9. EXTRACTING THE SINGULAR PART 57
(2) The functionals I[P ] satisfy the following locality axiom. For any
(i, k), the support of
Supp Ii,k[P ] M
k
can be made as close as we like to the diagonal by making the
parametrix P small. More precisely, we require that, for any open
neighbourhood U of the small diagonal M M
k,
we can find some
P such that
Supp Ii,k[P ] U
for all P P .
(3) Finally, the functionals I[P ] all have smooth first derivative.
Theorem 8.2.2. This definition of a quantum field theory is equivalent
to the previous one, presented in definition 7.1.2.
More precisely, if I[L] is a set of effective interactions satisfying the
heat-kernel definition of a QFT, then if we set
I[P ] = W (P P (0,L),I[L])
for any parametrix P , the functionals I[P ] satisfy the parametrix definition
of quantum field theory presented in this section. Further, every collection
I[P ] of functionals satisfying the parametrix definition arises uniquely in
this way.
The proof of this theorem will be presented in Section 12, after the proof
of Theorem A.
9. Extracting the singular part of the weights of Feynman graphs
9.1. In order to construct the local counterterms needed for theorem A,
we need a method for extracting the singular part of the finite-dimensional
integrals wγ(P (ε, L),I) attached to Feynman graphs. This section will de-
scribe such a method, which relies on an understanding of the behaviour of
of the functions wγ(P (ε, L),I) as ε 0. We will see that wγ(P (ε, L),I) has
a small ε asymptotic expansion
wγ(P (ε, L),I)(a) gi(ε)Φi(L, a),
where the Φi(L, a) are well-behaved functions of the field a and of L. Fur-
ther, the Φi(L, a) have a small L asymptotic expansion in terms of local
action functionals.
The functionals gi(ε) appearing in this expansion are of a very special
form: they are periods of algebraic varieties. For the purposes of this book,
the fact that these functions are periods is not essential. Thus, the reader
may skip the definition of periods without any loss. However, given the
interest in the relationship between periods and quantum field theory in the
mathematics literature (see (KZ01), for example) I felt that this point is
worth mentioning.
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