knot theory and quantum gravity

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1、UU-REL-93/1/9hep-th/9301028January9th1992KnotTheoryandQuantumGravityinLoopSpace:APrimerJorgePullinDepartmentofPhysics,UniversityofUtahSaltLakeCity,UT84112USAAbstractThesenotessummarizethelecturesdeliveredintheVMexicanSchoolofParticlePhysics,attheUni

2、versityofGuanajuato.WegiveasurveyoftheapplicationofAshtekar'svariablestothequantizationofGeneralRelativityinfourdimensionswithspecialemphasisontheapplicationoftechniquesofanalyticknottheorytothelooprepresentation.WediscusstherolethattheJonesPolynomia

3、lplaysasageneratorofnondegeneratequantumstatesofthegravitationaleld.1QuantumGravity:whyandhow?Iwishtothanktheorganizersforinvitingmetospeakhere.Thismaywellbeasignofourtimes,thatapersongenerallyperceivedasaGeneralRelativist"wouldbeinvitedtospeakataP

4、articlePhysicsSchool.Itjustre ectsthehigherdegreeofinterplaythesetwoeldshaveenjoyedoverthelastyears.Intheselectureswewillseemorereasonsforthisenhancedinterplay.WewillseeseveralnotionsfromGaugeTheories,asWilsonarXiv:hep-th/9301028v224Oct2000Loopsfori

5、nstance,playingacentralroleingravitation.AnevengreaterinterplaytakesplacewithTopologicalFieldTheories.WewillseetheimportantrolethattheChern-Simonsform,theJonesPolynomialandothernotionsofknottheoryseemtoplayinGeneralRelativity.ThequantizationofGeneral

6、Relativityisaproblemthathasdeedresolutionforthelastsixtyyears.Inspiteofthelongtimethathasbeeninvestedintryingtosolveit,webelievethatseveralpeopledonotnecessarilyfullyappreciatethereasonsofourfailureandthemagnitudeoftheproblem.Itisageneralperception{

7、especiallyamongparticlephysicists{thatGeneralRelativityisnonrenormalizable"andthatisthebasicproblemwiththetheory.Thisstatementismisleadinginthreeways:a)Thefactthatatheoryisnonrenormalizabledoesnotnecessarilymeanthatthetheoryhasanintrinsicproblemoris

8、bad"inanyway.ItmerelysaysthatperturbationToappearinProceedingsoftheVthMexicanSchoolofParticlesandFields",J.L.Lucio,editor,WorldScientic,Singapore(1993).1theorydoesnotapplytotheprobleminquestion.Aswewillseeinc)thereareactu-allygoodreasonstobelieve