In this talk, we briefly review the basic concepts of anomalous gauge
theories. It has been known for some time how theories with local anomalies can
be handled. Recently it has been pointed out that global anomalies, which
obstruct the quantization of certain gauge theories in the temporal gauge, get
bypassed in canonical quantization.Comment: Invited Talk at DAE Symposium, Santiniketan, 9 pages, LATE

Mitra, P.

English

arXiv

In this talk, we briefly review the basic concepts of anomalous gauge theories. It has been known for some time how theories with local anomalies can be handled. Recently it has been pointed out that global anomalies, which obstruct the quantization of certain gauge theories in the temporal gauge, get bypassed in canonical quantization

Mitra, P

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hep-th/9506158
QUANTIZATION OF GAUGE THEORIES WITH ANOMALIES1
P. Mitra2
Saha Institute of Nuclear Physics
Block AF, Bidhannagar
Calcutta 700 064, INDIA
Abstract
In this talk, we briefly review the basic concepts of anomalous gauge
theories. It has been known for some time how theories with local anomalies
can be handled. Recently it has been pointed out that global anomalies,
which obstruct the quantization of certain gauge theories in the temporal
gauge, get bypassed in canonical quantization.
Anomalies of two dierent types can be involved in the quantization of
gauge theories. The existence of divergence anomalies has been known for a
long time [1]: certain classical theories have symmetry currents which cease
to be conserved after quantization. In case the current is associated with a
symmetry which is gauged, there appears to be a problem in the quantization
of the theory because the equations of motion of the gauge elds require the
current to be conserved. Fortunately, these apparently contradictory features
{ nonconservation due to the anomaly and conservation due to the gauging
{ can be ironed out because the anomaly itself can be made to vanish by
going to a submanifold of the classical phase space before quantization. Of
course, there is a dierence from theories with nonanomalous gauge currents.
In those theories, there is gauge freedom, which means that the theories can
be described in any of an innite variety of gauges. This is not possible
in a straightforward manner in anomalous gauge theories, where the gauge
is, as it were, xed by the anomaly. However, we shall see below how an
enlargement of the eld content can lead to the usual kind of gauge freedom.
1Invited Talk at DAE Symposium, Santiniketan, Jan 1, 1995
2e-mail mitra@saha.ernet.in
1
A second kind of anomaly - the so-called global, as opposed to the more
common local, kind - was discovered in the early eighties [2]. Here the gauge
current is conserved, but the group of time-independent gauge transforma-
tions is not simply connected. This has serious consequences for Dirac quan-
tization in the so-called temporal gauge. One obtains a representation of the
Lie algebra of the group of time-independent gauge transformations in the
Hilbert space of states, but this provides in general only projective (multiple-
valued) representations of the group itself. When the fermion content is such
that the representation is not a true one, there is no state in the Hilbert space
which is invariant under the group, so that the subspace of states obeying
Gauss’s law is trivial. Fortunately, this problem can be avoided by fully x-
ing the gauge. The dierence between theories with global anomalies and
anomaly-free theories is very slight, as we shall see.
We shall review theories with both kinds of anomalies. If one follows
the canonical procedure of quantization, it is easy to see that there is no
conceptual diculty in quantizing these theories. So we shall rst present this
line of argument. But most high energy theorists nowadays think in terms of
functional integrals, so we shall also explain the dierence between anomaly-
free theories, locally anomalous ones and those with global anomalies in the
context of functional integrals.
First we discuss the case of common i.e., local anomalies. The canonical
method of quantization requires the determination of momenta corresponding
to the dierent eld variables. As usual, A0 has no canonical conjugate,
so there is a primary constraint 0 = 0. To preserve this constraint in
time, it is necessary to have a further constraint, and this is how Gauss’s
