Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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limit theorem’’ does not apply since our generating
function is of the form
ðe
i
Þ¼
ð1
1
e
i
Þð1 e
i
Þ
ð1
1
e
i
Þð1 e
i
Þ
1=2
½16
In 1968, Fisher and Hartwig raised a conjecture
about D
n
() for nonsmooth which included the
above example. They considered generating func-
tions of the form
ðe
i
Þ¼ ðe
i
Þ
Y
R
j¼1
j
;
j
ðe
ið
j
Þ
Þ½17
where
;
ðe
i
Þ¼ð2 2 cos Þ
e
iðÞ
; 0 <<2
<>1=2, and is not an integer. The function
is assumed to be a smoot h function. Using the
Fisher–Hartwig notation, the symbol of interest in
the Ising model from eqn [16] can be written as
ðe
i
Þ
0;1=2
ðe
i
Þ
where
ðe
i
Þ¼
1
1
e
i
1
1
e
i
1=2
The conjecture of Fisher and Hartwig for general
symbols of this type stated that
D
n
ðÞGð Þ
n
n
p
E
where
p ¼
X
R
r¼1
ð
2
r
2
r
Þ
and E
is a constant whose value they did not
identify. The constant was later computed to be
E
ðÞ¼Eð Þ
Y
R
j¼1
þ
ðe
i
j
Þ
j
þ
j
ðe
i
j
Þ
j
j
Y
1s6¼rR
ð1 e
ið
s
r
Þ
Þ
ð
s
þ
s
Þð
r
r
Þ
Y
R
j¼1
Gð1 þ
j
þ
j
ÞGð1 þ
j
j
Þ
Gð1 þ 2
j
Þ
where G(z) is the Barnes G-function satisfying
Gð1 þ zÞ¼ðzÞGðzÞ
and is defined by
Gð1 þ zÞ¼ð2Þ
z=2
e
ðzþ1Þz=2z
2
=2
Y
1
k¼1
1 þ
z
k
k
e
zþz
2
=2k
For the above factors, we normalize so that the
geometric mean is 1. Then we may assume that
the factors
þ
,
(
þ
= )are1atzeroand
infinity, respectively, and this defines the loga-
rithms for the first product. The E( )termisthe
constant in Szego¨ ’s theorem, and the argument of
atermoftheform(1 e
i(
s
r
)
)istakenbetween
=2and=2.
In the case where R = 1, the conjecture is known
to hold if <>1=2 and the function b satisfies the
conditions of Szego¨ ’s theorem and is infinitely
differentiable. The theorem also has an extension
to the case where <<1=2, with 2 not an
integer, as long as the Fourier coefficients are
defined as the coefficients of a distri bution.
If we apply the theorem to the generating fun ction
from [16]
ðe
i
Þ
0;1=2
ðe
i
Þ¼
1
1
e
i
1
1
e
i
1=2
0;1=2
ðe
i
Þ
we see that the asymptotic expansi on is given by
n
1=4
1 þ
1
1
1
1=4
Gð1=2ÞGð3=2Þ
This last formula shows that, at the critical
temperature,
lim
n!1
h
0;0
0;n
i¼lim
n!1
D
n
ðÞ¼0
thus, M = 0, an d hence there is no correlation
between distant lattice points .
It should be remarked here that the diagonal
correlation at the critical temperature is also given
by a singular Toeplitz determinant,
h
0;0
n;n
i¼D
n
ð
0;1=2
Þn
1=4
Gð1=2ÞGð3=2Þ
and thus this limit is also zero.
The proof of the Fisher–Hartwig conjecture is
much more complicated than the proof of the
‘‘strong Szego¨ limit theorem.’’ For an indication of
how it is proved, note t hat if we consider the
generating function
0,
, the Fourier coefficients
are (sin )=[(n )] an d hence the matrix is
Cauchy and the determinant can be computed
exactly. From this the asymptotics can be derived
and they yield a special case of the Fisher–Hartwig
conjecture. The main idea in extending the result to
a symbol of the form
ðe
i
Þ
0;
ðe
i
Þ
is to prove that the limit of
D
n
ð
0;
Þ
D
n
ð ÞD
n
ð
0;
Þ
248 Toeplitz Determinants and Statistical Mechanics
exists. The proof uses much of the same trace-c lass
approach used in proving the ‘‘strong Szego¨ limit
theorem,’’ although the results are more compli-
cated. These ideas are then extended for R > 1 and
also more general and .
It should be noted that in this article the Fisher–
Hartwig conjecture does not always hold. If we
consider the function
ðe
i
Þ¼
1; <<0
1; 0 <<
then
n
¼
0, if k is even
2i/ðkÞ,ifk is odd
The matrix T
n
() is antisymmetric and, if n is odd,
D
n
() = 0. If n is even, using elementary row and
column operations, the determinant can be put in
block form with each block of Cauchy type. The
determinant can then be evaluated to find
D
n
ðÞðiÞ
n
n
1=2
K
where K is a certain constant.
It is instructive to note that
ðe
i
Þ¼
0;1=2
ðe
i
Þ
0;1=2
ðe
iðÞ
Þ
¼
0;1=2
ðe
i
Þ
0;1=2
ðe
iðÞ
Þ
and thus that this particular symbol has two
representations of the type give n in [17] and each
would give a differe nt asymptotic expansion of the
determinant if the conjecture were true for this set of
parameters. Hence, it is clear that the conjecture
must fail to hold in this case.
