Riemann-Hypothesis-and-Short-Distance-Fermionic-Gr

arXiv:math-ph/0504035v110Apr2005RiemannHypothesisandShortDistanceFermionicGreen’sFunction[e-mail]Green’sfunctionofatwodimensionalfermionwithamodifieddispersionrelationandshortdistanceparameteraisgivenbytheLerchzetafunction.TheGreen’sfunctionisdefinedonacylinderofradiusRandweshowthattheconditionR=ayieldstheRiemannzetafunctionasaquantumtransitionamplitudeforthefermion.WeformulatetheRiemannhypothesisphysicallyasanonzeroconditiononthetransitionamplitudebe-tweentwospecialstatesassociatedwiththepointoforiginandapointhalfwayaroundthecylindereachofwhicharefixedpointsofaZ2transformation.Bystudyingpartialsumsweshowthatthatthetransitionamplitudeformulationisanalogoustoneutrinomixinginalowdimensionalcontext.Wealsoderivethethermalpartitionfunctionofthefermionictheoryandthethermaldivergenceattemperature1/a.Inanalternativeharmonicoscillatorformalismwediscusstherelationtothefermionicdescriptionoftwodimensionalstringtheoryandmatrixmodels.FinallywederivevariousrepresentationsoftheGreen’sfunc-tionusingenergymomentumintegrals,pointparticlepathintegrals,andstringpropagators.1IntroductionTheRiemannHypothesis(thattheRiemannzetafunctionhasnontrivialzerosonlywhentherealpartofitsargumentisonehalf)embodiesoneofthedeepestmysteriesinmathematics.Mathematicsandphysicshavesimilaroriginsandmanyseeminglyabstractresultsinmathematicsturnouttohavephysicalrealizations.BecauseofthedepthoftheRiemannhypothesisonemayhopenotonlythataphysicalrealizationofthehypothesismayleadtoitssolution,butalsothatitsformulationinphysicaltermsmayuncoverphysicalideasperhapsrelatedtoshortdistancephysicsandstringtheory.Thereareseveralexamplesofaphysicalrealizationofmathematicaldevelopmentsleadingtophysicaladvances.TherelationoftheAtiyah-Singerindex1theoremtoanomalies,instantons,andchiralfermionsisoneexample,therelationofChern-Simonsgaugetheoryandknottheorytothreemanifoldtheoryisanother,alsoonehastheFrenkel-KacvertexoperatorconstructionofinfiniteLiealgebrasandtherelationtotheheteroticstring.ThusonecanexpectthatthephysicalrealizationofmathematicalproblemsliketheRiemannhypothesishasbenefitforbothmathematicsandphysics.ThesearchforaphysicalrealizationoftheRiemannHypothesishasalonghis-torygoingbacktoHilbertandPolyawholookedforaquantumhermitianoperatorwitheigenvaluesattheRiemannzeros.MorerecentapproachesincludetherelationoftheRiemannzerostoaJostfunctioninscatteringtheory[1],therelationofthezetafunctiontoquantumchaos[2],andtherelationoftheRiemannhypothesistononcommutativegeometry[3].MoreinformationaboutrecentapproachestotheRiemannhypothesisaredescribedin[4].InthispaperthiswegiveanewphysicalrealizationoftheRiemannhypothesisintermsofa1+1dimensionalfermionictheorywhoseshortdistancedynamicsaremodifiedinadefiniteway.Theshortdistancedynamicsintroduceanewparameterofdimensionlengthorinverseenergy.WithinthistheorywederivethepropagatororGreen’sfunctionwhichgivesthetransitionamplitudetoobservethefermionatanypointinspace-time(x,t)giventhatitsinitialpositionandtime.WhenthespatialcoordinatexlivetakesitsvaluesonacircleS1ofcircumferenceL=2πRweshowthatthisGreen’sfunctionisproportionaltotheLerchzetafunctionwitharguments(x2πR,ita,Ra).OnlywhentheratioL2πa=Ra=1,thatiswhentheradiusoftheuniverseequalsthenewshortdistancescale,doweobtaintheRiemannzetafunction,sointhissensetheRiemannzetafunctionprobesshortdistancebehavioroftheGreen’sfunction.Byconsideringinitialstatesofthefermionicparticlethatarestatisticallydistributed,(essentiallyBoltzmanndistributedwithrespecttoenergyandamixingparameterσ)weshowthatthetransitionamplitudeismodifiedbytheLerchzetafunctionwithparameters(x2πR,ita+σa,Ra).TheRiemannhypothesisisformulatedphysicallyassayingthatthetransitionprobabilitytoapositionL2halfwayacrossthe1+1dimensionaluniverseisstrictlynonzeroforalltimesandvaluesofthemixingparameterintherange12σa1.Thispaperisorganizedasfollows.Insection2wegiveareviewofcertainDirich-letseriesandzetafunctionsthatwilloccurinthispaper.Insection3wediscussthetransitionamplitudesandGreen’sfunctionofthe1+1dimensionalfermionictheoryinthefirstquantizedandsecondquantizedfieldtheorypointofviewandexplaintherelationtothezetafunction.Insection4weintroducethenotionofaBoltzmanndistributedinitialstateandintroducethemixingparameterσ.WeformulatetheRiemannhypothesisassayingthatthetransitionrateisneverzeroforthetransitionfrominitialstateparameterizedbyσandx=0toafinalposition2stateatx=L2=πR.Wediscussasimplifiedanalysisintermsofpartialsumsandshowhowthebehaviorofthetransitionprobabilitiesisreflectedinthephenomenaofneutrinomixing.Wealsodiscusstheσparameterfromseveralpointsofviewincludingtheintegraloverthetafunctions,targetspaceduality,andspace-timere-flections.Insection5wediscussthestatisticalmechanicsofthenonstandard1+1dimensionalfermionictheoryanditsrelationtomultiplicativenumbertheoryandthermaldivergence.Insection6wediscussanalternateformulationintermsanon-standardoscillatoranddiscusstherelationtothefermionicformoftwodimensionalstringtheoryandmatrixmodels.Insection7wediscusstherelation