Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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Symmetry Breaking in Field Theory
T W B Kibble, Imperial College, London, UK
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
Spontaneous symmetry breaking in its simplest form
occurs when there is a symmetry of a dynamical
system that is not manifest in its ground state or
equilibrium state. It is a common feature of many
classical and quantum systems. In quantum field
theories, in the infinite-volume limit, there are new
features, the appearance of unitarily inequivalent
representations of the canonical commutation
relations, and the possibility of a true phase
transition – a point in the phase space where the
thermodynamic free energy is nonanalytic. The
spontaneous breaking of a continuous global sym-
metry implies the existence of massless particles, the
Goldstone bosons, while in the local-symmetry case
some or all of these may be eliminated by the Higgs
mechanism. Spontaneous symmetry breaking in
gauge theories is however a more elusive concept.
Breaking of Global Symmetries
In a quantum-mechanical system a (time-independent)
symmetry is represented by a unitary operator
^
U
acting on the Hilbert space of quantum states which
commutes with the Hamiltonian
^
H. If the ground state
j0i of the system in not invariant under
^
U,then
j0
0
i=
^
Uj0i 6¼ cj0i is also a ground state. In other
words, the ground state is degenerate.
For a system with a finite number of degrees
of freedom, whose states are represented by vectors
in a separable Hilbert space H, symmetry breaking
of an abelian symmetry group G is impossible,
unless there are additional accidental symmetries.
Consider, for example, a particle in a double-well
potential
V ¼
m!
2
4a
2
ðx
2
a
2
Þ
2
½1
which has the discrete symmetry group G = Z
2
; the
inversion symmetry operator
^
U satisfies
^
U
2
=
^
1.
There are then two approximate ground states j0i
and j0
0
i=
^
Uj0i, with wave functions proportional to
exp[ð1/2Þm!(x a)
2
]. However, there is an over-
lap between these, and the off-diagonal matrix
element h0j
^
Hj0
0
i is nonzero, although exponentially
small, so the true energy eigenstates are, approxi-
mately, j0
i= (1=
ffiffiffi
2
p
)(j0ij0
0
i). (More accurate
energy eigenfunctions and eigenvalues may be
found by using the WKB approximation.)
Of course, if the symmetry group is nonabelian,
and the ground state belongs to a nontrivial
representation, then degeneracy is unavoidable. For
example, if G is the rotation group SO(3) (or SU(2))
198 Symmetry Breaking in Field Theory
and the ground state has angular momentum j 6¼ 0,
then it is (2j þ 1)-fold degenerate.
The situation is different, however, in a quantum
field theory. In the infinite-volume limit, even abelian
symmetries can be spontaneously broken. Take, for
example, a real scalar field with Lagrangian
L¼
1
2
@
@
V ¼
1
2
_
2
1
2
ðrÞ
2
V ½2
(where we set c =h = 1), again with a double-well
potential
V ¼
1
8
ð
2
2
Þ
2
½3
exhibiting a Z
2
symmetry under which
(x) 7!(x).
At least in the semiclassical or tree approxi-
mation, there are two degenerate vacuum states j0i
and j0
0
i, with
h0j
^
ðxÞj0i and h0
0
j
^
ðxÞj0
0
i ½4
If we quantize the system in a box of finite volume
V, then, as earlier, there is an off-diagonal matrix
element of the Hamiltonian connecting the two
states, so the true ground state is (approximately)
(1=
ffiffiffi
2
p
)(j0iþj0
0
i). However, this matrix element
goes to zero exponentially as V!0. Even for large
but finite volume, the rate of transitions from j0i to
j0
0
i is exponentially slow.
Similarly, we can consider a complex scalar field
theory with a sombrero potential:
L¼j
_
j
2
jrj
2
V
V ¼
1
2
jj
2
1
2
2
2
½5
This model is invariant under the U(1) group of phase
transformations, (x) 7!(x)e
i
, so we now have a
continuously infinite set of degenerate vacuum states
j0
i labeled by an angle , and satisfying
h0
j
^
ðxÞj0
i
1
ffiffiffi
2
p
e
i
½6
Once again, one finds that in the infinite-volume
limit there are no matrix elements connecting the
different vacuum states. Moreover, in this limit no
polynomial formed from the field operators
ˆ
(x)in
a finite volume can have nonzero matrix elements
between j0
i and j0
i for 6¼ . Applying the
operators
ˆ
(x) to any one of these vacuum states
j0
i, we can construct a Fock space H
, and the
representations of the canonical commutation rela-
tions on these separate Hilbert spaces are unitarily
inequivalent. Formally, we can introduce operators
^
U
that perform the symmetry transformations:
^
U
^
ðxÞ
^
U
1
¼
^
ðxÞe
i
½7
However, these are not unitary operators on the
spaces H
, but rather maps from one space to
another:
^
U
: H
!H
þ
– or, alternatively, opera-
tors on the nonseparable Hilbert space H=
L
H
.
So far, our discussion has been restricted to the
tree approximation. For a full quantum treatment,
V() must be replaced by the effective potential
V
eff
(), which may be defined as the minimum value
of the mean energy density in all states in which the
field
ˆ
has the uniform expectation value h
ˆ
(x)i= .
V
eff
may be computed by summing vacuum loop
diagrams.
A point to note is that although the degenerate
vacua j0
i are mathematically distinct, in the
absence of any external definition of phase, they
are physically identical. There is no internal obser-
vational test that will distinguish them.
