74 2. THEORIES, LAGRANGIANS AND COUNTERTERMS
The adjointness properties of the differential operators D , DE1 imply
that D Kt is symmetric.
The propagator for the theory is
P (ε, L) =
L
ε
D Kt ∈ E
⊗2
⊗ A
which is again symmetric.
Note that unless we impose additional positivity conditions on the oper-
ator DE1 , the heat kernel Kt may not exist at t = ∞; thus, the propagator
P (ε, ∞) may not exist. In almost all examples, however, the operator DE1
is positive, and so the heat kernel K∞ does exist.
If we specialize the case of the free scalar field theory on a Riemannian
manifold (M, g0), then, as we have seen, E1 =
C∞(M),
DE1 =
g0
is the
non-negative Laplacian for the metric g0. In this example the operator D
is the identity. Thus, the propagator prescribed by this general definition
coincides with the propagator presented in our earlier analysis of the free
field theory.
13.4. As before, we can define the algebra
O(E , A ) = Hom(E
⊗n,
A )Sn
of all functionals on E with values in A . Here Hom denotes the space of
continuous linear maps. The properties of the symmetric monoidal category
of nuclear spaces, as detailed in Appendix 2, show that
O(E , A ) =
n
Symn
E
∨
⊗ A .
There is a subspace
Ol(E , A ) ⊂ O(E , A )
of A -valued local action functionals, defined as follows.
Definition 13.4.1. A functional Φ ∈ O(E , A ) is a local action func-
tional if, when we expand Φ as a sum Φ =
∑
Φn of its homogeneous com-
ponents, each
Φn : E
⊗n
→ A
can be written in the form
Φn(e1,...,en) =
k
j=1
M
(D1,je1) · · · (Dn,jen)dμ
where
dμ ∈ Densities(M)
is some volume element on M, and each
Di,j : E ⊗ A →
C∞(M)
⊗ A
is an A -linear differential operator.