However, this example indicates that there might
be a generalization of the original conjecture of
Fisher and Hartwig. If
;
ðe
iðÞ
Þ¼
;;
then
¼
Y
R
j¼1
j
;
j
;
j
it is also the case that
¼
Y
R
j¼1
j
;
j
þn
j
;
j
where
X
R
j¼1
n
j
¼ 0 and
¼
Y
R
j¼1
ðe
i
j
Þ
n
j
In the example above,
1
= 1=2,
2
= 1=2,
1
= 0,
2
= , n
1
= 1, and n
2
= 1. The result for
the counterexample, combined with what is known
for the case of integer values of and , leads to the
following generalized con jecture. Suppose
ðe
i
Þ¼
k
Y
R
j¼1
k
j
;
k
j
;
j
for some set of indices k.DefineQ(k) =
P
R
j = 1
(
k
j
)
2
(
k
j
)
2
.LetQ = max
k
<(Q(k)) and
K¼fk j<ðQðkÞÞ ¼ Q g
The generalized asymptotic formula is conjectured
to be
D
n
ð Þ¼
X
k2K
Gð
k
Þn
QðkÞ
E
k
þ oðjGðÞj
n
n
Q
Þ
It may turn out that there is only one element in K
and for these symbols there is a unique representa-
tion that yields the highest power in the exponent of
the asymptotic expansion. These are the symbols for
which the original Fisher–Hartwig conjecture should
be true and it is now confirmed in these cases. For
example, the conjecture is known to hold for R > 1
when j<
r
j < 1=2 and j<
r
j < 1=2.
Symbols with Nonzero Index or T > T
c
The last possibility in computing the correlation
asymptotics is the case where
2
> 1. Note that, for
fixed E
1
and E
2
, there is exactly one value of
= 1=kT where
2
¼ z
1
1
1 z
2
1 þ z
2
¼ 1
For values of T > T
c
, we have that the symbol
ð1
1
e
i
Þð1
2
e
i
Þ
ð1
1
e
i
Þð1
2
e
i
Þ
1=2
is the same as
e
i
ð1
1
e
i
Þð1 ð1/
2
Þe
i
Þ
ð1
1
e
i
Þð1 ð1/
2
Þe
i
Þ
1=2
with the argument chosen so that the symbol is
positive at . Except for the extra factor of e
i
, this
is the same type of smooth symbol that was
consi dered earl ier (see the sect ion ‘‘Strong Szego¨
limit theore m’’). How ever, a factor of e
i
can change
the asymptotics considerably as can be seen by
considering the simple example of the 1.
Fortunately, a variation of the Szego¨ theorem, first
Toeplitz Determinants and Statistical Mechanics 249
considered by Fisher and Hartwig, holds for this
case of smooth, nonvanishing index.
Theorem 2 Suppose that =
þ
satisfies the
condition of the ‘‘strong Szego¨ limit theorem’’ and
in addition is at least once continuously differenti-
able. Then, if b =
1
þ
and c =
1
þ
,
D
n
ðe
im
Þ
ð1Þ
ðn1þmÞm
GðÞ
n
EðÞGðcÞ
m
½18
det
b
n
b
nmþ1
.
.
.
.
.
.
.
.
.
b
n1þm
b
n
0
B
B
@
1
C
C
A
þ Oðn
3
Þ
0
B
B
@
1
C
C
A
ð1 þ Oðn
1
ÞÞ ½19
Applying this to the symbol
e
i
ðe
i
Þ¼e
i
ð1
1
e
i
Þð1 ð1/
2
Þe
i
Þ
ð1
1
e
i
Þð1 ð1/
2
Þe
i
Þ
1=2
we have that m = 1, G() = 1; G(c) = 1, and
EðÞ¼ 1
2
1
1
1
2
2
1
1
2
2
"#
1=4
½20
The determinant in the above formula is the
constant
b
n
¼
1
2
Z
2
0
ð1
1
e
i
Þ 1
1
2
e
i
ð1
1
e
i
Þ 1
1
2
e
i
1=2
e
in
d
The last integral can be deformed to a segment of
the real line and evaluated asymptotically to find
that the leading term is
1
ffiffiffi
p
n
2
1
1
2
ð1
1
2
Þ 1
1
2
2
1=2
ðn þ 1=2Þ
ðn þ 1Þ
Putting this toget her with the above constants, we
have, for T > T
c
,
h
0;0
0;n
i
1
ffiffiffiffiffiffi
n
p
n
2
1
2
1
1=4
1
1
2
2
1=4
ð1
1
2
Þ
1=2
"#
This implies that the correlation tends to zero very
rapidly as n !1.
Further Remarks
The interaction between statistical mechanics and
the theory of Toeplitz determinants has a long
history, and much of the motivation to describe the
asymptotics of the determinants was spurred by the
question of spontaneous magnetization in the two-
dimensional Ising model. The previous three sections
attempt to show how the very different physical
situations – T < T
c
, T = T
c
, and T > T
c
– all
correspond to very different behavior in the symbols
of the generating functions. Critical systems predict
qualitatively different Szego¨ type theorems. For
example, the phase transition at T
c
predicts that
the asymptotics for singular symbols cannot be
predicted by the smoot h symbols, that is, one cannot
use continuous functions to approximate the results
for singular symbols.
Onsager (1971) was the first to understand that
the correlation function could be expressed as a
Toeplitz determinant. This was made explicit by
Montroll et al. (1963). For more information about
the Ising model, the reader is referred to McCoy
and Wu (1973), where a clear and complete
description of the Ising model (and most of the
notation used here in reference to this model) can
be found.
