2 -? The photon propagator
We have used choice of a specific gauge transformaRon to modify the equaRon of moRon. The quesRon is how do you modify the Lagrangian to get this equaRon of
6. Quantum Electrodynamics
Note: This is the propagator we found when quantizing in Lorentz gauge (using the. Feynman gauge parameter). In general quantizing the Lagrangian (6.37) in
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We have used choice of a specific gauge transformaRon to modify the equaRon of moRon The quesRon is how do you modify the Lagrangian to get this equaRon of
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Now consider photon propagators for different gauge conditions for the EM potential Consequently in the new gauge the Feynman propagator becomes
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Recent studies of QCD Green's functions and their applications in hadronic physics are reviewed We discuss the definition of the generating functional in
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with two vertices and one gauge boson propagator These are of order g2 On the other hand the last Feynman rule is a contribution to the amplitude in and
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The most common use of the propagator is in calculating probability amplitudes for particle interactions using Feynman diagrams. These calculations are usually carried out in momentum space.Is Feynman propagator symmetric?
It follows that the causal propagator ? ? ? + ? ? ? is skew-symmetric in its arguments: ? S ( x ? y ) = ? ? S ( y ? x ) .- Propagator is a model whose objective is to determine the position of satellite at any instance of time, with given acceleration and initial velocity.
6.Quant umElectrodynamics
Inth issectionwefinally gettoquantu melectrod ynamics(QED),thetheor yofl ight interactingwithchargedmatter. Ourpathtoquantization willbeasbef ore:westart withthefree theoryoftheel ectromagneti cfieldandseehowt heq uantumtheory gives risetoaphoton witht wop olarization states.W ethendescribe howtocouplethe photontofermionsand tobos ons.6.1Maxwel l'sEquations
TheLagran gianforMaxwell'sequationsinthe absenceof anysourcesissimply L=! 1 4 F F (6.1) wherethefieldstre ngthisdefin edby F A A (6.2) Theequat ionsofmotionwhichfollowfro mthisLa grangianare !L A F =0(6 .3)Meanwhile,fromthedefini tionofF
,thefieldstrengthalsosatisfiestheBianchi identity F F F =0(6 .4) Toma kecontactwi ththeformofMaxwell's equationsyou learnabou tinhighscho ol, wene edsome3-vecto rnotation.I fwedefineAA),the ntheelectric field
Eand magneticfieldBaredefinedb y
E=!""!
A t and B="#A(6.5)
which,intermsofF ,becomes F0ExEyEz
!Ex0!BzBy !EyBz0!Bx !Ez!ByBx0 (6.6) TheBianch iidentity(6.4)thengivestwoofMaxwell'sequations,B=0a nd
B tE(6.7)
-124- Theseremain trueeveninthep resenceofelectricsour ces.Meanwhil e,theequ ations ofmotion givethe remainingtwoMaxwell equations,E=0a nd
E tB(6.8)
Aswe wills eeshortly,in thepresen ceofchargedmatterthesee quati onspickup extra termsontherigh t-handside .6.1.1GaugeSymmet ry
Themasslessv ectorfieldA
has4c omponen ts,whichwouldnaivelyseemtotellus that thegaugefie ldhas 4degreesof free dom.Ye tweknowthatthephotonhasonlyt wo degreesoffreedomwhich wecal litspolarizationstat es.Howarewegoing toreso lve thisdisc repancy?Therearetworelatedcomme ntswhichwille nsurethat quantizing thegaugefie ldA givesriseto2degrees offreedom, ratherthan4. •Thefield A 0 hasno kineticte rm A 0 intheLagrangian: itis notdynamical. This meansthatifwe aregive nsome initialdataA i and A i atatime t 0 ,the nthefield A 0 isfully determinedby theequationofmotion"·E=0which,expandingout,
reads 2 A 0 A t =0(6 .9)Thishasthe solution
A 0 (#x)= d 3 x A/ t)(#x4$|#x!#x
(6.10) SoA 0 isnotinde pende nt:wedon'tgettospecifyA 0 onthe initialtimeslice. It lookslikew ehaveonly3 degreesoff reedominA ratherthan4.Butthis iss till onetoo many. •TheLagrang ian(6.3)hasaverylargesymme trygroup,actingonthevector potentialas A (x)$A (x)+! %(x)(6.11) foran yfunction %(x).We 'llaskonlythat%(x)dieso!suitablyquicklyatspatial x$%.Wecallthisagaugesymmetry .Th efieldstre ngthisinvari antunderthe gaugesymmetry: F (A (A %)=F (6.12) -125- Sowhat arewetomake ofthis? Wehavea theory withaninfinitenum ber of symmetries,oneforeachfunction%(x).Pre viouslyweonlyencounteredsymme - trieswhichac tthesameatallpoin tsinspacetime ,for example&$e i &fora complexscalarfield.Noe ther'stheoremtold usthatthesesymm etriesgiverise tocons ervationlaws.Dowenowhave aninfinitenumberofconse rvationla ws?Theanswe risno!Gaugesymmetri es haveavery di
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