6. THE MAIN THEOREM 17
scheme is a way to extract the singular part of certain functions of one
variable. We construct a certain subalgebra
P((0, 1))
C∞((0,
1))
consisting of functions f(ε) of a “motivic” nature. Functions in P((0, 1))
arise as the periods of families of algebraic varieties over Zariski open subsets
U AQ,
1
such that U(R) contains (0, 1). (For more details, see Chapter 2,
Section 9).
Definition 6.0.1. A renormalization scheme is a subspace
P((0, 1))
0
P((0, 1))
of “purely singular” functions, complementary to the subspace
P((0, 1))≥0 P((0, 1))
of functions whose r limit exists.
The choice of a renormalization scheme gives us a way to extract the
singular part of functions in P((0, 1)).
The variant theorem is the following.
Theorem B. The choice of a renormalization scheme leads to a bijec-
tion between the space of theories and the space of local action functionals
S :
C∞(M)
R[[ ]].
Equivalently, there is a bijection between the space of theories and the
space of Lagrangians up to the addition of a Lagrangian which is a total
derivative.
Theorem B implies theorem A, but theorem A is the more natural for-
mulation.
There are certain caveats:
(1) Like the effective actions
Seff
[Λ], the local action functional S is a
formal power series both in φ
C∞(M)
and in .
(2) Modulo , we require that S is of the form
S(φ) =
1
2
M
φ D φ + cubic and higher terms.
There is a more general formulation of this theorem, where the space of
fields is allowed to be the space of sections of a graded vector bundle. In
the more general formulation, the action functional S must have a quadratic
term which is elliptic in a certain sense.
6.1. Let me sketch how to prove theorem A. Given the action S, we
construct the low-energy effective action
Seff
[Λ] by renormalizing a certain
functional integral. The formula for the functional integral is
Seff
[Λ](φL) = log
φH ∈C∞(M)(Λ,∞)
eS(φL+φH )/
.
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