6. THE GEOMETRIC INTERPRETATION OF FEYNMAN GRAPHS 49
6.2. We should interpret these identities as follows. We should think of
particles moving through space-time as equipped with an “internal clock”;
as the particle moves, this clock ticks at a rate independent of the time pa-
rameter on space-time. The world-line of such a particle is a parameterized
path in space-time, that is, a map f : R M. This path is completely
arbitrary: it can go backwards or forwards in time. Two world-lines which
differ by a translation on the source R should be regarded as the same. In
other words, the internal clock of a particle doesn’t have an absolute starting
point.
If I = [0,τ] is a closed interval, and if f : I M is a path describing
part of the world-line of a particle, then the energy of f is, as before,
E(f) =
[0,τ]
df, df .
In quantum field theory, everything that can happen will happen, but with
a probability amplitude of
eiE
where E is the energy. Thus, to calculate
the probability that a particle starts at the point x in space-time and ends
at the point y, we must integrate over all paths f : [0,τ] M, starting
at x and ending at y. We must also integrate over the parameter τ, which
is interpreted as the time taken on the internal clock of the particle as it
moves from x to y. This leads to the expression (in Lorentzian signature)
we discussed earlier,
P (x, y) =

t=0
f:[0,t]→M
f(0)=x,f(t)=y
eiE(f).
6.3. We would like to have a similar picture for interacting theories. We
will consider, for simplicity, the
φ3
theory, where the fields are φ
C∞(M),
and the action is
S(φ) =
M

1
2
φ D φ +
1
6
φ3.
We will discuss the heuristic picture first, ignoring the difficulties of renor-
malization. At the end, we will explain how the renormalization group flow
and the idea of effective interactions can be explained in the world-line point
of view.
The fundamental quantities one is interested in are the correlation func-
tions, defined by the heuristic functional integral formula
E(x1, . . . , xn) =
φ∈C∞(M)
eS(φ)/
φ(x1) · · · φ(xn).
We would like to express these correlation functions in the world-line point
of view.
6.4. The
φ3
theory corresponds, in the world-line point of view, to a
theory where three particles can fuse at a point in M. Thus, world-lines
in the
φ3
theory become world-graphs; further, just as the world-lines for
the free theory are parameterized, the world-graphs arising in the
φ3
theory
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