Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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defined for all times t 0(cf.Kenig et al. (1996))so
that the KdV model cannot be used to shed light on the
wave breaking phenomenon.
Whitham (1980) suggested the equation
t
þ
x
þ
Z
R
kðx yÞ
y
ðt; yÞdy ¼ 0 ½9
for the free surface profile x 7!(t, x), with the
singular kernel
kðxÞ¼
1
2
Z
R
tanh ðÞ
1=2
e
ix
d
to model wave breaking. It can be shown
(see Constantin and Escher (1998) and references
therein) that [9] describes wave breaking: there are
smooth initial profiles x 7!(0, x) such that the
resulting unique solution of [9] exists on a maximal
time interval [0, T) with
sup
ðt;xÞ2½0;TÞR
fð t; xÞg < 1
inf
x2R
f
x
ðt; xÞg ! 1 as t " T
(the solution remains bounded but its slope becomes
infinite in finite time). However, in contrast to the KdV
model, eqn [9] is not integrable and does not possess
soliton solutions. As emphasized by Whitham (1980),
it is intriguing to find models for water waves which
exhibit both soliton interaction and wave breaking.
The Camassa–Holm equation
t
txx
þ 3
x
¼ 2
x
xx
þ
xxx
½10
was first obtained by Fokas and Fuchssteiner (1981/
82) as a nonlinear partial differential equation with
infinitely many conservation laws. Camassa and Holm
(1993) derived [10] as a model for shallow water
waves, established that the equation possesses soliton
solutions and found that it is formally integrable (for
a discussion of the integrability issues we refer
to Constantin (2001),andLenells (2002)). Moreover,
the solitons of [10] are stable (Constantin and Strauss
2003). An astonishing plentitude of structures is
tied into the Camassa–Holm equation: [10] is a re-
expression of geodesic flow on the diffeomorphism
group (Constantin 2000, Kouranbaeva 1999), a
property that can be used to show that the least action
principle holds in the sense that there is a unique flow
transforming a wave profile into a nearby profile
within the class of flows that minimize the kinetic
energy (see the discussion in Constantin (2000) and
Constantin and Kolev (2003)). Interestingly, the
Camassa–Holm equation also models wave breaking.
More precisely (see the discussion in Constantin
(2000)), for any initial data x 7!
0
(x) = (0, x)in
H
3
(R) there is a unique solution of [10] defined on
some maximal time interval [0, T) and the solution
stays uniformly bounded on [0, T)with
lim
t "T
inf
x2R
f
x
ðt; xÞgðT tÞ
¼2ifT < 1
In addition to this, for a large class of initial data, there
is precisely one point where the slope of the wave
becomes infinite at breaking time (Constantin 2000): if
0
6 0 is odd and such that
0
(x)
00
0
(x) 0 for all
x 0, then the corresponding wave t 7![x 7!(t, x)]
will break in finite time T < 1 and
lim
t"T
x
ðt;0Þ¼1
whereas
j
x
ðt; xÞj K þ K
coshðxÞ
jsinhðxÞj
t 2½0; TÞ; x 6¼ 0
for some constant K > 0. Thus, the Camassa–Holm
model is an integrable infinite-dimensional Hamil-
tonian system with stable solitons and eqn [10]
admits also breaking waves as local solutions (see
Constantin an d Escher (1998) and McKean (1998)
and references therein for further results on wave
breaking for the Camassa–Hol m equation).
We conclude our discussion by pointing out that it
is possible to continue solutions of the Camassa–
Holm equation past the breaking time. For this
purpose it is convenient to rewrite [10] as the
nonlinear nonlocal conservation law
t
þ
x
þ
1
2
@
x
Z
R
e
jxyj
2
þ
2
x
2
dy ¼ 0 ½11
reminiscent to some extent to the form of [7] and [9]
and obtained by formally applying the operator
(1 @
2
x
)
1
to [10] in view of the fact that
ð1 @
2
x
Þ
1
f ¼ P f for f 2L
2
ðRÞ
the kernel of the convolution being
PðxÞ¼
1
2
e
jxj
; x 2R
By introducing a new set of independent and depen-
dent variables it is possible to resolve all singularities
due to wave breaking in the sense that [11] is
transformed into a semilinear system, the unique
solution of which can be obtained as a fixed point of
a contractive operator (Bressan and Constantin 2005).
In terms of [11], a semigroup of global conservative
solutions (in the sense that the total energy
1
2
Z
R
ð
2
þ
2
x
Þdx
Breaking Water Waves 385
equals a constant, for almost every time), depending
continuously on the initial data (0, ) 2 H
1
(R), is
thus constructed.
See also: Compressible Flows: Mathematical Theory;
Dynamical Systems in Mathematical Physics:
An Illustration from Water Waves; Integrable Systems:
Overview; Interfaces and Multicomponent Fluids.
Further Reading
Acheson DJ (1990) Elementary Fluid Dynamics. New York:
Oxford University Press.
