86 2. THEORIES, LAGRANGIANS AND COUNTERTERMS

14.7. One can ask how the definition of theory depends on the choice

of fake heat kernel. It turns out that there is no dependence. Let Kt, Kt

be two heat fake heat kernels, with associated propagators P (ε, L), P (ε, L).

We have seen that Kt − Kt vanishes to all orders at t = 0. It follows

that P (0,L) − P (0,L) is smooth, that is, an element of E ⊗ E ⊗ A . Also,

P (0,L) − P (0,L) has proper support. Further, as L → 0, P (0,L) and all of

its derivatives vanish faster than any power of L.

Thus, the renormalization group operator

W P (0,L) − P (0,L), − : Op

+(Ec,

A )[[ ]] → Op

+(Ec,

A )[[ ]]

is well-defined.

Lemma 14.7.1. Let {I[L]} be a collection of effective interactions defin-

ing a theory for the propagator P (ε, L). Then,

I[L] = W P (0,L) − P (0,L), I[L]

defines a theory for the propagator P (ε, L).

Further, the small L asymptotic expansion of I[L] is the same as that of

I[L].

Proof. The renormalization group equation

W (P (ε, L),I[ε]) = I[L]

is a corollary of the general identity,

W

(

P, W

(

P , I

))

= W

(

P + P , I

)

for any P, P ∈ E ⊗ E ⊗ A of proper support, and any I ∈ Op

+(E

, A )[[ ]].

The statement about the small L asymptotics of I[L] – and hence the locality

axiom which says that I[L] defines a theory – follows from the locality axiom

for I[L] and the fact that P (0,L) and all its derivatives tend to zero faster

than any power of L, as L → 0.

14.8. Now let U ⊂ M be any open subset. The free field theory on M –

defined by the vector bundle E, together with certain differential operators

on it – restricts to define a free field theory on U ⊂ M. One can ask if a

theory on M will restrict to one on U as well.

The following proposition, which will be proved later, shows that one

can do this.

Proposition 14.8.1. Let U ⊂ M be an open subset.

Given any theory

{I[L] ∈ Op

+(E

, A )[[ ]]}

on M (defined using any fake heat kernel on M), there is a unique theory

{IU [L] ∈ Op

+(Γ(U,

E |U ), A )[[ ]]}