Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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for example y is an isospinor field, then local
isospin rotations are given by
yðxÞ!exp i
t
2
yðxÞ
no
yðxÞ¼UðxÞ yðxÞ½6
where t are the Pauli matrices: t=2 are the generators
of SU(2). The gauge field then has three components
A
i
(i = 1, 2, 3) which may be written as a matrix
A
¼ A
i
i
2
transforming as
A
! A
0
¼UðxÞA
U
1
ðxÞ
i
g
ð@
UðxÞÞU
1
ðxÞ½7
where g is the coupling constant, analogous to
electric charge. The problem with this idea was that
the isospin gauge field, analogous to the photon in
electrodynamics, should, like the photon, be mass-
less and have polarization states 1 (commonly, but
inaccurately – see the work of Wigner (1939) –called
spin 1); whereas the Yukawa particle, identified as the
meson, was massive and had spin 0, so could not act
as the isospin gauge field.
The Yang–Mills idea really came into its own
with the standard model (SM) of particle physics.
This (gauge) model has an invariance group SU(2)
U(1) SU(3), the first two groups corresponding to
electroweak interactions (a unification of weak
interactions and electromagnetism) and the final
SU(3) to quantum chromodynamics (QCD), the
gauge theory describing quark interactions, which
‘‘glues’’ them together to make hadrons – protons,
neutrons, pions, etc. This model is a dramatically
successful one. The QCD sector of the theory
requires essentially no further elaboration on the
Yang–Mills idea than replacing the group SU(2) by
SU(3). This is a straightforward matter of replacing
the generators t=2 of SU(2) with the eight generators
(3 3 matrices) of SU(3). U(x) then also becomes a
3 3 matrix. The three degrees of freedom are the
three quark ‘‘colors,’’ for which there is good
experimental evidence, and the gluons, the quanta
of the gauge fields, are indeed massless and have
good experimental support. In the electroweak
sector, however, the gauge fields, the W and Z
bosons, were found with the predicted masses of
80.3 and 91.2 GeV respectively (the proton mass, for
comparison, is 0.98 GeV). They are certainly not
massless, as the straightforward Yang–Mills theory
would require, and the explanation for this requires
the introduction of the concept of spontaneous
symmetry breaking.
Spontaneous Symmetry Breaking
The general idea of spontaneous symmetry breaking
is that the vacuum – the state of lowest energy – is
not invariant under the symmetry in question. A
simple and common illustration is a pencil balanced
vertically on its tip on a horizontal plane. The pencil
is in unstable equilibrium but the system has a
symmetry under rotations in the plane about the
axis coincident with the pencil. Eventually, the
pencil will fall into its lowest-energy state (vacuum ),
lying on the table in some direction – and the
rotational symmetry is then lost. In fact, under
rotations the actual lowest-energy (vacuum) state
will be changed into another such state. There is a
degenerate vacuum.
A similar scenario may be constructed in a
complex scalar field theory. Consider such a theory
with a Lagrangian given by
L ¼ð@
Þð@
Þm
2
ð
Þ
2
½8
that is, with a potential energy function given by
Vð;
Þ¼m
2
þ ð
Þ
2
½9
where m is the mass of the field (quantum) and is the
coupling of its self-interaction. The ground state is
obtained by minimizing V,hence@V=@ = 0, giving
(assuming that m
2
> 0) a minimum at =
= 0.
If, however, m
2
< 0, there is a local maximum at
= 0 and a minimum at jj
2
= m
2
=2>0. In
quantum theory language, the vacuum expectation
value < 0jj0 > of the field is nonzero. Goldstone
showed that this implied the presence of a massless
scalar particle – a Goldstone boson. There was some
interest in this result in particle physics, where the
hypothesis of ‘‘partial conservation of the axial vector
current’’ (PCAC) might result in a Goldstone boson
that could be identified with the pion; although not
massless, the pion is the lightest hadron, so ‘‘almost’’
massless.
Higgs analyzed what happens to the Goldstone
model if electromagnetism is included. The Lagran-
gian [8] is invariant under the global transformation
[4], but if this is made local, as in [5], a gauge field
must be introduced and it is found that the massless
Goldstone boson disappears and the massless gauge
field (photon) becomes massive. Thus, spontaneous
symmetry breaking of a gauge theory results in the
appearance of a massive, rather than massless, gauge
particle. (It is relevant to remark that a massless
photon possesses two polarization states, but a
massive one possesses three, so the number of spin-
polarization states is preserved – the massless
photon ‘‘eats’’ the Goldstone boson and becomes
massive.) The Higgs model was generalized to the
168 Symmetries and Conservation Laws
case of a nonabelian symmetry group by Guralnik,
Hagen, and Kibble and invoked by Weinberg in his
1971 model for the electroweak interact ion in which
the gauge quanta were massive.
Higgs’ work was motivated by the theory of
superconductivity, where the Meissner effect (expul-
sion of magne tic flux from a superconductor), when
relativistic, implies that the effective mass of a
photon in a superconductor is nonzero – this is,
the ‘‘reason’’ that the flux does not penetrate. In the
theory of Bardeen, Cooper, and Schrieffer (BCS), a
superconductor is described by an effective scalar
field, a composite of electron pairs (though paired in
momentum space rather than coordinate space), and
this provides a physical analogy with the model
above. The SM of particle physics postulates a Higgs
scalar field analogous to the BCS composite scalar
field. If this field exists, Higgs particles should also
exist, but they have not yet been found. This is an
outstanding problem for the SM.
