8. AN ALTERNATIVE DEFINITION 55
this is a choice that only has to be made once, and then it applies to all
theories on all manifolds.
Theorem B. Let us fix a renormalization scheme.
Then, we find a section of each torsor T
(n+1)
T
(n),
and so a bijection
between the set of perturbative quantum field theories and the set of local
action functionals I
Oloc(C∞(M +
))[[ ]]. (Recall the superscript + means
that I must be at least cubic modulo .)
7.2. We will first prove theorem B, and deduce theorem A (which is the
more canonical formulation) as a corollary.
In one direction, the bijection in theorem B is constructed as follows. If
I
Oloc(C∞(M +
))[[ ]] is a local action functional, then we will construct a
canonical series of counterterms
ICT
(ε). These are local action functionals,
depending on a parameter ε (0, ∞) as well as on . The counterterms
are zero modulo , as the tree-level Feynman graphs all converge. Thus,
ICT
(ε)
Oloc(C∞(M
))[[ ]]
C∞((0,
∞)) where denotes the completed
projective tensor product.
These counterterms are constructed so that the limit
lim
ε→0
W
(
P (ε, L),I
ICT
(ε)
)
exists. This limit defines the scale L effective interaction I[L].
Conversely, if we have a perturbative QFT given by a collection of ef-
fective interactions I[l], the local action functional I is obtained as a cer-
tain renormalized limit of I[l] as l 0. The actual limit doesn’t exist;
to construct the renormalized limit we again need to subtract off certain
counterterms.
A detailed proof of the theorem, and in particular of the construction of
the local counterterms, is given in Section 10.
8. An alternative definition
In the previous section I presented a definition of quantum field theory
based on the heat-kernel cut-off. In this section, I will describe an alter-
native, but equivalent, definition, which allows a much more general class
of cut-offs. This alternative definition is a little more complicated, but is
conceptually more satisfying. One advantage of this alternative definition is
that it does not rely on the heat kernel.
As before, we will consider a scalar field theory where the quadratic term
of the action is
1
2
φ(D
+m2)φ.
Definition 8.0.1. A parametrix for the operator D
+m2
is a distribu-
tion P on M ×M, which is symmetric, smooth away from the diagonal, and
is such that
((D
+m2)
1)P δM
C∞(M
× M)
is smooth; where δM refers to the delta distribution along the diagonal in
M × M.
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