QMGreensite_merged

24.2. STATIONARYPHASEANDTHEFUNCTIONALDERIVATIVE 373

andfinally,usingT=N!,wehaveintheN→∞limit

GT(x,y)=

√

m

2 πi ̄hT

eim(x−y)

(^2) /(2 ̄hT)
(24.34)
whichisinperfectagreementwiththeexpressionobtainedusingeq. (24.3).
24.2 StationaryPhase and the Functional Deriva-
tive
Afunctionfisakindofinput/outputdevice.Inputisanumberx,outputisanother
numberf(x).Ageneralizationoftheconceptoffunctionisthefunctional,whichis
alsoaninput/outputdevice. Inputisafunctionf(x),outputisanumberF[f(x)].
AnexampleofafunctionalistheactionS[x(t)]. Inputistrajectory, x(t),between
timest 1 andt 2 .Outputisanumber,theactionofthetrajectory
S[x(t)]=
∫t 2
t 1
dt

{ 1

2

mx ̇^2 −V(x(t))

}

(24.35)

Anordinaryintegralsums thevaluesof afunctionoverallpoints inacertain
range. Afunctionalintegralsumsthevaluesofafunctionaloverallfunctionsina
certainrange.TheFeynmanpathintegral
∫
Dx(t)eiS[x(t)]/ ̄h (24.36)

isafunctionalintegral.
Ifthereissucha thingas integration ofa functional, oneexpects alsoacor-
responding conceptofdifferentiation, whichwouldexpresshow fastthefunctional
changeswhenitsargument(thefunction),changesinacertainway. Thefunctional
derivativeisdefined,inanalogytoordinaryderivatives,inthefollowingway:

δF

δf(x)

≡lim

!→ 0

F[f(x′)+!δ(x−x′)]−F[f(x′)]

!

(24.37)

Thisquantity isessentiallyameasureofhow fastthefunctionalchanges whenits
argument(the function) is changed inthe region of pointx′ = x. Mostof the
usualloreaboutordinaryderivatives,e.g. thechainrule,holdstrueforfunctional
derivativesaswell.Aparticular,veryimportantcaseiswhenthefunctionaldepends
onlyonthevalueoftheinputfunctionatoneparticularpoint,e.g. F(f)=f(y).In
thiscase

δ

δf(x)

f(y) = lim!→ 0

(f(y)+!δ(y−x))−f(y)

!

= δ(x−y) (24.38)