law appears. In anomaly-free theories, no further constraint arises, and the
above two constraints have vanishing Poisson brackets, i.e., are rst class.
In anomalous theories, two things can happen. The preservation in time
of Gauss’s law may require further constraints, or alternatively the Poisson
bracket of 0 and the Gauss law operator may be nonvanishing. In the former
case, the new constraints turn out to have nonvanishing Poisson brackets with
the above two, so that one always has second class constraints and there is
no gauge invariance; thus, the new constraints are analogous to the gauge
conditions that one has the freedom to choose in ordinary gauge theories, so
one may say that the gauge is determined by the anomalous theory itself. In
the other case, the closure of the set of second class constraints at the level of
Gauss’s law indicates the occurrence of additional degrees of freedom. Both
2
situations have been seen in the two dimensional chiral Schwinger model,
where there is a regularization parameter a [3]. For a = 1, the number of
constraints is exactly as in the anomaly-free case, so it is a case of the gauge
being automatically xed. For a > 1, there are additional degrees of freedom
in the form of massless particles. These may be thought of as would-be gauge
degrees of freedom which have become physical because of the loss of gauge
invariance. Whatever happens, the set of constraints has to be identied
and imposed on the phase space, quantization being carried out thereafter
in terms of the reduced degrees of freedom. A clarication has to be made
here about the occurrence of eects of the anomaly. The quantization that
was referred to above was the quantization of the gauge eld. The fermions
are understood to have been quantized earlier, otherwise the anomaly would
not arise.
Next we consider the problem arising in the case of global anomalies.
Observe that the argument given above (impossibility of imposing Gauss’s
law) is specic to Dirac’s method of quantization, where quantization is done
prior to the removal of gauge degrees of freedom, and is to be contrasted with
canonical quantization [4], where all constraints and gauge conditions are im-
posed at the classical level and quantization is carried out on the nonsingular
theory. The imposition of Gauss’s law and the gauge condition reduces the
phase space. The dynamical system that remains can be quantized as usual.
As Gauss’s law becomes an operator equation in the Hilbert space, this space
does not carry any nontrivial representation of either the gauge group or its
Lie algebra, so that there is no question of any complication involving projec-
tive representations in canonical quantization. The enforcement of Gauss’s
law in this approach may seem to be done by brute force when compared to
Dirac quantization, but the point is that it works [5].
We pass on to the functional integral formulation of the theories. The
full partition function of a gauge theory with fermions will be written as
Z =
Z
DAZ[A]; (1)
where Z[A] is the exponential of the negatived eective action, obtained by
functionally integrating the exponential of the negatived classical action over
the fermion elds.
In an anomaly-free theory, Z[A] is gauge invariant. The presence of an
3
anomaly makes Z[A] vary with gauge transformations of A:
Z[Ag] = ei(A;g
−1)Z[A]; (2)
where , which is an integral representation of the anomaly [6], has to satisfy
some consistency conditions:
(A; g−12 g
−1
1 ) = (A
g1; g−12 ) + (A; g
−1
1 );
(A; g−1) = −(Ag; g): (3)
The case of a global anomaly involves a special form of . One way of
characterizing a theory with a global anomaly is to say that the full group
of time-dependent gauge transformations is disconnected. Thus there is a
possibility of distinguishing between transformations not connected to the
identity and ones obtainable from the identity by a sequence of innitesimal
transformations. It is only under the former, i.e., the large gauge transfor-
mations, that Z[A] does not stay invariant in these theories. To be precise,
the transformation is given by
Z[Ag] = eiγ(g)Z[A]; (4)
where γ(g) cannot be taken to vanish except for gauge transformations g
connected to the identity.
In anomaly-free theories, the full partition function factorizes into the
volume of the gauge group and a gauge-xed partition function:
Z =
Z
DAZ[A]
