4.3. THEDIRACDELTAFUNCTION 53
Itisreasonabletoaskwhyoneshouldgotothetroubleofintroducingsequences.
WhynottaketheL→∞limitrightaway,ineq. (4.40),anddefine
δ(x−y)={
∞ if x=y
0 if x+=y(4.42)
inanalogytotheKroneckerdeltaδij?TheansweristhattheDiracdeltafunctionis
not,strictlyspeaking,afunction,anymorethan∞isanumber,andtheintegralof
therighthandsideof(4.42)isill-defined.Ifonetreats∞asanordinarynumber,it
iseasytoproducecontradictions,e.g.
1 ×∞= 2 ×∞ =⇒ 1 = 2 (4.43)Instead,onemustthinkof∞asthelimitofasequenceofnumbers,butwhichisnot
itselfanumber.Therearemanydifferentsequenceswhichcanbeused,e.g.
1 , 2 , 3 , 4 , 5 ,....
or 2 , 4 , 6 , 8 , 10 ,....
or 1 , 4 , 9 , 16 , 25 ,... (4.44)allofwhichhave∞astheirlimit. Similarly,therearemanydifferentsequencesof
functionstendingtothe samelimit,whichisnotafunctionintheordinarysense.
Suchlimitsareknownas”generalizedfunctions,”ofwhichtheDiracdeltafunction
isoneveryimportantexample.
Letuscheckthat(4.40)and(4.41)satisfythedefiningproperty(4.39)fordelta
sequences. Forthesequenceofgaussians(4.40)
lim
L→∞∫∞−∞dy√
L
πe−L(y−x)2
f(y)= lim
L→∞∫∞−∞dz√
L
πe−Lz2
f(x+z)= lim
L→∞∫∞−∞dz√
L
πe−Lz2[
f(x)+df
dxz+1
2
d^2 f
dx^2z^2 +...]= lim
L→∞√
L
π[√
π
Lf(x)+ 0 +1
4 L
√πLd^2 f
dx^2+...
]= f(x) (4.45)wherewehavechangedvariablesz =y−x, expandedf(x+z)inaTaylorseries
aroundthepointx,andusedthestandardformulasofgaussianintegration:
∫∞
−∞dze−cz
2
=√
π∫ c
∞
−∞dzze−cz2
= 0
∫∞−∞dzz^2 e−cz
2
=1
2 c√
π
c