Baryon and Lepton Numbers
The fact that the proton p does not decay into
positron plus photon, e
þ
þ , or muon plus photon,
þ
þ , implies a conservation law of baryon
number B (the proton possessing B = 1 and the
others B = 0). Furthermore, the stability of
and
t
against decay into e
þ implies conservation of
lepton numbers L
e
, L
,andL
t
. These are regarded
as global, not local, symmetries, so there are no
associated gauge fields or interactions . Interestingly,
however, these symmetries are not built into the SM,
so are not guaranteed by it. More interestingly, these
symmetries are actually destroyed in one attempt to
go beyond the SM. This is the hypothesis that QCD
may be unified with electroweak interactions to
produce a ‘‘grand unified’’ theory (GUT). The
simplest GUT is the one in which the SU(2) U(1)
SU(3) symmetry is assumed to be a subgroup of the
much tighter symmetry SU(5), and in that theory the
proton is unstable:
p ! e
þ
þ
0
½10
The predicted lifetime is 10
301
years, while a recent
estimate of the lifetime for this decay mode is >
5 10
32
years. It may be that GUTs do not exist in
nature, but since the decay [10] violates conserva-
tion of the quantities B and L
e
, even entertaining the
idea that the decay might take place begs the
question, ‘‘are these conservation laws sacros anct?’’
Another rece nt development which leads to the
same question is the subject of neutrino oscillations.
A strong motivation for this is the solar neutrino
problem; this is the problem that the number of
electron neutrinos detected on Earth, originating in
the Sun, is less than the number predicted, by a
factor close to 3. The mismatch could be at least
partly, and perhaps completely, explained if electron
neutrinos ‘‘oscillated’’ into muon and/or tau neutri-
nos on their passage from the Sun to the Earth, since
the reaction which detects the neutrinos on Earth is
sensitive only to electron neutrinos, and not to the
other species. But oscillation is only permitted if
L
e
, L
, and L
t
are not separately conserved quan-
tities. Oscillation can also only take place if the
masses of the differe nt neutrinos are different – the
oscillation rate depends on m
2
– hence not all
the neutrinos may be massless.
Discrete Symmetries
Ever since parity violation was discovered in weak
interactions (nuclear beta decay) by Wu in 1957, the
whole subject of discrete symmetries has presented
problems which are still not resolved. The symme-
tries in question are
P (space inversion): (x, y, z) ! (x, y, z)
T (time reversal): t !t
C (particle–antiparticle conjugation): particle $
antiparticle
Are the laws of physics invariant under these
operations? The Wu experiment revealed that weak
interactions are not invariant under P, but what
about other interactions and other operations? In
this context, the CPT theorem is highly important.
According to this theorem (based on very general
assumptions), all laws of nature must be invariant
under the combined operation CPT, so that, for
example, the fact that weak interactions are not
invariant under P means that they are not invariant
under the product CT either.
The violation of P invariance in beta decay was
soon related to the fact that the neutrino involved
(the electron neutrino – or, to be precise, antineu-
trino) was massless. Spin-1/2 particles like the
electron and neutrino obey the Dirac equation,
which may be written out as a pair of coupled
equations for left- and right-handed states. In the
case m = 0, however, these equ ations decouple so it
is possible to have a massless spin-1/2 particle which
is either left-handed or right-handed. Any interac-
tion involving this particle would automatically
violate parity (which turns a left-handed state into
a right-handed one). Experiments have verified that
the neutrino is indeed left-handed. The SM incorpo-
rates this in the sense that the left-handed electron
e
L
and the electron neutrino
e
are assigned to a
Symmetries and Conservation Laws 169
weak isospin SU(2) doublet, while the right-handed
electron e
R
transforms as a singlet. A similar
pattern is repeated for the and t particles and
their neutrinos. The phenomenon of neutrino oscil-
lations, on the other ha nd, does not allow all the
neutrino states also to be purely left-handed (since
they cannot be massless). This poses a potential
problem for the SM.
For a few years after 1957 it was believed that beta
decay violated C as well as P, but conserved the
product CP; and indeed that all weak interactions
were CP invariant. In 1964, however, it was found
that there is a small element of CP violation in K
0
decay. CP-violating effects are also expected in B
0
decays. The physical origin of CP violation is still not
understood, but its importance is that it implies T
violation, so that in (at least some) weak interactions,
there is an ‘‘arrow of time’’ on the subnuclear scale.
(Such an arrow of time is, of course, familiar in
thermodynamics.) This is used in a cosmological
context to explain baryon–antibaryon asymmetry in
the Universe.
Baryon–Antibaryon Asymmetry
In the standard model of cosmology it is shown that
applying the known laws of physics to the early
Universe (the first few minutes) leads to the
conclusion that at an age of 226 s nuclear fusion
reactions took place resulting in a mixture of 74%
protons and 26% particles, so that, hundreds of
thousands of years later, when galactic condensation
took place, it would involve precisely this admixture
of hydrogen and helium gases. Just this amount of
helium has been found in the Sun, giving great
confidence to the ‘‘big bang’’ model. Assuming that
at extremely small times the baryon number of the
Universe was zero, B = 0, and assuming also (a big
assumption, but one nevertheless made by cosmol-
ogists) that the Universe is made of matter and not
antimatter, we may then ask, why is this – where
has the antimatter gone?
Surprisingly, this question was addressed as early
as 1966 by Sakharov, who showed that, starting
with an initial state with B = 0, it would be possible
to reach a state with B 6¼ 0 as long as three
conditions obtained: B violating interactions, CP
and C violating inte ractions, and lack of thermal
equilibrium. GUTs and ordinary weak interactions
already provide possibilities for the first two of these
conditions. Breakdown of thermal equilibrium will
be expected to occur as the Universe expands.
