Quantum Knots and Riemann Hypothesis

1、arXiv:math/0603275v1[math.GM]12Mar2006QuantumKnotsandRiemannHypothesisSzeKuiNgDepartmentofMathematics,HongKongBaptistUniversity,HongKongszekuing@yahoo.com.hkAbstractInthispaperweproposeaquantumgaugesystemfromwhichweconstructgeneralizedWilsonloopswhichwillbeasquantumknots.Fromquantumknotswegiveaclassiﬁcationtableofknotswhereknotsareone-to-oneassignedwithanintegersuchthatprimeknotsarebijectivelyassignedwithprimenumbersandtheprimenumber2correspondstothetrefoilknot.Thenbyconsideringthequantumknotsa。

2、speriodicorbitsofthequantumsystemandbytheidentityofknotswithintegersandanapproachwhichissimilartothequantumchaosapproachofBerryandKeatingwederiveatraceformulawhichmaybecalledthevonMangoldt-Selberg-Gutzwillertraceformula.FromthistraceformulawethengiveaproofoftheRiemannHypothesis.ForourproofoftheRiemannHypothesisweshowthattheHilbert-Polyaconjectureholdsthatthereisaself-adjointoperatorforthenontrivialzerosoftheRiemannzetafunctionandthisoperatoristheVirasoroenergyoperatorwithcentralchargec=12.Ourapp。

3、roachforprovingtheRiemannHypothesiscanalsobeextendedtoprovetheExtendedRiemannHypothesis.WealsoinvestigatetherelationofourapproachforprovingtheRiemannHypothesiswiththeRandomMatrixTheoryforL-functions.MathematicsSubjectClassiﬁcation:57M27,11M26,11N05,11P32.1IntroductionItiswellknownthattheJonespolynomialasaknotinvariantcanbederivedfromaquantumChern-Simongaugeﬁeldtheory[1][2].Inspiredbythisworkinthispaperweshallalsoproposeaquantumgaugemodel.InthisquantummodelwegeneralizethewayofdeﬁningWilsonloopsto。

4、constructgeneralizedWilsonloopswhichwillbeasquantumknots.Fromquantumknotswegiveaclassiﬁcationtableofknotswhereknotsareone-to-oneassignedwithanintegersuchthatprimeknotsarebijectivelyassignedwithprimenumbersandtheprimenumber2correspondstothetrefoilknot.ThenbyconsideringthequantumknotsasperiodicorbitsofthequantummodelandbytheidentityofknotswithintegersandanapproachwhichissimilartothequantumchaosapproachofBerryandKeatingwederiveatraceformulawhichmaybecalledthevonMangoldt-Selberg-Gutzwillertraceformu。

5、la.FromthistraceformulawethengiveaproofoftheRiemannHypothesis[3]-[17].FromthequantumgaugemodelweﬁrstdeﬁnetheclassicalWilsonloopandWilsonline.ThenfromthequantumgaugemodelwederiveadeﬁnitionforthegeneratoroftheWilsonline.ThenwederivetwoquantumKnizhnik-Zamolodchikov(KZ)equationswhicharedualtoeachotherfortheproductofquantumWilsonlines.ThisquantumKZequationindualformmayberegardedasaquantumYang-MillequationasanalogoustotheclassicalYang-MillequationderivedfromtheclassicalYang-MilltheorysincethisquantumK。

6、Zequationisasthebasicquantumequationderivedfromthequantumgaugemodel.SolutionsofthisquantumYang-MillequationarethenusedtoconstructgeneralizedWilsonloopswhichareasquantumknots(ThesequantumknotsmayberegardedassolitonsassimilartotheinstantonsoftheclassicalYang-Millequation).InderivingthisquantumKZequationweﬁrstderiveaconformalﬁeldtheoryconsistingoftheKac-MoodyalgebraandtheVirasoroenergyoperatorandVirasoroalgebra.Thenfromthequantumknotswederiveaknotinvariant.Fromthisknotinvariantwegivetheclassiﬁcatio。

7、ntableofknots.1ThenthequantumknotsastheperiodicorbitsofthequantumgaugesystemandtheidentityofprimeknotswithprimenumbersareasthetwobasicingredientsforprovingtheRiemannHypothesis.ForourproofoftheRiemannHypothesisweshowthattheHilbert-Polyaconjectureholdsthatthereisaself-adjointoperatorforthenontrivialzerosoftheRiemannzetafunctionandthisoperatoristheVirasoroenergyoperatorwithcentralchargec=12[18]-[19].OurapproachforprovingtheRiemannHypothesiscanalsobeextendedtoprovetheExtendedRiemannHypothesis.Wealso。

8、investigatetherelationofourapproachforprovingtheRiemannHypothesiswiththeRandomMatrixTheoryforL-functions[20]-[30].Thispaperisorganizedasfollows.Insection2wegiveabriefdescriptionofaquantumgaugemodelofelectrodynamicsanditsnonabeliangeneralization.InthispaperweshallconsideranonabeliangeneralizationwithaSU(2)gaugesymmetry.Withthisquantummodelinsection3weintroducethedeﬁnitionofclassicalWilsonloopandWilsonline.Insection4wederivethedeﬁntionofthegeneratoroftheWilsonline.Fromthisdeﬁnitioninsection4and5we。

9、deriveaconformalﬁeldtheorywhichincludestheVirasoloenegryoperatorandVirasoloalgebra,theaﬃneKac-MoodyalgebraandthequantumKZequationindualform.Insection6wecomputethesolutionsofthequantumKZequationindualform.Insection7wecomputethequantumWilsonlines.Insection8werepresentthebraidingoftwopiecesofcurvesbydeﬁningthebraidingoftwoquantumWilsonlines.Bythisrepresentationinsection10wedeﬁnethegeneralizedWilsonloopwhichwillbeasaquantumknot.Insection9wecomputethequantumWilsonloop.Insection10wedeﬁnegeneralizedWil。

10、sonloopswhichwillbeshowntohavepropertiesofthecorrespondingknotdiagramandwillberegardedasquantumknots.Insection11wegivesomeexamplesofgeneralizedWilsonloopsandshowthattheyhavethepropertiesofthecorrespondingknotdiagramandthusmayberegardedasquantumknots.Insection12weshowthatthisgeneralizedWilsonloopisacompletecopyofthecorrespondingknotdiagramandthuswemaycallageneralizedWilsonloopasaquantumknot.FromquantumknotswehaveaknotinvariantoftheformTrR−mW(z,z)whereW(z,z)denotesaquantumWilsonloopandRisthebraidin。