Benjamin TB (1992) The stability of solitary waves. Proceedings
of the Royal Society of London Series A 328: 153–183.
Bressan A and Constantin A (2005) Global conservative
solutions of the Camassa–Holm equation, Preprints on
Conservation Laws 2005-016 (www.math.ntnu.no/conserva-
tion/2005/016) .
Camassa R and Holm DD (1993) A new integrable shallow water
equation with peaked solitons. Physical Review Letters
71: 1661–1664.
Constantin A (2000) Existence of permanent and breaking waves
for a shallow water equation: a geometric approach. Annales
de l’Institut Fourier (Grenoble) 50: 321–362.
Constantin A (2001) On the scattering problem for the Camassa–
Holm equation. Proceedings of the Royal Society of London
Series A 457: 953–970.
Constantin A and Escher J (1998) Wave breaking for nonlinear
nonlocal shallow water equations. Acta Mathematica
181: 229–243.
Constantin A and Kolev B (2003) Geodesic flow on the
diffeomorphism group of the circle. Commentarii Mathematici
Helvetica 78: 787–804.
Constantin A and Strauss WA (2000) Stability of peakons. Commu-
nications on Pure and Applied Mathematics 53: 603–610.
Fokas AS and Fuchssteiner B (1981/82) Symplectic structures,
their Ba¨cklund transformations and hereditary symmetries.
Physica D 4: 47–66.
Gesztesy F and Holden H (2003) Soliton Equations and their
Algebro-Geometric Solutions. Cambridge: Cambridge Univer-
sity Press.
Johnson RS (1997) A Modern Introduction to the Mathematical
Theory of Water Waves. Cambridge: Cambridge University Press.
Johnson RS (2002) Camassa–Holm, Korteweg–de Vries and
related models for water waves. Journal of Fluid Mechanics
455(2002): 63–82.
Kenig CE, Ponce G, and Vega LA (1996) A bilinear estimate with
applications to the KdV equation. Journal of the American
Mathematical Society 9: 573–603.
Kouranbaeva S (1999) The Camassa–Holm equation as a geodesic
flow on the diffeomorphism group. Journal of Mathematical
Physics 40: 857–868.
Lenells J (2002) The scattering approach for the Camassa–Holm
equation. Journal of Nonlinear Mathematical Physics
9: 389–393.
McKean HP (1979) Integrable systems and algebraic curves. In:
Global Analysis, Lecture Notes in Mathematics, vol. 755,
pp. 83–200. Berlin: Springer.
McKean HP (1998) Breakdown of a shallow water equation.
Asian Journal of Mathematics 2: 867–874.
Whitham GB (1980) Linear and Nonlinear Waves. New York:
Wiley.
BRST Quantization
M Henneaux, Universite
´
Libre de Bruxelles, Bruxelles,
Belgium
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
The BRST symmetry was originally introduced in the
seminal papers by Becchi et al. (1976) and Tyutin (1975)
for Yang–Mills gauge theories as a tool for controlling
the renormalization of the models in a consistent (gauge-
independent) way. This symmetry was discovered as a
residual symmetry of the gauge-fixed action. It was
realized later that, in fact, the BRST construction is quite
general, in the sense that it covers arbitrary gauge
theories and not just Yang–Mills gauge models.
Furthermore, it is intrinsic, in that no gauge choice is
actually necessary to define it.
The purpose of this review is to explain the general,
intrinsic features of the BRST formalism applicable to
‘ ‘any’’ gauge theory. The proper setting for discussing
these issues is that of homological algebra (Stasheff
(1998), and references therein). This article first explains
the necessary algebraic material underlying the con-
struction and then illustrates it in the cases of the
Hamiltonian BRST formalism and the Lagrangian
BRST formalism.
A Result from Homological Algebra
The main result of homological algebra needed in
the BRST construction deals with a differential
complex C with two gradings. The first grading is
an N-degree and is called the ‘‘resolution degree,’’ or
‘‘r-degree.’’ The second grading is a Z-degree and is
called the total ghost number. It is denoted by gh.
We assume that there are two odd derivations and
s
0
that have the following properties:
rðÞ¼1; ghðÞ¼1
rðs
0
Þ¼0; ghðs
0
Þ¼1
½1
and
2
¼ 0; s
0
þ s
0
¼ 0; s
2
0
¼½; s
1
½2
386 BRST Quantization
for some derivation s
1
of r-degree 1 and ghost
number 1. The bracket [ ,] is the graded commu-
tator – in this specific case, the anticommutator. We
also assume that the homology of vanishes at
nonzero value of the r-degree, both in the original
complex C,
H
k
ð; CÞ ¼ 0; k > 0 ½3
(which is equivalent to a ¼ 0, rðaÞ > 0 ) a ¼ b)
and in the space of derivations,
½; ¼0; rðÞ 6¼ 0 ) ¼½; ½4
where and are both derivations in C. The
r-degree of a homogeneous linear operator
is defined through r((x)) = r() þ r(x) for any
element x 2Cand is negative when decreases the
r-degree.
