QMGreensite_merged

390 CHAPTER25. AGLIMPSEOFQUANTUMFIELDTHEORY

aninfiniteamountof energy. Thegroundstate(25.57)isthelowestenergy com-
promise,consistentwiththeUncertainty Principle,betweentheenergydueto the
”potential”termB%^2 ,andtheenergyduetothe”kinetic”termE%^2 ,inthequantized
Hamiltonian.
Tofindtheexcitedstatewavefunctionalsandenergies,weneedtointroduceraising
andloweringoperators. Sincethedegreesoffreedom,bothintheHamiltonianandin
thephysicalstates,arethetransversecomponentsofthevectorpotential,weneedto
extractthesefromthefullvectorpotential.Tothisend,foragiven%k,letusintroduce
twounit(polarization)vectorswhichareorthogonalto%kandtoeachother:

%!λ(k)·%!λ

′

(k) = δλλ′

%!λ(k)·%k = 0 (25.59)

wherethesupersciptsareλ= 1 ,2.ThenwecanalwayswriteatransversevectorA%T
orE%T asasuperpositionofvectorsinthe%!^1 and%!^2 directions. Nowintroducethe
creation/destructionoperators

a(k,λ) =

1

√

2 |k|

!λi(k)

[

|k|Ai(k)+

δ

δAi(−k)

]

=

1

√

2 |k|

!λi(k)

[

|k|ATi(k)+

δ

δATi(−k)

]

a†(k,λ) =

1

√

2 |k|

!λi(k)

[

|k|Ai(−k)−

δ

δAi(k)

]

=

1

√

2 |k|

!λi(k)

[

|k|ATi(−k)−

δ

δATi(k)

]

(25.60)

Itisstraighforwardtoverifythat

1.theseoperatorshavethestandardraising/loweringcommutationrelations

[a(k,λ),a(k′,λ′)]=δλλ′δ^3 (x−x′) (25.61)

2.thedestructionoperatorsallannihilatethegroundstate

a(k,λ)Ψ 0 = 0 forall k (25.62)

3.theHamiltoniancanbeexpressed,intermsoftheseoperators,as

H=

∫

d^3 k|k|

∑

λ=1, 2

[a†(k,λ)a(k,λ)+

1

2

δ^3 (0)] (25.63)