Self-Consistent Field Theory and Its Applications by M. W. Matsen

1.11 Appendix: The Calculus of Functionals 77
which is derived using Fourier transforms, Eqs. (1.80) and (1.81), combined with the sifting
property from Eq. (1.329). It is in fact the sifting property that is the essential characteristic
by which the delta function is defined. The functional version of this sifting property is
∫
Df δ[f − g]F[f] =F[g] (1.333)
where F[f] is an arbitrary functional. The generalization of Eq. (1.332) is arrived at by first
constructing the approximate discrete version,
δ[f] ≈ δ(f 0 )δ(f 1 ) ···δ(f M )
∫
( )
1
M∑
=
dk
(2π) (M+1) 0 dk 1 ···dk M exp i k m f m (1.334)
m=0
Then taking the limit M →∞gives
∫ ( ∫
)
δ[f] ∝ Dk exp i dx k(x)f(x)
(1.335)
where now we have a functional integral over k(x). Notice that there is a slight difficulty
in taking the limit; as M increases, the proportionality constant approaches zero, although
this is compensated for by the increasing number of integrations. In practice, this is not a
problem, because functional integrals are always evaluated for finite M, but it does prevent
us from specifying a proportionality constant in Eq. (1.335). Nevertheless, the constant is
unimportant in the applications to SCFT, and so Eq. (1.335) is sufficient for our purposes.