In H
0
(, C), the (odd) derivation s
0
defines a
differential. The cohomolo gy of s
0
modulo ,
denoted H
k
(s
0
, H
0
(, C)), is the cohomology of s
0
in
H
0
(, C). It is explicitly defined through the cocycle
condition
s
0
a ¼ m ½5
with coboundaries of the form
s
0
b þ n ½6
The central result underlying the BRST construc-
tion is:
Theorem 1 Given the above setting, there exists
an odd derivation s in C with the following
properties:
s ¼ þ s
0
þ s
1
þ ½7
rðs
k
Þ¼k; ghðs
k
Þ¼1 ½8
s
2
¼ 0 ½9
Furthermore, one has
H
k
ðs; CÞ ¼ H
k
ðs
0
; H
0
ð; CÞÞ ½10
The proof is straightforward (see, e.g.,
Henneaux and Teitelboim (1992)). In particular,
the proof of [10] is a standard spectral sequence
argument with a sequence that collapses after the
second step. It is interesting to note that, contrary
to s
0
, which is only a differential modulo ,s is a
true diff erential. The cons truction of s provides a
model for H
k
(s
0
, H
0
(, C)). The differential s is not
unique, but this does not affect the subsequent
discussion.
In physical applications, the total ghost number is
a derived quantity. The primary gradings are the
resolution degree and the ‘‘filtration degree’’ called
the pure ghost number and denoted pgh. It is an
N-degree and one has
gh ¼ pgh r ½11
The r-degree is known as the antighost or antifield
number, depending on the context (see below).
When r(x) = 0, one has gh(x) = pgh(x). Since the
pure ghost number is non-negative, this implies that
H
k
ðs; CÞ ¼ 0; k < 0 ½12
A Geometric Application
Geometric Setting
Theorem 1 is relevant to the following situation.
Consider a surface in a manifold M, defined by
equations
f
a
¼ 0 ½13
which may or may not be independent. (We assume
for definiteness that the variables in M are bosonic,
that is, that M is an ordinary mani fold – as opposed
to a supermanifold. The graded case can be covered
without difficulty by including appropriate sign
factors at the relevant places.) Assume that is
partitioned by orb its generated by vector fields X
defined everywhere in M, tangent to and closing
on in the Lie bracket,
½X
; X
¼C
X
þ ‘‘more’’ ½14
where ‘‘more’’ denotes terms that vanish on .We
assume, for simplicity, that the vector fields X
are
linearly independent of , although this is not
necessary. The formalism can be developed in the
nonindependent case, but it then requires more vari-
ables. We are interested in the quotient space =O of
the surface by the orbits. To guide the geometrical
intuition, we shall assume that this quotient space is a
smooth manifold (the fiber of the orbits, etc.), and we
shall suggestively adopt notations adapted to this best
possible case. The approach, being purely algebraic, is
in fact more general. (Accordingly, the notations
should be understood with a liberal mind.)
The aim here is to describe the algebra of
‘ ‘observables,’’ that is, the algebra C
1
(=O)of
functions on the quotient space =O. The terminology
‘ ‘observables’’ anticipates the physical situation dis-
cussed below, where the orbits are the ‘‘gauge orbits.’’
In order to describe algebraically the algebra of
observables, one observes that this algebra is obtained
BRST Quantization 387
through a two-step procedure. First, one restricts the
functions from M to . Second, one imposes the
invariance condition along the orbits. To each of these
steps corresponds a separate differential.
Longitudinal Complex
The longitudinal complex is associated with the
second step. One can consider on an ‘‘exterior
derivative operator D along the gauge orbits.’’ This
operator is defined on functions on as
Df ¼ X
ðf ÞC
½15
where the 1-forms C
dual to the X ’s are called
ghosts. In the physical context, the form-degree is
the pgh described earlie r, and so pgh(C
) = 1. The
action of D on the ghosts is given by
DC
¼
1
2
C
C
C
½16
The longitudinal complex L
is the complex of
exterior forms along the gauge orbits. In our
representation used here, it i s given by the space
of polynomials in the ghosts C
with coefficients
that are functions on . The exterior derivative D
is defined on this space by extending the formulas
[15] and [16] so that it is an odd derivation. One
clearly has (on )
D
2
¼ 0 ½ 17
The functions on the quotient space =O are just the
elements of the zeroth cohomological group
H
0
(D, L
),
H
0
ðD; L
Þ¼C
1
ð=OÞ ½18
In general, H
k
(D, L
) 6¼ 0.
Koszul–Tate Differential
The Koszul–Tate differential implements the first
step in the reduction procedure. More precisely, it
provides an algebraic resolution of the algebra
C
1
() of the smooth functions on the surface .
That algebra can be identified with the quotient
algebra
C
1
ðÞ¼C
1
ðMÞ=N½19
where N is the ideal of functions that vanish on .
