knot theory and quantum gravity

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

2、eyoftheapplicationofAshtekar'svariablestothequantizationofGeneralRelativityinfourdimensionswithspecialemphasisontheapplicationoftechniquesofanalyticknottheorytothelooprepresentation.WediscusstherolethattheJonesPolynomialplaysasageneratorofnondegeneratequantumstatesofthegravitation

3、aleld.1QuantumGravity:whyandhow?Iwishtothanktheorganizersforinvitingmetospeakhere.Thismaywellbeasignofourtimes,thatapersongenerallyperceivedasaGeneralRelativist"wouldbeinvitedtospeakataParticlePhysicsSchool.Itjustre ectsthehigherdegreeofinterplaythesetwoeldshaveenjoyedoverthela

4、styears.Intheselectureswewillseemorereasonsforthisenhancedinterplay.WewillseeseveralnotionsfromGaugeTheories,asWilsonarXiv:hep-th/9301028v224Oct2000Loopsforinstance,playingacentralroleingravitation.AnevengreaterinterplaytakesplacewithTopologicalFieldTheories.Wewillseetheimportantr

5、olethattheChern-Simonsform,theJonesPolynomialandothernotionsofknottheoryseemtoplayinGeneralRelativity.ThequantizationofGeneralRelativityisaproblemthathasdeedresolutionforthelastsixtyyears.Inspiteofthelongtimethathasbeeninvestedintryingtosolveit,webelievethatseveralpeopledonotnece

6、ssarilyfullyappreciatethereasonsofourfailureandthemagnitudeoftheproblem.Itisageneralperception{especiallyamongparticlephysicists{thatGeneralRelativityisnonrenormalizable"andthatisthebasicproblemwiththetheory.Thisstatementismisleadinginthreeways:a)Thefactthatatheoryisnonrenormaliz

7、abledoesnotnecessarilymeanthatthetheoryhasanintrinsicproblemorisbad"inanyway.ItmerelysaysthatperturbationToappearinProceedingsoftheVthMexicanSchoolofParticlesandFields",J.L.Lucio,editor,WorldScientic,Singapore(1993).1theorydoesnotapplytotheprobleminquestion.Aswewillseeinc)ther

8、eareactu-allygoodreasonstobelieve