Symmetry-Breaking Phase Transitions
Spontaneous symmetry breaking often occurs in the
context of a phase transition. At high temperature,
T , there are large fluctuations in and the
central hump of the potential is unimportant. Then
the equilibrium state is symmetric, with h
ˆ
i= 0.
However, as the temperature falls, it becomes less
probable that the field will fluctuate over the top of
the hump. It will tend to fall into the trough, and
acquire a nonzero average value h
ˆ
i – the order
parameter for the phase transition – thus breaking
the symmetry. The direction of symmetry breaking
(e.g., the phase of in the U(1) model) is random,
determined in practice by small preexisting fluctua-
tions or interactions with the environment.
One way of studying this process is to compute
the temperature-dependent effective potential
V
eff
(, T). In the one-loop approximation, at high
temperature, the leading corrections to the zero-
temperature effective potential V
eff
(, T) are of the
form
V
eff
ð; TÞ¼V
eff
ð; 0Þ
2
90
N
T
4
þ
1
24
M
2
ðÞT
2
þOðTÞ½8
where N
is the total number of helicity states of light
particles (those with masses T), and M
2
,which
depends on , is the sum of their squared masses.
(Fermions if present contribute to N
with a factor of
7/8 and to M
2
with a factor of 1/2.) In the simplest
case, where we have only a multiplet f = (
a
)
a = 1, ...; N
of real scalar fields, N
= N and M
2
= M
2
aa
Symmetry Breaking in Field Theory 199
(summation over a implied), where the mass-squared
matrix is
M
2
ab
¼
@
2
V
@
a
@
b
½9
For example, in an O(N) theory, with V = (1/8)
(f
2
2
)
2
, where f
2
=
a
a
, one has
M
2
ab
¼
1
2
ðf
2
2
Þ
ab
þ
a
b
½10
whence
V
eff
ðf; TÞ
1
8
ðf
2
2
Þ
2
2
90
NT
4
þ
1
48
T
2
½ðN þ 2Þf
2
N
2
½11
It is then easy to see that the minimum occurs at
f = 0 for T > T
c
, where in this approximation
T
c
2
= 12
2
=(N þ 2), while below the critical tem-
perature the minimum is at
f
2
¼
2
eq
ðTÞ
2
N þ 2
12
T
2
½12
As T ! 0, the equilibrium state approaches one of
the vacuum states j0
n
i, labeled by an N-dimensional
unit vector n, such that h0
n
j
ˆ
fj0
n
i= n.
It is often convenient to introduce a classical
symmetry-breaking potential. For example, in the
O(N) model, we may take V
sb
= j f(x), where j
is a constant N-vector. This has the effect of tilting the
potential, thus removing the degeneracy. A character-
istic of spontaneous symmetry breaking is that the
limits j ! 0 and V!1do not commute. If (for
T < T
c
) we take the infinite-volume limit first, and
then let j ! 0, we get different equilibrium states,
depending on the direction from which j approaches
zero; if we fix n and let j = jn, j ! 0, then we find
lim
j!0
lim
V!1
h
ˆ
fðxÞi
jn
¼
eq
ðTÞn ½13
We may also regard j as representing an interac-
tion with the external environment (e.g., other
fields). If such a term is present during the cooling
of the system through the phase transition, it will
constrain the direction of the spontaneous symmetry
breaking. Note that one always arrives in this way
at one of the degenerate vacua j0
n
i, not a linear
combination of them.
Goldstone Bosons
The Goldstone theorem states that spontaneous
breaking of any continuous global symmetry leads
inevitably (except, as we discuss later, in the
presence of long-range forces) to the appearance of
massless modes – the Goldstone bosons.
The proof is straightforward. Associated with any
continuous symmetry there is a Noether current
satisfying the continuity equation @
^
j
= 0 and such
that infinitesimal symmetry transformations are
generated by the spatial integral of
^
j
0
. The fact that
the symmetry is broken means that there is some
scalar field
ˆ
(x) whose vacuum expectation value
h0j
ˆ
(0)j0i is not invariant under the symmetry
transformation. Hence,
lim
V!0
i
Z
V
d
3
xh0j½
^
j
0
ðxÞ;
^
ð0Þj0ij
x
0
¼0
6¼ 0 ½14
Moreover, the time derivative of this integral is
lim
V!0
i
Z
V
d
3
xh0j½@
0
^
j
0
ðxÞ;
^
ð0Þj0ij
x
0
¼0
¼lim
V!0
i
Z
@V
dS
k
h0j½
^
j
k
ðxÞ;
^
ð0Þj0ij
x
0
¼0
¼ 0 ½15
where @V is the bounding surface of V. This vanishes
because the surface integral is zero – in a relativistic
theory, because the commutator vanishes at space-
like separation, and more generally in the absence of
long-range interactions because it tends rapidly to
zero at large spatial separation.
Now, inserting a complete set of momentum
eigenstates jn, pi in [14], we can see that there must
exist states such that hn, pj
ˆ
(0)j0i 6¼ 0, with p
0
! 0
in the limit jpj!0, that is, massless modes.
