16 1. INTRODUCTION
for all Λ (0, ∞). More precisely,
Seff
[Λ] should be a formal
power series both in the field φ
C∞(M)[0,Λ]
and in the variable
.
(2) Modulo , each
Seff
[Λ] must be of the form
Seff
[Λ](φ) =
1
2
M
φ D φ + cubic and higher terms.
where D is the positive-definite Laplacian. (If we want to consider
a massive scalar field theory, we can replace D by D
+m2).
(3) If Λ Λ,
Seff
] is determined from
Seff
[Λ] by the renormaliza-
tion group equation (which makes sense in the formal power series
setting).
(4) The effective actions
Seff
[Λ], when translated into a solution to
the length-scale version of the RGE, satisfy the asymptotic locality
axiom.
Since solutions to the energy and length-scale versions of the RGE are
equivalent, one can base this definition entirely on the length-scale version
of the RGE. We will do this in the body of the book.
Earlier we sketched a very general form of the RGE, which uses an
arbitrary parametrix to define a “scale” of the theory. In Chapter 2, Section
8, we will give a definition of a quantum field theory based on arbitrary
parametrices, and we will show that this definition is equivalent to the one
described above.
6. The main theorem
Now we are ready to state the first main result of this book.
Theorem A. Let T
(n)
denote the set of theories defined modulo
n+1.
Then, T
(n+1)
is a principal bundle over T
(n)
for the abelian group of local
action functionals S :
C∞(M)
R.
Recall that a functional S is a local action functional if it is of the form
S(φ) =
M
L (φ)
where L is a Lagrangian. The abelian group of local action functionals is
the same as that of Lagrangians up to the addition of a Lagrangian which
is a total derivative.
Choosing a section of each principal bundle T
(n+1)
T
(n)
yields an
isomorphism between the space of theories and the space of series in whose
coefficients are local action functionals.
A variant theorem allows one to get a bijection between theories and
local action functionals, once one has made an additional universal (but
unnatural) choice, that of a renormalization scheme. A renormalization
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