QMGreensite_merged

25.2. THEQUANTIZATIONOFLIGHT 389

ButEL=−iδ/δAL,andwehaveseenthatΨ[A]doesn’tdependonAL,bytheGauss
Lawconstraint.Sothetime-independentSchrodingerequationbecomes

1

2

∫ d (^3) k
(2π)^3
[
−
δ^2
δATi(k)δATi(−k)
+k^2 ATi(k)ATi(−k)
]
Ψ[AT]=EΨ[AT] (25.54)
Onceagain,theHamiltonianhastheformofasumoverharmonicoscillatorterms.
Theground-statesolution,whichmustbeagaussianforanysystemof uncoupled
harmonicoscillators,canbewrittendownalmostbyinspection:
Ψ 0 [A]=Nexp
[
−

1

2

∫

d^3 k|k|ATi(k)ATi(−k)

]

(25.55)

andthiscaneasilybeconvertedbacktoafunctionalofA(x):

Ψ 0 = Nexp

[

−

1

2

∫

d^3 k

1

|k|

(k×AT(k))·(k×AT(−k))

]

= Nexp

[

−

1

2

∫

d^3 k

(

4 π

∫ d (^3) z
(2π)^3

1

z^2

eikz

)(∫

d^3 x 1 i∇x 1 ×A(x 1 )eikx^1

)

·

(∫

d^3 x 2 i∇x 2 ×A(x 2 )e−ikx^2

)]

(25.56)

Integrationoverkandzleads,finally,tooneofmyfavoriteequationsinphysics,the
groundstatewavefunctionaloftheelectromagneticfield

Ψ 0 [A]=Nexp



−^1

4 π

∫

d^3 xd^3 y

B%(x)·B%(y)

|x−y|^2



 (25.57)

SubstitutingthisstatebackintotheSchrodingerequation,weeasilyfindthezero-
pointgroundstateenergy

E 0 =

1

2

∫ d (^3) k
(2π)^3
|k|δ^3 (0) (25.58)
whichisinfinite.Thisinfinityisnotsurprising,becausewearequantizinganinfinite
numberofdegreesoffreedom,eachofwhichhasafinitezero-pointenergy.
Thegroundstateofaquantumfieldtheoryisknown,quiterightly,asthe”vac-
uum”state;itisthelowestenergystateattainable.Italsoshowsusthatthevacuum
isnot”nothing.”Evenintheabsenceofanysources,eveninthelowestenergystate
possible,therearestillquantumfluctuationsoftheelectromagneticfield.Intheend,
thisisaconsequenceoftheUncertaintyPrinciple.Ifthevectorpotentialwereevery-
wherevanishing,sothattheuncertainty∆A=0,theuncertaintyoftheelectricfield
(theconjugatemomentum)wouldbeinfinite.Thiswouldgiveanenergyexpectation
valueevenmorestronglydivergentthanE 0 ;eachdegreeoffreedomwouldcontribute