22 1. INTRODUCTION

Now let us state how one classifies scalar field theories, in general.

Theorem 8.1.1. Let

R(k)(Rn)

denote the space of renormalizable scalar

field theories on

Rn,

invariant under translation, defined modulo

n+1.

Then,

R(k+1)(Rn)

→

R(k)(Rn)

is a torsor for the vector space of local action functionals S(φ) which are a

sum of terms of non-negative dimension.

Further,

R(0)(Rn)

is canonically isomorphic to the space of local action

functionals of the form

S(φ) = −

1

2

Rn

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

R(k+1)(Rn)

→

R(k)(Rn),

and so to a bijection between

the space of renormalizable scalar field theories and the space of series

−

1

2

φ D φ +

iSi

where each Si is a translation invariant local action functional of non-

negative dimension, and S0 is at least cubic.

Applying this to

R4,

we find the following.

Corollary 8.1.2. Renormalizable scalar field theories on

R4,

invariant

under SO(4)

R4

and under φ → −φ, are in bijection with Lagrangians of

the form

L (φ) = aφ D φ +

bφ4

+

cφ2

for a, b, c ∈ R[[ ]], where a = −

1

2

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.