When the particle density is high, reactions such as
p þ
p ! þ will ensure an eq ual population of
baryons and antibaryons, even in the presence of B
violating interactions, but as the density increases
and this reaction rate becomes less than the
expansion rate, thermal equilibr ium can no longer
be maintained. Thus, GUTs offer an explanation of
why there is no antimatter in the Universe. It might
be thought that this sort of explanation is implau-
sible, since the B-violating and CP-violating forces
are so weak, but actually this is not a problem, since
the ratio of baryon number to photon number in the
Universe is of the order N
B
=N
10
9
; so we may
conjure up a scenar io in which the B and CP
violating forces give rise to a volume of space in
which there are, say, 10
9
antibaryons, 10
9
þ 1
baryons and approximately the same number of
photons. Then, all the antibaryons become annihi-
lated leaving one baryon and 10
9
photons – as
observed.
A recent development in the area of discrete
symmetries has been the suggestion by Kostelecky
and coworkers that there might exist spontaneous
violation of CPT and Lorentz symmetry.
Topological Charges
Conserved quantities of a quite different type have
received a lot of attention in recent decades. Their
conservation is a consequence of nontrivial bound-
ary conditions for the fields. A famous example is
the sine-Gordon ‘‘kink.’’ The sine-Gordon equation
@
2
@t
2
@
2
@x
2
þ
1
b
2
sinðbÞ¼0 ½11
describes a scalar field in one space and one time
dimension. It is a nonlinear equation which pos-
sesses, among others, the interesting solution
f ðÞ¼
4
b
arctan exp ½ð=
p
bÞ
where = x vt and = (1 v
2
)
1=2
. This corre-
sponds to a solitary wave which moves, preserving
its shape and size – in distinction to usual waves,
which spread out and dissipate. Waves of this type
are called solitons, and solitons have in fact been
observed moving along canals. In this case, they are
solutions to the Korteveg de Vries equation. Equa-
tion [11] clearly possesses the constant solutions
¼
2n
b
; n ¼ 0; 1; 2; ...
which, it may be shown, all have zero energy. We
may then construct a solution of the above type, but
with n = 0asx !1and n = N as x !þ1. This
so-called ‘‘kink’’ solution has finite energy and is not
continuously deformable into a solution with n = 0
everywhere, since this would involve overcoming an
170 Symmetries and Conservation Laws
infinite energy barrier. The ‘‘kink number ’’ may be
characterized as a charge: defining the current
J
¼
b
2
"
@
with "
the totally antisymmetric symbol, it is clear
that this is identically conserved, @
J
= 0. This is a
consequence of the definition of "
; it is not a
consequence of invariance of the sine-Gordon
Lagrangian under a symmetry operation, so the
current J
is not a Noether current. The associated
conserved charge is
Q ¼
Z
J
0
dx ¼
b
2
Z
@
@x
dx
¼
b
2
½ð1Þ ð1Þ ¼ N
Models of the above type may be written down in
a spacetime with more than two dimensions. In that
case the above solution depends only on one
coordinate, so represents an infinite planar ‘‘domain
wall,’’ on the two sides of which the field assumes
different values. Such domain walls, as well as
‘‘cosmic strings,’’ are considered as serious possibi-
lities in cosmology.
Nonabelian gauge theories and the sigma model
also provide a fertile ground for topological excita-
tions – field configurations which for topological
reasons do not decay. Gauge theories with sponta-
neous symmetry breaking have two-dimensional
solutions corresponding to vortex lines and three-
dimensional solutions corresponding to magnetic
monopoles. In spacetime (3 þ 1 dimensions), there
is a solution to the gauge field equations, with no
spontaneous symmetry breaking, corresponding to
an ‘‘instanton,’’ a finite-energy field configuration,
localized in time as well as in space (hence the name).
The gauge group here is SU(2), whose group space is
S
3
. Spacetime is ‘‘Euclideanized’’ into R
4
, whose
boundary is then S
3
. Asymptotic field configurations
may then be characterized by mappings of S
3
in field
space into S
3
in parameter space, and since the third
homotopy group of S
3
is nontrivial,
3
(S
3
) = Z, these
field configurations belong to different classes and
are not deformable into each other. These define
‘‘degenerate vacua’’ of the gauge field equations. In
quantum theory, tunneling between these vacua is
allowed and ’t Hooft has shown how this may give
rise to deuteron decay d ! e
þ
þ
. Other exam-
ples of topolog ically nontrivial configurations are
so-called sphalerons, which may also contribute to
baryon number violation in the early Universe, and
skyrmions, constructs in the nonlinear sigma model
which serve as a model for baryon number.
Supersymmetry
Supersymmetry is a fermion–boson symmet ry, pos-
tulating that mul tiplets of fundamental particles
contain both fermions and bosons. Thus, for
example, since electrons exist there should also be
‘‘selectrons’’ – ‘‘scalar’’ electrons, with spin 0. There
should also be photinos, with spin 1/2, to take their
place alongside photons, and so on. If supersymme-
try were exact, these particles would have the same
mass as their partners and woul d have all been
found, but in fact none have yet been discovered, so
presumably supersymmetr y is a broken symmetry.
The feature that makes sup ersymmetry attractive is
that it holds some promise for solving divergence
problems in quantum field theory, since the radia-
tive corrections from fermion and boson loops are
opposite in sign and may exactly cancel. Super-
symmetric models can also help to solve the
so-called hierarchy problem in quantum field theory.
