Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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That is, in the GNS (Gelfand–Naimark–Segal) repre-
sentation of the state !, the sequence S
(B) =
1=jj
P
x2
x
B converges weakly to a multiple of
the identity: S(B) !(B)1.Thistheorem,calledthe
mean ergodic theorem, characterizes the class of
states yielding a weak law of large numbers. Clearly,
these limits {S(A)jA 2B} form a trivial abelian algebra
of macroscopic observables.
Now we go a step further and consider space
fluctuations. Define the local fluctuation of an
observable A in a homogeneous (spatial invariant)
state ! by
F
ðAÞ¼
1
jj
1=2
X
x 2
x
A !ðAÞðÞ½5
The problem is to give a rigorous meaning to
lim F
(A) for tending to Z
in the sense of
extending boxes. When does such a limit exist?
What are the properties of the fluctuations or the
limits F(A) = lim F
(A), etc.? Again, the F(A) are
macroscopic variables of the microsystem.
Already we remark the following: if A, B are
strictly local elements, A, B 2A
L
, then
X
y 2Z
½A;
y
B2A
L
and an easy computation yields, by [4],
weak lim
F
ðAÞ; F
ðBÞ½
¼ weak lim
1
jj
X
x 2
x
X
y 2
½A;
yx
B
!
¼ weak lim
1
jj
X
x 2
x
X
y 2Z
½A;
y
B
!
¼
X
y 2Z
! ½A;
y
B
iðA; BÞ1
that is, if the F( A) and F(B) limits do exist, then
FðAÞ ; FðBÞ½¼iðA; BÞ1 ½6
This property indicates that fluctuations should have
the same commutation relations as boson fields. If
fluctuations can be characterized as macroscopic
observables, they must sati sfy the canonical com-
mutation relations (CCRs). Therefore, in the next
section we introduce the essentials on CCR
representations.
CCR Representations
We present the abstract Weyl CCR C
-algebra.
More details can be found in Brattelli and
Robinson (1979, 2002) and in particular in
Manuceau et al. (1973),wherethecaseofareal
test function space (H, ) with a possibly degen-
erate symplectic form is treated. Hence, H is a
real vector space and a bilinear, antisymmetric
form on H.
Denote by W(H, ) the complex vector space
generated by the functions W(f ), f 2H, defined by
Wðf Þ : H !C : g !Wðf Þg
¼
0iff 6¼ g
1iff ¼ g
W(H , ) becomes an algebra with unit W(0) for the
product
Wðf ÞWðgÞ¼Wðf þ gÞe
ði=2Þðf ;gÞ
; f ; g 2H
and a
-algebra for the involution
Wðf Þ!Wðf Þ
¼ Wðf Þ
It becomes a C
-algebra C
(H, ) following the
construction of Verbeure and Zagrebnov (1992).
A linear functional ! of a C
-algebra C
(H, )is
called a state if !(I) = 1 and !(A
A) 0 for all
A 2C
(H, )andI = W(0). Every state gives rise to a
representation through the GNS construction
(Brattelli and Robinson 1979, 2002). In particular,
! is a state if for any choice of A =
P
j
c
j
W(f
j
)we
have
X
j
k
c
j
c
k
! Wðf
j
f
j
Þ
e
iðf
j
;f
k
Þ
0
! Wð0ÞðÞ¼1
A remark abou t the special case that is degenerate
is in order. Denote by H
0
the kernel of :
H
0
¼ff 2Hjðf ; gÞ¼0 for all g 2Hg
If H = H
0
H
1
with
1
a nondegenerate symplectic
form on H
1
and
1
equal to the restriction of to
H
1
, we have that C
(H, ) is a tensor pr oduct:
C
ðH;Þ¼C
ðH
0
; 0ÞC
ðH
1
;
1
Þ
Note that C
(H
0
, 0) is abelian and that each
positive-definite normali zed functional ’,
’ : h 2H
0
!’ðWðhÞÞ
defines a state !(W(h)) = ’(W(h)) on C
(H
0
, 0).
Let be any character of the abelian additive
group H, then the map
,
Wðf Þ¼ðf ÞWðf Þ
extends to a
-automorphism of C
(H, ). Let s be a
positive symmetric bilinear form on H such that for
all f , g 2H:
1
4
jðf ; gÞj
2
sðf ; f Þsðg; gÞ½7
132 Quantum Central-Limit Theorems
and let !
s,
be the linear functional on C
(H, )
given by
!
s;
ðWðhÞÞ ¼ ðhÞe
ð1=2Þsðh;hÞ
½8
then it is straightforward (Bratt elli and Robinson
1979, 2002) to check that !
s,
is a state on C
(H, ).
All states of the type [8] are called quasifree states
on the CCR algebra C
(H, ).
