11
Perturbation Theory and Feynam Diagrams
We now turn our attention to the dynamics of a quantum field theory. All of
the results that we will derive in this section apply equally to both relativistic
and nonrelativistic theories with only minor changes. Here we will use the path
integrals approach we developed in previous chapters.
The properties of any field theory can be understood if the
N
point Green
functions are known
G
N
(
x
1
, . . . , x
N
) =
(
0

T φ
(
x
1
)
. . . φ
(
x
N
)

0
)
(1)
Much of what we will do below can be adapted to any field theory of interest.
We will discuss in detail the simplest case, the relativistic selfinteracting scalar
field theory. It is straightforward to generalize this to other theories of interest.
we will only give a summary of results for the other cases.
11.1
The Generating Functional in Perturbation Theory
The
N
point function of a scalar field theory,
G
N
(
x
1
, . . . , x
N
) =
(
0

T φ
(
x
1
)
. . . φ
(
x
N
)

0
)
,
(2)
can be computed from the generating functional
Z
[
J
]
Z
[
J
] =
(
0

T e
i
integraldisplay
d
D
xJ
(
x
)
φ
(
x
)

0
)
(3)
In
D
=
d
+ 1dimensional Minkowski spacetime
Z
[
J
] is given by the path
integral
Z
[
J
] =
integraldisplay
D
φ e
iS
[
φ
] +
i
integraldisplay
d
D
xJ
(
x
)
φ
(
x
)
(4)
where the action
S
[
φ
] is the action for a relativistic scalar field. The
N
point
function, Eq.(1), is obtained by functional differentiation,
i.e.,
G
N
(
x
1
, . . . , x
N
) = (
−
i
)
N
1
Z
[
J
]
δ
N
δJ
(
x
1
)
. . . δJ
(
x
N
)
Z
[
J
]
vextendsingle
vextendsingle
vextendsingle
J
=0
(5)
Similarly, the Feynman propagator
G
F
(
x
1
−
x
2
), which is essentially the 2point
function, is given by
G
F
(
x
1
−
x
2
) =
−
i
(
0

T φ
(
x
1
)
φ
(
x
2
)

0
)
=
i
1
Z
[
J
]
δ
2
δJ
(
x
1
)
δJ
(
x
2
)
Z
[
J
]
vextendsingle
vextendsingle
vextendsingle
J
=0
(6)
Thus, all we need to find is to compute
Z
[
J
].
We will derive an expression for
Z
[
J
] in the simplest theory, the relativistic
real scalar field with a
φ
4
interaction, but the methods are very general. We
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
will work in Euclidean spacetime (
i.e.,
in imaginary time) where the generating
function takes the form
Z
[
J
] =
integraldisplay
D
φ e
−
S
[
φ
] +
integraldisplay
d
D
xJ
(
x
)
φ
(
x
)
(7)
where
S
[
φ
] now is
S
[
φ
] =
integraldisplay
d
D
x
bracketleftbigg
1
2
(
∂φ
)
2
+
m
2
2
φ
2
+
λ
4!
φ
4
bracketrightbigg
(8)
In the Euclidean theory the
N
point functions are
G
N
(
x
1
, . . . , x
N
) =
(
φ
(
x
1
)
. . . φ
(
x
N
)
)
=
1
Z
[
J
]
δ
N
δJ
(
x
1
)
. . . δJ
(
x
N
)
Z
[
J
]
vextendsingle
vextendsingle
vextendsingle
J
=0
(9)
Let us denote by
Z
0
[
J
] the generating action for the free scalar field, with action
S
0
[
φ
]. Then
Z
0
[
J
]
=
integraldisplay
D
φ e
−
S
0
[
φ
] +
integraldisplay
d
D
xJ
(
x
)
φ
(
x
)
=
bracketleftbig
Det
(
−
∂
2
+
m
2
)bracketrightbig
−
1
/
2
e
1
2
integraldisplay
d
D
x
integraldisplay
d
D
yJ
(
x
)
G
0
(
x
−
y
)
J
(
y
)
(10)
where
∂
2
is the Laplacian operator in
D
dimensional Euclidean space, and
G
0
(
x
−
y
) is the free field Euclidean propagator (
i.e.,
the Green function)
G
0
(
x
−
y
) =
(
φ
(
x
)
φ
(
y
)
)
0
=
(
x

1
−
∂
2
+
m
2

y
)
(11)
where the subindex label 0 denotes a free field expectation value.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Leigh
 Quantum Field Theory, Perturbation theory, twopoint function

Click to edit the document details