The Koszul–Tate complex K is defined by adding
one new generator for each equation f
a
= 0 defining
, den oted t
a
and assigned r-degree 1. In the algebra
C
1
(M) ^(t
a
) (where ^(t
a
) is the exterior algebra
on t
), one defines through
f ¼ 0 8f 2 C
1
ðMÞ;t
a
¼ f
a
½20
and extends it as an odd derivation. It is clear
that r() = 1andthat
2
= 0. Because the
functions on M are annihilated by ,theyare
clearly cycle s at r-degree zero. Bec ause the left-
hand side f
a
of the equations f
a
= 0areexact
(equal to t
a
), the ideal N coincides with the set
of boundaries in degree zero.
Thus,
H
0
ð; KÞ ¼ C
1
ðÞ½21
We see accordingly that successfully enforces the
restriction to the surface through its homology in
degree zero.
However, if the equations f
a
= 0 are not indepen-
dent, this is not the end of the story. Indeed, any
identity Z
a
A
f
a
= 0 on the functions f
a
leads to a
nontrivial cycle Z
a
A
t
a
in r-degree 1, (Z
a
A
t
a
) = 0. This
is undesirable. To cure this drawback, one intro-
duces further generators t
A
in r-degree 2, one for
each identity Z
a
A
f
a
= 0, and defines
t
A
¼ Z
a
A
t
a
; rðt
A
Þ¼2 ½22
in order to ‘‘kill’’ the unwanted cycles Z
a
A
t
a
. The
Koszul complex K is thus enlarged to contain these
new (even) variables and redefined as
K¼C
1
ðMÞ^ðt
a
ÞSðt
A
Þ½23
where S(t
A
) is the symmetric algebra in t
A
. The
operator is extended to K as an odd derivation.
One has
2
= 0 and the property [21] is unaffected
by the inclusion of the new generators. Furthermore,
by construction,
H
1
ð; KÞ ¼ 0 ½24
If there is no ‘‘identity on the identities,’’ we shall
assume that the process stops. Otherwise, one needs
to introduce further generators in r-degree 3 and
possibly higher. When all the appropriate variables
are included, there is no homology at higher
r-degree. Thus,
H
k
ð; KÞ ¼ 0; k > 0 ½25
Combining with D
We now turn to the problem of combining the
Koszul–Tate complex with the longitudinal com-
plex, so as to implement the full reduction. To that
end, we define C by adding the ghosts to K,
C¼K^ðC
Þ¼0 ½26
We then extend the action of the Koszul–Tate
differential in the simplest way which preserves all
gradings, namely
C
¼ 0 ½27
388 BRST Quantization
It is clear that the homology of in C is given by
H
0
ð; CÞ ¼ L
; H
k
ð; CÞ ¼ 0 ðk > 0Þ½28
One can also extend the longitudinal derivative
D to the whole complex C because the vector fields
X
are defined through out M and so , the defini-
tions [15] and [16] make sense i n C. O ne defines
the action of D on the generators t
by requiring
that
D þ D ¼ 0 ½29
This is easily verified to be possible. However, the
(odd) derivation so obtained fails to be a differential
in C when the vector fields X
do not close off the
surface . In that case, the gauge transformations
are not integrable off ; one says that they form an
‘‘open algebra.’’ One has then D
2
= 0 only on , or,
more precisely,
D
2
¼s
1
s
1
½30
for some (odd) derivation s
1
(that vanis hes in the
‘‘closed algebra’’ case). But this situation is precisely
the one discussed earlier, with the Koszul–Tate
differential being indeed , as anticipated by the
notation, and the longitudinal differential D playing
the role of s
0
(the degrees also match). Applying the
theorem discussed there, we can conclude:
Theorem 2 There exists a differential s in C,
s ¼ þ D þ s
1
þ; s
2
¼ 0 ½31
such that
H
0
ðs; CÞ ¼ C
1
ð=OÞ ½32
This is an immediate consequence of Theorem 1
and eqns [18] and [28]. The differential s is known
in the physical applications described below as the
BRST differential.
Hamiltonian BRST Construction
As a first application of the above setting, we
consider the Hamiltonian description of gauge
systems. As already known, gauge systems are
characterized in the Hamiltonian description by
constraints and, for this reason, are called ‘‘con-
strained Hamiltonian systems .’’ Furthermore, the
gauge transformations generate gauge orbits on the
constraint surface and the physical observables are
the functions on the quotient space of the constraint
surface by the gauge orbits.
A further important feature arises in the Hamilto-
nian formalism: the gauge transformations are
canonical transformations that are generated by the
first-class constraint s. Assuming that all the second-
class constraints have been eliminated and that the
bracket being used is the Dirac bracket, one sees
that there is a vector field X
for each constraint
function f
a
, a. (The functions f
a
are thus
assumed to be independent since the vector fields
X
are assumed to be so. If not, further variable s are
needed, but the analysis proceeds along the same
ideas.)
