Pictures Paths Particles Processes

80 November 7, 2013
For large m|⃗x|, the propagator therefore decreases exponentially, while for small
m|⃗x|, we have
Π(⃗x) ≈ ¯h ( ) 1
2π log , D = 2 ,
m|⃗x|
Π(⃗x) ≈ ¯hΓ ( D
2 − 1) x 2−D , D ≥ 3 . (2.67)
4 π D/2
In every dimension, the propagator is normalized in the same way :
(
∫
Π(⃗x) d D x = ¯h ∫ ∫ exp i ⃗ )
k · ⃗x
(2π) D d D x d D k
| ⃗ k| 2 + m 2
=
¯h
(2π) D
2.4.3 Three examples
∫
d D k (2π)D δ D ( ⃗ k)
| ⃗ k| 2 + m 2 = ¯h m 2 . (2.68)
We may consider where the evolution in Feynman rules has taken us so far.
We can best illustrate this by inspecting three examples. In the first place, we
of course have the lowest-order (no-loop) two-point function, the propagator,
given by the diagram
A 1 = x 1
x 2
(2.69)
which equals
A 1 = Π(⃗x 1 − ⃗x 2 ) . (2.70)
Next, we we can look at the lowest-order contributions to the four-point function:
A 2 = 〈ϕ(⃗x 1 )ϕ(⃗x 2 )ϕ(⃗x 3 )ϕ(⃗x 4 )〉 in ϕ 4 theory. According to the standard
rules, we can obtain this Green’s function by writing down all Feynman diagrams
with four external lines, and no source vertices. In lowest order of the
loop expansion, this Green’s function contains four diagrams :
x 1
x 2
x 1
x
A 2 =
+
x x 4
3
x 1
x
2
x
x
+
4
+
3
x x 4
3
and, upon implementation of the Feynman rules, evaluate to
A 2 = Π(⃗x 1 − ⃗x 2 ) Π(⃗x 3 − ⃗x 4 )
+ Π(⃗x 1 − ⃗x 3 ) Π(⃗x 2 − ⃗x 4 )
x
x
2
1
2
y
x
x
3
4
, (2.71)
+ Π(⃗x 1 − ⃗x 4 ) Π(⃗x 3 − ⃗x 2 )
− λ ∫
4
d D ⃗y Π(⃗x 1 − ⃗y) Π(⃗x 2 − ⃗y) Π(⃗x 3 − ⃗y) Π(⃗x 4 − ⃗y) . (2.72)
¯h