Definition(Retarded propagator).The retarded propagator is given by
U+(t,qt−q 0 ) =
{
0 t < 0
U(t,qt−q 0 ) t > 0
This can also be written
U+(t,qt−q 0 ) =θ(t)U(t,qt−q 0 )
whereθ(t) is the step-function
θ(t) =
{
1 t > 0
0 t < 0
We will use an integral representation ofθ(t) given by
θ(t) = lim
→ 0 +
i
2 π
∫+∞
−∞
1
ω+i
e−iωtdω (12.10)
To derive this, note that as a distribution,θ(t) has a Fourier transform given by
lim
→ 0 +
i
√
2 π
1
ω+i
since the calculation
1
√
2 π
∫+∞
−∞
θ(t)eiωtdω=
1
√
2 π
∫+∞
0
eiωtdω
=
1
√
2 π
(
−
1
iω
)
makes sense forωreplaced by lim→ 0 +(ω+i) (or, for real boundary values of
ωcomplex, taking values in the upper half-plane). Fourier inversion then gives
equation 12.10.
Digression.The integral 12.10 can also be computed using methods of complex
analysis in the variableω. Cauchy’s integral formula says that the integral about
a closed curve of a meromorphic function with simple poles is given by 2 πitimes
the sum of the residues at the poles. Fort < 0 , sincee−iωtfalls off exponentially
ifωhas a non-zero positive imaginary part, the integral along the realωaxis will
be the same as for the semi-circleC+closed in the upper half-plane (with the
radius of the semi-circle taken to infinity).C+encloses no poles so the integral
is 0.