This implies, in turn, that there is a pairing between
the ghosts C
a
associated with the longitudinal exterior
derivative and the generators t
a
of the Koszul–Tate
complex. This pairing enables one to extend the
bracket structure defined on the phase space to the
pairs (C
a
, t
a
) by declaring that these are canonically
conjugate. The variables t
a
are the momenta conjugate
to the ghosts, [t
a
,C
b
] =
b
a
. Accordingly, the complex C
relevant to the Hamiltonian situation,
C¼C
1
ðPÞ^ðC
a
Þ^ðt
a
Þ½33
has a phase-space structure (here, P M is the
manifold obtained after eliminating the second-class
constraints, equipped with the Dirac bracket). The
space C is known as the ‘‘extended phase space.’’
The r-degree is called ‘‘antighost number’’ in the
Hamiltonian context.
By the general theorem described in the previous
section, one knows that the cohomology at gh = 0of
the BRST differential is isomorphic to the algebra of
the observables. Thus, there are two alternative
ways to describe this physical algebra, either
through reduction, by eliminating the redundant
(gauge) variables, or cohomologically in an extended
space containing additional variables, the ghosts,
and their momenta.
There is an additional interesting feature of the
BRST construction in the Hamiltonian case: the
BRST transformation is a canonical transformation
in the extended phase space, in the sense that
sF ¼½; F½34
for some ‘‘BRST generator’’ of ghost number 1
(F, 2C). The nilpotency s
2
of the BRST differen-
tial is equivalent to
½; ¼0 ½35
That s is canonically generated implies that the
cohomological BRST groups come with a natural
bracket structure: the Poisson bracket of the extended
phase space passes on to the BRST cohomological
groups. In particular, H
0
(s, C), equipped with this
bracket structure, is isomorphic (as Poisson algebra)
to the algebra of physical observables.
BRST Quantization 389
Lagrangian BRST Construction
The analysis of the Lagrangian BRST construc-
tion, due to Batalin and Vilkovisky (1981) (‘‘anti-
field formalism’’), proceeds in the same way because
the covariant description of the space of observables
involves also the same geometric ingredients. The
surface is now the ‘‘stationary surface,’’ that is,
the space of solutions to the equations of motion.
The space M in which it is embedded is the space of
all field histories. The gauge symmetry acts on this
space. Furthermore, the gauge vect or fields are
tangent to since a solution is mapped on a
solution by a gauge transformation. The integral
submanifolds are the gauge orbits. The observables
are the functions on the quotient space.
Since the equations of motion follow from an
action principle, there are as many equations as
there are fields ’
i
. The corresponding genera tors t
a
in the Koszul–Tate complex (at degree 1) are called
‘‘antifields conjugate to the fields’’ and are denoted
’
i
. The r-degree is known as ‘‘antifield’’ (or also
‘‘antighost’’) number. The gauge symmetry of the
action imp lies Noether identities on the equations of
motion. These are, therefore, not independent.
According to the above general discussion, there
are further generators in the Koszul–Tate complex,
at degree 2. More precisely, there are as many new
generators in degree 2 as there are Noether identities
or independent gauge symmetries. These are called
antifields conjugate to the ghosts and denoted C
.
In the longitudinal complex, one has the ghosts C
,
with as many ghosts as there are gauge symmetries.
Thus, the BRST complex is the space
C¼C
1
ðMÞ^ðC
Þ^ð’
i
ÞSðC
Þ½36
where M is the space of all field histories. There is
now a natural pairing between the original field
variables ’
i
and the antifields ’
i
, as well as between
the ghosts C
and the antifields C
. One thus defines
a bracket in which the fields ’
i
and the ghosts C
on
the one hand, and the antifields ’
i
and C
on the
other, are declared to be conjugate. This bracket is
denoted by parentheses,
ð’
i
;’
j
Þ¼
i
j
; ðC
; C
Þ¼
½37
However, since the bracket pairs variables with
degrees that add up to 1, it is in fact an ‘‘odd
bracket,’’ called the ‘‘antibracket.’’
The BRST differential is again canonically gener-
ated, but this time in the antibracket,
sF ¼ðS; FÞ; F 2C ½38
where the generator S is an even function of the
fields, the ghosts and the antifields, with gh = 0 (the
ghost number is carried by the odd antibracket).
The nilpotency s
2
= 0 of the BRST differential is
equivalent to the crucial ‘‘master equation,’’
ðS; SÞ¼0 ½39
Because the BRST differential is canonically
generated, there is a natural bracket in cohomology.
This bracket is not the Poisson bracket of observa-
bles (at gh = 0) because it changes the ghost number
by one unit. One can, however, relate it to the
Poisson bracket of observables (Barnich and Hen-
neaux 1996); furthermore, it plays an important role
in the study of the consistent deformations of the
action.
Spacetime Locality
In the context of local field theory, one is often
interested in a particular class of functions of the
field histories, namely the so-called space of local
functionals. A local functional is, by definition, the
integral of a local n-form (where n is the spacetime
dimension). A local n-form reads, in local
coordinates,
! ¼ f ðxÞ d
n
x ½40
where f (x) depends on the fields at x as well as on a
finite number of their derivatives. When the ghosts
and the antifields are included, the local functions
depend on them in the same way.