Szego¨ (1915, 1952) had originally proved a weak
form of the ‘‘limit’’ theorem and he understood that
it was desirable to extend to a second-order term.
Szego¨ first proved the ‘‘strong Szego¨ limit theorem’’
for positive generating functions and this was later
extended to the nonpositive case.
The first to understand that a different asymptotic
behavior was expected at the critical temperature
was Fisher and this resulted in the conjecture for the
class of determinants generated by what is now
known as Fisher–Hartwig symbols (Fisher and
Hartwig 1968). Progress on the conjecture was
made by many authors. Bo¨ ttcher and Silbermann
(1998) have provided general results concerning
Toeplitz operators and determinants. Additional
information about the conjectures of Fisher and
Hartwig can be found in Bo¨ ttcher and Silbermann
(1990, 1998), Ehrhardt (2001), and Ehrhardt and
Silbermann (1997).
Toeplitz determinants are also important in many
other applications. One more recent area of interest
is the connection between random-matrix theory
and Toep litz determinants. Many statistical quanti-
ties for the circular unitary ensemble can be
described as a Toeplitz determinant. For example,
the probability of finding no eigenvalues in an
interval can be expressed as a Toeplitz determinant.
It is also the case that many of the most interesting
250 Toeplitz Determinants and Statistical Mechanics
statistics correspond to singular symbols. For basic
random-matrix theory information see Mehta
(1991), and for connections between the circular
unitary ensemble and Toeplitz determinants,
see Hughes (2001), Tracy and Widom (1993), and
Widom (1994).
See also: Integrable Systems in Random Matrix Theory;
Two-Dimensional Ising Model.
Further Reading
Bo¨ ttcher A and Silbermann B (1990) Analysis of Toeplitz
Operators. Berlin: Akademie-Verlag.
Bo¨ ttcher A and Silbermann B (1998) Introduction to Large
Truncated Toeplitz Matrices. Berlin: Springer.
Ehrhardt T (2001) A status report on the asymptotic behavior of
Toeplitz determinants with Fisher–Hartwig singularities.
Operator Theory: Advances and Applications 124: 217–241.
Ehrhardt T and Silbermann B (1997) Toeplitz determinants with
one Fisher–Hartwig singularity. Journal of Functional Analysis
148: 229–256.
Fisher ME and Hartwig RE (1968) Toeplitz determinants: some
applications, theorems, and conjectures. Advances in Chemical
Physics 15: 333–353.
Hughes C, Keating JP, and O’Connell N (2001) On the
characteristic polynomial of a random unitary matrix. Com-
munications in Mathematical Physics 220: 429–451.
McCoy BM and Wu TT (1973) The Two-Dimensional Ising
Model. Cambridge, MA: Harvard University Press.
Metha ML (1991) Random Matrices. San Diego: Academic Press.
Montroll EW, Potts RB, and Ward JC (1963) Correlations and
spontaneous magnetization of the two-dimensional Ising
model. Journal of Mathematical Physics 4(2): 308–322.
Onsager L (1971) The Ising model in two dimensions. In: Mills
RE, Ascher E, and Jaffee RI (eds.) Critical Phenomena in
Alloys, Magnets, and Superconductors, pp. 3–12. New York:
McGraw-Hill.
Szego¨ G (1915) Ein Grenzwertsatz u¨ ber die Toeplitzschen
Determinanten einer reellen positiven Funktion. Mathema-
tische Annalen 76: 490–503.
Szego¨ G (1952) On Certain Hermitian Forms Associated with the
Fourier Series of a Positive Function, pp. 222–238. Lund:
Festschrift Marcel Riesz.
Tracy CA and Widom H (1993) Introduction to random
matrices. In: Helminck GF (ed.) Geometric and Quantum
Aspects of Integrable Systems, (Scheveningen, 1992), Lecture
Notes in Physics, vol. 424, pp. 103–130. Berlin: Springer.
Widom H (1994) Random Hermitian matrices and (nonrandom)
Toeplitz matrices. In: Basor E and Gohberg I (eds.) Toeplitz
Operators and Related Topics (Santa Cruz, CA, 1992), Oper.
Theory Adv. Appl., vol. 71, pp. 9–15. Basel: Birkhuser.
Tomita–Takesaki Modular Theory
S J Summers, University of Florida, Gainesville, FL,
USA
ª 2006 Elsevier Ltd. All rights reserved.
Basic Structure
The origins of Tomita–Takesaki modular theory lie
in two unpublished papers of M Tomita in 1967 and
a slim volume by Takesaki (1970). It has developed
into one of the most important tools in the theory of
operator algebras and has found many app lications
in mathematical physics.
Although the modular theory has been formulated
in a more general setting, it will be presented in the
form in which it most often finds application in
mathematical physics (for generalizations, details,
and further references concer ning the material
covered in this article, the reader is referred to the
Further Reading section). Let M be a von Neumann
algebra on a Hilbert space H containing a vector
which is cyclic and separating for M. Define the
operator S
0
on H as follows:
S
0
A ¼ A
; for all A 2M
This operator extends to a closed antilinear operator
S defined on a dense subset of H. Let be the
unique positive, self-adjoint operator and J the
unique antiunitary operator occurring in the polar
decomposition
S ¼ J
1=2
¼
1=2
J
is called the modular operator and J the modular
conjugation (or modular involution) associated with
the pair (M, ). Note that J
2
is the identity operator
and J = J
. Moreover, the spectral calculus may be
applied to so that
it
is a unitary operator for
each t 2 R and {
it
jt 2 R} forms a strongly con-
tinuous unitary group. Let M
0
denote the set of all
bounded linear operators on H which commute with
all elements of M. The modular theory begins with
the following remarkable theorem.
