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.