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,