Quantum ElectroDynamics II

Transcript

1 Quantum ElectroDynamcs II Dr.arda Tahr Physcs department CIIT, Islamabad

2 Photon Coned by Glbert Lews n In Greek Language Phos meanng lght

3 The Photons A What do you know about Photon?

4 Photon Dscrete bundle or quantum of electromagnetc or lght energy. Massless spn 1 partcle & behaves lke both wave and partcle

5 Relatvstc Mechancs very fast Photon A Quantum eld Theory E 2 2 p c 2 0 Quantum Mechancs zero mass p rame work E p h h t h h x A P h P, E p x

6 Results due to slght Modfcaton Relatvstc Energy-momentum relaton for massless partcle E p c In four vector notaton P P 0 Soluton s A 0 2 h A 0 A 0 p. x h ae ε p h x Polarzaton vector 4-comp., but not all ndependent

7 Electromagnetc Waves Maxwell

8 Quck Revew Maxwell s equaton Unfed descrpton of electrcty and magnetsm 1864 B E t. E 4π 4π J ρ homogeneous nhomogeneous. B 0 E + B t 0

9 Quck Revew Charge conservaton comes from the contnuty equaton ρ +. J 0 t J. J ρ t ρ, J ; 0,1, 2, 3 Prove by usng S.E 4-vector notaton Lorentz nvarant form Local charge conservaton J 0 0 J + 0 J 0

10 Quck Revew rom homogeneous Maxwell equaton one can get scalar and vector potental φ,a or A B A and E A φ t Maxwell equaton remans satsfed e.g. 0 B. A E Nothng new A φ t B A E φ t E 0 t A

11 Why To Introduce scalarφ and vector A potental

12 Just for the sake of convenent mathematcal nventons. or Due to some concrete reason f yes! Then what that reason s?

13 defect arbtrarness χ φ φ φ + t A A A χ Peruse the defnton of these potentals χ χ t, x Maxwell equatons stll satsfed Change of potentals has no effect on the feld Gauge transformaton

14 Verfcaton consder B A A. B..B. A χ A A A χ χ. B. A.. B. B. B 0 0 Magnetc feld remans nvarant under the local gauge transformaton

15 Assgnment B E E t. E 4π B + t 4π J ρ 0 Remans nvarant under local gauge transformaton? gauge freedom

16 Enjoy gauge freedom Exploted t and beneft from t how

17 Covarant form of Maxwell s Eq Relatvstcally E and B can be represented by antsymmetrc 2 nd rank tensor, the feld strength ν tensor, ν 0,1,2, E E E x y z 0 B E z B x y E 0 B B x y z E B 0 y B z x In the form of Potentals ν A ν φ A A, ν A Explot gauge freedom to mpose constrant on potental A 0 Lorentz condton

18 Elegant form Covarant form of Maxwell eqs. ν + ν +,, ν 0,1, 2,3 ν J ρ, J J ν ν ν A J ν ν A 0 In terms of 4-vector potental. ν A J 0 ν, t Homogeneous In Homogeneous Verfy. E ρ f ν J 0,1,2,3

19 or free photon empty space J 0 KG eqs. for massless partcle ν A 0 A 0 0,. A 0 Coulomb gauge A p. x h ae ε p P P 0 Polarzaton vector 4-comp.,but not all ndependent Lorentz condton requres that P ε 0

20 Informaton from Coulomb gauge In coulomb gauge ε 0 0, ε. p 0 ree photon s Transversely polarzed Polarzaton three vector ɛ s perpendcular to the drecton of propagaton. Coulomb gauge s Transverse gauge

21 The eynman rules for QED Photon A ae h. P ε s ree ε P ε ε 0 * 1 2 * ε ε 1 0 Lorentz condton Orthogonal Normalzed s 1,2 ε s ε s δj pˆ pˆ j j Completeness

22 Lagrangan densty Lagrangan densty for photon feld 1 L 4 ν ν J A K.E term for photonc feld Externally specfed current J s coupled to photon feld

23 Covarant Gauge Transformaton Gauge transform Lagrangan densty usng 1 ν L ν J A 4 ν A ν ν A usng 1 ν ν A A 4 A A J A ν ν A A A χ Gauge T n covarant notaton usng ν χ ν χ Order of dff. s unmport. for scalar

24 Smplfcaton after substtutons 1 ν ν J 4 L ν ν A A A A J A χ 1 4 ν ν J A J χ why? Invarant Physcs remans Invarant

25 Drac Equaton n electromagnetc feld

26 Drac Equaton descrbng a spn ½ fermons of mass m n free space m 0 The correspondng Lagrangan densty L ψ mcψ What happened to Drac equaton under U1 gauge transformaton.e. α x x e x

27 U1 n n UU + U + U 1 Matrx U s untary f Product of two untary Matrx U s untary. n n untary Matrces form a group under Matrx multplcaton, denoted by Un. Un has n 2 generators. + detu det I det UU detu detu detu or n1 detu e detu nα e α

28 If α s just a number L L 1 m m U e m e α α m L L Invarant Global gauge transformaton

29 Local gauge transformaton If α α L L 1 m m U e m e α α [ ] m e e e α α α { } + m e e e e α α α α

30 x x + α L L + m α Not Invarant L L Local gauge transformaton

31 Requrement of local gauge transformaton enforces the ntroducton of electromagnetc feld descrbe by the 4-vector potental A P α A P A qa [ ] qa m 0 qχ e q χ χ

32 bmn [ ] [ ] m qa m qa U 1 { } [ ] 0 + q m e A q χ χ 0 + e m q A q e q q χ χ χ e m q A q e e q q q q χ χ χ χ χ

33 After ntroducng the gauge transformaton extra term wll exactly cancel out [ ] 0 + m e q A q q qχ χ χ [ ] 0 m e A q qχ [ ] 0 m qa

34 Complete Lagrangan for f & Lagrangan densty descrbng the fermonc feld n the presence of an electromagnetc feld s L [ ] ν qa m ν J A 1 4 L 1 4 [ ] ν m ν J + q A Current produce by Drac partcle

35 Couplng to photon feld consst of two parts 1. Wth external current densty J.e J A 2. Wth fermon feld J q When ths current coupled to A, descrbe the nteracton vertex

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