=
Z
DAZ[A]
Z
Dg(f(Ag))f(A)
=
Z
Dg
Z
DAZ[Ag
−1
](f(A))f(A)
=
Z
Dg
Z
DAZ[A](f(A))f(A)
= (
Z
Dg)Zf : (5)
Here standard Faddeev-Popov notation has been used, with (f) implement-
ing a gauge-xing condition and f the corresponding Faddeev-Popov de-
terminant. In deriving the fourth equality, the invariance of Z[A] under a
gauge transformation has been used.
4
The above decoupling of the gauge degrees of freedom does not occur if a
(local) anomaly is present. In this case, one has [7]
Z =
Z
Dg
Z
DAei(A;g)Z[A](f(A))f(A); (6)
in which g and A are seen to be coupled because of the anomaly term . As
indicated above, there are two possibilities: rst, it may happen that there is
a gauge function f such that  vanishes in the special gauge f = 0 and so
the gauge degrees of freedom decouple in this gauge; alternatively, if there is
no such gauge, the would-be gauge degrees of freedom become physical. In
this latter case one can still x a gauge, but only in an enlarged theory where
the eld g in the above expressions is also taken as a dynamical eld. By using
the consistency condition (3) for , the product Z[A; g]  ei(A;g)Z[A] can be
seen to be invariant under gauge transformations of A if g is appropriately
transformed at the same time:
Z[Ah; gh] = Z[A; g]: (7)
This is Faddeev’s idea [6] of making the anomalous theory gauge invariant
by introducing a Wess-Zumino eld to cancel the gauge variation. The new
theory with
Z =
Z Z
DgDAZ[A; g](f(A))f(A); (8)
can be sought to be treated by standard methods. It is not obvious whether
unitarity and renormalizability will hold. Wisdom gained from experience in
two dimensions suggests that there will be some theories, or more precisely
some regularizations, for which unitarity is violated. Other regularizations
have to be used. (It has to be recognized that the regularization enters the
picture through the form of the anomaly.) The issue of renormalization is
less well understood, because two dimensions cannot help us here.
What about theories with global anomalies? The partition function seems
to factorize:
Z =
Z
Dge−iγ(g)
Z
DAZ[A](f(A))f(A): (9)
But one has to be careful. The phase factors form a representation of the
group, so Z
Dge−iγ(g) =
Z
D(gh)e−iγ(gh) = e−iγ(h)
Z
Dge−iγ(g); (10)
5
where a xed element h of the gauge group has been used. If it is not
connected to the identity, the left and right hand sides seem to dier by a
phase factor, indicating that
R
Dge−iγ(g) must vanish. This implies that the
partition function Z vanishes. In fact, this was given as one of the arguments
against the denability of such theories [2]. However, one is really interested
in the expectation values of gauge invariant operators:R
DAZ[A]OR
DAZ[A]
=
R
Dge−iγ(g)
R
DAZ[A](f(A))f(A)OR
Dge−iγ(g)
R
DAZ[A](f(A))f(A)
: (11)
The right hand side is of the form 0=0 because of the presence of the factorR
Dge−iγ(g) in the numerator and the denominator. Although it is formally
meaningless, one can hope to interpret this ratio in a sensible way by remov-
ing this common vanishing factor. One thus expects
< O >=
R
DAZ[A](f(A))f(A)OR
DAZ[A](f(A))f(A)
: (12)
Now (12) is precisely what one gets in the canonical approach to quan-
tization. We have considered above the Lagrangian functional integral: the
singular nature of the Lagrangian has been ignored and all degrees of free-
dom, physical or unphysical, integrated over. In the canonical approach, on
the other hand, the gauge degrees of freedom are removed by xing the gauge
at the classical level [4] and only the physical part of the theory quantized.
The functional integration is then over only the physical elds. There are
both ordinary elds and conjugate momenta, but the latter are easily inte-
grated over, resulting in functional integrals leading to (12). This is achieved
without making use of the full partition function which was used in the La-
grangian approach and caused all the problem in this case by happening to
vanish.
This simple resolution of the problem does not mean that there is no
trace whatsoever of the global anomaly. An interesting consequence of the
disconnectedness of the gauge group is that gauge-xing functions f can be
classied. Two functions f and f 0 belong to the same class if one can nd
a gauge transformation connected to the identity to go from a gauge eld
conguration satisfying one gauge condition to one satisfying the other. In