Theorem 1 Let M be a von Neumann algebra
with a cyclic and separating vector . Then
J==, and the following equalities hold:
JMJ ¼M
0
and
it
M
it
¼M; for all t 2 R
Note that if one defines F
0
A
0
=A
0
, for all A
0
2
M
0
, and takes its closure F, then one has the relations
¼ FS;
1
¼ SF; F ¼ J
1=2
Tomita–Takesaki Modular Theory 251
Modular Automorphism Group
By Theorem 1, the unitaries
it
, t 2 R, induce a one-
parameter automorphism group {
t
}ofM by
t
ðAÞ¼
it
A
it
; A 2M; t 2 R
This group is called the modular automorphism
group of M (relative to ). Let ! denote the faithful
normal state on M induced by :
!ðAÞ¼
1
kk
2
h; Ai; A 2M
From Theorem 1 it follows that ! is invariant under
{
t
}, that is, !(
t
(A)) = !(A) for all A 2Mand t 2 R.
The modular automorphism group contains infor-
mation about both M and !. For example, the
modular automorphism group is an inner auto-
morphism on M if and only if M is semifinite. It is
trivial if and only if ! is a tracial state on M. Indeed,
for any B 2M, one has
t
(B) = B for all t 2 R if and
only if !(AB) = !(BA) for all A 2M. Let M
denote the set of all such B in M.
The KMS Condition
The modular automorphism group satisfies a condi-
tion which had already been used in mathematical
physics to characterize equilibrium temperature
states of quantum systems in statistical mechanics
and field theory – the Kubo–Martin–Schwinger
(KMS) condition. If M is a von Neumann algebra
and {
t
jt 2 R}isa-weakly continuous one-
parameter group of automorphisms of M, then the
state on M satisfies the KMS condition at (inverse
temperature) (0 <<1) with respect to {
t
}if
for any A,B 2M there exists a complex function
F
A,B
(z) which is analytic on the strip {z 2 C j0 <
Im z <} and continuous on the closure of this strip
such that
F
A;B
ðtÞ¼ð
t
ðAÞBÞ
F
A;B
ðt þ iÞ¼ðB
t
ðAÞÞ
for all t 2 R. In this case, (
i
(A)B) = (BA), for all
A, B in a -weakly dense, -invariant -subalgebra
of M. Such KMS states are -invariant, that is,
(
t
(A)) = (A), for all A 2M, t 2 R, and are stable
and passive (cf. Bratteli and Robinson (1981) and
Haag (1992)).
Every faithful normal state satisfies the KMS
condition at = 1 (henceforth called the modular
condition) with respect to the corresponding mod-
ular automorphism group.
Theorem 2 Let M be a von Neumann algebra
with a cyclic and separating vector . Then the
induced state ! on M satisfies the modular condi-
tion with respect to the modular automorphism
group {
t
jt 2 R} associated to the pair (M, ).
The modular automorphism group is, therefore,
endowed w ith the analyticity associated with the
KMS condition, and this is a powerful tool in
many applications of the m odular theory to
mathematical physics. In addition, the physical
properties and interpretations of KMS states are
often invoked when applying modular theory to
quantum physics.
Note that while the nontriviality of the modular
automorphism group gives a measure of the non-
tracial nature of the state, the KMS condition for the
modular automorphism group provides the missing
link between the values !(AB) and !(BA), for all
A,B 2M(hence the use of the term ‘‘modular,’’ as
in the theory of integration on locally compact
groups).
The modular condition is quite restrictive. Only
the modular group can satisfy the modular condition
for (M, ), and the modular group for one state can
satisfy the modular condition only in states differing
from the original state by the action of an element in
the center of M.
Theorem 3 Let M be a von Neuman n algebra
with a cyclic and separating vector , and let {
t
}
be the corresponding modular automorphism
group. If the induced state ! satisfies the modular
condition with respect to a group {
t
} of auto-
morphisms of M, then {
t
} must coincide with {
t
}.
Moreover, a normal state on M satisfies the
modular condition with respect to {
t
} if and only
if ( ) = !(h ) = !(h
1=2
h
1=2
) for some unique
positive injective operator h affiliated with the
center of M.
Hence, if M is a factor, two distinc t states cannot
share the same modular automorphism group. The
relation between the modu lar automorphism groups
for two different states will be described in more
detail.
One Algebra and Two States
Consider a von Neumann algebra M with two
cyclic and separating vectors and , and denote
by ! and , respectively, the induced states on M.
Let {
!
t
} and {
t
} denote the corresponding modular
groups. There is a general relation between the
modular automorphism grou ps of these states.
252 Tomita–Takesaki Modular Theory
Theorem 4 There exists a -strongly contin uous
map R 3 t 7!U
t
2Msuch that
(i) U
t
is unitary for all t 2 R;
(ii) U
tþs
= U
t
!
t
(U
s
) for all s,t 2 R; and
(iii)
t
(A) = U
t
!
t
(A)U
t
for all A 2Mand t 2 R.
