XV Quantum Electrodynamics
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1 XV Qnt lctrdynics Fynn Rls fr QD An xl: Sry: iht Sts f Fynn Tchnis Fr rfrnc s: Hlzn&Mrtin s 86,8,9 Intrdctin t Prticl Physics ctr XV Cntnts R. Or Srin 005
2 Fynn rls sin 0 ty dl sin sin htn xtrnl lin in in ε t t ε * Vrtx i i ± πα fr Intrnl lin i i i Nt: Thr is n dditinl rl: W h t incld ins sin twn dirs which diffr in th xchn f tw incin r tin lctrns, r n incin lctrn nd tin sitrn, s rird y th Fri Dirc sttistics. Intrdctin t Prticl Physics ctr XV Fynn rls R. Or Srin 005
3 t s tk th rcss s n xl. im i i i i rntz ctr nr rntz sclr Dirc sinr x trix rntz ctr Adjint Dirc sinr Nt tht th clictd strctr is jst nr. If w scify th incin nd tin sins, w cn lt it. Hwr, w rrly knw th intil nd finl sins, nd w h t r r th initil sins nd s r th finl sins. 3 Intrdctin t Prticl Physics ctr XV R. Or Srin 005
4 This is 6 trs t lt, ch tr hin s r, iin 6 trs in ll. Frtntly, thr is trick which silifis ths clcltins cnsidrly. W nd t clclt M, nd It is cnnint t writ this s th rdct f tw tnsrs: In nrl cs with lctrn nd : Intrdctin t Prticl Physics ctr XV R. Or Srin 005 * * M * * whr, M n lctrn n lctrn *
5 5 It is sy t shw tht if: thn th rrs rdr, thrfr: Nw s r sins: y th cltnss rltin! Intrdctin t Prticl Physics ctr XV 3 R. Or Srin 005 * whr, trix is it sinc... σ σ... { } sins
6 6 Intrdctin t Prticl Physics ctr XV R. Or Srin 005 hnd nttin. slly writtn s shrt is th trix jst th trc f This is Sin r sins : With th indics writtn xlicitly : trix t, sins ji ij ji ij j ij i sins Q Q Q Q
7 7 If ithr r hd s instd f s, thr wld instd f. Nt: Th trcs cn sily ltd y sin trc thrs, which r ll in in th txt ks.. Hlzn & Mrtin. Fr r xl rctin: : Tkin th liit >>: Intrdctin t Prticl Physics ctr XV 5 R. Or Srin 005 * * M λ σ λ σ λ λ σ σ r r initil sins
8 Fr: Fr: Bysin: M M 8 [ αa] [ A] whr α is nr [ σ λ σλ λ σ ] σ λ σ λ σ λ σλ λ σ n 0 Intrdctin t Prticl Physics ctr XV 6 R. Or Srin 005 8
9 9...cntind T t th crss sctin, Nt: W knw fr insctin f th Fyn dir tht σ. Thrfr, ll w t fr th lst thr lctrs ws th fctr π3... θ Intrdctin t Prticl Physics ctr XV 7 R. Or Srin 005 Θ Θ Θ r r Θ Θ Θ M α d d πα π α φ α σ Θ Θ
10 Fynn Rls fr QD Sry f tins ndd fr ssi sin frins nd sslss sin sns s, r th tw ssil sin stts: Fnctin Dirc Adjint Klin rdn Orthnl Nrlisd Clt Ψ x lctrns Psitrns Phtns s s i x s Ψ x s i x s s ε A ε * ε x * ε 0 ε i x 0 ε s Intrdctin t Prticl Physics ctr XV Fynn rls fr QD R. Or Srin 005 0
11 Fynn Rls fr QD Th iht Sts. Nttin l incin nd tin nt,, 3,..., n nd th crrsndin sinss,s,s 3,...,s n l intrnl lins,, 3,..., n Arrws n xtrnl lins indict whthr it is rticl r nti rticl Arrws n intrnl lins indict th dirctin f flw. Sinrs: xtrnl lins cntrit fctrs: Incin Prticl: Otin Prticl: Incin Anti Prticl: Otin Anti Prticl: Incin Phtn: ε Otin Phtn: ε * 3. Vrtx Fctrs: ch rtx cntrits fctr: i. Prtrs: ch Intrnl lin cntrits fctr s fllws: Sin 0: i Sin frin: i Sin htn: i Intrdctin t Prticl Physics ctr XV Fynn rls fr QD R. Or Srin 005
12 Fynn Rls fr QD Th Sn Sts Cnsrtin f nry nd Mnt: Fr ch rtx writ dlt fnctin: π δ 3 whr r th nt cin int th rtx. If th lin is in t, thn is ins th nt xct fr xtrnl nti rticls. 6. Intrt r Intrnl Mnt: Fr ch intrnl nt writ fctr: d π nd intrt! 7. Cncl th dlt fnctin: Th rslt f th will incld fctr: π δ... This crrsnds t rll nry nt cnsrtin. Cncl this fctr nd wht is lft, is th litd im. 8. Antisytristin: Incld ins sin twn dirs tht diffr nly in. n Intrchn f tw incin r tin rticls f th s ty Intrchn f n incin rticl nd tin nti rticl f th s ty i.. chr cnjts Intrdctin t Prticl Physics ctr XV Fynn rls fr QD 3 R. Or Srin 005
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