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by: Dr. Simeon Wiza


Dr. Simeon Wiza
GPA 3.96

Dam Son

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Dam Son
Class Notes
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This 4 page Class Notes was uploaded by Dr. Simeon Wiza on Wednesday September 9, 2015. The Class Notes belongs to PHYS 557 at University of Washington taught by Dam Son in Fall. Since its upload, it has received 22 views. For similar materials see /class/192438/phys-557-university-of-washington in Physics 2 at University of Washington.




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Date Created: 09/09/15
Notes on eld theory 1 Conventions Unit h c 17 GeV 109 gtlt1610 19 J Metric 77 diag17 717 717 71 2 Scalar eld theory Action for a real scalar 1 s g d4z Mam e m2 2l lt1 Field equation 5 m2 0 2 Expansion to plane wave modes dgk 7239 w 239 w 15W Wak k 116 k 3 Where Ek k2 m212 Canonical quantization ak7 am 2713932Ek63k i k 4 other commutators are zero Hamiltonian dk H all ak const 5 The vacuum l0gt is the ground state of H akl0gt 0 The Fock space is constructed by applying creation operators on l0gt all0gt lkgt7 aha lk17k2gt 6 These are one particle7 two particle77 states Complex scalar eld theory 3 maw imam lt7 1 Where W2 ow Plane wave expansion 27T12E kak iikz blez kw 8 Where ak7 aL 2713932Ek63k 7 k 7 bk7 bL 27f32Ek63k 7 k 7 other commutators are zero Noether theorem Assume the Lagrangian is invariant under a continous symmetry 5 66 5 9 then there exist a conserved current7 i B aw In the case of the complex scalar eld7 the symmetry is g5 a e iezz 7 or do 72 6W 72 The conserved current is at 07 339 6 10 J Mama 7 BWW 11 the conserved charge is dk Q d3x0 W am 7 bLbk 12 T The operator 1k increases Q by one unit7 While bl decreases Q by one unit One can think about bl as the operator that creates an antiparticle7 While 1 creates a particle 3 Spin1 elds 31 Massless spin1 elds An example of a massless spin 1 eld is the electromagnetic eld photons It is described by a 4 vector AM Its components are the scalar potential and the vector potential7 AM Q57 7A The action is s 7 d4z FWFW 13 Where FM is the eld strength tensor7 FM BMAV i BVAM 14 The eld equation is BMFW 0 15 These constitute two of the four Maxwell7s equations The other two are the Bianchi identity7 8FVA 8117M BAFM 0 FM is invariant under gauge transformations Al aAM8po 16 By chosing an appropriate or one can impose the Lorentz gauge condition 8M4 0 The eld equation then becomes DAV 0 17 and the plane wave solution is AM Mpe ip39z 18 where p2 0 Here EM is a polarization vector The Lorentz gauge condition requires that p 6 0 This means that there are only three independent polarization vectors possible Even after imposing the Lorentz gauge7 one still can make a residual gauge tranformation AM a AM am where Dd 0 This amounts to changing 5M a EM app with arbitrary constant 1 Thus one has to impose one additional constraint to x EM For example7 one can impose 60 0 The EM is a spatial vector perpendicular to p For example7 if p p70707p then there are two independent polarization vectors 07 17 07 0 and 07 07 17 0 These correspond to linearly polarized plane waves Circularly polarized waves corresponds to EM 07 17 2397 or 07 17 7239 Denote the two independent polarizations as p7 A 172 We can normalize the M vectors so that EA E V 76M 19 The quantum eld AM can then be expanded as dp iipz T A4lt ipz A WWMWU V apx t WE 20 p Quantization of the Maxwell7s theory leads to the commutation relation am7 ILN 27r32Ep66p 7 p 21 So aLAlO is the state with one photon with momentum p and polarization EA 32 Massive spin1 eld Lagrangian 1 W m2 M iZFWF 7AMA 22 The action is no longer gauge invariant The eld equation is a w mZAquot 0 23 Taking 8 of that equation7 we nd BVAquot 0 Hence the eld equation is D m2AM 0 The plane wave solution is AM epe ip39w 24 where p2 m2 and p 6 0 There is 3 possible polarization vectors 6 A massive spin 1 particle has 3 possible polarizations


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