The 1-cocycle {U
t
} is commonly called the cocycle
derivative of with respect to ! and one writes
U
t
= (D : D!)
t
. There is a chain rule for this
derivative, as well: If , , and are faithful normal
states on M, then (D : D)
t
= (D : D)
t
(D : D)
t
,
for all t 2 R. More can be said about the cocycle
derivative if the states satisfy any of the conditions
in the following theorem.
Theorem 5 The following conditions are
equivalent:
(i) is {
!
t
}-invariant;
(ii) ! is {
t
}-invariant;
(iii) there exists a unique positive injective operator
h affiliated with M
!
\M
such that !( ) ¼
(h ) = (h
1=2
h
1=2
);
(iv) there exists a unique positive injective operator
h
0
affiliated with M
!
\M
such that ( ) ¼
!(h
0
) = !(h
01=2
h
01=2
);
(v) the norms of the linear functionals ! þ i and
! i are equal; and
(vi)
!
t
s
=
s
!
t
, for all s, t 2 R.
The conditions in Theorem 5 turn out to be
equivalent to the cocycle derivative being a
representation.
Theorem 6 The cocycle {U
t
} intertwining {
!
t
} with
{
t
} is a group representation of the additive group
of reals if and only if and ! satisfy the conditions
in Theorem 5. In that case, U(t) = h
it
.
The operator h
0
= h
1
in Theorem 5 is called the
Radon–Nikodym derivative of with respect to !
(often denoted by d=d!), due to the following
result, which, if the algebra M is abelian, is the
well-known Radon–Nikodym theorem from mea-
sure theory.
Theorem 7 If and ! are normal positive linear
functionals on M such that ( A) !(A), for all
positive elements A 2M, then there exists a unique
element h
1=2
2Msuch that ( ) = !(h
1=2
h
1=2
) and
0 h
1=2
1.
The analogies with measure theory are not
accidental, although these are not discussed in detail
here. Indeed, any normal trace on a (finite) von
Neumann algebra M gives rise to a noncommuta-
tive integration theory in a natural manner. Mod-
ular theory affords an extension of this theory to the
setting of faithful normal functional s on von
Neumann algebras M of any type, enabling the
definition of noncommutative L
p
spaces, L
p
(M, ).
Modular Invariants and the Classification
of von Neumann Algebras
As already ment ioned, the modular structure carries
information about the algebra. This is best evi-
denced in the structure of type III factors. As this
theory is rather involved, only a sketch of some of
the results can be given.
If M is a type III algebra, then its crossed
product N = Mo
!
R relative to the modular
automorphism group of any faithful normal state
! on M is a type II
1
algebra with a faithful
semifinite normal trace such that
t
= e
t
,
t 2 R, where is the dual of
!
on N. Moreover,
the algebra M is isomorphic to the cross product
N o
R, and this decomposition is unique in a very
strong sense. This structure theorem entails the
existence of important algebraic invariants for M,
which has many consequences, one of which is made
explicit here.
If ! is a fai thful normal state of a von Neumann
algebra M induced by , let
!
denote the modular
operator associated to (M, ) and sp
!
denote the
spectrum of
!
. The intersection
S
0
ðMÞ ¼ \ sp
!
over all faithful normal states ! of M is an algebraic
invariant of M.
Theorem 8 Let M be a factor acting on a
separable Hilbert space. If M is of type III, then
0 2 S
0
(M); otherwise, S
0
(M) = {0,1} if M is of type
I
1
or II
1
and S
0
(M) = {1} if not. Let M now be a
factor of type III.
(i) M is of type III
,0<<1, if and only if
S
0
(M) = {0} [ {
n
jn 2 Z}.
(ii) M is of type III
0
if and only if S
0
(M) = {0, 1}.
(iii) M is of type III
1
if and only if S
0
(M) = [0, 1).
In certain physically relevant situations, the
spectra of the modular operators of all faithful
normal states coincide, so that Theorem 8 entails
that it suffices to compute the spect rum of any
conveniently chosen modular operator in order to
determine the type of M. In other such situations,
there are distinguished states ! such that
S
0
(M) = sp
!
. One such example is provide d by
asymptotically abelian systems. A von Neumann
algebra M is said to be ‘‘asymptotically abelian’’ if
there exists a sequence {
n
}
n2N
of automorphisms of
M such that the limit of {A
n
(B)
n
(B)A}
n2N
in
Tomita–Takesaki Modular Theory 253
the strong operator topolog y is zero, for all A, B 2
M. If the state ! is
n
-invariant, for all n 2 N, then
sp
!
is contained in sp
, for all faithful normal
states on M, so that S
0
(M) = sp
!
. If, moreover,
sp
!
= [0, 1), then sp
!
= sp
, for all as
described.
Self-Dual Cones
Let j : M!M
0
denote the antilinear -isomorphism
defined by j(A) = JAJ,A 2M. The natural positive
cone P
\
associated with the pair (M, ) is defined as
the closure, in H, of the set of vectors
fAjðAÞ jA 2Mg
Let M
þ
denote the set of all positive elements of M.
The following theorem collects the main attributes
of the natural cone.
Theorem 9
(i) P
\
coincides with the closure in H of the set
{
1=4
A jA 2M
þ
}.
(ii)
it
P
\
= P
\
for all t 2 R.
(iii) J= for all 2P
\
.
(iv) Aj(A)P
\
P
\
for all A 2M.
(v) P
\
is a pointed, self-dual cone whose linear
span coincides with H.
(vi) If 2P
\
, then is cyclic for M if and only if
is separating for M.
