Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets

Thisisthefifth,expandededitionofthecomprehensivetextbookpublishedin1990onthetheoryandapplicationsofpathintegrals.Itisthefirstbooktoexplicitlysolvepathintegralsofawidevarietyofnontrivialquantum-mechanicalsystems,inparticularthehydrogenatom.Thesolutionshavebeenmadepossiblebytwomajoradvances.ThefirstisaneweuclideanpathintegralformulawhichincreasestherestrictedrangeofapplicabilityofFeynman'stime-slicedformulatoincludesingularattractive1/r-and1/r2-potentials.Thesecondisanewnonholonomicmappingprinciplecarryingphysicallawsinflatspacetimetospacetimeswithcurvatureandtorsion,whichleadstotime-slicedpathintegralsthataremanifestlyinvariantundercoordinatetransformations.Inadditiontothetime-sliceddefinition,theauthorgivesaperturbative,coordinate-independentdefinitionofpathintegrals,whichmakestheminvariantundercoordinatetransformations.Aconsistentimplementationofthispropertyleadstoanextensionofthetheoryofgeneralizedfunctionsbydefininguniquelyproductsofdistributions.ThepowerfulFeynman-Kleinertvariationalapproachisexplainedanddevelopedsystematicallyintoavariationalperturbationtheorywhich,incontrasttoordinaryperturbationtheory,producesconvergentresults.Theconvergenceisuniformfromweaktostrongcouplings,openingawaytopreciseevaluationsofanalyticallyunsolvablepathintegralsinthestrong-couplingregimewheretheydescribecriticalphenomena.Tunnelingprocessesaretreatedindetail,withapplicationstothelifetimesofsupercurrents,thestabilityofmetastablethermodynamicphases,andthelarge-orderbehaviorofperturbationexpansions.Avariationaltreatmentextendstherangeofvaliditytosmallbarriers.Acorrespondingextensionofthelarge-orderperturbationtheorynowalsoappliestosmallorders.Specialattentionisdevotedtopathintegralswithtopologicalrestrictionsneededtounderstandthestatisticalpropertiesofelementaryparticlesandtheentanglementphenomenainpolymerphysicsandbiophysics.TheChern-Simonstheoryofparticleswithfractionalstatistics(anyons)isintroducedandappliedtoexplainthefractionalquantumHalleffect.Therelevanceofpathintegralstofinancialmarketsisdiscussed,andimprovementsofthefamousBlack-Scholesformulaforoptionpricesaredevelopedwhichaccountforthefact,recentlyexperiencedintheworldmarkets,thatlargefluctuationsoccurmuchmorefrequentlythaninGaussiandistributions.