Quantum Mechanics for Mathematicians

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.