Master Dissertation

and for each a > 0,
ρ(aδ)
ρ(δ)
= a2−n
Thus in R 4 , ω = −2. Since the singular order is negative the splitting may
be done by multiplication by the Θ-function (7.16). Hence the retarded
part of the first term of (8.3) is
R2(x1, x2) = −ie 2 : ψ(x1)γ µ ψ(x1)ψ(x2)γ ν ψ(x2) : ΘD0(x1 − x2)
Further R ′ (x1, x2) is given by (8.2) by inspecting each term. By (5.12)
T2(x1, x2)
= R2(x1, x2) − R ′ (x1, x2)
= −ie 2 : ψ(x1)γ µ ψ(x1)ψ(x2)γ ν ψ(x2) :
ΘD0(x1 − x2) + D (+)
0 (x2 − x1) .
It is not the aim of this project to introduce Feynman propagators but the
reader with knowledge of these might notice that the expression in
brackets is actually a Feynman propagator describing the exchange of a
photon between electrons.
Figure 8.1: Photon exhange.
8.2 The Adiabatic Limit
In the introduction of the S-matrix we used test-functions g ∈ S(R 4 ),
so-called switching functions. As the name infers they switch off
long-range interaction to prevent infrared divergences 3 . Of course this is
not a good model and we need to consider the so-called adiabatic limit
g → 1 to take long-range interaction like the Coulomb-potential into
account. We show how we may carry out the adiabatic limit. In practise it
can me done by taking the so-called scaling limit.
Let g0 ∈ S(R 4 ) be a fixed test-function such that g0(0) = 1. Then we let
g(x) := g0(ɛx) and take the scaling limit, that is, we let ɛ → 0.
Calculations are in practise usually done in momentum space. With this in
mind we note that
ˆg(k) =
g(x)e −ix·k d 4 x =
g0(ɛx)e −ik·x d 4 x = 1
ɛ
k
ˆg0( ). (8.4)
4 ɛ
3 By infrared divergence we simply mean a divergence due to physical phenomena at
very long distances or because of contributions from objects with very small energy
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