this situation, Zf and Zf 0 are equal. In general, however, the transformation
that is needed will not be connected to the identity. To see what happens in
6
this situation, we can go through the argument which is used, in anomaly-
free theories, to show that the gauge-xed partition function is the same for
dierent gauge functions. Thus,
Zf =
Z
DAZ[A](f(A))f(A)
=
Z
DAZ[A](f(A))f(A)
Z
Dg(f 0(Ag))f 0(A)
=
Z
Dg
Z
DAZ[A](f(A))f(A)(f
0(Ag))f 0(A)
=
Z
Dg
Z
DAZ[Ag
−1
](f(Ag
−1
))f(A)(f
0(A))f 0(A)
=
Z
DAZ[A](f 0(A))f 0(A)
Z
Dge−iγ(g)(f(Ag
−1
))f(A) (13)
Were it not for the phase factor e−iγ(g), the last integral would be the identity
and the right hand side would reduce to the gauge-xed partition function for
the gauge function f 0. The two integrals appear to be coupled here. But that
is not really the case. Although dierent gauge eld congurations have to
be integrated over, only those are relevant for which both f 0(A) and f(Ag
−1
)
vanish, and the second condition picks out one g for each A satisfying the
rst condition. As A changes continuously { the spacetime is taken to be
compactied { g varies in a xed homotopy class, so that γ(g), which depends
only on the class, remains unchanged. Consequently, the factor can be pulled
out and one can write
Zf = e
−iγ(g0)Zf 0; (14)
where g0 is an element of the relevant homotopy class, which is determined
by the gauge functions f and f 0. It is through these factors that theories
with global anomalies dier from anomaly-free theories. But these factors
occur only in the partition functions and clearly cancel out in the expecta-
tion values of gauge invariant operators, so that Green functions of gauge
invariant operators are fully gauge independent.
One may wonder whether these theories are unitary and renormalizable.
They indeed are. This may be understood by recalling that the gauge current
is conserved in these theories { if it is not, one is dealing with a theory with
the rst (local) kind of anomaly instead of the second (global) kind! The
invariance under innitesimal gauge transformations which exists in these
theories implies a BRS invariance [8] within each gauge-xed version. This,
7
together with the global gauge invariance of Green functions of formally
gauge invariant objects, may be used as usual to demonstrate unitarity and
renormalizability [9]. Indeed, all of standard perturbative gauge theory relies
only on invariance under innitesimal gauge transformations, i.e., under the
group of gauge transformations connected to the identity. The theories under
discussion do possess this invariance: γ(g) is zero for all g connected to the
identity.
There is one mathematical question which has to be addressed: the pos-
sibility of choosing a gauge condition in these theories. There is, in fact, a
problem, but it applies to all gauge theories irrespective of whether they are
aicted by anomalies of any kind. There is a general theorem [10] asserting
that gauges cannot be chosen in a smooth way. But it is also known [10] that
for the construction of functional integrals, it is sucient to have piecewise
smooth gauges. It bears repetition that this has to be done even for theories
without disconnected gauge groups.
To sum up, the problems with gauge theories suering from anomalies can
be avoided by canonical quantization, where the singular nature of the gauge
eld Lagrangian is recognized and all the constraints properly imposed. Even
the Lagrangian functional integral approach can be used for both local and
global anomalies if factors of 0=0 are interpreted in the most natural way.
References
[1] S. Adler, Phys. Rev. 177,2426(1969);
J. S. Bell and R. Jackiw, Nuov. Cim. 60A,47(1969)
[2] E. Witten, Phys. Lett. 117B,324(1982)
[3] R. Jackiw and R. Rajaraman, Phys. Rev. Lett. 54,1219(1985);
R. Rajaraman, Phys. Lett. 162B,148(1985);
H. Girotti, H. Rothe and K. Rothe, Phys. Rev. D33,514(1986)
[4] L. D. Faddeev, Theo. Math. Phys. 1,1(1969)
[5] P. Mitra, Lett. Math. Phys. 31,111(1994)
8
[6] L. D. Faddeev, Phys. Lett. 145B,81(1984);
L. D. Faddeev and S. L. Shatashvili, Phys. Lett. 167B,225(1986)
[7] K. Harada and I. Tsutsui, Phys. Lett. 183B,311(1986)
[8] C. Becchi, A. Rouet and R. Stora, Ann. Phys. 98,287(1976)
[9] C. Itzykson and J. B. Zuber, Quantum Field Theory, McGraw-Hill(1980)
[10] I. M. Singer, Comm. Math. Phys. 60,7(1978)
9


Quantization of Gauge Theories with Anomalies

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