(vii) If 2P
\
is cyclic, and hence separating, for
M, then the modular conjugation and the
natural cone associated with the pair (M, )
coincide with J and P
\
, respectively.
(viii) For every normal positive linear functional
on M, there exists a unique vector
2P
\
such that (A) = h
, A
i for all A 2M.
In fact, the algebras M and M
0
are uniquely
characterized by the natural cone P
\
[4]. In light of
(viii), if is an automorphism of M, then
VðÞ
¼
1
defines an isometric operator on P
\
, which by (v)
extends to a unitary operator on H. The map
7!V() defines a unitary representation of the
group of automorphisms Aut(M)onM in such a
manner that V()AV()
1
= (A) for all A 2Mand
2 Aut(M). Indeed, one has the following:
Theorem 10 Let M be a von Neumann algebra
with a cyclic and separating vector . The group V
of all unitaries V satisfying
VMV
¼M; VJV
¼ J; VP
\
¼P
\
is isomorphic to Aut(M) under the above map
7!V(), which is called the ‘‘standard implemen-
tation’’ of Aut(M).
Often of particular physical interest are (anti-)auto-
morphisms of M leaving ! invariant. They can only
be implem ented by (anti)unitaries which leave
the pair (M, ) invariant. In fact, if U is a unitary
or antiunitary operator satisfying U= and
UMU
= M, then U commutes with both J and .
Two Algebras and One State
Motivated by applications to quantum field theory,
the study of the modular structures associated with
one state and more than one von Neumann algebra
has begun (see Borchers (2000) for references and
details). Let NMbe von Neumann algebras
with a common cyclic and separating vector ,
and
N
, J
N
and
M
, J
M
denote the corresponding
modular objects. The structure (M, N, ) is called
a -half-sided modular inclusion if
it
M
N
it
M
N,forallt 0.
Theorem 11 Let M be a von Neumann algebra
with cyclic and separating vector . The following
are equivalent:
(i) There exists a proper subalgebra NMsuch that
(M, N, ) is a -half-sided modular inclusion.
(ii) There exists a unitary group {U(t)} with positive
generator such that
UðtÞMUðt Þ
1
M; for all t 0;
UðtÞ ¼ ; for all t 2 R
Moreover, if these conditions are satisfied, then the
following relations must hold:
it
M
Uðs Þ
it
M
¼
it
N
Uðs Þ
it
N
¼ Uðe
2t
sÞ
and
J
M
Uðs ÞJ
M
¼ J
N
Uðs ÞJ
N
¼ UðsÞ
for all s,t 2 R. In addition, N = U(1)MU(1)
1
,
and if M is a factor, it must be type III
1
.
The richness of this structure is further suggested
by the next theorem.
Theorem 12
(i) Let (M, N
1
, ) and (M, N
2
, ) be -half-sided,
resp. þ-half-sided, modular inclusions satisfy-
ing the condition J
N
1
J
N
2
= J
M
J
N
2
J
N
1
J
M
. Then
the modular unitaries
it
M
,
is
N
1
,
iu
N
2
, s, t, u 2 R,
generate a faithful continuous unitary repre-
sentationoftheidentitycomponentofthe
254 Tomita–Takesaki Modular Theory
group of isometries of two-dimensional Min-
kowski space.
(ii) Let M, N, N\M be von Neumann algebras
with a common cyclic and separating vector . If
(M, M\N, ) and (N, M\N, ) are -half-
sided, resp. þ-half-sided, modular inclusions such
that J
N
MJ
N
= M, then the modular unitaries
it
M
,
is
N
,
iu
N\M
, s, t, u 2 R, generate a faithful
continuous unitary representation of SL
(2, R)=Z
2
.
This has led to a further useful notion. If NM
and is cyclic for N\M, then (M, N, ) is said to
be a ‘‘-modular intersection’’ if both (M, M\N, )
and (N, M\N, )are-half-sided modular inclu-
sions and
J
N
lim
t!1
it
N
it
M
J
N
¼ lim
t!1
it
M
it
N
where the existence of the strong operator limits is
assured by the preceding assumptions. An example
of the utility of this structure is the following
theorem.
Theorem 13 Let N, M, L be von Neumann alge-
bras with a common cyclic and separating vector . If
(M, N, ) and (N
0
, L, ) are –-modular intersections
and (M, L, ) is a þ-modular intersection, then the
unitaries
it
M
,
is
N
,
iu
L
, s, t, u 2 R, generate a faithful
continuous unitary representation of SO
"
(1, 2).
These results and their extensions to larger
numbers of algebras were developed for application
in algebraic quantum field theory, but one may
anticipate that half-sided modular inclusions will
find wider use. Modular theory has also been
applied fruitfully in the theory of inclusions NM
of properly infinite algebras with finite or infinite
index.
Applications in Quantum Theory
The Tomita–Takesaki theory has found many
applications in quantum field theory and quantum
statistical mechanics. As mentioned earlier, the
modular automorphism group satisfies the KMS
condition, a property of physical significance in the
quantum theory of many-particle systems, which
includes quantum statistical mechanics and quantum
field theory. In such settings, for a suitab le algebra
of observables M and state !, an automorphism
group {
t
} representing the time evolution of the
system satisfies the modular condition. Hence, on
the one hand, {
t
} is the modular automorphism
group of the pair (M, ), and, on the other, ! is an
equilibrium state at inverse temperature , with all
the consequences which both of these facts have.
