13. VECTOR-BUNDLE VALUED FIELD THEORIES 73

13.2. The space E2 of non-propagating fields is introduced into this

definition with an eye to future applications: none of the examples treated

in this book will have non-propagating fields. Thus, the reader will lose

nothing by ignoring the space of non-propagating fields.

For those who are interested, however, let me briefly explain the reason

for considering non-propagating fields. Let us consider a free scalar field

theory on a Riemannian manifold M with metric g0. Let us consider per-

turbing the metric to g0 + h. The action of the scalar field theory is given

by S(φ) = φ g0+hφ, as usual. Notice that the action is not just local as a

function of the field φ, but also local as a function of the perturbation h of

the metric. However, h is not treated as a quantum field, only as a classical

field: we do not consider integration over the space of metrics.

In this situation, the tensor h is said to be a background field, or (in the

terminology adopted here) a non-propagating field.

In this example, the space of fields is

E =

C∞(M)

⊕ Γ(M,

Sym2

TM)

where the space E1 of propagating fields is

C∞(M),

and the space E2 of

non-propagating fields is Γ(M,

Sym2

TM).

The action of the theory is S(φ) =

M

φ g0+hφ. This action can, as

usual, be split into quadratic and interacting parts:

S(φ) =

M

φ

g0

φ +

M

φ (

g0+h

−

g0

) φ.

The quadratic part of the action is the only part relevant to the definition

of a free field theory as presented above. As we see, the quadratic part only

depends on the propagating field φ, and not on h. However, the interaction

term depends both on φ and h. It is a general feature of the interaction

terms that they must have some dependence on φ; they canot be functions

just of the non-propagating field h. We will build this into our definition of

interactions in the presence of non-propagating fields shortly.

If we take the free theory associated to this example – given by discarding

the interacting terms in the action S – we fit it into the general definition as

follows. The operator DE1 is the Laplacian

g0

:

C∞(M)

→

C∞(M).

The

operator D is the identity map

C∞(M)

→

C∞(M).

13.3. Now that we have the general definition of a free field theory, we

can start to define the concept of effective interaction in this context. First,

we have to define the heat kernel.

If the manifold M is compact, there is a unique heat kernel

Kt ∈ E1

!

⊗ E1 ⊗

C∞(R

0) ⊗ A

for the operator DE1 .

Composing with the operator D gives an element

D Kt ∈ E1 ⊗ E1 ⊗

C∞(R

0) ⊗ A .

We will view this as an element of E ⊗ E ⊗

C∞(R

0).