One can see this more directly in the U(1) model
above. Consider a vacuum state j0i such that
h0j
ˆ
j0i= =
ffiffiffi
2
p
is real. Then it is useful to shift the
origin of by writing
ðxÞ¼
1
ffiffiffi
2
p
½ þ ’
1
ðxÞþi’
2
ðxÞ ½16
where ’
1
and ’
2
are real. Then the Lagrangian
becomes
L¼
1
2
_
’
2
1
ðr’
1
Þ
2
þ
_
’
2
2
ðr’
2
Þ
2
2
’
2
1
h
’
1
’
2
1
þ ’
2
2
1
4
’
2
1
þ ’
2
2
2
i
½17
Evidently, the field ’
1
, corresponding to radial
oscillations in , is massive, with mass
ffiffiffi
p
. But
there is no term in ’
2
2
,so’
2
is massless.
In the case of spontaneous symmetry breaking of
nonabelian symmetries, there may be several Gold-
stone bosons, one for each broken component of the
continuous symmetry. In our theory with symmetry
group G = O(N), the possible values of the vacuum
expectation value at T = 0 are h0
n
j
ˆ
(0)j0
n
in,
200 Symmetry Breaking in Field Theory
where n is an arbitrary unit vector. In this case, for
given n, there is an unbroken symmetry subgroup
H ¼fR 2 OðNÞ : Rn ¼ ng¼OðN 1Þ½18
and the number of broken symmetries is
dim G dim H ¼ N 1 ½19
Thus, the radial component of is massive, and
there are N 1 Goldstone bosons, the N 1
transverse components.
Spontaneously Broken Gauge Theories
As we shall see, symmetry breaking in gauge
theories is a more problematic concept but, for the
moment, these complications are ignored and the
present discussion will continue with an approach
similar to that used above.
The simplest local gauge symmetry theory is a
U(1) Higgs model, a model of a complex scalar field
(x) interacting with a gauge potential A
(x),
described by the Lagrangian
L¼D
D
1
4
F
F
VðjjÞ ½20
where V is a sombrero potential as in [5], while the
covariant derivative D
and gauge field F
are
given by
D
¼ @
þ ieA
; F
¼ @
A
@
A
½21
The model is invariant under the local U(1) gauge
transformations
ðxÞ7!ðxÞe
iðxÞ
A
ðxÞ7!A
ðxÞ
1
e
@
ðxÞ
½22
The Goldstone theorem does not apply to local-
symmetry theories. The problem is that to have a
Hilbert space containing only physical states one
must eliminate the gauge freedom by choosing a
gauge condition (e.g., in the U(1) case the Coulomb
gauge @
k
A
k
(x) = 0, which has the effect of restricting
the number of polarization states of photons to
two). This necessarily breaks manifest Lorentz
invariance, although the theory is, of course, still
fully Lorentz invariant. The proof of the theorem
fails because the current is no longer local; the long-
range Coulomb interaction makes the commutator
fall off only like 1=r
2
, so the surface integral no
longer vanishes in the infinite-volume limit. (The
theorem also fails for nonrelativistic models with
long-range forces.)
Again, consider a vacuum state j0i in which
h0j
ˆ
j0i= =
ffiffiffi
2
p
, and make the same decomposition,
[16]. Then, if we set
A
0
¼ A
þ
1
e
@
’
2
½23
we find that the kinetic term for ’
2
has been
absorbed into a mass term (1/2)e
2
2
A
0
A
0
for the
vector field. We have a model with only massive
fields: the ‘‘Higgs field’’ ’
1
with mass
ffiffiffi
p
and the
gauge field A
0
with mass e. The Goldstone bosons
have been ‘‘eaten up’’ by the vector field to provide
its longitudinal mode. This is the Higgs mechanism,
first noted by Anderson in the context of the photon
in a plasma becoming a massive plasmon.
A more elegant way of seeing this is to note that
we can always make a gauge transformation to
ensure that is real (at least so long as 6¼ 0; where
it is zero, there may be problems). This means that
(x) = ð1/
ffiffiffi
2
p
Þ( þ ’
1
); ’
2
disappears altogether, and
its kinetic term reduces to ð1/2Þe
2
A
A
( þ ’
1
)
2
,
which includes the mass term for A
as well as cubic
and quartic interaction terms.
As before, the discussion can be generalized to
nonabelian theories, although there are additional
problems to be discussed later. If we have a local
symmetry group G that breaks spontaneously to
leave an unbroken subgroup H, then the gauge fields
associated with H remain massless. Each of the
(dim G dim H) complementary fields ‘‘eats up’’
one of the Goldstone bosons, becoming massive in
the process. We are left only with other, ‘‘radial’’
components of f, the massive Higgs fields.
Consider, for example, a local SO(3) model,
with scalar fields f = (
a
)
a=1,2,3
and gauge potentials
A
= (A
a
). The infinitesimal gauge transformations are
f ¼ dw f;A
¼ dw A
1
e
@
dw ½24
where dw is the gauge parameter. The Lagrangian is
L¼
1
2
D
f D
f
1
4
F
F
1
8
ðf
2
2
Þ
2
½25
where the covariant derivative and gauge field are
D
f ¼ @
f þ eA
f
F
¼ @
A
@
A
þ eA
A
½26
If we take h
ˆ
fi in the 3-direction, the fields A
1
and
A
2
absorb the Goldstone fields
1
,
2
to become
massive. As in the abelian case, we can use the local
SO(3) invariance to rotate f everywhere to the
3-direction, and write f = (0, 0, þ ’
3
). In this
gauge the kinetic term ð1/2Þ(eA
f)
2
gives a mass
e to the fields A
1
, A
2
while A
3
remains massless,
and the Higgs field ’
3
again has mass
ffiffiffi
p
.