But it has become increasingly clear that the
modular objects
it
, J, of certain algebras of
observables and states encode additional physical
information. In 1975, it was discovered that if one
considers the algebras of observables associated with
a finite-component quantum field theory satisfying
the Wightman axioms, then the modular objects
associated with the vacuum state an d algebras of
observables localized in certain wedge-shaped
regions in Minkowski space have geometric content.
In fact, the unita ry group {
it
} implements the group
of Lorentz boosts leaving the wedge region invariant
(this property is now called modular covariance),
and the modular involution J implement s the space-
time reflection about the edge of the wedge, along
with a charge conjugation. This discovery caused
some intense research activity (see Baumgartel and
Wollenberg 1992, Borchers 2000, Haag 1992).
Positive Energy
In quantum physics the time development of the
system is often represented by a strongly continuous
group {U(t) = e
itH
jt 2 R} of unitary operators, and
the generator H is interpreted as the total energy of
the system. There is a link between modular
structure and positive energy, which has found
many applications in quantum field theory. This
result was crucial in the development of Theorem 11
and was motivat ed by the 1975 discovery mentioned
above, now commonly called the Bisognano–
Wichmann theorem.
Theorem 14 Let M be a von Neumann algebra
with a cyclic and separating vector , and let {U(t)}
be a continuous unitary group satisfying U(t)MU
( t) M, for all t 0. Then any two of the
following conditions imply the third:
(i) U(t) = e
itH
, with H 0;
(ii) U(t)=, for all t 2 R; and
(iii)
it
U(s)
it
= U(e
2t
s) and JU(s)J = U(s), for
all s, t 2 R.
Modular Nuclearity and Phase Space Properties
Modular theory can be used to express physically
meaningful properties of quantum ‘‘phase spaces’’
by a condition of compactness or nuclearity of
certain maps. In its initial form, the condition was
formulated in terms of the Hamiltonian, the global
energy operat or of theories in Minkowski space.
The above indications that the modular operators
carry information about the energy of the system
were reinforced when it was shown that a
Tomita–Takesaki Modular Theory 255
formulation in terms of modular operators was
essentially equivalent.
Let O
1
O
2
be nonempty bounded open subregions
of Minkowski space with corresponding algebras of
observables A(O
1
) A(O
2
) in a vacuum representa-
tion with vacuum vector ,andlet be the modular
operator associated with (A(O
2
), ) (by the Reeh–
Schlieder theorem, is cyclic and separating for
A(O
2
)). For each 2 (0, 1=2) define the mapping
: A(O
1
) !Hby
(A) =
A. The compactness
of any one of these mappings implies the compactness
of all of the others. Moreover, the l
p
(nuclear) norms of
these mappings are interrelated and provide a measure
of the number of local degrees of freedom of the
system. Suitable conditions on the maps in terms of
these norms entail the strong statistical independence
condition called the split property. Conversely, the split
property implies the compactness of all of these maps.
Moreover, the existence of equilibrium temperature
states on the global algebra of observables can be
derived from suitable conditions on these norms in the
vacuum sector.
The conceptual advantage of the modul ar com-
pactness and nuclearity conditions compared to
their original Hamiltonian form lies in the fact that
they are meaningful also for quantum systems in
curved spacetimes, where global energy operators
(i.e., generators corresponding to global timelike
Killing vector fields) need not exist.
Modular Position and Quantum Field Theory
The characterization of the relative ‘‘geometric’’
position of algebras based on the notions of modular
inclusion and modular intersection was directly
motivated by the Bisognano–Wichmann theorem.
Observable algebras associated with suitably chosen
wedge regions in Minkowski space provided exam-
ples whose essential structure could be abstracted
for more general application, resulting in the notions
presented in the preceding sections.
Theorem 12(ii) has been used to construct, from
two algebras and the indicated half-sided modular
inclusions, a conformal quantum field theory on the
circle (compactified light ray) with positive energy.
Since the chiral part of a conformal quantum field
model in two spacetime dimensions naturally yields
such half-sided modular inclusions, studying the
inclusions in Theorem 12(ii) is equivalent to study-
ing such field theories. Theorems 12(i) and 13
and their generalizations to inclusions involving up
to six algebras have been employed to construct
Poincare´-covariant nets of observable alge bras (the
algebraic form of quantum field theories) satisfying
the spectrum condition on (d þ 1)-dimensional
Minkowski space for d = 1, 2, 3. Conversely, such
quantum field theories naturally yield such systems
of algebras.
This intimate relation would seem to open up the
possibility of constructing interacting quantum field
theories from a limited number of modular inclu-
sions/intersections.
Geometric Modular Action
The fact that the modular objects in quantum field
theory associated with wedge-shaped regions and the
vacuum state in Minkowski space have geometric
significance (‘‘geometric modular action’’) was origin-
ally discovered in the framework of the Wightman
axioms. As an algebraic quantum field theory (AQFT)
does not rely on the concept of Wightman fields, it was
natural to ask (i) when does geometric modular action
hold in AQFT and (ii) which physically relevant
consequences follow from this feature?
There are two approaches to the study of
geometric modular action. In the first, attention is
focused on modular covariance, expressed in terms of
the modular groups associated with wedge algebras
and the vacuum state in Minkowski space. Modular
covariance has been proven to obtain in conformally
invariant AQFT, in any massive theory satisfying
asymptotic completeness, and also in the presence of
other, physically natural assumptions. To mention
only three of its consequences, both the spin–statistics
theorem and the PCT theorem, as well as the
existence of a continuous unitary representation of
the Poincare´ group acting covariantly upon the
observable algebras and satisfying the spectrum
condition follow from modular covariance.
