50 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

have a metric, that is, a length along each edge. This length on the edge of

the graph corresponds to time traversed by the particle on this edge in its

internal clock.

Measuring the value of a field φ at a point x ∈ M corresponds, in the

world-line point of view, to observing a particle at a point x. We would like to

find an expression for the correlation function E(x1, . . . , xn) in the world-line

point of view. As always in quantum field theory, one should calculate this

expectation value by summing over all events that could possibly happen.

Such an event is described by a world-graph with end points at x1,...,xn.

Since only three particles can interact at a given point in space-time, such

world-graphs are trivalent. Thus, the relevant world-graphs are trivalent,

have n external edges, and the end points of these external edges maps to

the points x1,...,xn.

Thus, we find that

E(x1, . . . , xn) =

γ

1

|Aut γ|

−χ(γ)

g∈Met γ f:γ→M

e−E(f).

Here, the sum runs over all world-graphs γ, and the integral is over those

maps f : γ → M which take the endpoints of the n external edges of γ to

the points x1,...,xn.

The symbol Met(γ) refers to the space of metrics on γ, in other words,

to the space

RE0γ) ( T (γ)

where E(γ) is the set of internal edges of γ, and

T (γ) is the set of tails.

If γ is a metrized graph, and f : γ → M is a map, then E(f) is the sum

of the energies of f restricted to the edges of γ, that is,

E(f) =

e∈E(γ)

l(e)

0

df, df .

The space of maps f : γ → M is given a Wiener measure, constructed from

the usual Wiener measure on path space.

This graphical expansion for the correlation functions is only a formal

expression: if γ has a non-zero first Betti number, then the integral over

Met(γ) will diverge, as we will see shortly. However, this graphical expansion

is precisely the expansion one finds when formally applying Wick’s lemma

to the functional integral expression for E(x1, . . . , xn). The point is that we

recover the propagator when we consider the integral over all possible maps

from a given edge.

In Lorentzian signature, of course, one should use

eiE(f)

instead of

e−E(f).

6.5. As an example, we will consider the path integral f : γ → M where

γ is the metrized graph