C Quantum Field Theory
403
In the previous discussion, the BRST symmetry was assumed to originate from
the Faddeev–Popov gauge fixing. But, in fact, it is possible to start directly with an
action of the form
S[φ, b, c, B] = S 0 [φ] + Q B [φ, b, c, B],
(C.63)
where has the ghost number of −1. It can be proven that this is the most general
action invariant under the BRST transformations (C.45). This can describe gauge
fixed action that cannot be described by the Faddeev–Popov procedure: in particular,
the latter yields actions that are quadratic in the ghost fields (by definition of the
Gaussian integral representation of the determinant), but this does not exhaust all
the possibilities. For example, the background field method applied to Yang–Mills
theory requires using an action quartic in the ghosts.
In this section, several hypotheses have been implicit (off-shell closure, irreducibility and constant structure coefficients). If one of them breaks, then it is
necessary to employ the more general BV formalism.
C.3
BV Formalism
The Batalin–Vilkovisky (BV or also field–antifield) formalism is the most general
framework to quantize theories with a gauge symmetry. While the BRST formalism
(Appendix C.2) is sufficient to describe simple systems, it breaks down when the
structure of the gauge symmetry is more complicated, for example, in systems
implying gravity. The BV formalism is required in the following three cases (which
can occur simultaneously):
1. the gauge algebra is open (on-shell closure);
2. the structure coefficients depend on the fields;
3. the gauge symmetry is reducible (not all transformations are independent).
The BV formalism is also useful for standard gauge symmetries to demonstrate
renormalizability and deal with anomalies.
As explained in the previous section, the ghosts and the BRST symmetry are
crucial to ensure the consistency of the gauge theory. The idea of the BV formalism
is to put on an equal footing the physical fields and all the required auxiliary
and ghost fields (before gauge fixing). The introduction of antifields—one for
each of the fields—and the description of the full quantum dynamics in terms
of a quantum action (constrained by the quantum master equation) ensure the
consistency of the system. Additional benefits are the presence of a (generalized)
BRST symmetry, the existence of a Poisson structure (which allows to bring
concepts from the Hamiltonian formalism), the covariance of the formalism and
the simple interpretation of counter-terms as corrections to the classical action.
403
In the previous discussion, the BRST symmetry was assumed to originate from
the Faddeev–Popov gauge fixing. But, in fact, it is possible to start directly with an
action of the form
S[φ, b, c, B] = S 0 [φ] + Q B [φ, b, c, B],
(C.63)
where has the ghost number of −1. It can be proven that this is the most general
action invariant under the BRST transformations (C.45). This can describe gauge
fixed action that cannot be described by the Faddeev–Popov procedure: in particular,
the latter yields actions that are quadratic in the ghost fields (by definition of the
Gaussian integral representation of the determinant), but this does not exhaust all
the possibilities. For example, the background field method applied to Yang–Mills
theory requires using an action quartic in the ghosts.
In this section, several hypotheses have been implicit (off-shell closure, irreducibility and constant structure coefficients). If one of them breaks, then it is
necessary to employ the more general BV formalism.
C.3
BV Formalism
The Batalin–Vilkovisky (BV or also field–antifield) formalism is the most general
framework to quantize theories with a gauge symmetry. While the BRST formalism
(Appendix C.2) is sufficient to describe simple systems, it breaks down when the
structure of the gauge symmetry is more complicated, for example, in systems
implying gravity. The BV formalism is required in the following three cases (which
can occur simultaneously):
1. the gauge algebra is open (on-shell closure);
2. the structure coefficients depend on the fields;
3. the gauge symmetry is reducible (not all transformations are independent).
The BV formalism is also useful for standard gauge symmetries to demonstrate
renormalizability and deal with anomalies.
As explained in the previous section, the ghosts and the BRST symmetry are
crucial to ensure the consistency of the gauge theory. The idea of the BV formalism
is to put on an equal footing the physical fields and all the required auxiliary
and ghost fields (before gauge fixing). The introduction of antifields—one for
each of the fields—and the description of the full quantum dynamics in terms
of a quantum action (constrained by the quantum master equation) ensure the
consistency of the system. Additional benefits are the presence of a (generalized)
BRST symmetry, the existence of a Poisson structure (which allows to bring
concepts from the Hamiltonian formalism), the covariance of the formalism and
the simple interpretation of counter-terms as corrections to the classical action.