If supers ymmetry is made into a local symmetry,
rather than simply a global one, extra fields must be
introduced (as the photon field was introduced
above), and it turns out that one of these is a spin-2
field, which may be identified with the graviton.
Local supersymmetry thus becomes supergravity.
General Relativity
Symmetries and conservation laws take on new aspects
when general relativity is considered. Einstein’s field
equations relate the energy–momentum tensor of
matter (and radiation) to the Ricci tensor of spacetime.
The Ricci tensor has vanishing covariant divergence,
which means that the energy–momentum tensor
possesses the same property, but conservation of
energy and momentum requires that it is the ordinary
derivative, not the covariant one, of this tensor that
should vanish. It might be expected that this problem
could be alleviated by including the contribution of the
gravitational field itself in energy–momentum tensor.
This is quite reasonable, but then problems of
interpretation arise, since at any one point in a general
spacetime, a coordinate system might be found which
is inertial (this is the force of the equivalence principle),
corresponding to no gravitational field, and therefore
no energy. The usual procedure is to introduce an
energy–momentum ‘‘pseudotensor,’’ and to conclude
that energy in a gravitational field is not localizable.
The role of symmetries in general relativity is rather
different from its role in particle physics, which is set in
Minkowski spacetime. In a general spacetime there are
no symmetries, but many examples of particular
spacetimes with their own symmetries are now
known. The symmetry operations involved are
Symmetries and Conservation Laws 171
isometries, with corresponding groups of motion (so
that the isometry group of Minkowski space is the
Poincare´ group). These groups are an important
subject of study in cosmology; for example, there is a
classification of homogeneous cosmological models,
labeled according to the Bianchi classification.
See also: Cotangent Bundle Reduction; Effective Field
Theories; Electroweak Theory; General Relativity:
Overview; Infinite-Dimensional Hamiltonian Systems;
Noncommutative Geometry and the Standard Model;
Quantum Field Theory: A Brief Introduction;
Quasiperiodic Systems; Sine-Gordon Equation;
Supergravity; Symmetries in Quantum Field Theory of
Lower Spacetime dimensions; Symmetry and Symplectic
Reduction; Symmetry Classes in Random Matrix Theory;
Topological Defects and Their Homotopy Classification.
Further Reading
Aitchison IJ and Hey AJ (1981) Gauge Theories in Particle
Physics. Bristol: Adam Hilger.
Cheng T-P and Li L-F (1984) Gauge Theory of Elementary
Particle Physics. Oxford: Clarendon Press.
Eguchi T, Gilkey PB, and Hanson AJ (1980) Gravitation, gauge
theories and differential geometry. Physics Reports 66: 213.
Huang K (1998) Quantum Field Theory: From Operators to Path
Integrals. New York: Wiley.
Kostelecky VA (ed.) (2004) CPT and Lorentz Symmetry:
Proceedings of the Third Meeting. Singapore: World Scientific.
Landau LD and Lifshitz EM (1971) The Classical Theory of
Fields. Oxford: Pergamon Press.
Manton N and Sutcliffe P (2004) Topological Solitons.
Cambridge: Cambridge University Press.
Perkins DH (2000) Introduction to High Energy Physics, 4th edn.
Cambridge: Cambridge University Press.
Review of Particle Properties (2002), Physical Review D 66:
01002, July 2002, Part 1.
Rubakov V (2002) Classical Theory of Gauge Fields. Princeton:
Princeton University Press.
Ryder LH (1996) Quantum Field Theory, 2nd edn. Cambridge:
Cambridge University Press.
Stephani H (2004) Relativity: An Introduction to Special and General
Relativity, 3rd edn. Cambridge: Cambridge University Press.
Weinberg S (1983) The First Three Minutes. London: Fontana.
Wess J and Bagger (1983) Supersymmetry and Supergravity.
Princeton: Princeton University Press.
Wigner E (1939) On unitary representations of the inhomoge-
neous Lorentz group. Annals of Mathematics 40: 149.
Symmetries in Quantum Field Theory of Lower Spacetime
Dimensions
J Mund, Universidade de Sa
˜
o Paulo, Sa
˜
o Paulo, Brazil
K-H Rehren, Universita
¨
tGo
¨
ttingen, Go
¨
ttingen,
Germany
ª 2006 Elsevier Ltd. All rights reserved.
Symmetries in Quantum Field Theory
Symmetries have proved to be one of the most
powerful concepts in quantum theory, and in
quantum field theory in particular. From the
beginnings of quantum mechanics, it is well known
that the presence of a symmetry allows one to
predict relations between different measurements, to
classify spectra (energy or other), and to understand
the Pauli exclusion principle, to name only a few
applications. Much more remarkably, in modern
relativistic quantum field theory, designed to
describe the interactions of elementary particles,
fundamental interactions have been found to be
induced by the principle of local gauge invariance.
One distinguishes spacetime symmetries (Poincare´
or conformal transformations), which change the
position and orientation of the system in space and
time, and internal symmetries, which preserve the
localization, acting on certain internal degrees of
freedom. The Coleman–Mandula (1967) theorem
states that internal and spacetime symmetries cannot
be mixed, in the sense that the generators of internal
symmetries must be Lorentz scalars, hence the total
group of symmetries factorizes into a direct product.
Supersymmetries are an exception of this theorem
because their generators do not form a Lie algebra,
and they were in fact designed to circumvent the
Coleman–Mandula theorem.
