QMGreensite_merged

24.2. STATIONARYPHASEANDTHEFUNCTIONALDERIVATIVE 375

casejustmeansthedisplacementofthestringatpointxandtimet,byφ(x,t).The
actionofthestringisknowntobe

S[φ]=μ

∫

dt′

∫

dx′

{

1

2

(∂t′φ(x′,t′))^2 −

v^2

2

(∂x′φ(x′,t′))^2

}

(24.44)

whereμisthe stringmassperunitlength. Then the equationofmotionfor the
stringisfoundfromtheconditionthattheactionisstationarywithrespecttosmall
variationsinφ(x,t),i.e.

0 =

δ

δφ(x,t)

S[φ(x′,t′)]

= lim

!→ 0

S[φ(x′,t′)+!δ(x−x′)δ(t−t′)]−S[φ(x′,t′)]

!

= μ

∫

dt′

∫

dx′

{

1

2

δ

δφ(x,t)

(∂t′φ(x′,t′))^2 −

v^2

2

δ

δφ(x,t)

(∂x′φ(x′,t′))^2

}

= μ

∫

dt′dx′

{

∂t′φ(x′,t′)∂t′

δφ(x′,t′)

δφ(x,t)

−v^2 ∂x′φ(x′,t′)∂x′

δφ(x′,t′)

δφ(x,t)

}

(24.45)

andusing
δφ(x′,t′)
δφ(x,t)

=δ(x−x′)δ(t−t′) (24.46)

andtheproperty
f(x)∂xδ(x−y)=−[∂xf(x)]δ(x−y) (24.47)

weobtain

0 =μ

∫

dx′dt′

{

−∂^2 t′φ(x′,t′)+v^2 ∂x^2 ′φ(x′,t′)

}

δ(x−x′)δ(t−t′) (24.48)

whichfinallyleadsustotheclassicalequationofmotion,namely,thewaveequation

∂^2 φ

∂t^2

−v^2

∂^2 φ

∂x^2

= 0 (24.49)

ItsnoweasytounderstandtherelationshipbetweentheFeynmanpathintegral
andclassicalphysics.Letsaskthequestionofwhichpaths,outofallpossiblepaths
between two points(x 1 ,t 1 )and (x 2 ,t 2 ), givethe largest contribution to thepath
integral ∫

Dx(t)eiS[x(t)]/ ̄h (24.50)

Nowthecontributionofeachpathtotheintegralhasthesamemagnitude,sinceeiS/ ̄h
isjustacomplexnumberofmodulus1. However,ifS>> ̄h,thenformostpathsa
smallvariationδx(t)inthepathwillcauseanenormouschangeinthephaseS/ ̄hthe
integrand. Therefore,thecontributionsfromnearbypathsoscillatewildlyinphase,