Symmetry Breaking in Field Theory 201
Elitzur’s Theorem; the Role of
Gauge Fixing
The concept of spontaneous symmetry breaking in
the context of a local symmetry requires further
discussion, in particular because of Elitzur’s theo-
rem, proved in 1975, which states in essence that
‘‘spontaneous breaking of a local symmetry is
impossible.’’ In the light of this theorem, it may
seem that a ‘‘spontaneously broken gauge theory’’ is
an oxymoron. In fact, it means something rather
different, although even that is not unproblematic.
The theorem was proved in the context of lattice
gauge theory, where the spatial continuum is
replaced by a discrete lattice. The scalar field is
then represented by values f
x
at each lattice site, and
the gauge potential by values A
x,
on the links of the
lattice. This is significant because on the lattice one
can use a manifestly gauge-invariant formalism.
Expectation values of gauge-invariant physical
variables can be found, for example, by a Monte
Carlo algorithm that effectively averages over all
possible gauges. In this context, it is possible to
show that the expectation value of any gauge-
noninvariant operator (such as
^
f
x
) necessarily
vanishes identically.
To be more specific, suppose we incorporate a
symmetry-breaking term of the form j
P
x
f
x
, and
consider the limits V!1followed by j ! 0. In the
global-symmetry case, as we noted earlier, this yields
the nonzero result [13]. However, in the case of a
local gauge symmetry, one can show rigorously that
lim
j!0
lim
V!1
h
^
f
x
i
jn
¼ 0 ½27
The essential reason for this is that we can make a
gauge transformation in the neighborhood of the
point x to make f
x
have any value we like without
changing the energy by more than a very small
amount that goes to zero as j ! 0. Within this
manifestly gauge-invariant formalism, it is clear that
the expectation value of a gauge-noninvariant
operator such as
ˆ
f is not an appropriate order
parameter. One must instead look for a gauge-
invariant order parameter.
It is important to note, however, that this result
applies only in the context of a manifestly gauge-
invariant formalism. But, in general, gauge theories
cannot be quantized in a manifestly gauge-invariant
way. In a path-integral formalism, the action
functional, which appears in the exponent, is
constant along the orbits of the gauge-group action.
Consequently, the integral contains an infinite
factor, the volume of the (infinite-dimensional)
gauge group. There are corresponding divergences
in the perturbation series. As is well known, this
problem can be dealt with by introducing a gauge-
fixing term, which explicitly breaks the gauge
symmetry, and renders Elitzur’s theorem inapplic-
able. But this procedure leaves a global symmetry
unbroken, and it is in fact that global symmetry that
is broken spontaneously.
One example is the Landau–Ginzburg model of a
superconductor, which is essentially just the non-
relativistic limit of the abelian Higgs model,
although there is one significant difference: here
the field
ˆ
annihilates a Cooper pair, a bound pair
of electrons with equal and opposite momenta and
spins, so e above is replaced by the charge 2e of a
Cooper pair. The appearance of a condensate of
Cooper pairs in the low-temperature superconduct-
ing phase corresponds to a state in which h
ˆ
i is
nonzero. This would not be possible without fixing
a gauge. In the nonrelativistic context, the obvious
gauge to choose is the Coulomb gauge, defined by
the condition @
k
A
k
= 0. This gauge-fixing condition
breaks the local symmetry explicitly, but it leaves
unbroken the global symmetry (x) ! (x)e
i
with
constant . It is that global symmetry that is
spontaneously broken when h
ˆ
i 6¼ 0.
For a model with nonabelian local symmetry the
standard procedure used to derive a perturbation
expansion is that of Faddeev and Popov. Consider,
for example, the SO(3) gauge theory discussed in the
preceding section. To fix the gauge, we can choose a
set of functions F = (F
a
) of the fields, and introduce
into the path integral a gauge-fixing term of the form
L
gf
¼
1
2
F
2
½28
where is an arbitrary real constant. However, to
ensure that this does not bias the integral, so that the
gauge-fixed theory is at least formally equivalent to
the original gauge-invariant theory, one must also
include the determinant of the Jacobian matrix
J
ab
ðx; yÞ¼
F
a
ðxÞ
!
b
ðyÞ
½29
The easiest way to do this is to introduce Faddeev–
Popov ghost fields
C, C, which are scalar Grassmann
variables, and an appropriate term in the
Lagrangian
L
FP
¼
C J C
½30
For the SO(3) model, a convenient choice of gauge is
the R
gauge defined by
F ¼ @
A
en f ½31
202 Symmetry Breaking in Field Theory
where n is an arbitrarily chosen unit vector. It is
clear that the full Lagrangian LþL
gf
þL
FP
is no
longer invariant under the full SO(3) gauge group,
although there is a residual U(1) gauge invariance
corresponding to rotations about n. In this gauge,
the arbitrary choice of n means that the global
SO(3) symmetry is also broken. However, for other
choices, such as the Lorentz gauge F ¼ @
A
or
axial gauge F ¼ A
3
, the Lagrangian is invariant
under global SO(3) rotations of all the fields. This
global symmetry is then spontaneously broken, with
ˆ
f acquiring as before a nonzero expectation value of
the form h
ˆ
f(x)i= n.
It is interesting to look again at the particle
content of this model. By setting f(x) = n þ j(x)
with n = (0, 0, 1), one finds that in the quadratic part
of the Lagrangian, the cross-terms between A
and j
combine to form a total divergence which can be
dropped. As before, ’
3
is the Higgs field, with
m
2
=
2
, A
3
is the massless gauge field corres-
ponding to the unbroken gauge symmetry, and the
three transverse components of A
1
and A
2
represent the massive vector fields, with m
2
= e
2
2
.