It is well known that the structure of symmetries
of quantum systems in low-dimensional spacetime
differs significantly from that in four-dimensional
spacetime. (‘‘Low’’ means in our context two or
three, depending on the type of charge localization,
c.f. below.) To name some examples :
Two-dimensional quantum systems may have much
higher symmetries than four-dimensional ones:
– In two dimensions, there exist massive integr-
able models with infinitely many conservation
laws and facto rizable scattering matrices (see
Integrability and Quantum Field Theory).
These models exhibit solitonic superselection
sectors, c.f. below.
– The conformal group of two-dimensional
spacetime is infinite dimensional, allowing for
172 Symmetries in Quantum Field Theory of Lower Spacetime Dimensions
the exact computation of correlation functions
by the help of Ward identities (Belavin,
Polyakov, and Zamolodchikov 1984). Only
the finite-dimensional Mo¨ bius group, however,
is also a symmetry of the vacuum state.
Mo¨ bius covariance implies that the theory
contains two subtheories of chiral fields
defined on the light rays t x = constant,
resp. t þ x = constant, and that these can be
extended to fields defined on a circle, by
adding a ‘‘point at infinity’’ to the light ray
(Lu¨ scher and Mack 1976). One arrives thus at
one-dimensional chiral qua ntum field theories
on a circle, which will play an important role
in the discussion below.
Continuous symmetries cannot be spontaneously
broken in two dimensions. The latter is true not
only for relativistic quantum field t heory (Cole-
man 1973), but also in quantum statistical
mechanics (Mermin and Wagner 1966) where
it is responsible for the absence of ferromagnet-
ism ( see Symmetry Breaking in Field Theory).
Spontaneous symmetry breakdown requires
long-range order which is overcome by thermal
fluctuations down to zer o temperature , because
these diverge logarithmically (in the thermody-
namical limit) in two dimensions. This theorem
thus illustrates how the spacetime dimension-
dependent size of phase space has an effect on
internal symmetries of quantum systems. A
detailed mathematical analysis of the balance
between phase space (thermal f luctuations) and
long-range order (symmetry breakdown) has
been given in a recent discussion of the Gold-
stone theorem (Buchholz, Doplicher, Longo and
Roberts 1992).
The Coleman–Mandula theorem, excluding a
mixing between internal and spacetime symme-
tries (see above), is valid only in higher
dimensions.
In more recent times, it has become apparent that
low-dimensional quan tum systems do not only
admit more symmetries, but they may exhibit
internal symmetries of an entirely new type, not
describable by groups of transformations. In this
article, we shall focus on the various ways in which
the new symmetries can arise, and how they can be
understood. In order to properly appreciate these
issues, let us first recall some basic symmetry
concepts in the conventional case.
In the traditional setting, symmetries arise in the
form of groups of transformations of the quantum
system which leave observable quantities (e.g.,
vacuum expectation values and correlation
functions) invariant. The symmetries form a group
of -automorphisms of the algebra of fields:
g
ð
1
2
Þ¼
g
ð
1
Þ
g
ð
2
Þ
g
ðÞ
¼
g
ð
Þ
g
1
g
2
¼
g
1
g
2
½1
(typically given by linear transformations of field
multiplets). In the strongest case, the automorphisms
are implemented by unitary operators on the state
space
UðgÞUðgÞ
¼
g
ðÞ½2
The imp lementers form a representation of the
group of automorphisms,
Uðg
1
ÞUð g
2
Þ¼Uðg
1
g
2
Þ½3
and there is an invariant vector state (a ground state,
or the vacuum state in relativistic quantum field
theory),
UðgÞ ¼ ½4
However, depending on the dynamics of the
quantum system, these relations cannot always be
fully realized. One therefore considers several
weaker or more general notions of symmetries
relevant in four dimensions:
Spontaneously broken symmetries. The transfor-
mations are given as automorphisms of an
algebra, but which are not unitarily implemented
in a given irreducible representation of the
algebra. Invariant pure states do not exist.
Projective representations. The symmetries are
unitarily implemented, but the implementers fail
to satisfy the group law [3]. They give rise to ray
(projective) representations or representations of a
covering group. In particular, an invariant state
vector as in [4] cannot exist in an irreducible
representation.
Infinitesimal symmetries. Lie algebras of infinite-
simal transformations, given as derivations of an
algebra, which cannot be integrated to finite
transformations. Derivations may or may not be
implemented in a given representation of the algebra
by commutators with self-adjoint generators.
Supersymmetry. The infinitesimal transforma-
tions form a graded Lie algebra.
Local gauge symmetries form an infinite-
dimensional group which are, however, not
realized as automorphisms of the quantum alge-
bra. Quantization of classical gauge interactions
usually proceeds by breaking the gauge invariance
in some way and restoring it at a later stage.
Symmetries in Quantum Field Theory of Lower Spacetime Dimensions 173
The Connection between Symmetry
and Superselection Sectors
It is often convenient to describe a model in terms of
localized fields which do not represent an observable
(in the sense of quantum mechanics that an operator
corresponds to some measurement prescription). For
example, Fermi fields which violate the principle of
causality because they anticommute with each other
at spacelike distance rather than commute are not
observables. Only fields which are quadratic in the
Fermi fields (densities of charge, current, energy) are
observables. This means that an internal symmetry
is used in order to distinguish the observables as
those operators which are invariant under the
symmetry: in the example, the symmetry transfor-
mation multiplies each Fermi field by 1 (by the
spin-statistics theorem, this transformation coincides
with the univalence of the Lorentz group). We
characterize this situatio n by writing
AðOÞ¼FðOÞ
G
½5
where A (O) and F(O) stand for the algebras of
observables and fields localized in some spacetime
region O, respectively, G is the internal symmetry
group acting by automorphisms on each F(O)
without affecting the localization, and F(O)
G
{a 2
F(O),
g
(a) = a for all g 2 G} denotes the subalgebra
of invariant s. The internal symmetry group G which
distinguishes the observables according to [5] is
usually called the ‘‘(global) gauge group.’’