The previous general cohomological result was
derived in the space of all function(al)s, without locality
restriction. When changing the space of cochains, one
may change the cohomology. For instance, a local
functional which is BRST-trivial in the space of all
functionals may become nontrivial in the space of local
functionals. This indeed happens here because the
homology of the Koszul–Tate differentials usually no
longer vanishes at strictly positive r-degree in the space
of local functionals, where it is related to local
conservation laws. As a result, the analysis of the
BRST cohomology in the space of local functionals is
an interesting and nontrivial problem. In particular, the
cohomological groups H
k
(s) in the space of local
functionals may not vanish at negative ghost numbers.
BRST Quantization
The quantization of a dynamical system can proceed
along different lines. For gauge models, the path-
integral approach is most efficiently pursued in the
context of the antifield formalism. We shall briefly
outline here the general principles underlying the
390 BRST Quantization
operator approach, which is based on the Hamiltonian
formalism.
In the operator approach, all the variables,
including the ghosts and the conjugate momenta,
are realized as operators in a space endowe d with a
nonpositive-definite inner product (because of the
ghosts and the gauge modes). Real dynamical
variables become formally Her mitian operators.
Ignoring anomalies, the BRST generator becomes
an operator that fulfills the conditions
¼ ;
2
¼ 0 ½41
(which allows for nontrivial solutions 6¼ 0 because
the inner product is not positive definite). The
second relation is a consequence of the classical
Poisson bracket relation [,] = 0 and the fact that
the graded Poisson bracket of two odd objects
becomes the anticommutator.
To remove the ghost and gauge redundancy, which
has no physical content, one must impose a condition
that selects physical states. The appropriate condition
is motivated by the general cohomological result
connecting the BRST cohomology with the algebra of
physical observables. One imposes the condition
j i¼0 ½42
Because of [41], states of the form ji are solutions
of [42], but they have a vanishing inner product with
any other physical states, including themselves. They
are called null states. The physical states are given by
the BRST state cohomology. The physical operators
are given by the BRST operator cohomology at
gh = 0 and induce a well-defined action in the state
cohomology. In particular, the Hamiltonian, being
gauge invariant in the original theory, is represented
by a BRST cohomological class, so that the time
evolution maps physical states on physical states.
The whole scheme is (formally) consistent because
exact BRST operators have vanishing matrix elements
between states annihilated by the BRST operator ,
while null states jiare such that h jAji= 0whenever
A is a BRST-closed operator, [A, ] = 0, and j i a
physical state. Problems may arise, however, if the
classical relations [, ] = 0and[H,] = 0 are not
satisfied in presence of extra terms of order h,that is,
2
6¼ 0orH þ H 6¼ 0 ½43
In such cases, one says that they are anomalies. These
are usually fatal to the consistency of the theory.
Some Applications
The number of applications of the BRST formalism
is so large that it would be out of place to try being
exhaustive here. Some of its main successes are
outlined here, with suggesti ons for ‘‘Furt her readi ng.’’
Renormalization of Gauge Theories
First, there is the original context of perturbative
renormalization and anomalies for gauge theories of
the Yang–Mills type. The relevant cohomology here
is the BRST cohomology in the space of local
functionals involving the fields, the ghosts, and the
antifields. The antifields are also known in this
context as Zinn-Justin sources for the BRST varia-
tions of the fields and ghosts, since Zinn-Justin was
the first to introduce them (with that meaning).
Many authors have contributed to the full computa-
tion of the local BRST cohomology. A review is
given in Barnich et al. (2000), where extensions to
other theories are also indicated.
String Theory
Modern string theory would be inconceivable with-
out the BRST formalism. This started with the
pioneering paper by Kato and Ogawa (1983), where
the critical dimension of the bosonic string was
derived from the condition that
2
should vanish
(quantum mechanically), and where it was shown
that the string physical states could be identified
with the state BRST cohomology. The reader is
referred to excellent monogra phs on modern string
theory (see ‘‘Furt her readi ng’’).
Deformations of Gauge Models
The study of consistent deformations of a given
gauge theory (i.e., the problem of introducing
consistent couplings) is also efficiently dealt with in
the BRST context. References to applications may
be found in Henneaux (1998).
See also: Anomalies; Batalin–Vilkovisky Quantization;
BF Theories; Constrained Systems; Functional
Integration in Quantum Physics; Graded Poisson
Algebras; Indefinite Metric; Perturbative Renormalization
Theory and BRST; Quantum Chromodynamics; Quantum
Field Theory: A Brief Introduction; Renormalization:
General Theory; String Field Theory; Supermanifolds;
Topological Sigma Models.
Further Reading
Barnich G, Brandt F, and Henneaux M (2000) Local BRST
cohomology in gauge theories. Physics Reports 338: 439.
Barnich G and Henneaux M (1996) Isomorphisms between the
Batalin–Vilkovisky antibracket and the Poisson bracket.
Journal of Mathematical Physics 37: 5273.