A state ! of C
(H, ) is called a regular state if, for
all f , g 2H, the map 2 R !!(W(f þ g)) is con-
tinuous. The regularity property of a state yields the
existence of a Bose field as follows. Let (H, , )be
the GNS representation (Brattelli and Robinson
1979, 2002) of the state w, then the regularity of
w implies that there exists a real linear map
b : H !L(H) (linear operators on H) such that
8f 2H: b(f )
= b(f ) and
ðWðf ÞÞ ¼ exp ibðxÞðÞ
The map b is called the Bose field satisfying the Bose
field commutation relations:
½bðf Þ; bðgÞ ¼ iðf ; gÞ½9
Note that the Bose fields are state dependent. Note
also already that if is a continuous character of H,
then any quasifree state [8] is a regular state
guaranteeing the existence of a Bose field.
Normal Fluctuations
In this section we develop the theory of normal
fluctuations for -dimensional quantum lattice sys-
tems with a quasilocal structure (see the section
‘‘Quantum lattice systems’’) and for technical simpli-
city we assume that the local C
-algebra A
x
, x 2 Z
,
are copies of the matrix algebra M
n
(C)ofn n
complex matrices. Most of the results stated can be
extended to the case where A
x
is a general C
-algebra
(Goderis et al. 1989, 1990, Goderis and Vets 1989).
We consider a physical system (B, !) where ! is a
translation-invariant state of B, that is, !
x
= ! for
all x 2 Z
. Later on we extend the situation to a
C
-dynamical system (B, !,
t
) and analyze the
properties of the dynamics
t
under the central limit.
For any local A we introduced its local fluctuation
in the state ! of the system:
F
ðAÞ¼
1
jj
1=2
X
x 2
ð
x
A !ðAÞÞ ½10
The main problem is to give a rigorous mathema-
tical meaning to the limits
lim
!1
F
ðAÞFðAÞ
where the limit is taken for any increasing
Z
-absorbing sequence {}
of finite volumes of
Z
. The limits F(A) are called the macroscopic
fluctuation operators of the system (B, !).
Already earlier work (Cushen and Hudson 1971,
Sewell 1986) suggested that the fluctuations behave
like bosons. We complete this idea by proving that
one gets a well-defined representation of a CCR C
-
algebra of fluctuations uniquely defined by the
original system (B, !).
Denote by A
L, sa
and B
sa
the real vector space of
the self-adjoint elements of A
L
, respectively, B.
Definition 1 An observable A 2B
sa
satisfies the
central-limit theorem if
(i) lim
!(F
(A)
2
) s
!
(A, A) exists and is finite, and
(ii) lim
!(e
itF
(A)
) = e
(t=2)
2
s
!
(A, A)
for all t 2 R.
Clearly, our definition coincides with the notion in
terms of characteristic functions, for classical systems (A
abelian) equivalent with the notion of convergence in
distribution. For quantum systems, there does not exist
a standard notion of ‘ ‘convergence in distribution.’’
Only the concept of expectations is relevant. This does
not exclude the notion of central-limit theorem in terms
of the moments, which is the analog of the moment
problem (Giri and von Waldenfels 1978).
Definition 2 The system (B, !) is said to have
normal fluctuations if ! is translation invariant and if
(i) 8A, B 2A
L
X
x 2Z
j!ðA
x
BÞ!ðAÞ!ðBÞj < 1
(ii) the central-limit theorem holds for all A 2A
L, sa
.
Note that (i) implies that the state ! is mixing for
space translations. Also by (i), one can define a
sesquilinear form on A
L
:
hA; Bi
!
¼ lim
! F
ðA
ÞF
ðBÞðÞ
¼
X
!ðA
x
BÞ!ðA
Þ!ðBÞðÞ
and denote
s
!
ðA; BÞ¼RehA; Bi
!
!
ðA; BÞ¼2ImhA; Bi
!
For A, B 2A
L, sa
one has
!
ðA; BÞ¼i
X
x 2Z
!ð½A;
x
BÞ ½ 11
s
!
ðA; AÞ¼hA; Ai
!
½12
Clearly, (A
L, sa
,
!
) is a symplectic space and s
!
a
non-negative symmetric bilinear form on A
L, sa
.
Quantum Central-Limit Theorems 133
Follo wing the discu ssion in the section ‘‘CCR
repre sentati ons’’ we get a natural CCR C
-algebra
C
(A
L, sa
,
!
) defined on this symplectic space. The
following theorem is an essential step in the
construction of a macroscopic physical system of
fluctuations of the microsystem (B, !).
Theorem 1 If the system (B, !) has normal
fluctuations, then the limits { lim
!(e
iF
(A)
) =
exp ((1=2)s
!
(A, A)), A 2A
L
} define a quasifree
state
˜
! on the CCR C
-algebra C
(A
L, sa
,
!
) by
~
!ðWðAÞÞ ¼ exp
1
2
s
!
ðA; AÞ
Proof The proof is clear from the definition [8] if
one can prove that the positivity condition [7] holds.
But the latter follows readily from
1
4
j
!
ðA; BÞj
2
¼ lim
jIm !ðF
ðAÞF
ðBÞÞj
2
lim
!ðF
ðAÞ
2
Þ!ðF
ðBÞ
2
Þ
¼ s
!
ðA; AÞ s
!
ðB; BÞ
by Schwarz inequality.