If the gauge symmetry G is unbroken in the
vacuum state, then there is a well-known connec-
tion between symmetry and superselection rules
(see Symmetries and Conservation La ws): namely,
the observables act reducibly on the vacuum
Hilbert space representation of F because they
commute with the unitary operators which imple-
ment the symmetry (or with their infinitesimal
generators, usually called charges). As a conse-
quence, the validity of the superposition principle is
restricted because two eigenstates of different
eigenvalues of the charges cannot exhibit interfer-
ence. In other words, they belong to different
superselection sectors. Wick, Wightman, and
Wigner (1952) were the first to point out this
relation. We therefore call this scenario the ‘‘WWW
scenario’’ for brevity.
In the WWW scenario, the decomposition of the
Hilbert space is determined by the central decom-
position of the internal symmetry group (the
eigenvalues of the Casim ir operators). In this way,
the superselection sectors are in one-to-one corre-
spondence with the irreducible representations of
the internal symmetry group.
Superselection sectors of two-dimensional models
do not follow this scheme expected by the WWW
scenario (see below). This was most strikingly
demonstrated through the classification of the
unitary highest-weight representations of the
Virasoro algebra (Friedan, Qiu, and Shenker)
which is nothing other than the classification of the
superselection sectors of the observable algebra
generated by the chiral stress–energy tensor, and
through the determination of their fusion rules by
Belavin, Polyakov, and Zamolodchikov (1984).
In two dimensions, one is therefore lacking a
compelling a priori ansatz, like the WWW scenario,
for describing the system in terms of auxiliary
nonobservable charged fields. At this point, one
may argue that from an operational point of view, a
quantum field theory, and in particular its symme-
tries, should be understood entirely in terms of its
observables. (This viewpoint is emphasized in the
algebraic approach to QFT, see Algebraic Approach
to Quantum Field Theory.) We shall therefore now
ask the opposite question: suppose we are given an
algebra A of local observables (without knowledge
of a field algebra and its gauge group). We define
the superselection sect ors intrinsically as (the unitary
equivalence classes of) the positive-energy represen-
tations of A. Then the question is: do these sectors
arise through a WWW scenario from some field
algebra and a gauge symmetry, and if so, can the
latter be reconstructed from the given observables
alone?
The answer in four dimensions is positive, thanks
to a deep result due to Doplicher and Roberts
(1990). Let us sketch the line of reasoning leading to
this result in some detail, because it shows how the
connection between (global) gauge symmetry on the
one hand and spacetime geometry on the other hand
emerges through the principle of causality (locality)
of relativistic quantum field theory, and because it
makes apparent what is different in low-dimensional
spacetime.
The analysis is based on the gen eral structure
theory of superselection sectors due to Doplicher,
Haag, and Roberts (DHR, 1971) . The latter starts
with a selection criterion invoking the concept of a
localized charge : a superselection sector which by
measurements within the causal complement of
some spacetime region O cannot be distinguished
from the vacuum sector. The heuristic idea is, of
course, that the sector is obtained from the vacuum
sector by placing some charge in the region O (e.g.,
by the application of a localized charged field
operator to the vacuum vector).
It has been shown (Buchholz and Fredenhagen
1982) that positive-energy representations of
174 Symmetries in Quantum Field Theory of Lower Spacetime Dimensions
massive theories always satisfy this selection criter-
ion with a localization region O of the form of a
narrow cone extending in spacelike direction. (In
massless theories with long-range interactions, such
as QED, the situation is more complicated because
the charge creates an electric field whose flux at
infinity does not vanish (Gauss’ law) and is not
Lorentz invariant.) DHR assume that the localiza-
tion region is even compact, and can be chosen
arbitrarily within the unitary equivalence class of the
representation.
Exploiting a strong version of locality (Haag
duality) for the vacuum representation of the
observables, DHR proceed to define an a ssociative
composition (or fusion) law for pos itive-energy
representations. This law is commutative only up
to unitary equivalence. The crucial point is that the
unitary intertwiner establishing this equivalence (the
statistics operator) can be chosen in a unique way
provided any pair of spacelike disconnected locali-
zation regions can be continuously deformed into
any other such pair.
This point marks the separation between high and
low dimensions. In two dimensions, in each pair of
spacelike disconnected regions, one region is to the
left of the other, thus distinguishing the pair
(O
1
, O
2
) from (O
2
, O
1
). Consequently, they cannot
be deformed into each other, and there arise two
statistics operators. The same holds in three dimen-
sions when the localization regions are spacelike
cones, and O
1
, O
2
are taken within (the causal
complement of) some larger spacelike cone. If the
spacetime dimension is at least 4, or if in three
dimensions the localization regions are compact,
then the statistics operator is unique and, as a
consequence, coincides with its inverse.
The (non-)uniqueness of the statistics operator has
far-reaching consequences concerning our original
question about the underlying gauge symmetry.
Namely, the DHR analysis proceeds to show that
the set of positive-energy representations equipped
with the composition law, and the linear spaces of
inertwiners between different representation s,
together form the mathematical structure of a C
tensor category. The statistics operators which are
distinguished intertwiners give additional structure
to this category: this structure is called a (permuta-
tion) symmetry if the statistics operators coincide
with their inverse, and it is called a braiding
otherwise. (It gives rise to a representation of the
permutation group or the braid group, respectively.)
