Quantum Knots and Riemann Hypothesis

arXiv:math/0603275v1[math.GM]12Mar2006QuantumKnotsandRiemannHypothesisSzeKuiNgDepartmentofMathematics,HongKongBaptistUniversity,[e-mail]roposeaquantumgaugesystemfromwhichweconstructgeneralizedWilsonloopswhichwillbeasquantumknots.Fromquantumknotswegiveaclassificationtableofknotswhereknotsareone-to-oneassignedwithanintegersuchthatprimeknotsarebijectivelyassignedwithprimenumbersandtheprimenumber2correspondstothetrefoilknot.ThenbyconsideringthequantumknotsasperiodicorbitsofthequantumsystemandbytheidentityofknotswithintegersandanapproachwhichissimilartothequantumchaosapproachofBerryandKeatingwederiveatraceformulawhichmaybecalledthevonMangoldt-Selberg-Gutzwillertraceformula.FromthistraceformulawethengiveaproofoftheRiemannHypothesis.ForourproofoftheRiemannHypothesisweshowthattheHilbert-Polyaconjectureholdsthatthereisaself-adjointoperatorforthenontrivialzerosoftheRiemannzetafunctionandthisoperatoristheVirasoroenergyoperatorwithcentralchargec=12.OurapproachforprovingtheRiemannHypothesiscanalsobeextendedtoprovetheExtendedRiemannHypothesis.WealsoinvestigatetherelationofourapproachforprovingtheRiemannHypothesiswiththeRandomMatrixTheoryforL-functions.MathematicsSubjectClassification:57M27,11M26,11N05,11P32.1IntroductionItiswellknownthattheJonespolynomialasaknotinvariantcanbederivedfromaquantumChern-Simongaugefieldtheory[1][2].Inspiredbythisworkinthispaperweshallalsoproposeaquantumgaugemodel.InthisquantummodelwegeneralizethewayofdefiningWilsonloopstoconstructgeneralizedWilsonloopswhichwillbeasquantumknots.Fromquantumknotswegiveaclassificationtableofknotswhereknotsareone-to-oneassignedwithanintegersuchthatprimeknotsarebijectivelyassignedwithprimenumbersandtheprimenumber2correspondstothetrefoilknot.ThenbyconsideringthequantumknotsasperiodicorbitsofthequantummodelandbytheidentityofknotswithintegersandanapproachwhichissimilartothequantumchaosapproachofBerryandKeatingwederiveatraceformulawhichmaybecalledthevonMangoldt-Selberg-Gutzwillertraceformula.FromthistraceformulawethengiveaproofoftheRiemannHypothesis[3]-[17].FromthequantumgaugemodelwefirstdefinetheclassicalWilsonloopandWilsonline.ThenfromthequantumgaugemodelwederiveadefinitionforthegeneratoroftheWilsonline.ThenwederivetwoquantumKnizhnik-Zamolodchikov(KZ)equationswhicharedualtoeachotherfortheproductofquantumWilsonlines.ThisquantumKZequationindualformmayberegardedasaquantumYang-MillequationasanalogoustotheclassicalYang-MillequationderivedfromtheclassicalYang-MilltheorysincethisquantumKZequationisasthebasicquantumequationderivedfromthequantumgaugemodel.SolutionsofthisquantumYang-MillequationarethenusedtoconstructgeneralizedWilsonloopswhichareasquantumknots(ThesequantumknotsmayberegardedassolitonsassimilartotheinstantonsoftheclassicalYang-Millequation).InderivingthisquantumKZequationwefirstderiveaconformalfieldtheoryconsistingoftheKac-MoodyalgebraandtheVirasoroenergyoperatorandVirasoroalgebra.Thenfromthequantumknotswederiveaknotinvariant.Fromthisknotinvariantwegivetheclassificationtableofknots.1ThenthequantumknotsastheperiodicorbitsofthequantumgaugesystemandtheidentityofprimeknotswithprimenumbersareasthetwobasicingredientsforprovingtheRiemannHypothesis.ForourproofoftheRiemannHypothesisweshowthattheHilbert-Polyaconjectureholdsthatthereisaself-adjointoperatorforthenontrivialzerosoftheRiemannzetafunctionandthisoperatoristheVirasoroenergyoperatorwithcentralchargec=12[18]-[19].OurapproachforprovingtheRiemannHypothesiscanalsobeextendedtoprovetheExtendedRiemannHypothesis.WealsoinvestigatetherelationofourapproachforprovingtheRiemannHypothesiswiththeRandomMatrixTheoryforL-functions[20]-[30].Thispaperisorganizedasfollows.Insection2wegiveabriefdescriptionofaquantumgaugemodelofelectrodynamicsanditsnonabeliangeneralization.InthispaperweshallconsideranonabeliangeneralizationwithaSU(2)gaugesymmetry.Withthisquantummodelinsection3weintroducethedefinitionofclassicalWilsonloopandWilsonline.Insection4wederivethedefintionofthegeneratoroftheWilsonline.Fromthisdefinitioninsection4and5wederiveaconformalfieldtheorywhichincludestheVirasoloenegryoperatorandVirasoloalgebra,theaffineKac-MoodyalgebraandthequantumKZequationindualform.Insection6wecomputethesolutionsofthequantumKZequationindualform.Insection7wecomputethequantumWilsonlines.Insection8werepresentthebraidingoftwopiecesofcurvesbydefiningthebraidingoftwoquantumWilsonlines.Bythisrepresentationinsection10wedefinethegeneralizedWilsonloopwhichwillbeasaquantumknot.Insection9wecomputethequantumWilsonloop.Insection10wedefinegeneralizedWilsonloopswhichwillbeshowntohavepropertiesofthecorrespondingknotdiagramandwillberegardedasquantumknots.Insection11wegivesomeexamplesofgeneralizedWilsonloopsandshowthattheyhavethepropertiesofthecorrespondingknotdiagramandthusmayberegardedasquantumknots.Insection12weshowthatthisgeneralizedWilsonloopisacompletecopyofthecorrespondingknotdiagramandthuswemaycallageneralizedWilsonloopasaquantumknot.FromquantumknotswehaveaknotinvariantoftheformTrR−mW(z,z)whereW(z,z)denotesaquantumWilsonloopandRisthebraidin