BRST Quantization 391
Batalin IA and Vilkovisky GA (1977) Relativistic S-matrix of
dynamical systems with boson and fermion constraints.
Physics Letters B69: 309.
Batalin IA and Vilkovisky GA (1981) Gauge algebra and
quantization. Physics Letters B102: 27.
Becchi C, Rouet A, and Stora R (1976) Renormalization of gauge
theories. Annals of Physics, NY 98: 287.
Fradkin ES and Vilkovisky GA (1975) Quantization of relativistic
systems with constraints. Physics Letters B55: 224.
Green MB, Schwarz JH, and Witten E (1987) Superstring Theory,
vols. 1 and 2. Cambridge: Cambridge University Press.
Henneaux M (1998) Consistent interactions between gauge fields:
the cohomological approach. Contemporary Mathematics
219: 93.
Henneaux M and Teitelboim C (1992) Quantization of Gauge
Systems. Princeton: Princeton University Press.
Kato M and Ogawa K (1983) Covariant quantization of strings
based on BRS invariance. Nuclear Physics B212: 443.
Kugo T and Ojima I (1979) Local covariant operator formalism
of nonabelian gauge theories and quark confinement problem.
Progress of Theoretical Physics (Suppl.) 66: 1.
Polchinski J (1998) String Theory,vols. 1 and 2. Cambridge:
Cambridge University Press.
Stasheff JD (1998) The (secret?) homological algebra of the
Batalin–Vilkovisky approach. Contemporary Mathematics
219: 195.
Tyutin IV (1975) Gauge invariance in field theory and statistical
physics in the operator formalism. Preprint Lebedev-75-39.
392 BRST Quantization
C
C
-Algebras and their Classification
G A Elliott, University of Toronto, Toronto, Canada
ª 2006 Elsevier Ltd. All rights reserved.
The study of algebras of Hilbert space operators, closed
under the adjoint operation and in the weak operator
topology, was begun by John von Neumann shortly
after the discovery of quantum mechanics, and partly
with the aim of understanding the monolithic ideas
proposed by Heisenberg and Schro¨dinger.
Seventy-five years later, the theory of these
algebras has become a monolith in its own right
(see von Neumann Algebras: Introduction, Modular
Theory and Classification Theory; von Neumann
Algebras: Subfactor Theory), with more internal
structure and with more external reference to physics
and, as it turns out, to other areas of mathematics
than could possibly have been imagined at the outset.
(The most striking example of an application to
mathematics is perhaps the discovery of the Jones
knot polynomial (see The Jones Polynomial); note
that this has also had repercussions for physics.)
Twenty-five years after the beginning of the
theory of von Neumann algebras, as these algebras
are now called, Gelfand and Naimark noticed that a
second class of algebras of operators on a Hilbert
space, closed under the adjoint ope ration, was
worthy of study, namely those closed in the norm
topology. Gelfand and Naimark made two impor-
tant discoveries co ncerning this class of operator
algebras, now called C
-algebras.
First, Gelfand and Naimark showed that, in the
commutative case, at least when the C
-algebra is
considered only up to isomorphism – with its
identity as a concrete algebra of operators sup-
pressed – the information contained in a C
-algebra
is purely topological. More precisely, Gelfand and
Naimark showed that the category of unital
commutative C
-algebras, with unit-preserving
algebra homomorphisms (these necess arily prese rve
the adjoint operation), is equivalent in a contra-
variant way (i.e., with reversal of arrows) to the
category of compact Hausdorff spaces, with con-
tinuous maps. The compact space associated with a
unital commutative C
-algebra under the Gelfand–
Naimark correspondence may be viewed as the
space of maximal proper ideals, with a natural
topology (the hull-kernel , or Jacobson, topology),
and is called t he spectrum. This space may also be
viewed as the set of (unital, linear, multiplicative)
maps from the algebra into the complex numbers,
in which case the topology is that of pointwise
convergence.
Second, using this result, Gelfand and Naimark
proved that arbitrary C
-algebras could be axioma-
tized in a simple way abstractly, as
-algebras – that
is, as algebras over the complex numbers with a
conjugate linear anti-automorphism of order 2 – with
certain special properties. It is now known that the
only property that needs to be assumed is the
existence of a (necessarily unique) Banach space
norm related to the
-algebra structure by means of
the so-called C
-algebra identity:
x
x
kk
¼ x
kk
x
kk
½1
This is clearly related to – and in fact implies – the
normed algebra inequality
xy
kk
x
kk
y
kk
½2
One reason that the Gelfand–Naimark axiomati-
zation of C
-algebras is important is that it under-
lines how natural it is to consider a C
-algebra
abstractly, i.e., independently of any particular
representation. Indeed, while one of the fundamen-
tal phenomena of von Neumann algebra theory
(discovered by Murray and von Neumann) is that,
essentially – in rather a strong sense – there is only
one way to represent a given von Neumann algebra
on a Hilbert space (and there is even a canonical
way, called the standard repre sentation!), it is an
equally fundamental phenomenon of C
-algebra
theory that, except in extremely special cases, this
is no longer true.
