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2-Thephotonpropagator

Pathintegralformalism-reminder

Classicallimit

!→0

Pathintegralformalism

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 k

Pathintegralformalism

Photonpropagator

A k =d 4 x e -ik.x A x ,A x 1 2π 4 d 4 k e ik.x A k

Projec7onmatrix:Pathintegralformalism

P (k)P (k)=P (k) k 2 P

Tocompletethesquareneedtoinvert

k 2 g -k k ≡k 2 P ...but P k =0, zero eigenvalue...not invertible

Photonpropagator

Tocompletethesquareneedtoinvert

k 2 g -k k ≡k 2 P ...but P k =0, zero eigenvalue...not invertible

Photonpropagator

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 =1

PhotonpropagatorIntheLorenzgauge

k 2 P

Theterm

iε fSi=dφ e iSφ fφ(t=+∞)φ(t=-∞)i (7meordering) n (q)=qn

H→(1-iε)H

Lim t'→-∞ q',t'=ψ 0 (q')0

Fixingthegauge

Wanttosolve:

+m 2 )ψ=-Vψ

Solu7on: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

Klein-Gordonpropagator

Wanttosolve:

+m 2 )ψ=-Vψ

Solu7on: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 AA

Thephotonpropagator

F A A )=j ≡g 2 A AA 2 AA

Choose as

1 (gauge fixing) A 1 2 22
1 (1)(1) pp i igpppg pp A -(1- 1 A )≡(g 2 -(1- 1 )A =j

Thephotonpropagator

F A A )=j ≡g 2 A AA 2 AA

Choose as

1 (gauge fixing) A A -(1- 1 A )≡(g 2 -(1- 1 )A =j

Fixingthegauge-pathintegralformalism

I=dgJ

Volumeofthegroup

Redundantintegra7on

Fixingthegauge-pathintegralformalism

I=dxdye

iSx,y ≡dxdye iSx 2 +y 2

I=dθ

J=2π

J where J=drre

iS(r) I=dg J

Redundantintegra7on

J

Faddeev,Popovgaugefixing

FaddeevPopovdeterminant

Dg''=Dg

i.e.Δ(A)=Δ(A g

Definewillbegaugefixingterm

Faddeev,Popovgaugefixing

FaddeevPopovdeterminant

Dg''=Dg

i.e.Δ(A)=Δ(A g changingandno7ngareinvariant

A→A

g -1

DA,S(A),Δ(A)

Definewillbegaugefixingterm

Noredundantvariables

I≡DA

e iS(A)

Fixingtheelectromagne7cgauge

A g =A

Choose f(A)=∂

A -σ(x)

Λ≡g

Fixingtheelectromagne7cgauge

A g =A

Choose f(A)=∂

A -σ(x)

Δ(A)

-1 =Dgδf(A g =DΛδ(∂A-∂ 2 -σ)"="DΛδ(∂ 2

Λ≡g

Fixingtheelectromagne7cgauge

A g =A

Choose f(A)=∂

A -σ(x)

Δ(A)

-1 =Dgδf(A g =DΛδ(∂A-∂ 2 -σ)"="DΛδ(∂ 2

Λ≡g

I≡DA

e iS(A) R gauge L gaugefixing 1 2 -1 Aquotesdbs_dbs14.pdfusesText_20
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