstringtachyonis
L1
p=
1p−1
−2
ϕ+
1
timederivativesfollowsfromtheexpansion
p
−
2
lnp
−lnp=
k≥0
k!
k.
Theequationofmotionfor(1)is
p−
n2
=
1
2
n−1
Ln
n+1
φ
n+1
φ
n≥1
n
−
,(3)
wherecoefficientsCn=
andCn=1,
butitseemstobelessnatural.Thusweretainthestringcouplingconstantg.ToemphasizethatLagrangian(3)describesanewfield,whichtakesintoaccountallp-adicfields,weintroducednotationφinsteadofϕ.ThetermC1L1=0,becauseofitstwoequalpartsofoppositesign,butthesepartsgivecontributiontokineticandpotentialtermsofthetotalLagrangianL.AccordingtothefamousEulerproductformulaonecanwrite
g2
g2
n2
n≥1
n−
1−p−
ζ(s)=
1
n≥1
1−
p−s
,s=σ+iτ,σ>1.(4)
Employingusualexpansionforthelogarithmicfunctionanddefinition(4)wecanrewrite(3)intheform
L=−
12φζ
2
actsaspseudodifferentialoperatorinthefollowingway(seealso
k2ζ−
[14]):ζ
(2π)D
e
ixk
2
φ=
1
2
˜(k)dk=φ
φ
g
[φ+ln(1−φ)]=−2
1
n
,(8)
4
whereV(φ)≤0for−1<φ<1:itincreasesfromV(φ→−1)=−0,31
2
φ(t)=
1
2+ε
e
−ik0t
ζ
k2
0
1−φ(t)
.(9)
Intheweakfieldapproximation(|φ(t)|≪1)theaboveexpressionφ/(1−φ)≈φand(9)becomesalinearequationwhichcanbewrittenintheform
k2
0−ik0t˜(k0)dk0=0,2−ε)−1φ(10)ζe
R
whereθistheHeavisidefunction.Sinceζ02the
˜(k0)=0.Thisalsomeanstheabsenceequation(10)hassolutiononlyforφ
ofmass.
k2
3Openandclosedscalarzetastrings
TheexactLagrangianforthecoupledopenandclosedp-adictachyonsispresentedin[2]anditreads
1p−11p2−1g2
111
11
−1)
L′p
=+
−
2
ϕ+
4
2
(ϕ
p+1
−
ψ+
Mp(ϕ,ψ)+
1
1
L′=
n2
Mn(φ,θ)+n4
Nn(θ)
=
1
2
φn−
+1
θ
n(n−1)
nh2
n≥1
−
1
4
θ+
1
g2
−
1
2
φ+
1
2
(φn+1
n≥1
−1)
+
1
2+1
2θζ
2+1
θ
n
n(13)
forthecoupledzetastringsφandθ,whichareopenandclosed,respectively.Theequationsofmotionare
ζ
(2π)D
eixkζ−k22φn,(14)ζ
ixk
k2(2π)D
e
ζ
−
+1)
θ
n(n−1)
2(n2
φ=φ,ζ
Thecorrespondingpotentialis
1
n≥1
n(n−1)
V(φ,θ)=
n+1
θ
1h2
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