whereU+(t,q−q 0 ) is the retarded propagator given by equations 12.12 and
12.13. Since
Dψ+(q,t) = (Dθ(t))ψ(q,t) +θ(t)Dψ(q,t) =iδ(t)ψ(q,t) =iδ(t)ψ(q,0)
ψ+(q,t) is a solution of 12.14 with
J(q,t) =iδ(t)ψ(q,0), Ĵ(ω,k) =
i
√
2 π
ψ ̃(k,0)
Using 12.16 to get an expression forψ+(q,t) in terms of the Green’s function
we have
ψ+(q,t) =
1
2 π
∫+∞
−∞
∫+∞
−∞
Ĝ(ω,k)√i
2 π
ψ ̃(k,0)e−iωteikqdωdk
=
(
1
2 π
) 2 ∫+∞
−∞
(∫+∞
−∞
∫+∞
−∞
iĜ(ω,k)e−iωteik(q−q
′)
dωdk
)
ψ(q′,0)dq′
Comparing this to equations 12.12 and 12.13, we find that the Green’s func-
tion that will give the retarded solutionψ+(q,t) is
Ĝ+(ω,k) = lim
→ 0 +
1
ω−k
2
2 m+i
and is related to the retarded propagator by
Û(ω,k) = i
2 π
Ĝ+(ω,k)
One can also define an “advanced” Green’s function by
Ĝ−= lim
→ 0 +
1
ω− 2 km^2 −i
and the inverse Fourier transform ofĜ−Ĵwill also be a solution to 12.14. Tak-
ing the difference between retarded and advanced Green’s functions gives an
operator
∆ =̂ i
2 π
(Ĝ+−Ĝ−)
with the property that, for any choice ofJ, ∆J will be a solution to the
Schr ̈odinger equation (since it is the difference between two solutions of the
inhomogeneous equation 12.14). The properties of ∆ can be understood by
using 12.11 to show that
∆ =̂ δ(ω−k
2
2 m