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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)