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).
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