Weuseascovariantderivative Dµ = ∂µ +igAa µT a withcoupling g> 0,fieldstrength
F a µν = ∂µAa ν ∂ν Aa µ gf abcAb µAc ν andgenerators T a satisfying[T a,T b]=if abcT c for allgaugegroups.SpecialcasesusedintheSMarethegroupsUem(1),UY(1),SUL(2) andSU(3)with g = {q,g′,g,gs} and T a = {1, 1,τ a/2,λa/2} inthefundamental representation.Inparticular,theelectricchargeofthepositronis q = e> 0.
Employingdimensionalregularisation(DR),wechangethedimensionofloopintegrals from d =4to d =2ω =4 2ε
Afunctional F [f (x)]isamapfromacertainspaceoffunctions f (x)intotherealor complexnumbers.Wewillconsidermainlyfunctionalsfromthespaceof(atleast)twice differentiablefunctionsbetweenfixedpoints a and b.Morespecifically,Hamilton’s principleusesasfunctionaltheaction S definedby
where L isafunctionofthe2n independentfunctions qi and˙ qi =dqi/dt aswellasof theparameter t.Inclassicalmechanics,wecall L theLagrangefunctionofthesystem, qi areits n generalisedcoordinates,˙ qi thecorrespondingvelocitiesand t isthetime. Theextremaofthisactiongivethosepaths q(t)from a to b whicharesolutionsofthe equationsofmotionforthesystemdescribedby L. Howdowefindthosepathsthatextremizetheaction S?Firstofall,wehaveto prescribewhichvariablesarekeptconstant,whicharevariedandwhichconstraintsthe variationshavetoobey.Dependingonthevariationprinciplewechoose,theseconditionsandthefunctionalformoftheactionwilldiffer.Hamilton’sprinciplecorresponds toasmoothvariationofthepath, q i(t,ε)= q i(t, 0)+ εη i(t), thatkeepstheendpointsfixed, ηi(a)= ηi(b)=0butisotherwisearbitrary.Thescale factor ε determinesthemagnitudeofthevariationfortheone-parameter familyof paths εηi(t).Thenotation S[qi]stressesthatweconsidertheactionasafunctional onlyofthecoordinates qi.Thevelocities˙ qi arenotvariedindependentlybecause ε istime-independent.Sincethetime t isnotvariedinHamilton’sprinciple,varying thepath qi(t,ε)requiresonlytocalculatetheresultingchangeoftheLagrangian L Followingthisprescription,theactionhasanextremumif
forasystemdescribedby n generalisedcoordinates.Wecaneliminate˙ ηi infavourof ηi,integratingthesecondtermbyparts,arrivingat
Theboundaryterm[...]b a vanishesbecausewerequiredthatthefunctions ηi arezero attheendpoints a and b.Sincethesefunctionsareotherwisearbitrary,eachindividual terminthefirstbrackethastovanishforanextremalcurve.The n equationsresulting fromthecondition ∂S[qi(t,ε)]/∂ε =0arecalledthe(Euler–)Lagrangeequationsof theaction S
Weclosethisparagraphwiththreeremarks.First,wenotethatHamilton’sprinciple isoftencalledtheprincipleofleastaction.Thisnameissomewhatmisleading,since theextremumoftheactioncanbealsoamaximumorasaddle-point.Second,observe thattheLagrangian L isnotuniquelyfixed.Addingatotaltimederivative, L → L′ = L +df (q,t)/dt,doesnotchangetheresultingLagrangeequations,
sincethelasttwotermsvanishvaryingtheactionwiththerestrictionoffixedendpoints a and b.Finally,notethatweusedaLagrangianthatdependsonlyonthecoordinatesandtheir first derivatives.SuchaLagrangianleadstosecond-orderequations ofmotionandthustoamechanicalsystemspecifiedbythe2n piecesofinformation {qi, qi}.Ostrogradskyshowed1850thatastableground-stateisimpossible,ifthe Lagrangiancontainshigherderivatives¨q,q(3) ,...,cf.problem1.3.Thereforesuchtheoriescontradictourexperiencethatthevacuumisstable.ConstructingLagrangians forthefundamentaltheoriesdescribingNature,weshouldrestrictourselvesthusto Lagrangiansthatleadtosecond-orderequationsofmotion.
Lagrangefunction. Weillustratenowhowonecanusesymmetriestoconstrain thepossibleformofaLagrangian L.Asexample,weconsiderthecaseofafreenonrelativisticparticlewithmass m subjecttotheGalileanprincipleofrelativity.More precisely,weusethatthehomogeneityofspaceandtimeforbidsthat L dependson x and t,whiletheisotropyofspaceimpliesthat L dependsonlyonthenormofthe velocityvector v,butnotonitsdirection.ThustheLagrangefunctionofafreeparticle canbeonlyafunctionof v2 , L = L(v2).