In a second approach to geometric modular action,
the modular involutions are the primary focus. Here,
no aprioriconnection between the modular objects
and isometries of the spacetime is assumed. The central
assumption, given the state vector and the von
Neumann algebras of localized observables {A(O)} on
the spacetime, is that there exists a family W of subsets
of the spacetime such that J
W
1
R(W
2
)J
W
1
2
{R(W) jW 2W}, for every W
1
,W
2
2W.Thiscondi-
tion makes no explicit appeal to isometries or other
special attributes and is thus applicable, in principle, to
quantum field theories on general curved spacetimes.
It has been shown for certain spacetimes, including
Minkowski space, that under certain additional
technical assumptions, the modular involutions
encode enough information to determine the
dynamics of the theory, the isometry group of the
spacetime, and a continuous unitary representation of
the isometry group which acts covariantly upon the
observables and leaves the state invariant. In certain
256 Tomita–Takesaki Modular Theory
cases including Minkowski space, it is even possible
to derive the spacetime itself from the group J
generated by the modular involutions {J
W
jW 2W}.
The modular unitaries
it
W
enter in this approach
through a condition which is designed to assure the
stability of the theory, namely that
it
W
2J, for all
t 2 R and W 2W. In Minkowski space, this addi-
tional condition entails that the derived representation
of the Poincare´ group satisfies the spectrum condition.
Further Applications
As previously observed, through the close connec-
tion to the KMS condition, modular theory enters
naturally into the equilibrium thermodynamics of
many-body systems. But in recent work on the
theory of nonequilibrium thermodynamics it also
plays a role in making mathematical sense of the
notion of quantum systems in local thermodynamic
equilibrium. Modular theory has also proved to be
of utility in recent developments in the theory of
superselection rules and their attenda nt sectors,
charges and charge-carrying fields.
See also: Algebraic Approach to Quantum Field Theory;
Axiomatic Quantum Field Theory; Quantum Central-Limit
Theorems; Symmetries in Quantum Field Theory of
Lower Spacetime Dimensions; Thermal Quantum Field
Theory; Positive Maps on C
-Algebras; Two-Dimensional
Models; von Neumann Algebras: Introduction, Modular
Theory, and Classification Theory.
Further Reading
Baumga¨rtel H and Wollenberg M (1992) Causal Nets of Operator
Algebras. Berlin: Akademie-Verlag.
Borchers HJ (2000) On revolutionizing quantum field theory with
Tomita’s modular theory. Journal of Mathematical Physics 41:
3604–3673.
Bratteli O and Robinson DW (1981) Operator Algebras and
Quantum Statistical Mechanics II. Berlin: Springer.
Connes A (1974) Caracte´risation des alge` bres de von Neumann
comme espaces vectoriels ordonne´s. Annales de l’Institut
Fourier 24: 121–155.
Haag R (1992) Local Quantum Physics. Berlin: Springer.
Kadison RV and Ringrose JR (1986) Fundamentals of the Theory
of Operator Algebras, vol. II. Orlando: Academic Press.
Pedersen GK (1979) C
-Algebras and Their Automorphism
Groups. New York: Academic Press.
Stratila S (1981) Modular Theory in Operator Algebras. Tun-
bridge Wells: Abacus Press.
Takesaki M (1970) Tomita’s Theory of Modular Hilbert Algebras
and Its Applications, Lecture Notes in Mathematics, vol. 128.
Berlin: Springer.
Takesaki M (2003) Theory of Operator Algebras II. Berlin:
Springer.
Topological Defects and Their Homotopy Classification
T W B Kibble, Imperial College, London, UK
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
Symmetry-breaking phase transitions occur in a wide
variety of systems – from condensed matter to the
early universe. One of the common features of such
transitions is the appearance, in the broken-symmetry
phase, of topological defects, trapped regions in
which the symmetry is restored, or at least changed.
Examples are vortices in superfluids, domain walls in
ferromagnets, and disclination lines in liquid crystals.
Often these defects are stable for topological reasons,
and play an important role in the dynamics of the
system. An astonishingly rich variety of defects can be
found in various systems. They can usefully be
classified using the tools of homotopy theory.
Spontaneous Symmetry Breaking
Let us consider a quantum-mechanical system with a
symmetry group G. This means that each g 2 G is
represented on the Hilbert space of quantum states
by a unitary operator
ˆ
U(g), which commutes with
the Hamiltonian. Spontaneous symmetry breaking
occurs if this symmetry is not shared by the ground
state or vacuum state j0i of the system. In other
words, for some g 2 G,
ˆ
U(g)j0i 6¼j0i. Then the
ground state is necessarily degenerate:
ˆ
U(g)j0i must
have the same energy as j0i.
Spontaneous symmetry breaking is usually
describable in terms of an order-parameter field,
which vanishes above the transition and is nonzero
below it. We can find a scalar field
ˆ
(r), or multiplet
of fields
ˆ
= (
ˆ
i
, i = 1, ..., n) transforming according
to some representation D of G (assumed not to
contain the trivial represent ation), whose expecta-
tion value in the ground state is nonzero:
h0j
^
ðrÞj0i¼
0
6¼ 0 ½1
This is the order parameter. Since
h0j
^
U
y
ðgÞ
^
ðrÞ
^
UðgÞj0i¼Dð g Þ
0
½2
it follows that the only elements of G that can be
symmetries of the ground state are those in the
Topological Defects and Their Homotopy Classification 257