2 1. INTRODUCTION
phenomena occurring at energies less than Λ, we can compute any quantity
we would like in terms of the effective action
Seff
[Λ].
If Λ Λ, then the energy Λ effective field theory can be deduced
from knowledge of the energy Λ effective field theory. This leads to an
equation expressing the scale Λ effective action
Seff
] in terms of the
scale Λ effective action
Seff
[Λ]. This equation is called the renormalization
group equation.
If we do have a continuum quantum field theory (whatever that is!)
we should, in particular, have a low-energy effective field theory for every
energy. This leads to our definition : a continuum quantum field theory is
a sequence of low-energy effective actions Seff [Λ], for all Λ ∞, which are
related by the renormalization group flow. In addition, we require that the
Seff
[Λ] satisfy a locality axiom, which says that the effective actions
Seff
[Λ]
become more and more local as Λ ∞.
This definition aims to be as parsimonious as possible. The only as-
sumptions I am making about the nature of quantum field theory are the
following:
(1) The action principle: physics at every energy scale is described by a
Lagrangian, according to Feynman’s sum-over-histories philosophy.
(2) Locality: in the limit as energy scales go to infinity, interactions
between fields occur at points.
1.2. In this book, I develop complete foundations for perturbative quan-
tum field theory in Riemannian signature, on any manifold, using this defi-
nition.
The first significant theorem I prove is an existence result: there are as
many quantum field theories, using this definition, as there are Lagrangians.
Let me state this theorem more precisely. Throughout the book, I will
treat as a formal parameter; all quantities will be formal power series in
. Setting to zero amounts to passing to the classical limit.
Let us fix a classical action functional
Scl
on some space of fields E ,
which is assumed to be the space of global sections of a vector bundle on a
manifold M
1.
Let T
(n)(E
,
Scl)
be the space of quantizations of the classical
theory that are defined modulo
n+1.
Then,
Theorem 1.2.1.
T
(n+1)(E
,
Scl)
T
(n)(E
,
Scl)
is a torsor for the abelian group of Lagrangians under addition (modulo those
Lagrangians which are a total derivative).
Thus, any quantization defined to order n in can be lifted to a quan-
tization defined to order n + 1 in , but there is no canonical lift; any two
lifts differ by the addition of a Lagrangian.
1The
classical action needs to satisfy some non-degeneracy conditions
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