Undergraduate Lecture Notes in Topological Quantum

arXiv:0810.0344v1[math-ph]2Oct2008UndergraduateLectureNotesinTopologicalQuantumFieldTheoryVladimirG.Ivancevic∗TijanaT.Ivancevic†AbstractThesethird–yearlecturenotesaredesignedfora1–semestercourseintopologicalquantumfieldtheory(TQFT).Assumedbackgroundinmathematicsandphysicsareonlystandardsecond–yearsubjects:multivariablecalculus,introductiontoquantummechanicsandbasicelectromagnetism.Keywords:quantummechanics/fieldtheory,pathintegral,Hodgedecomposition,Chern–SimonsandYang–MillsgaugetheoriesContents1Introduction21.1Basicsofnon-relativisticquantummechanics..................31.1.1Quantumstatesandoperators......................31.1.2Threequantumpictures..........................81.1.3State–spacefornnon-relativisticquantumparticles..........101.2Transitiontoquantumfields............................121.2.1Amplitude,relativisticinvarianceandcausality.............121.2.2Gaugetheories...............................151.2.3Freeandinteractingfieldtheories.....................181.2.4DiracQED.................................191.2.5AbelianHiggsModel............................21∗LandOperationsDivision,DefenceScience&TechnologyOrganisation,P.O.Box1500,EdinburghSA5111,Australia([e-mail])†SchoolofElectricalandInformationEngineering,UniversityofSouthAustralia,MawsonLakes,S.A.5095,Australia([e-mail])12FeynmanPathIntegral232.1Theaction–amplitudeformalism.........................232.2Correlationfunctionsandgeneratingfunctional.................282.3Quantizationoftheelectromagneticfield.....................293Path–IntegralTQFT303.1Schwarz–typeandWitten–typetheories.....................303.2Hodgedecompositiontheorem...........................323.3Hodgedecompositionandgaugepathintegral..................353.3.1Functionalmeasureonthespaceofdifferentialforms..........353.3.2AbelianChern–Simonstheory.......................364Non-AbelianGaugeTheories384.1Introtonon-Abeliantheories...........................384.2Yang–Millstheory.................................384.2.1Yang–Millsaction.............................404.2.2Gaugetransformations...........................414.3QuantizationofYang–Millstheory........................434.3.1Faddeev–Popovdeterminant.......................445Appendix465.1Manifoldsandbundles...............................465.2Liegroups......................................485.3DifferentialformsandStokestheorem......................495.4DeRhamcohomology...............................501IntroductionThereisanumberofgoodtextbooksinquantumfieldtheory(QFT,see[1,2,3,4,5,6,7,8,9,10].However,theyarealldesignedforthegraduate-levelstudyandwecanonlyhopethatundergraduatestudentscanreadsomeeasypartsofthem.Moreover,therearecertainlynoundergraduate-leveltextbooksforTQFT,sopurestudentsareforcedtotrytoreadtheoriginalpapersfromitsinventors,EdWitten[11]andMichaelAtiyah[12].Thegoalofthepresentlecturenotesistotrytofillinthisgap,togivethetalentedundergraduatestheveryfirstglimpseofthemathematicalphysicsoftheXXICentury.Throughouttheselecturenoteswewillusethefollowingconventions:(i)naturalunits,inwhich(someorallof)thefollowingdefinitionsareused:c=~=m=1;(ii)i=√−1,˙z=dz/dt,∂z=∂/∂z;(iii)Einstein’ssummationconventionoverrepeatedindices,whilenDmeansn−dimensional.21.1Basicsofnon-relativisticquantummechanics1.1.1QuantumstatesandoperatorsNon-relativisticquantum-mechanicalsystemshavetwomodesofevolutionintime[13,14].Thefirst,governedbystandard,time–dependentSchr¨odingerequation:i~∂t|ψi=ˆH|ψi,(1)describesthetimeevolutionofquantumsystemswhentheyareundisturbedbymeasure-ments.‘Measurements’aredefinedasinteractionsofthequantumsystemwithitsclassicalenvironment.Aslongasthesystemissufficientlyisolatedfromtheenvironment,itfollowsSchr¨odingerequation.Ifaninteractionwiththeenvironmenttakesplace,i.e.,ameasurementisperformed,thesystemabruptlydecoheresi.e.,collapsesorreducestooneofitsclassicallyallowedstates.Atime–dependentstateofaquantumsystemisdeterminedbyanormalized,complex,wavepsi–functionψ=ψ(t).InDirac’swords[13],thisisaunitketvector|ψi,whichisanelementoftheHilbertspaceL2(ψ)≡H,withacoordinatebasis(qi).1Thestateket–vector1Thefamilyofallpossiblestates(|ψi,|φi,etc.)ofaquantumsystemconfiturewhatisknownasaHilbertspace.Itisacomplexvectorspace,whichmeansthatcanperformthecomplex–number–weightedcombinationsthatweconsideredbeforeforquantumstates.If|ψiand|φiarebothelementsoftheHilbertspace,thensoalsoisw|ψi+z|φi,foranypairofcomplexnumberswandz.Here,weevenaloww=z=0,togivetheelement0oftheHilbertspace,whichdoesnotrepresentapossiblephysicalstate.Wehavethenormalalgebraicrulesforavectorspace:|ψi+|φi=|φi+|ψi,|ψi+(|φi+|χi)=(|ψi+|φi)+|χi,w(z|ψi)=(wz)|ψi,(w+z)|ψi=w|ψi+z|ψi,z(|ψi+|φi)=z|ψi+z|φi0|ψi=0,z0=0.AHilbertspacecansometimeshaveafinitenumberofdimensions,asinthecaseofthespinstatesofaparticle.Forspin12,theHilbertspaceisjust2D,itselementsbeingthecomplexlinearcombinationsofthetwostates|↑iand|↓i.Forspin12n,theHilbertspaceis(n+1)D.However,sometimestheHilbertspacecanhaveaninfinitenumberofdimensions,ase.g.,thestatesofpositionormomentumofaparticle.Here,eachalt