There are, however, also unphysical fields with
-dependent masses: ’
1, 2
, C
1, 2
,
C
1, 2
, and the long-
itudinal components @
A
1, 2
all have m
2
= e
2
2
.We
can now compute the effective potential V
eff
(T, j).
One point that should be noted in performing this
calculation is that the ghost fields C,
C contribute
negatively. Obviously, V
eff
, being -dependent, is
not itself physically meaningful. Nevertheless, it can
be shown that the stationary points of V
eff
are
physical, and correspond to the possible equilibrium
states of the theory. Moreover, the extremal values
of V
eff
are independent of and give correctly the
thermodynamic potential in the corresponding equi-
librium states. The negative contributions from the
ghost fields to N
and M
2
ensure that the
dependence cancels out, and we find as expected
N
= 9 and M
2
= ( þ 6e
2
)
2
.
Phase Transitions and Crossovers
Our discussion so far has for the most part been
restricted to a semiclassical or mean-field approx-
imation. It is important to bear in mind, however,
that this approximation does not suffice to deter-
mine whether a phase transition (where the thermo-
dynamic free energy is nonanalytic) exists, or what
its nature is. Determining the detailed characteristics
of phase transitions requires other methods, such as
the renormalization group or lattice simulations. In
many cases, it is far from trivial to establish the
order of the transition, or even whether a true phase
transition actually exists.
Gauge theories pose particular problems because
of the infrared divergences in the thermal field
theory at high temperature, and because in asymp-
totically free nonabelian theories the coupling
becomes large at very low energy. Even when they
appear to exhibit spontaneous symmetry breaking,
they do not necessarily undergo a true phase
transition. Lattice gauge theory calculations have
led to the conclusion that in nonabelian gauge
theories with the Higgs field in the fundamental
representation, there are values of the coupling
constants for which there is no phase transition,
only a rapid but smooth crossover from one type of
behavior to another, so that the high- and low-
temperature phases are analytically connected. If the
coupling constant is small, there is a first-order
phase transition, and for moderate values the theory
exhibits a very rapid crossover that looks quite
similar to a symmetry-breaking phase transition.
Nevertheless, the analytic connection between the
two phases implies that there cannot exist an order
parameter that is strictly zero above the transition
and nonzero below it.
In particular, it appears that for physical values
of the Higgs mass, the electroweak theory does not
undergo in fact undergo a true phase transition. It is
somewhat ironic that the most famous example of a
spontaneously broken gauge theory probably does
not, strictly speaking, exhibit a symmetry-breaking
phase transition!
Conclusions
We have discussed the main features of spontaneous
symmetry breaking in both the global- and local-
symmetry cases, especially the appearance of Gold-
stone bosons when a continuous global symmetry
breaks, and their elimination in the local-symmetry
case by the Higgs mechanism, as well as the
problems attaching to the concept of spontaneous
symmetry breaking in gauge theories.
See also: Abelian Higgs Vortices; Effective Field
Theories; Electroweak Theory; Finite Group Symmetry
Breaking; Lattice Gauge Theory; Noncommutative
Geometry and the Standard Model; Phase Transitions in
Continuous Systems; Quantum Central Limit Theorems;
Quantum Spin Systems; Symmetries in Quantum Field
Theory of Lower Spacetime Dimensions; Topological
Defects and their Homotopy Classification.
Further Reading
Anderson PW (1963) Plasmons, gauge invariance and mass.
Physical Review 130: 439–442.
Symmetry Breaking in Field Theory 203
Coleman S (1985) Aspects of Symmetry, ch. 5. Cambridge:
Cambridge University Press.
Elitzur S (1975) Impossibility of spontaneously breaking local
symmetries. Physical Review D 12: 3978–3982.
Englert F and Brout R (1964) Broken symmetry and the mass of
gauge vector bosons. Physical Review Letters 13: 321–323.
Fradkin E and Shenker SH (1979) Phase diagrams of lattice gauge
theories with Higgs fields. Physical Review D 19: 3682–3697.
Goldstone J, Salam A, and Weinberg S (1962) Broken symmetries.
Physical Review 127: 965–970.
Guralnik GS, Hagen CR, and Kibble TWB (1964) Global
conservation laws and massless particles. Physical Review
Letters 13: 585–587.
Higgs PW (1964) Broken symmetries, massless particles and
gauge fields. Physics Letters 12: 132–133.
Kibble TWB (1967) Symmetry-breaking in non-abelian gauge
theories. Physical Review 155: 1554–1561.
Weinberg S (1996) The Quantum Theory of Fields, Vol. II.
Modern Applications, chs. 19 and 21. Cambridge: Cambridge
University Press.
Symmetry Classes in Random Matrix Theory
M R Zirnbauer, Universita
¨
tKo
¨
ln, Ko
¨
ln, Germany
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
A classification of random matrix ensembles by
symmetries was first established by Dyson, in an
influential 1962 paper with the title ‘‘the threefold
way: algebraic structure of symmetry groups and
ensembles in quantum mechanics.’’ Dyson’s three-
fold way has since become fundamental to various
areas of theoretical physics, including the statistical
theory of complex many-body systems, mesoscopic
physics, disordered electron systems, and the field of
quantum chaos.
