25.2. THEQUANTIZATIONOFLIGHT 385
ofAμ,whose stationarityconditionsuppliesthe othertwoof Maxwell’s equations,
namelyGauss’andAmpere’sLaw
(Note: From here on, I am going to adopt a system of units which is
popularamongparticlephysicists,inwhich ̄h=c=1.)
Thisactionisgivenby
S =
∫
dtL[A,A ̇]=1
2
∫
dt∫
d^3 x[E%^2 −B%^2 ]=1
2
∫
dt∫
d^3 x[(∇A 0 −∂tA%)^2 −(∇×A%)^2 ] (25.32)ItseasytocheckthatthetworemainingMaxwellequations(Gauss’LawandAmpere’s
Law)aregivenbythestationaryphasecondition
δS
δA 0= 0 =⇒ ∇·E%= 0
δS
δAk= 0 =⇒ ∂tE−∇×B= 0 (25.33)ThisactioncanbeusedforFeynmanpath-integralquantizationoftheelectromagnetic
field,butletusinsteadderiveaSchrodingerequation.Forthatpurposewewouldlike
togotheHamiltonianformulation,butatthisstageweencounterastupidtechnical
complication. InordertogofromtheactiontotheHamiltonian,wehavetodefine
momenta
Pμ=δL
δA ̇μ(25.34)
TheproblemisthattheLagrangiancontainsnotimederivativeofthescalarpotential
A 0 .ThismeansthatP 0 =0!
Inthiscourse,thefactthatP 0 = 0 isjustanannoyingtechnicality,whichwehave
todealwithinsomewaybeforeproceeding.Thereisactuallyaconsiderablebodyof
theoryonthispreciselythissortofphenomenoninwhatareknownas”constrained
dynamicalsystems,”but,tomakealongstoryshort,whatitindicatesisthatnotall
fieldconfigurationsAμarephysicallydistinguishable.LetAμbeafieldconfiguration,
andA′μbeanotherfieldconfiguration.Ifonecanfindafunctionφ(x,t)suchthatthe
twofieldconfigurationsarerelatedbygaugetransformation
A′μ(x,t)=Aμ(x,t)+∂μφ(x,t) (25.35)thenAandA′haveexactlythesameelectricandmagneticfields.Theyare,therefore,
physicallyequivalent.Fourdegreesoffreedomperpointisstilltoomuch,atleastone