It is worth noting that the propagator, and therefore the solution, depends only on the propagator
, which comes from the kinetic term. This is, of course, because we are perturbing about a solution
to the free theory.
We can be more precise by defining 1
/
as
x
Π(
x, y
) =

δ
4
(
x

y
)
(2.8.27)
38
so that
Π(
x, y
) =
Z
d
4
k
(2
π
)
4
e
ik
(
x

y
)
1
k
2
(2.8.28)
Note that really Π(
x, y
) = Π(
y, x
) = Π(
x

y
).
Now Π allows us to write
h
0
(
x
) =

Z
d
4
y
Π(
x, y
)
J
(
y
)
(2.8.29)
as we can check immediately that this satisfies the equation of motion. Thus our more general
solution is
h
(
x
)
=

Z
d
4
y
Π(
x, y
)
J
(
y
)
(2.8.30)
+
λ
Z
d
4
w
Π(
x, w
)
Z
d
4
y
Π(
w, y
)
J
(
y
)
Z
d
4
z
Π(
w, z
)
J
(
z
) +
· · ·
which is what we meant earlier when we wrote a formal solution in terms of 1
/
.
This can be represented by pictures where the propagator Π(
x, y
) looks like a line, while the
currents
J
look like insertions that come together at points such as
x
and
w
. These are our first
example of
Feynman diagrams
. The rules for associating mathematical expressions with the pictures
are called
Feynman rules
. These diagrams give some physical intuition, and they also allow us to
generate all the mathematical expressions allowed to any order in
λ
, via the rules
1. Draw a point
x
and a line from
x
to a new point
x
i
.
2.
Either truncate a line at a source
J
or let the line branch into two lines, adding a new point a
factor of
λ
.
3. Repeat the previous until all lines truncate at sources.
4.
The final value for
h
(
x
) is given by summing up graphs with lines associated with propagators,
internal points integrated over, and all points external points except
x
associated with
J
.
One can solve the EoM for a classical field by drawing these pictures. When we move to QFT, the
main difference will be that lines can loop back on themselves.
2.9
Overview of Scattering and Perturbation Theory
We learned about symmetries, canonical quantization, and the quantization of a free quantum
scalar field, which describes a Fock space of free relativistic particles. We saw examples in classical
field theory where we obtained a Feynman diagram perturbation series that solves the classical
field equations, and we studied dimensional analysis to see what interactions are important at long
distances.
39
Now we would like to develop the perturbative description of QFT. What sort of processes should
we study?
For a variety of reasons we will study
Scattering
in this course. If this were primarily a condensed
matter physics course we might study other questions... because the best motivation for studying
scattering is that if we are going to do ‘particle physics’ or ‘highenergy physics’, the whole point is
to pursue reductionism to the extreme. We would like to ‘see what stuff is made of’. To do that we
need a microscope. But due to the uncertainty principle
δx
·
δp
≥
~
2
(2.9.1)
we cannot look at matter at very short distances, or ‘take it apart’, without very large momenta,
and thus very large energies (hence the name highenergy physics). But if we study matter using
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 Fall '12
 Halperin
 Physics, mechanics, Quantum Field Theory