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.