baez j. (ed.) knots and quantum gravity (proc. riverside, oxford, 1994)(241s)_pqgr_

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1、PrefaceAfundamentalproblemwithquantumtheoriesofgravity,asopposedtotheotherforcesofnature,isthatinEinstein’stheoryofgravity,generalrelativity,thereisnobackgroundgeometrytoworkwith:thegeometryofspacetimeitselfbecomesadynamicalvariable.Thisislooselysum-marizedbysayingthatt

2、hetheoryis‘generallycovariant’.Thestandardmodelofthestrong,weak,andelectromagneticforcesisdominatedbythepresenceofafinite-dimensionalgroup,thePoincar´egroup,actingasthesymmetriesofaspacetimewithafixedgeometricalstructure.Ingeneralrelativity,ontheotherhand,thereisaninfinite

3、-dimensionalgroup,thegroupofdiffeomorphismsofspacetime,whichpermutesdifferentmathe-maticaldescriptionsofanygivenspacetimegeometrysatisfyingEinstein’sequations.Thiscausestwodifficultproblems.Ontheonehand,onecannolongerapplytheusualcriterionforfixingtheinnerproductintheHilbert

4、spaceofstates,namelythatitbeinvariantunderthegeometricalsymmetrygroup.Thisistheso-called‘innerproductproblem’.Ontheotherhand,dynamicsisnolongerencodedintheactionofthegeometricalsymmetrygrouponthespaceofstates.Thisistheso-called‘problemoftime’.Theprospectsfordevelopingam

5、athematicalframeworkforquantizinggravityhavebeenconsiderablychangedbyanumberofdevelopmentsinthe1980sthatatfirstseemedunrelated.Ontheonehand,Jonesdiscov-eredanewpolynomialinvariantofknotsandlinks,promptinganintensivesearchforgeneralizationsandaunifyingframework.Itsoonbeca

6、meclearthatthenewknotpolynomialswereintimatelyrelatedtothephysicsof2-dimensionalsystemsinmanyways.Atiyah,however,raisedthechallengeoffindingamanifestly3-dimensionaldefinitionofthesepolynomials.ThischallengewasmetbyWitten,whoshowedthattheyoccurnaturallyinChern–Simonstheory

7、.Chern–Simonstheoryisaquantumfieldtheoryin3dimensionsthathasthedistinctionofbeinggenerallycovariantandalsoagaugetheory,thatisatheoryinvolvingconnectionsonabundleoverspacetime.Anyknotthusdeterminesaquantitycalleda‘Wilsonloop,’thetraceoftheholonomyoftheconnectionaroundthek

8、not.TheknotpolynomialsaresimplytheexpectationvaluesofWilsonloopsinthevac-uumstateofChern–Simonstheory,andtheir