For instance, although the C
-algebra of compact
operators on a given Hilbert space has, up to unitary
equivalence, only a single irreducible representation –
this is what underlies the fact, proved by von
Neumann, referred to as the uniqueness of the
Heisenberg commutation relations for a quantum-
mechanical system with finitely many degrees of
freedom – as soon as one considers a physical system
with infinitely many degrees of freedom, one finds that
the naturally associated C
-algebra has infinitely
many – indeed, uncountably many – unitary equiva-
lence classes of irreducible representations, and it is
impossible to parametrize these in any reasonable way.
This striking dichotomy presents itself also in
other contexts, more elementary perhaps than the
physics of infinitely many degrees of freedom.
Consider the dynamical system consisting of a circle
and a fixed rotation acting on it. If the rotation is of
finite order – i.e., if the angle is a rational multiple
of 2 – then the naturally associated C
-algebra is
relatively easy to study. In the case of angle zero, it
is the unital commutative C
-algebra with Gelfand–
Naimark spectrum the torus. In the general case of a
rational angle, the space of unitary equivalence
classes of irreducible representations is still naturally
parametrized by the torus. (And this is the same as
the space of primit ive ideals – the kernels of the
irreducible representations – with the Jacobson
topology.)
In the irrational case – the case of a rotation by an
irrational multiple of 2 (still elementary from a
geometrical point of view; note that the calendar is
based on such a system!) – the irreducible represen-
tations are no longer parametrized up to unitary
equivalence by the torus – and the space of primitive
ideals consists of a single point – the C
-algebra is
simple. (But it is decidedly not simple to study!)
This fundamental dichotomy in the classification
of C
-algebras – conjectured by Gaarding and
Wightman in the quantum-mechanical setting and
by Mackey in the geometrical one – was established
by Glimm. Glimm proved (in the setting of separ-
ability; most of his results were generalized later
to the nonseparable case) that a large number of
a priori different ways that a C
-algebra could
behave well were in fact one and the same behavio r:
either all present for a given C
-algebra, or all
catastrophically absent!
Some of the properties considered by Glimm, and
shown to be equivalent (for a separable C
-algebra)
were as follows. First of all, every representation of
the C
-algebra on a Hilbert space shou ld be of type
I, i.e., should generate a von Neumann algebra of
type I. (A von Neumann algebra was said by Murray
and von Neumann to be of type I if it contained a
minimal projection of central support one, i.e., a
projection not contained in a proper direct sum-
mand and minimal with this property.) Second, in
every irreducible representation (not necessarily
injective) on a Hilbert space, the image of the
C
-algebra should contain the compact operators.
Third, any two irreducible representations with the
same kernel should be unitarily equivalent. Fourth,
it should be possible to parametrize the unitary
equivalence classes of irreducible representations by
a real number in a natural way (respecting the
natural Borel structure introduced by Mackey).
The first of the equivalent properties listed above,
that all representation s of a C
-algebra should be of
type I, suggested a name for the property – that the
C
-algebra itself should be of type I. This property
of a C
-algebra, identified by Glimm – or, rather, its
opposite, which as mentioned above is much more
common (just as irrational numbers are more
common than rationals, or systems with infinitely
many degrees of freedom are, at least in theory,
much more common than those with finitely many
degrees of freedom) – is a fundamental unifying
principle of nature.
Besides commutat ive C
-algebras – as mentioned
above, just another way of looking at topological
spaces (compact Hausdorff spaces, that is) – and
besides the C
-algebra associated to a rotation or to
a physical system with infinitely many degrees of
freedom, what are some of the naturally occurring
examples of C
-algebras – of type I or not!
First, let us take a closer look at what arises from
a system with infinitely many degrees of freedom –
in the fermion case. As shown by Jordan and
Wigner, one obtains what, as a C
-algebra, is very
easy to describe, namely, just the infinite tensor
product in the category of unital C
-algebras of
copies of the algebra of 2 2 matrices over the
complex numbers. As it happens, in work earlier
than that referred to above, Glimm had considered
such infinite tensor product C
-algebras, also allow-
ing the components to be matrix algebras of order
different from two. This raised a problem of
classification – for those C
-algebras, all of which
were simple and not of type I. (The only simple
unital C
-algebra of type I is a single matrix algebra,
or a finite tensor product of matrix algeb ras!)
In a pioneering classification paper (the first paper
on the classification of C
-algebras being perhaps
that of Gelfand and Naimark, in which the commu-
tative case was described), Glimm obtained the
classification of infini te tensor products of matrix
algebras, showing that it was a direct extension of
the classification of finite tensor products, i.e., just
of the matrix algebras themselves. As described later
by Dixmier, Glimm’s classification was as follows.
Given a sequence n
1
, n
2
, ... of natural numbers
(equal to one or more), form the infinite product in
a natural way – just by keeping track of the total
number of times each prime number appears in the
394 C
-Algebras and their Classification