386 CHAPTER25. AGLIMPSEOFQUANTUMFIELDTHEORY
degreeoffreedomisredundant;agivensetofE,Bfieldsdoesnotuniquelyspecify
thevectorpotential. Soletususethegaugefreedomtoeliminateonemoredegree
of freedom,byimposing anadditional conditionF[A] = 0 onthe A-fields. Some
popularexamplesare
A 0 = 0 temporalgauge
A 3 = 0 axialgauge
∇·A%= 0 Coulombgauge
∂μAμ= 0 Landaugauge
(25.36)
Forourpurposes,themostconvenientgaugeistemporalgauge,A 0 =0,becausethis
immediatelysolvesthe P 0 = 0 problem. Ifwe removeA 0 fromthestart, thenof
courseithasnocorrespondingmomentum. However,wethenlosetheGaussLaw,
becausethisisderivedbyvaryingS withrespecttoA 0. Sowhatwehavetodois
imposetheGaussLaw∇·E= 0 asasupplementaryconditiononourinitialdata,
beforesolvingtheotherequationsofmotion. Inquantumtheory,onerequiresthat
Gauss’Lawissatisfiedasanoperatorequation
(∇·E)Ψ= 0 (25.37)onallphysicalstates.
Allofthisbusinesswithgaugeinvariancejustlookslikeanastytechnicalitywhich
complicatesthequantizationofelectromagnetism,andsoitis. Itisonlyinamore
advancedcourse,onquantumfieldtheory,thatonelearnsthattheprincipleofgauge
invarianceisoneofthejewelsoftheoreticalphysics; itisourguidinglightincon-
structingtheoriesofthethestrong,weak, electromagnetic(andevengravitational)
interactions.
Havingdecidedto usethegauge transformationtoset A 0 =0, weproceedto
constructtheHamiltonian. Themaindifferenceinprocedure,comparedtothatin
Lecture1,isthatordinaryderivativesarereplacedbyfunctionalderivatives.Thus
Pi =δL
δA ̇i
= A ̇i
= Ei (25.38)and
H =
{∫
d^3 xPi(x)A ̇i(x)}
−L=
1
2
∫
d^3 x[E%^2 +B%^2 ]=1
2
∫
d^3 x[P%^2 +(∇×A%)^2 ] (25.39)