Over the last decade, a number of random matrix
ensembles beyond Dyson’s classification have come
to the fore in physics and mathematics. On the
physics side, these emerged from work on the low-
energy Dirac spectrum of quantum chromodynamics
(QCD) and from the mesoscopic physics of low-
energy quasiparticles in disordered superconductors.
In the mathematical research area of number theory,
the study of statistical correlations in the values of
Riemann zeta and similar functions has prompted
some of the same generalizations.
In this article, Dyson’s fundamental result will be
reviewed from a modern perspective, and the recent
extension of Dyson’s threefold way will be moti-
vated and described. In particular, it will be
explained why symmetry classes are associated
with large families of symmetric spaces.
The Framework
Random matrices have their physical origin in the
quantum world, more precisely in the statistical
theory of strongly interacting many-body systems
such as atomic nuclei. Although random matrix
theory is nowadays understood to be of relevance to
numerous areas of physics – see Random Matrix
Theory in Physics – quantum mechanics is still
where many of its applications lie. Quantum
mechanics also provides a natural framework in
which to classify random matrix ensembles.
Following Dyson, the mathematical setting for
classification consists of two pieces of data:
A finite-dimensional complex vector space V with
a Hermitian scalar product h, i, called a ‘‘unitary
structure’’ for short. (In physics applications,
V will usually be the truncated Hilbert space of
a family of quantum Hamiltonian systems.)
On V there acts a group G of unitary and
antiunitary operators (the joint symmetry group
of the multiparameter family of quantum systems).
Given this setup, one is interested in the linear space
of self-adjoint operators on V – the Hamiltonians H
– with the property that they commute with the
G-action. Such a space is reducible in general, that
is, the matrix of H decomposes into blocks. The goal
of classification is to list all of the irreducible blocks
that occur.
Symmetry Groups
Basic to classification is the notion of a symmetry
group in quantum Hamiltonian systems, a notion
that will now be explained.
In classical mechanics, the symmetry group G
0
of
a Hamiltonian system is understood to be the group
of canonical transformations that commute with the
phase flow of the system. An important example is
the rotation group for systems in a central field.
In passing from classical to quantum mechanics,
one replaces the classical phase space by a quantum-
mechanical Hilbert space V and assigns to the
symmetry group G
0
a (projective) representation by
unitary C-linear operators on V. Besides the one-
parameter continuous subgroups, whose significance
is highlighted by Noether’s theorem, the compo-
nents of G
0
not connected with the identity play an
204 Symmetry Classes in Random Matrix Theory
important role. A prominent example is provided by
the operator for space reflection. Its eigenspaces are
the subspaces of states with positive and negative
parity; these reduce the matrix of any reflection-
invariant Hamiltonian to two blocks.
Not all symmetries of a quantum-mechanical
system are of the canonical, unitary kind: the
prime counterexample is the operation of inverting
the time direction, called time reversal for short. In
classical mechanics, this operation reverses the sign
of the symplectic structure of phase space; in
quantum mechanics, its algebraic properties reflect
the fact that inverting the time direction, t 7!t ,
amounts to sending i =
ffiffiffiffiffiffi
1
p
to i. Indeed, time t
enters in the Dirac, Pauli, or Schro¨ dinger equation
as ihd=dt. Therefore, time reversal is represented in
the qua ntum theory by an antiunitary operator T,
which is to say that T is complex antilinear:
TðzvÞ¼
zTv ðz 2 C; v 2 VÞ
and preserves the Hermitian scalar product or
unitary structure up to complex conjugation:
Tv
1
; Tv
2
hi
¼
v
1
; v
2
hi
¼ v
2
; v
1
hi
Another operation of this kind is charge conjugation
in relativistic theories such as the Dirac equation.
By the symmetry group G of a quantum-mechanical
system with Hamiltonian H, one then means the group
of all unitary and antiunitary transformations g of V
that leave the Hamiltonian invariant: gHg
1
= H.We
denote the unitary subgroup of G by G
0
,andthesetof
antiunitary operators in G by G
1
(not a group). If V
carries extra structure, as will be the case for some
extensions of Dyson’s basic scheme, the action of G on
V hastobecompatiblewiththatstructure.
The set G
1
may be empty. When it is not, the
composition of any two elements of G
1
is unitary, so
every g 2 G
1
canbeobtainedfromafixedelementof
G
1
,sayT, by right multiplication with some U 2 G
0
:
g = TU. In other words, when G
1
is nonempty the
coset space G=G
0
consists of exactly two elements, G
0
and T G
0
= G
1
. We shall assume that T represents
some inversion symmetry such as time reversal or
charge conjugation. T must then be a (projective)
involution, that is, T
2
= z Id with z acomplex
number of unit modulus, so that conjugation by T
2
is
the identity operation. Since T is complex antilinear,
the associative law T
2
T = T T
2
forces z to be real,
and hence T
2
= Id.
Finding the total symmetry group of a Hamiltonian
system need not always be straightforward, but this
complication will not be an issue here: we take the
symmetry group G and its action on the Hilbert
space V as fundamental and given, and then ask
what are the corresponding symmetry classes,
meaning the irreducible spaces of Hamiltonians on
V that commute with G.
For technical reasons, we assume the group G
0
to
be compact; this is an assumption that covers most
(if not all) of the cases of interest in physics. The
noncompact group of space translations can be
incorporated, if necessary, by wrapping the system
around a torus, whereby translations are turned into
compact torus rotations.
