22 1. INTRODUCTION
Now let us state how one classifies scalar field theories, in general.
Theorem 8.1.1. Let
denote the space of renormalizable scalar
field theories on
invariant under translation, defined modulo
is a torsor for the vector space of local action functionals S(φ) which are a
sum of terms of non-negative dimension.
is canonically isomorphic to the space of local action
functionals of the form
S(φ) = −
φ D φ + cubic and higher terms, of non-negative dimension.
As before, the choice of a renormalization scheme leads to a section of
each of the torsors
and so to a bijection between
the space of renormalizable scalar field theories and the space of series
φ D φ +
where each Si is a translation invariant local action functional of non-
negative dimension, and S0 is at least cubic.
Applying this to
we find the following.
Corollary 8.1.2. Renormalizable scalar field theories on
and under φ → −φ, are in bijection with Lagrangians of
L (φ) = aφ D φ +
for a, b, c ∈ R[[ ]], where a = −
modulo and b = 0 modulo .
More generally, there is a finite dimensional space of non-free renormal-
izable theories in dimensions n = 3, 4, 5, 6, an infinite dimensional space in
dimensions n = 1, 2, and none in dimensions n 6. ( “Finite dimensional”
means as a formal scheme over Spec R[[ ]]: there are only finitely many
R[[ ]]-valued parameters).
Thus we find that the scalar field theories traditionally considered to be
“renormalizable” are precisely the ones selected by the Wilsonian definition
advocated here. However, in this approach, one has a conceptual reason for
why these particular scalar field theories, and no others, are renormalizable.
9. Gauge theories
We would like to apply the Wilsonian philosophy to understand gauge
theories. In Chapter 5, we will explain how to do this using a synthesis of
Wilsonian ideas and the Batalin-Vilkovisky formalism.