Quantization of Nonabelian Gauge Fields and BRST Formalism
Quantization of Nonabelian Gauge Fields and BRST Formalism
Quantization of Nonabelian Gauge Fields and BRST Formalism
BRST Formalism
Volker Bach
FB Mathematik; Johannes Gutenberg-Universitat;
D-55099 Mainz; Germany;
email: vbach@mathematik.uni-mainz.de
May 24, 2002
Abstract
These lecture notes are a summary of my seminar on Quantization of
Nonabelian Gauge Fields and BRST Formalism given in the Hesselberg
workshop, 24.2.1.3.02.
It is a pleasure for me to thank Rydzard Nest, Florian Scheck, and Elmar
Vogt for organizing this excellent workshop and the Hesselberg team for
providing a very pleasant environment.
Contents
I
(I.1)
(I.2)
Note that if
. The quantization of quadratic theories is well-understood, as the corresponding field energy operator can be written as the quantization of a one-particle
operator.
In contrast, if is nonabelian, then is quadratic in
and in
is quartic in
. The corresponding quantized field is genuinely self-interacting
and its quantization is difficult.
by using a path integral quantization. Setting
(II.1)
, the vacuum
where
(II.2)
(II.3)
is a normalization factor.
is a path integral measure, and
do not exist, and Formula (II.2) is
Unfortunately, the measure (II.3) and
formal in various senses. One obvious source of divergence is the redundancy
related by a
in (II.2) coming from connections
and
gauge transformation . This observation leads us to consider the gauge orbits
(II.4)
for
. Since both and are gauge invariant, we have
,
, and
(II.5)
linking two neighboring sites in . Thus, there are only finitely many integrations to perfom, and, moreover, each integration is over the Haar measure of the
compact Lie group . The difficulties arisising from (II.5) are shifted to making
sense of the limit of the integral in the limit . Furthermore, Lorenz
invariance is lost, and the lattice imposes an artificial anisotropy.
Yet, a third approach is furnished by gauge fixing. We assume to be given a
function
and a connection
such that the condition
defines a bijection between and the level set
.
In other words, for each
we require that there is a unique representant
,
such that
. The construction of and
is difficult and leads to the so-called Gribov problem. Locally, the bijectivity
requirement amounts to proving that the gauge orbits intersect
transversally,
i.e., that the Faddeev-Popov matrix
(II.6)
be invertible. The determinant of is then the appropriate Jacobian for the integral
(II.7)
which yields
(II.8)
(II.9)
using an (bosonic) auxiliary field (Nakanishi-Lautrup field). Furthermore, introducing (Lie algebra-valued) Grassmann fields,
, , called Faddeev-Popov
(II.10)
where
and
(II.11)
(II.12)
(II.13)
(II.14)
(II.16)
(III.1)
(III.2)
(III.3)
It implements gauge symmetry, i.e.,
is invariant under , but it additionally
involves the ghost and auxiliary fields and actually leaves the ghost Lagrangian
invariant, as well.
(III.4)
The BRST charge operator is the generator of on the level of operators,
(III.5)
, and denotes the graded
where is any of the quantized fields
, ,
, on
commutator. Explicitly,
Since
, we also have
(III.6)
(III.7)
The important facts that justify these definitions are the following:
(b)
(a)
(c)
is positive definite,
time translations),
(d) For any
"
"
(III.14)
Properties (a) and (b) insure that is a Hilbert space appropriate for quantum
mechanics: it is positive definite and carries no unphysical (ghost) degrees of
freedom. Moreover, Properties (c) and (d) insure that an initial state chosen in
evolves in this subspace under the physical dynamics. Thus, all transition
amplitudes are allowed to be computed with the aid of ghost fields, if this appears
to be convenient, and at the end of the day, one merely has to project all vectors
onto .