24 1. INTRODUCTION
In more pedestrian terms, the extended classical action functional en-
codes the following data:
(1) the original action functional on the original space of fields;
(2) the Lie bracket on the space of infinitesimal gauge symmetries,
(3) the way this Lie algebra acts on the original space of fields.
In order to construct a quantum theory, one asks that the extended action
satisfies the quantum master equation. This is a succinct way of encoding
the following conditions:
(1) The Lie bracket on the space of infinitesimal gauge symmetries
satisfies the Jacobi identity.
(2) This Lie algebra acts in a way preserving the action functional on
the space of fields.
(3) The Lie algebra of infinitesimal gauge symmetries preserves the
“Lebesgue measure” on the original space of fields. That is, the
vector field on the original space of fields associated to every infin-
itesimal gauge symmetry is divergence free.
(4) The adjoint action of the Lie algebra on itself also preserves the
“Lebesgue measure”. Again, this says that a vector field associated
to every infinitesimal gauge symmetry is divergence free.
Unfortunately, the quantum master equation is an ill-defined expression.
The 3rd and 4th conditions above are the source of the problem: the diver-
gence of a vector field on the space of fields is a singular expression, involving
the same kind of singularities as those appearing in one-loop Feynman dia-
9.4. This form of the quantum master equation violates our philosophy:
we should always express things in terms of the effective actions. The quan-
tum master equation above is about the original “infinite energy” action, so
we should not be surprised that it doesn’t make sense.
The solution to this problem is to combine the BV formalism with the
effective action philosophy. To give an effective action in the BV formalism
is to give a functional
[Λ] on the energy ≤ Λ part of the extended space
of fields (i.e., the space of ghosts, fields, anti-fields and anti-ghosts). This
energy Λ effective action must satisfy a certain energy Λ quantum master
The reason that the effective action philosophy and the BV formalism
work well together is the following.
Lemma. The renormalization group flow from scale Λ to scale Λ carries
solutions of the energy Λ quantum master equation into solutions of the
energy Λ quantum master equation.
Thus, to give a gauge theory in the effective BV formalism is to give
a collection of effective actions
[Λ] for each Λ, such that
fies the scale Λ QME, and such that
[Λ ] is obtained from