In other words, the spacetime topology, through the
intervention of the uniqueness of the stat istics
operator, causes the tensor category to be symmetric
in high dimensions, and braided in low dimensions.
At a more elementary level, one may think of
statistics operators as reflecting commutation rela-
tions between the searched-for charged fields. Mak-
ing an ansatz for the commutation relations at
spacelike separa tion, essentially the same topological
argument as before implies, together with Poincare´
invariance, that the coefficients appearing in this
relation should form a representation of the permu-
tation group, or of the braid group, respectively. The
DHR approach, however, is entirely intrinsic,
avoiding any a priori assumption of charged fields.
The duality theorem due to Doplicher and
Roberts (1990) now states that every symmetric C
tensor category (with some further qualifications
valid in the DHR setting) is isomorphic to the
category of unitary representations of a compact
group, in which the composition law is the tensor
product and the (permutation) symmetry is the
natural one. Moreover, the category uniquely
determines the group, and by a crossed product
construction (an action of the category on the
algebra A) one reconstruct s a field algebra F such
that [5] holds. If fermionic sectors are present, then
there is some arbitrariness in the commutation
relations among the corresponding fermionic fields,
which can be exploited to produce the normal
commutation relations (fermionic fields anticom-
mute among each other, and bosonic fields commute
with any field at spacelike separation). This fixes the
field algebra F up to unitary equivalence. The
conclusion is that the WWW scenario is the most
general in four dimensions (apart from the reserva-
tions due to long-range forces, see above).
Generalized Symmetries in Low
Dimensions
In view of the success of this program in four
dimensions and the advan tage of the WWW
scenario for model building, the obvious challenge
is to search for an analogous understanding of
superselection sectors (charges) in low dimensions in
terms of an algebra of charged fields and a gauge
symmetry distinguishing the observables. This gauge
symmetry cannot, in general, be a group for several
reasons:
As stated before, the tensor category of super-
selection sectors possesses only a braiding, rather
than a (permutation) symmetry, hence the duality
theorem fails.
One can associate a (statistical) dimension d
to
each superselection sector [] which is multi-
plicative under the composition law (fusion) , and
additive under direct sums. In a symmetric
Symmetries in Quantum Field Theory of Lower Spacetime Dimensions 175
category, the dimensions are necessarily positive
integers. Indeed, in the WWW scenar io, they
coincide with the naive dimension of the asso-
ciated representation of the gauge group. But in
the low-dimensional models, the dimensions turn
out to be nonintegers in general.
Moore and Seiberg (1988) have axiomatized the
superselection structure of chiral and two-
dimensional conformal field theories in terms
of a system of r ecoupling and braiding coeffi-
cients controlling the fusion of sectors and its
noncommutativity. (In fact, this system is
basically equivalent to the DHR category.) For
models such as SU(2) current algebras at level
k, these coefficients turn out to coincide with
the recoupling and braiding coefficients one can
associate with a quantum group deformation
(Drinfel’d 1986) of SU(2) with deformation
parameter q = exp i=k. Representations of
quantum groups (quasitriangular Hopf algebras,
see Hopf Algebras and q-Deformation Q uantum
Groups) have a tensor product defined in terms
of a noncocommutative coproduct. Moreover,
they possess a quantum dimension which is a
q-deformation of an integer. The quantum
dimensions precisely match t he statisti cal dimen-
sions of the superselection sectors. All this
strongly suggests that quantum groups appear as
generalized symm etries in two dimensions, at
least in a large class of models.
A natural testing ground for the search for
appropriate generalized symmetry concepts in low
dimensions is the abundance of models in chiral and
two-dimensional conformal QFT (see Two-
Dimensional Models). As mentioned before, confor-
mal symmetry in two dimensions has far-reaching
consequences, especially the existence of chiral quan-
tum fields which are defined on a one-dimensional
light ray. As a null direction in the two-dimensional
spacetime, this ray unites both the spacelike property
of carrying a causal structure, and the timelike
property that the generator of translations has positive
spectrum (energy). These two features together with
Mo¨ bius covariance are so powerful that they allow for
the exact construction of large classes of models. The
most elementary ones (minimal models) are
completely described by the chiral stress–energy
density field, that is, the local generator of the
conformal symmetry. Other models also contain
currents which are the local generators of internal
symmetries. These models exhibit many nontrivial
superselection structures, which illustrate the wide
range of possible deviations from higher-dimensional
QFT, and at the same time exhibit possible
approaches to appropriate symmetry concepts in
low dimensions.
Attempts to classify the possible algebraic struc-
tures of generalized internal symmetries in a model-
independent setting start from the idea that the
representation category of the internal symmetries of
a given model should be equivalent to the tensor
category of its superselection sectors. Several alge-
braic structures have been proposed as candidates,
complying with this idea. They all assume specific
modifications or deformation s of eqns [1]–[5] above,
highly constrained by self-consistency. Among these
proposals are:
quantum groups (see e.g., Fro¨ hlich and Kerler
1993),
weak quasiquantum groups (Mack and Schomerus
1992) and rational Hopf algebras (Fuchs et al.
1994),
weak C
Hopf algebras (Rehren 1997, Bo¨ hm and
Szlacha´nyi 1996) or quantum groupoids (Nik-
shych and Vainerman 1998), and
braided groups (Majid 1991).
