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