While the primary objects to classify are the
spaces of Hamiltonians H, we shall focus for
convenience on the spaces of time evolutions
U
t
= e
itH=h
instead. This change of focus results in
no loss, as the Hamiltonians can always be retrieved
by linearizing in t at t = 0.
Symmetric Spaces
We appropriate a few basic facts from the theory of
symmetric spaces.
Let M be a connected m-dimensional Riemannian
manifold and p a point of M. In some open subset
N
p
of a neighborhood of p there exists a map
s
p
: N
p
!N
p
, the geodesic inversion with respect to
p, which sends a point x 2 N
p
with normal
coordinates (x
1
, ..., x
m
) to the point with normal
coordinates (x
1
, ..., x
m
). The Riemannian mani-
fold M is called locally symmet ric if the geodesic
inversion is an isometry, and is called globally
symmetric if s
p
extends to an isometry s
p
: M !M,
for all p 2 M. A globally symmetric Riemannian
manifold is called a symmetric space for short.
The Riemann curvature tensor of a symmetric
space is covariantly constant, which leads one to
distinguish between three cases: the scalar curvature
can be positive, zero, or negative, and the symmetric
space is said to be of compact type, Euclidean type,
or noncompact type, respectively. (In mesoscopic
physics, each type plays a role: the first provides us
with the scattering matrices and time evolutions, the
second with the Hamiltonians, and the third with
the transfer matrices.) The focus in the current
article will be on compact type, as it is this type that
houses the unitary time evolution operators of
quantum mechanics. The compact symmetric spaces
are subdivided into two major subtypes, both of
which occur naturally in the present context, as
follows.
Type II
Consider first the case where the antiunitary
component G
1
of the symmetry group is empty, so
the data are (V, G) with G = G
0
. Let U (V) denote
the group of all complex linear transformations that
Symmetry Classes in Random Matrix Theory 205
leave the structure of the vector space V invariant.
Thus, U (V) is a group of unitary transformations if
V carries no more than the usual Hermitian scalar
product; and is some subgroup of the unitary group
if V does have extra structure (as is the case for the
Nambu space of quasiparticle excitations in a
superconductor). The symmetry group G
0
, by acting
on V and preserving its structure, is contained as a
subgroup in U (V).
Let now H be any Hamiltonian with the pre-
scribed symmetries. Then the time evolution
t 7!U
t
= e
itH=h
generated by H is a one-parameter
subgroup of U (V) which commutes with the
G
0
-action. The total set of transformations U
t
that
arise in this way is called the (connected part of the)
‘‘centralizer’’ of G
0
in U (V), and is denoted by Z.
This is the ‘‘good’’ set of unitary time evolutions –
the set compatible with the given symmetries of an
ensemble of quantum systems .
The centralizer Z is obviously a group: if U and
U
0
belong to Z, then so do their inverses and their
product. What can one say about the structure of
the group Z?
Since G
0
is compact by assumption, its group
action on V is completely reducible and V is
guaranteed to have an orthogonal decomposition
V ¼
M
V
where the sum runs over isomorphism classes of
irreducible G
0
-representations , and the vector
spaces V
are called the G
0
-isotypic components of
V. For example, if G
0
is the rotation group SO
3
, the
G
0
-isotypic component V
of V is the subspace
spanned by all the states with total angular
momentum .
Consider now any U 2 Z. Since U commutes with
the G
0
-action, it does not connect different
G
0
-isotypic components. (Indeed, in the example of
SO
3
-invariant dynamics, angular momentum is
conserved and transitions between different angular
momentum sectors are forbidden.) Thus, every
G
0
-isotypic component V
is an invariant subspace
for the action of Z on V, and Z decomposes as
Z =
Q
Z
with blocks Z
= Z j
V
.
To say more, fix a standard irreducible
G
0
-module R
of isomorphism class and consider
L
:¼ Hom
G
0
ðR
; V
Þ
the linear space of C-linear maps l : R
!V
that
intertwine the G
0
-actions on R
and V
. An element
of L
is called a G
0
-equivariant homomorphism.By
Schur’s lemma, L
ffi C if V
is G
0
-irreducible. More
generally, dimL
=: m
counts the multiplicity of
occurrence of R
in V
; for example, in the case of
G
0
= SO
3
we take R
to be the standard irreducible
module of dimension 2 þ 1; and m
then is the
number of times a multiplet of states with total
angular momentum occurs in V
.
The natural mapping L
R
!V
by l r 7!l(r)
is an isomorphism,
V
ffi L
R
and using it we can transfer the entire discussion
from V
to L
R
. The group G
0
acts trivially on
L
ffi C
m
and irreducibly on R
. Therefore, the
component Z
of the centralizer Z is the unitary
group
Z
ffi UðL
ÞffiU
m
if V is a unita ry vector space with no extra structure.
In the presence of extra structure (which, by
compatibility with the G
0
-action, restricts to every
subspace V
), the factor Z
is some subgroup of
U
m
. In all cases, Z is a direct product of connected
compact Lie groups Z
.
To make the connection with symmetric spaces, write
M := Z
.SinceM is a group, the operation of taking
the inverse, U 7!U
1
, makes sense for all U 2 M.
Moreover, being a compact Lie group, the manifold M
admits a left- and right-invariant Riemannian structure
in which the inversion U 7!U
1
is an isometry. By
translation, one gets an isometry s
U
1
: U 7!U
1
U
1
U
1
for every U
1
2 M.Allofthesemapss
U
1
are globally
defined, and the restriction of s
U
1
to some neighborhood
of U
1
coincides with the geodesic inversion with respect
to U
1
. Thus, M is a symmetric space by the definition
given above. Symmetric spaces of this kind are called
type II.
