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Chapter 24

The Feynman Path Integral

InclassicalphysicstheEuler-Lagrangeequationsarederivedfromtheconditionthat
theaction S[x(t)]shouldbe stationary. Thesesecondorderequations areequiva-
lenttothefirstorderHamiltonequationsofmotion,whichareobtainedbytaking
appropriatederivativesof theHamiltonianfunctionH[q,p]. NowtheHamiltonian
isaquantitythatwehaveencounteredfrequentlyinthesepages. Butwhatabout
theaction? Itwouldbesurprisingifsomethingsofundamentaltoclassicalphysics
hadnopart atall toplayinquantumtheory. Infact, theaction hasthecentral
roleinanalternativeapproachtoquantummechanicsknownasthe”path-integral
formulation.”
Letsstartwiththeconceptofapropagator. Giventhewavefunctionψ(x,t)at
timet,thewavefunctionatanylatertimet+Tcanbeexpressedas

ψ(x,t+T)=

∫

dyGT(x,y)ψ(y,t) (24.1)

whereGT(x,y)isknownasthepropagator,andofcourseitdependsontheHamilto-
nianoftheory. Infact,givenatime-independentHamiltonianwitheigenstates

Hφn(x)=Enφn(x) (24.2)

itseasytoseethat
GT(x,y)=

∑

n

φn(x)φ∗n(y)e−iEnT (24.3)

RichardFeynman,in1948,discoveredaverybeautifulexpressionforthepropagator
intermsofan integraloverallpathsthattheparticle canfollow,frompointyat
timet,topointxattimet+T,withanintegrand

eiS[x(t)]/ ̄h (24.4)

Aswewillsee,hisexpressioncanbetakenasanewwayofquantizingamechanical
system,equivalenttothe”canonical”approachbasedonexchangingobservablesfor
operators.

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