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
(x, y)dτ
where (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
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