PHYSICS 489/1489 LECTURE 7: QUANTUM ELECTRODYNAMICS

Transcription

1 PHYSICS 489/489 LECTURE 7: QUANTUM ELECTRODYNAMICS

2 REMINDER Problm st du today 700 in Box F

3 TODAY: W invstigatd th Dirac quation it dscribs a rlativistic spin /2 particl implis th xistnc of antiparticl stats drivd plan wav stats for th quation Today w xplor th lctromagntic fild wav quation, polarization how a chargd Dirac particl intracts with th fild Rcall th Goldn rul Rat ~ phas spac x matrix lmnt 2 W know how to dal with th phas spac today w s how to driv th matrix lmnt

4 APOLOGIES: This will contain (hopfully) th sktchist drivations and discussion of matrial Th book dos som dtaild handwaving to driv th matrix lmnt I will handwav ovr th handwaving A propr tratmnt rquirs quantum fild thory For th rst of th class, th important thing to gt out of this ar th Fynman ruls which giv us a systmatic way of dtrmining th amplitud driving th Fynman ruls is out of th scop of our class

5 THE PHOTON Maxwll s quations: r E =4 r B =0 r E + Ḃ =0 r B Ė =4 J Rcall that w can r-xprss th Maxwll quations using potntials: E = B = A ths can in turn b combind to mak a 4 vctor: Likwis for th sourc trms ρ and J: A µ =(, A) J µ =(, J)

6 MAXWELL S EQUATION All four quations can b xprssd as: µ µ A Th issu is that A is (far) from uniqu: Considr: ( µ A µ )= 4 c J A µ A µ + µ µ µ (A + ) ( µ (A µ + µ )= last trms cancl, so nw Aµ is also a solution to Maxwll s quation thy ar physically th sam, so w can mak som convntions: Lorntz gaug condition : Coulomb gaug µa µ =0 A 0 =0 F µ = µ A A µ = 0 E x E y E z E x 0 B z B y E y B z 0 B x E z B y B x 0 µ µ A ( µ (A µ )+ µ µ µ µ µ µ A = 4 c J A =0

7 FREE SOLUTIONS Fr mans no sourcs (chargs, currnts): J µ =0 Find solution by ansatz: A µ (x) =a ip x µ (p) Now chck: µ A (x) = ip µ a ip x (p) µ µ A =0 µ A µ =0 p µ µ (p) =0 µ µ A (x) =( i) 2 p µ p µ a ip x (p) =0 p 2 = m 2 c 2 =0 Conclusions: Photon is masslss A 0 =0 0 =0 p =0 Polarization ε is transvrs to photon dirction (in Coulomb gaug): it has two dgrs of frdom/polarizations

8 REMINDER OF DIRAC SPINORS Plan wav stats u = p E + m 0 0 p z /(E + m) (p x + ip y )/(E + m) lctrons positiv nrgy solutions C A u 2 = p E + m 0 0 (p x ip y )/(E + m) p z /(E + m) C A v 2 u 3 = p E + m 0 p z /(E + m) (p x + ip y )/(E + m) 0 C A v u 4 = p E + m 0 (p x ip y )/(E + m) p z /(E + m) 0 C A ngativ nrgy solutions positrons

9 TOWARDS AMPLITUDES: T fi = h f V ii + X j6=i E i E j h f V jih j V ii Prviously, w saw how transitions ( intractions ) in th Born approximation ar rlatd to matrix lmnts Considr th lctromagntic intraction btwn two chargd particls W can guss that: φi, φf dscrib th initial and final stats of ths particls for plan wavs, ths ar th u, v spinors w dscribd bfor. V dscribs th intraction of ths particls with th lctromagntic fild (i.. th photon). this will involv th polarization vctor for th photon

10 E&M WITH CHARGED DIRAC PARTICLES H 0 = 0 (~ p + m) V = q 0 µ A µ A µ =(, A) Can b drivd from minimal µ µ + iqa µ p µ ) p µ qa µ w ll s latr that this can aris from gaug invarianc for now, you can s: E ) E q p ) p qa i.. impact from scalar/vctor potntial on th nrgy of th particl W can thn xprss th intraction as follows i f h f V i i)u f Q 0 µ i u i ) Q ū f µ u µ

11 INTERACTION BETWEEN TWO CHARGED PARTICLES VIA EM FIELD h 3 V i)q ū 3 µ u µ 3 h 4 V 2 i)q ū 4 u 2 q 2 q X ( ) µ ( ) = g µ 2 4 Propagator factor for photon rflcts potntial nrgy of th fild Sum ovr polarizations of intrmdiat photon not that this rsults in contracting th Lorntz indics Accounts for both ordrings of th photon propagation M = Q2 q 2 [ū 3 µ u ][ū 4 µ u 2 ]

12 FEYNMAN RULES: EXTERNAL LINES Right down th Fynman diagram(s) for th procss and labl th momntum flow p s for xtrnal lins, q s for intrnal lins Not that thr ar two flows: particl/antiparticl p q p3 momntum Now th componnts of th xprssion p2 p4 Extrnal Lins: Elctrons: initial stat u s (p) final stat Positrons: initial stat v s (p) final stat Photons: initial stat final stat µ(p) ū s (p) v s (p) µ(p)

13 VERTICES AND PROPAGATORS For ach QED vrtx: ig µ (2 ) 4 4 (k + k 2 + k 3 ) momntum is + incoming, - outgoing from vrtx us a diffrnt Lorntz indx for ach γ matrix g is th lctromagntic coupling (Q) Intrnal lins: lctron/positron propagator i( µ q µ + mc) q 2 m 2 c 2 p2 q p3 Photon propagator ig µ indics match vrtics/polarization q 2 d 4 q (2 ) 4 Intgral ovr intrnal momntum: p p4 Finally: cancl th ovrall (2π) 4 δ 4 what rmains is -im

14 EXAMPLE: μ + +μ - p2 q p3 p p4 Ordr mattrs du to Dirac matrix structur (photon part dosn t car) Go backward along th frmion lins: In th final stat : ū(3) ig µ v(4) (2 ) 4 4 (q p 3 p 4 ) In th initial stat : v(2) ig u() (2 ) 4 4 (p + p 2 q) ig µ Throw in th intrnal photon propagator: (2 ) 4 d 4 q q 2 intgrat ovr intrnal momntum i(2 ) 4 4 g 2 (p + p 2 p 3 p 4 ) (p + p 2 ) 2 Problm 6.5 M = [ū(3) µ v(4)] g µ [ v(2) g 2 (p + p 2 ) 2 [ū(3) µ v(4)] [ v(2) µ u()] u()]

15 EXAMPLE: p q p3 p2 q p4 p p2 p4 ū(3) ig µ u() v(2) ig v(4) ū(3) ig v(4) v(2) ig u() p3 ig µ (p p 3 ) 2 ig (p + p 2 ) 2 (2 ) 4 4 (p + p 2 p 3 p 4 )

16 A FEW MORE: - + γ - + γ γ 3 γ 3 2 γ M = M 2 = 2 4 γ 4 g 2 h i (p p 3 ) 2 m 2 ū 4 / 2 (/p /p 3 + m)/ 3 u g 2 h i (p + p 2 ) 2 m 2 ū 4 / 3 ( /p + /p 2 + m)/ 2 u - + μ - - +μ - 3 M = g 2 (p p 3 ) 2 [ū 3 µ u ][ū 4 µ u 2 ] 2μ μ4

17 NEXT TIME: Plas rad 6. and 6.2 I would xplicitly work out spin summation procdur in 6.2. and 6.2.4