3. WILSONIAN LOW ENERGY THEORIES 9

action. Thus, P is a distribution on M

2.

Away from the diagonal in M

2,

P is a distribution. The value P (x, y) of P at distinct points x, y in the

space-time manifold M can be interpreted as the correlation between the

value of the field φ at x and the value at y.

Feynman realized that the propagator can be written as an integral

P (x, y) =

∞

τ=0

e−τm2

Kτ (x, y)dτ

where Kτ (x, y) is the heat kernel. The fact that the heat kernel can be

interpreted as the transition probability for a random path allows us to

write the propagator P (x, y) as an integral over the space of paths in M

starting at x and ending at y:

P (x, y) =

∞

τ=0

e−τm2

f:[0,τ]→M

f(0)=x,f(τ)=y

exp −

τ

0

df

2

.

(This expression can be given a rigorous meaning using the Wiener measure).

From this point of view, the propagator P (x, y) represents the proba-

bility that a particle starts at x and transitions to y along a random path

(the worldline). The parameter τ is interpreted as something like the proper

time: it is the time measured by a clock travelling along the worldline. (This

expression of the propagator is sometimes known as the Schwinger repre-

sentation).

Any reasonable action functional for a scalar field theory can be decom-

posed into kinetic and interacting terms,

S(φ) = Sfree(φ) + I(φ)

where Sfree(φ) is the action for the free theory discussed above. From

the space-time point of view on quantum field theory, the quantity I(φ)

prescribes how particles interact. The local nature of I(φ) simply says that

particles only interact when they are at the same point in space-time. From

this point of view, Feynman graphs have a very simple interpretation: they

are the “world-graphs” traced by a family of particles in space-time moving

in a random fashion, and interacting in a way prescribed by I(φ).

This point of view on quantum field theory is the one most closely related

to string theory (see e.g. the introduction to (GSW88)). In string theory,

one replaces points by 1-manifolds, and the world-graph of a collection of

interacting particles is replaced by the world-sheet describing interacting

strings.

3.5. Let us now briefly describe how to treat effective field theory from

the world-line point of view.

In the energy-scale picture, physics at scales less than Λ is described by

saying that we are only allowed fields of energy less than Λ, and that the

action on such fields is described by the effective action

Seff

[Λ].

In the world-line approach, instead of having an effective action

Seff

[Λ]

at each energy-scale Λ, we have an effective interaction

Ieff

[L] at each