Letusconsidertwoinertialframesmovingwiththeinfinitesimalvelocity ε relative toeachother.(Recallthataninertialframeisdefinedasacoordinatesystemwhere aforce-freeparticlemovesalongastraightline.)ThenaGalileantransformationconnectsthevelocitiesmeasuredinthetwoframesas v′ = v +ε.TheGalileanprincipleof relativityrequiresthatthelawsofmotionhavethesameforminboth frames,andthus theLagrangianscandifferonlybyatotaltimederivative.Expanding thedifference δL in ε giveswith δv
)
Since vi =dxi/dt,theterm ∂L/∂v2 hastobeindependentof v suchthatthedifference δL isatotaltimederivative.Hence,theLagrangianofafreeparticlehastheform
L = av2 + b.Theconstant b dropsoutoftheequationsofmotion,andwecanset itthereforetozero.Tobeconsistentwithusualnotation,wecall theproportionality constant m/2,andthetotalexpressionkineticenergy T ,
Forasystemofnon-interactingparticles,theLagrangefunction L isadditive, L = a 1 2 mav2 a.Ifthereareinteractions(assumedforsimplicitytodependonlyon the coordinates),thenwesubtractafunction V (x1, x2,...)calledpotentialenergy.One confirmsreadilythatthischoicefor L reproducesNewton’slawofmotion. Energy. TheLagrangianofaclosedsystemdoesnotdependontimebecause ofthe homogeneityoftime.Itstotaltimederivativeis
Wehavestilltoshowthat E coincidesindeedwiththeusualdefinitionofenergy. UsingasLagrangefunction L = T (q, ˙ q) V (q),wherethekineticenergy T isquadratic inthevelocities,wehave
andthus E =2T L = T + V . Conservationlaws. Inageneralway,wecanderivetheconnectionbetweena symmetryoftheLagrangianandacorrespondingconservationlaw asfollows.Letus assumethatunderachangeofcoordinates qi → qi + δqi,theLagrangianchangesat mostbyatotaltimederivative,
LegendretransformationandtheHamiltonfunction. IntheLagrangeformalism,wedescribeasystemspecifyingitsgeneralisedcoordinatesand velocitiesusing theLagrangian, L = L(qi , ˙ qi,t).Analternativeistousegeneralisedcoordinatesand theircanonicallyconjugatedmomenta pi definedas
Thepassagefrom {qi , qi} to {qi,pi} isaspecialcaseofaLegendretransformation:1 StartingfromtheLagrangian L wedefineanewfunction H(qi,pi,t)calledHamiltonian orHamiltonfunctionvia
Hereweassumethatwecaninvertthedefinition(1.17)andarethus abletosubstitute velocities˙ qi bymomenta pi intheLagrangian L
ThephysicalmeaningoftheHamiltonian H followsimmediatelycomparingits definingequationwiththeonefortheenergy E.Thusthenumericalvalueofthe Hamiltonianequalstheenergyofadynamicalsystem;weinsist,however,that H is expressedasfunctionofcoordinatesandtheirconjugatedmomenta.Acoordinate qi thatdoesnotappearexplicitlyin L iscalledcyclic.TheLagrangeequationsimply then ∂L/∂ ˙ qi =const.,sothatthecorrespondingcanonicallyconjugatedmomentum pi = ∂L/∂qi isconserved.
Palatini’sformalismandHamilton’sequations. Previously,weconsideredthe action S asafunctionalonlyof qi.Thenthevariationofthevelocities˙ qi isnot independentandwearriveat n second–orderdifferentialequationsforthecoordinates qi.Analternativeapproachistoallowindependentvariationsofthecoordinates qi andofthevelocities˙ qi.Wetradethelatteragainstthemomenta pi = ∂L/∂qi and rewritetheactionas
Theindependentvariationofcoordinates qi andmomenta pi gives
1TheconceptofaLegendretransformationmaybefamiliarfromthermodynamics,whereitisused tochangebetweenextensivevariables(e.g.theentropy S)andtheirconjugateintensivevariables(e.g. thetemperature T ).
Thefirsttermcanbeintegratedbyparts,andtheresultingboundarytermsvanishes byassumption.Collectingthenthe δqi and δpi termsandrequiringthatthevariation iszero,weobtain
Asthevariations δqi and δpi areindependent,theircoefficientsintheroundbracketshavetovanishseparately.ThusweobtaininthisformalismdirectlyHamilton’s equations,
Considernowanobservable O = O(qi,pi,t).Itstimedependenceisgivenby
whereweusedHamilton’sequations.IfwedefinethePoissonbrackets {A,B} between twoobservables A and B as
thenwecanrewriteEq.(1.23)as
Thisequationsgivesusaformalcorrespondencebetweenclassicalandquantum mechanics.Thetimeevolutionofanoperator O intheHeisenbergpictureisgiven bythesameequationasinclassicalmechanics,ifthePoissonbracketischangedtoa commutator.SincethePoissonbracketisantisymmetric,wefind
HencetheHamiltonian H isaconservedquantity,ifandonlyif H istime-independent.
Weillustratethismethodwiththeexampleoftheharmonicoscillatorwhichis theprototypeforaquadratic,andthusexactlysolvable,action. Inclassicalphysics, causalityimpliesthattheknowledgeoftheexternalforce J(t′)attimes t′ <t is sufficienttodeterminethesolution x(t)attime t.WedefinethereforetwoGreen functions G and GR by
Thustheoscillatorwasatrestfor t< 0,gotakickat t =0,andoscillatesaccording x(t)afterwards.Notethefollowingtwopoints:first,thefactthatthekickproceedsthe movementistheresultofourchoiceoftheretarded(orcausal)Greenfunction.Second, theparticularsolution(1.38)foranoscillatorinitiallyatrestcanbegeneralisedby addingthesolutiontothehomogeneousequation¨ x + ω2 x =0.
1.3Relativisticparticle
Inspecialrelativity,wereplacetheGalileantransformationsassymmetrygroupof spaceandtimebyLorentztransformations.Thelatterareallthosecoordinatetransformations x
thatkeepthesquareddistance
betweentwospacetimeevents xµ 1 and xµ 2 invariant.Thedistanceoftwoinfinitesimally closespacetimeeventsiscalledthelineelementds ofthespacetime.InMinkowski space,itisgivenby
