388 CHAPTER25. AGLIMPSEOFQUANTUMFIELDTHEORY
Onlystatessatisfyingthiscondition,inA 0 = 0 gauge,aretoberegardedasphysical
states.
ToseewhattheGaussLawconstraintimplies,letssubdividethevectorpotential
intoatransversepartAT andalongitudinalpartAL:
Ai(x)=ATi(x)+ALi(x) (25.46)
where
∇·AT= 0 ∇×AL= 0 (25.47)
Definealso
Ei(x) = EiT(x)+EiL(x)
= −i
δ
δAT(x)
−i
δ
δAL(x)
(25.48)
TheGaussLawconstraintbecomes
0 = ∇·EΨ
= ∇·ELΨ
= −i∂i
δ
δALi
Ψ (25.49)
whichissatisfiedifΨisindependentofthelongitudinaldegreeoffreedomAL,i.e.
Ψ[Ai]=Ψ[AT] (25.50)
Asinthecaseoftheone-dimensionalsolid,wewanttobeabletoseparatevari-
ables,andwritetheHamiltonianasasumofharmonicoscillators.Again,thisisdone
bygoingtotheFourier-transformedvariables
Ai(x) = ATi(x)+ALi(x)
=
∫ d (^3) k
(2π)^3
[ATi(k)+ALi(k)]eikx (25.51)
with
%k·A%T= 0 %k×A%L= 0 (25.52)
TheFouriercomponentA%(k)canberegardedastheamplitudeofawavetravelingin
the%kdirection. A%T isthecomponentperpendiculartothedirectionofpropagation,
whileA%Listhecomponentparalleltothedirectionofpropagation.
SubstitutetheFourier-transformedAL,T andEL,T intotheHamiltonian,andwe
findthatittakestheform
H=
1
2
∫ d (^3) k
(2π)^3
[ETi(k)EiT(−k)+EiL(k)ELi(−k)+k^2 ATi(k)ATi(−k)] (25.53)