48 2. THEORIES, LAGRANGIANS AND COUNTERTERMS
Recall that we can write the propagator of a massless scalar field theory
on a Riemannian manifold M as an integral of the heat kernel. If x, y ∈ M
are distinct points, then
P (x, y) =
∞
0
Kt(x, y)dt
where Kt ∈
C∞(M 2)
is the heat kernel. This expression of the propagator
is sometimes known as the Schwinger representation, and the parameter t as
the Schwinger parameter. One can also interpret the parameter t as proper
time, as we will see shortly.
The heat kernel Kt(x, y) is the probability density that a particle in
Brownian motion on M, which starts at x at time zero, lands at y at time
t. Thus, we can rewrite the heat kernel as
Kt(x, y) =
f:[0,t]→M
f(0)=x,f(t)=y
DWienerf
where DWienerf is the Wiener measure on the path space.
We can think of the Wiener measure as the measure for a quantum field
theory of maps
f : [0,t] → M
with action given by
E(f) =
t
0
df, df .
Thus, we will somewhat loosely write
Kt(x, y) =
f:[0,t]→M
f(0)=x,f(t)=y
e−E(f)
where we understand that the integral can be given rigorous meaning using
the Wiener measure.
Combining these expressions, we find the desired expression for the prop-
agator as a one-dimensional functional integral:
P (x, y) =
φ∈C∞(M)
eS(φ)φ(x)φ(y)
=
∞
t=0
f:[0,t]→M
f(0)=x,f(t)=y
e−E(f).
This expression is the core of the world-line formulation of quantum field
theory. This expression tells us that the correlation between the values of
the fields at points x and y can be expressed in terms of an integral over
paths in M which start at x and end at y.
If we work in Lorentzian signature, we find the (formal) identity
φ∈C∞(M)
eS(φ)iφ(x)φ(y)
=
∞
t=0
f:[0,t]→M
f(0)=x,f(t)=y
eiE(f).
This expression is difficult to make rigorous sense of; I don’t know of a
rigorous treatment of the Wiener measure when the target manifold has
Lorentzian signature.