Type I
Consider next the case of G
1
6¼;, where some
antiunitary symmetry T is present. As before, let Z
be the connected component of the centrali zer of G
0
in U (V). Conjugation by T,
U 7!ðU Þ :¼ TUT
1
is an automorphism of U (V) and, owing to T
2
= Id,
is involutive. Because G
0
G is a normal
subgroup, restricts to an involutive automorphism
(still denoted by )ofZ. Now recall that T is
complex antilinear and the good Hamiltonians are
subject to THT
1
= H. The good time evolutions
U
t
= e
itH=h
clearly satisfy (U
t
) = U
t
= U
1
t
. Thus,
the good set to consider is M := {U 2 Z jU = (U)
1
}.
The set M is a manifold, but in general is not a
Lie group.
Further details depend on what does with the
factorization Z =
Q
Z
.IfV
is a G
0
-isotypic
206 Symmetry Classes in Random Matrix Theory
component of V, then so is TV
, since T normalizes
G
0
. Thus, either V
\ TV
= 0, or TV
= V
. In the
former case, the involutive automorphism just
relates U 2 Z
with (U) 2 Z
TV
, whence no intrin-
sic constraint on Z
results, and the time evolutions
(U, (U)
1
) 2 Z
Z
TV
constitute a type-II sym-
metric space, as before.
A novel situation occurs when TV
= V
, in which
case restricts to an automorphis m of Z
. Let
therefore TV
= V
, put K Z
for short, and
consider
M :¼fU 2 KjU ¼ ðUÞ
1
g
Note that if two elements p, p
0
of K are in M,
then so is the product p
0
p
1
p
0
. The group K acts on
M K by
k U ¼ kUðkÞ
1
ðk 2 KÞ
and this group action is transitive, that is, every U 2 M
can be written as U = k(k)
1
with some k 2 K.
(Finding k for a given U is like taking a square root,
which is possible since exp : Lie K !K is surjective.)
There exists such a K-invariant Riemannian structure
for M that for all p
0
2 M the mapping s
p
0
: M !M
defined by
s
p
0
ðpÞ¼p
0
p
1
p
0
is the geodesic inversion with respect to p
0
2 M.
Thus, in this natural geometry M is a globally
symmetric Riemannian manifold and hence a sym-
metric space. The present kind of symmetric space is
called type I. If K
is the set of fixed points of in K,
the symmet ric space M is analytically diffeomorphic
to the coset space K=K
by
K=K
! M K; UK
7!UðU Þ
1
which we call the ‘‘Cartan embedding’’ of K=K
into K.
In summary, the solution to the problem of
finding the set of unitary time evolution operators
that are compatible with a given symmetry group G
and structure of Hilbert space V is always a
symmetric space. This is a valuable insight, as
symmetric spaces are rigid objects and have been
completely classified by Car tan.
If the dimension of V is kept variable, the
irreducible symmetric spaces that occur belong to
one of the large families listed in Table 1.
Dyson’s Threefold Way
Recall the goal: given a Hilbert space V and a
symmetry group G acting on it, one wants to classify
the (irreducible) spaces of time evolution operators
U that are ‘‘compatible’’ with G, meaning
U ¼ g
0
Ug
1
0
¼ g
1
U
1
g
1
1
ðfor all g
2 G
Þ
As we have seen, the spaces that arise in this way are
symmetric spaces of type I or II depending on the
nature of the time reversal (or other antiunitary
symmetry) T.
An even stronger statement can be made when
more information about the Hilbert space V is
specified. In Dyson’s classification, the Hermitian
scalar product of V is assumed to be the only
invariant structure that exists on V. With that
assumption, only three large families of symmetric
spaces arise; these correspond to what we call the
‘‘Wigner–Dyson symmetry classes.’’
Class A
Recall that in Dyson’s case, the connected part of the
centralizer of G
0
in U (V) is a direct product of
unitary groups, each factor being associated with one
G
0
-isotypic component V
of V. The type-II situation
occurs when the set G
1
of antiunitary symmetries is
either empty or else exchanges different V
. In both
cases, the set of good time evolution operators
restricted to one G
0
-isotypic component V
is a
unitary group U
m
,withm
being the multiplicity of
the irreducible G
0
-representation in V
.
The unitary groups U
N = m
or, to be precise, their
simple parts SU
N
, are called type-II symmetric spaces
of the A family or A series – hence the name class A.
The Hamiltonians H, the generators of time evolu-
tions U
t
= e
itH=h
, in this class are represented by
Table 1 The large families of symmetric spaces. The form of H
in the header applies to the last seven families
Family Symmetric
space
Form of H =
WZ
Z
y
W
A U
N
Complex Hermitian
AIU
N
=O
N
Real symmetric
AII U
2N
=USp
2N
Quaternion self-adjoint
C USp
2N
Z complex symmetric,
W = W
y
C I USp
2N
=U
N
Z complex symmetric,
W = 0
D SO
2N
Z complex skew,
W = W
y
DIII SO
2N
=U
N
Z complex skew,
W = 0
AIII U
pþq
=U
p
U
q
Z complex p q, W = 0
BDISO
pþq
=SO
p
SO
q
Z real p q, W = 0
CII USp
2pþ2q
=USp
2p
USp
2q
Z quaternion
2p 2q, W = 0
Symmetry Classes in Random Matrix Theory 207