In several cases, the respective ‘‘symmetry alge-
bra’’ can be reconstructed from the tensor category
of superselection sectors, and a field algebra with
linear transformation behavior can be constructed
which contains the observables as invariant ele-
ments as in [5]. However, the situation is unsatis-
factory for various reasons. First, the class of QFT
models for which these constructions have been
performed is quite restricted (most constructions
work only for rational models, i.e., models with a
finite set of charges); second, the reconstructed
symmetry algebra is not unique and finally, the
constructed field algebras have features which
diverge significantly from the WWW scenario. For
example, it is not always warranted that the
quantum symmet ries are consistent with the
-structure, indispensable for Hilbert space positiv-
ity (a necessary prerequisite for the probability
interpretation of quantum theory). Moreover, typi-
cally there are global gauge transformations which
are implemented by localized field operators, thus
exhibiting a mixing of local and global concepts. It
also happens that this holds for elements in the
center of the symmetry algebra, which implies that
the field algebra is not local relative to its gauge
invariant elements, that is, the charged fields do not
commute with the gauge-invariant elements at
spacelike separation. In other constructions, the
field algebra is not associative, or there are no finite
field multiplets.
Historically, the first candidate for a ‘‘symmetry
algebra’’ compatible with braid group statistics has
176 Symmetries in Quantum Field Theory of Lower Spacetime Dimensions
been the structure of a quantum group, as men-
tioned above. However, in physically interesting
models, the quantum group is not semisimple and
thus has too many (namely, indecomposable) repre-
sentations. Solutions to this problem have been:
1. A BRS approach in an indefinite-metric frame-
work (Hadjiivanov et al. 1991),
2. ‘‘Truncation,’’ that is, discarding the ‘‘unphysi-
cal’’ representations. Fro¨ hlich and Kerler (1993)
have done this consistently in a categorical
framework. In fact, they have given a complete
classification of the possible braided tensor
categories generated by a single irreducible object
with statistical dimension d satisfying 1 < d < 2,
in terms of categories constructed from the
‘‘truncated’’ representations of U
q
(sl
2
). Trunca-
tion can also be performed by dividing the
quantum group itself through the ideal which is
annihilated by all ‘‘physical’’ representation s,
leading to a weak quasiquantum group (Mack
and Schomerus 1992).
3. Relaxing the axioms, thus admitting the more
general structures mentioned above.
All the above approaches assume a given general-
ized symmetry concept and show to what extent
field algebras complying with it can be constructed.
They thus concern nonobservable objects, and it is
no contradiction if different symmetry concepts can
be associated with the same observable data.
A more radical concept of global gauge symmetry,
applicable to the low-dimensional case, has been
developed by Longo and Rehren (1995). Its point of
departure is the notion of a conditional expectation,
which has the same ab stract properties as a group
average. In the WWW scenario, the Haar measure
of the compact gauge group defines an average
: F 3 7!
Z
dðgÞ
g
ðÞ2A ½6
which is a positive linear map respecting the
localization, and the observables are invariant,
(a) = a. In fact , the observables are exactly the
image of this map, that is, [5] is equivalently
formulated, but without reference to the group
transformations, as
AðOÞ¼ðFðOÞÞ ½7
Turning to the observables A of a quantum field
theory in low dimensions, one looks for a quantum
field theory F, containing A and equipped with a
conditional expectation such that [7] holds, and
which preserves the vaccum state. F may not satisfy
local commutativity, but it should be local relative
to the observables in the sense mentioned before. In
rational chiral CFT, such extensions can be classi-
fied (and indeed constructed) in terms of the super-
selection category of A, giving direct access to the
decomposition of the vacuum Hilbert space of F into
superselection sectors of A. The advantage here is
that no problems with Hilbert space structure can
arise (because the approach is entirely in terms of
operator algebras); a drawback is that in genera l F is
not unique, and nonvacuum representations of F
also have to be considered in order to generate all
sectors of A.
The method can be used to classify and construct
both nonlocal chiral extensions as candidates for
sector-generating field algebras for a theory A of
chiral observables, and local two-dimensional quan-
tum field theories containing two given chiral
subtheories, that is, observable algebras of two-
dimensional models (Kawahigashi and Longo 2004).
The chiral sector structure of the latter models is
described by a ‘‘modular invariant.’’ In many cases,
this means that their thermal partition functions are
invariant under the group PSL(2, Z) of modular
transformations of the temperature (see below).
At this point, another link between spacetime and
internal symmetries may be noted. The modular
theory of von Neumann a lgebras (see Tom ita–
Takesaki Modular Theory) associates a one-para-
meter group of automorphisms (called the ‘‘modular
group’’) with a state and an algebra ‘‘in standard
position.’’ In quantum field theory, for the vacuum
state and an algebra of observables localized in
certain wedge regions of Minkowski spacetime, this
group can be identified with a boost subgroup of the
Lorentz group (Bisognano and Wichmann 1975).
Similarly, in chiral CFT on the circle, the modular
group associated with the observables in an interval
and the vacuum coincides with a subgroup of the
Mo¨ bius group. For nonlocal theories, there may be
an obstruction, however. On the other hand, if a
subalgebra is stable under the modular group of
some algebra, then there is a conditional expectation
from the larger algebra onto the smaller algebra.
Combining these general theorems, the Mo¨ bius
covariance of the inclusions A(O) F(O) implies
the existence of a conditional expectation, that is,
the above generalization of the average over the
internal symmetry. Moreover, assuming a general-
ized notion of compactness (‘‘finite index’’) for the
generalized internal symmetry, the Bisognano–Wich-
mann property holds also for nonlocal theories
(Longo and Rehren 2004).
Of course, there is also a WWW scenario in chiral
theories, that is, one may restrict a local theory to its
invariants under some group of internal gauge
Symmetries in Quantum Field Theory of Lower Spacetime Dimensions 177