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
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[PDF] 6 Quantum Electrodynamics - Department of Applied Mathematics
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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
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2-Thephotonpropagator
Pathintegralformalism-reminder
Classicallimit
!→0Pathintegralformalism
Genera7ngFunc7onal
Klein-Gordonpropagator
Z 0 0 =0|0 J=0 =Dχ exp- i 2 d 4 k 2π 4 k k 2 +m 2 -k =1(no interactions)Feynmanpropagator
Photonpropagator
A k =d 4 x e -ik.x A x ,A x 1 2π 4 d 4 k e ik.x A kPathintegralformalism
Photonpropagator
A k =d 4 x e -ik.x A x ,A x 1 2π 4 d 4 k e ik.x A kProjec7onmatrix:Pathintegralformalism
P (k)P (k)=P (k) k 2 PTocompletethesquareneedtoinvert
k 2 g -k k ≡k 2 P ...but P k =0, zero eigenvalue...not invertiblePhotonpropagator
Tocompletethesquareneedtoinvert
k 2 g -k k ≡k 2 P ...but P k =0, zero eigenvalue...not invertiblePhotonpropagator
In fact component of A
∝k doesn't appear ... sufficient to integrate DA over transverse components, A ,only ... k 2 P k -1 P k 2 -iε ... equivalentto k A =0...Lorentz gauge ∂ A =0 iden7tymatrixinsubspace A k =A k P J k k 2 -iε k 2 P k =A k P J k k 2 -iε Z 0 0 =0|0 J=0 =Dχ exp- i 2 d 4 k 2π 4 k k 2 P -k =1PhotonpropagatorIntheLorenzgauge
k 2 PTheterm
iε fSi=dφ e iSφ fφ(t=+∞)φ(t=-∞)i (7meordering) n (q)=qnH→(1-iε)H
Lim t'→-∞ q',t'=ψ 0 (q')0Fixingthegauge
Wanttosolve:
+m 2 )ψ=-VψSolu7on:where
ψ(x)=φ(x)+d
4 x'Δ F (x'-x)V(x')ψ(x') -m 2 F x'-x 4 x'-x and -m 2 )φx =0Klein-Gordonpropagator
Wanttosolve:
+m 2 )ψ=-VψSolu7on:where
ψ(x)=φ(x)+d
4 x'Δ F (x'-x)V(x')ψ(x') -m 2 F x'-x 4 x'-x and -m 2 )φx =0 22.(')24.(')44 11 22
ipxxipxx F emxxdxxexxdxx F (p)=- 1 (2π) 2 1 p 2 +m 2 -iε F (x)=- 1 (2π) 4 d 4 pe -ip.x 1 p 2 +m 2 -iε 2 1 (2)π 2 1 (2)π -p 2 +m 2 F p 1 2π 2
Reminder:Klein-Gordonpropagator
Thephotonpropagator
F A A )=j ≡g 2 A AA 2 AAThephotonpropagator
F A A )=j ≡g 2 A AA 2 AAChoose as
1 (gauge fixing) A 1 2 221 (1)(1) pp i igpppg pp A -(1- 1 A )≡(g 2 -(1- 1 )A =j