&
This theorem indicates that the quantum-mechan-
ical alternative for (classical) Gaussian measures are
quasifree states on CCR algebras. However, the
following basic question arises: is it possible to take
the limits of products of the form
lim
! e
iF
ðAÞ
e
iF
ðBÞ
and, if they exis t, do they preserve the CCR
structure? Clearly, this is a typical noncommutative
problem.
Using the following general bounds: for C
= C
and D
= D norm-bounded operators one has
e
iðCþDÞ
e
iC
kDk
½e
iC
; e
iD
k½C; Dk
e
iðCþDÞ
e
iC
e
iD
1
2
k½C; Dk
and by the expansion of the exponential function
one proves easily that
lim
e
iF
ðAÞ
e
iF
ðBÞ
e
iðF
ðAÞþF
ðBÞÞ
e
ð1=2Þ½F
ðAÞ;F
ðBÞ
¼ 0 ½13
if A and B are one-point observables, that is, if A, B 2
A
{0}
. For general local elements the proof is some-
what more technical and can be based on a Bernstein-
like argument (for details see Goderis and Vets
(1989)). The property [13] can be seen as a
Baker–Campbell–Hausdorff formula for fluctuations.
From [13], the mean ergodic theorem, and Theorem
1 we get:
Theorem 2 If the system (B, !) has normal
fluctuations then for A, B 2A
L, sa
:
lim
! e
iF
ðAÞ
e
iF
ðBÞ
¼ exp
1
2
s
!
ðA þ B; A þ BÞ
i
2
!
ðA; BÞ
¼
~
!ðWðAÞWðBÞÞ
with
˜
! a quasifree state on the CCR algebra C
(A
!
L, sa
).
Theorems 1 and 2 describe completely the
topological and analytical aspects of the quantum
central-limit theorem under the condition of normal
fluctuations (Definition 2). In fact, the quantum
central limit yields, for every microphysical system
(B, !), a macrophysical system (C
(A
L, sa
,
!
),
˜
!)
defined by the CCR C
-algebra of fluctuat ion
observables C
(A
L, sa
,
!
) in the representation
defined by the quasifree state
˜
!. As the state
˜
! is a
quasifree state, it is a regular state, that is, the map
2 R !
˜
!( W ( A þ B )) is contin uous. From in sec-
tion ‘‘C CR repre senta tions’’ we know that this
regularity property yields the existence of a Bose
field, that is, there exists a real linear map
F : A 2A
L;sa
!FðAÞ
where F(A) is a self-adjoint operator on the GNS
representation space
~
H of
˜
!, such that for all
A, B 2A
L, sa
:
½F ðAÞ; FðBÞ ¼ i
!
ðA; BÞ
Moreover, if one has a complex structure J on
(A
L, sa
,
!
) such that J
2
= 1 and for all A, B 2A
L, sa
:
!
ðJA; BÞ¼
!
ðA; JBÞ
!
ðA; JB Þ > 0
then one defines the boson creation and annihilation
operators
F
ðAÞ¼
1
ffiffiffi
2
p
ðFðA ÞiFðJAÞÞ
satisfying the usual boson commutation relations
½F
ðAÞ; F
þ
ðBÞ ¼
!
ðA; JB Þþi
!
ðA; BÞ
Finally, it is straightforward, nevertheless impor-
tant, to remark that Theorems 1 and 2 hold true if
the linear space of local observables A
L, sa
is replaced
by any of its subspaces. Some of them can have
greater physical importance than others. This means
that the quantum central-limit theorems can realize
several macrophysical systems of fluctuations. But
all of them are Bose field systems.
134 Quantum Central-Limit Theorems
It is also important to remark that these results
end up in giving a probabilistic canonical basis of
the canonical commutation relations.
Now we analyze the notion of coarse graining due
to the quantum central limit. Consider on A
L
the
sesquilinear form (see [11], [12]) again
hA; Bi
!
¼
X
x 2Z
ð!ðA
x
BÞ!ðAÞ!ðBÞÞ
¼ s
!
ðA; BÞþi
!
ðA; BÞ½14
This form defines a topology on A
L
which is not
comparable with the operator topologies induced by
!. In fact, this form is not closable in the weak,
strong, ultraweak, or ultrastrong operator topologies.
We call A and B in A
L
equivalent, denoted by
A B if hA B, A Bi
!
= 0. Clearly, this defines
an equivalence relation on A
L
. The property of
coarse graining is mathematically characterized by
the following: for all A, B 2A
L, sa
the relation A B
is equivalent with F(A) = F(B). Suppose first that
F(A) = F(B), then
½WðAÞ; WðBÞ ¼ 0
hence
!
(A, B ) = 0. Therefore, from Theorem 1:
1 ¼
~
!ðWðAÞWðBÞ
Þ¼
~
!ðWðAÞWðBÞÞ
¼
~
!ðWðA BÞÞ ¼ exp
1
2
s
!
ðA B; A BÞ
and from [12] and [14]: hA B, A Bi
!
= 0. The
converse is equally straightforward.
From this property, it follows immediately that, for
example, the action of the translation group is trivial
or that F(
x
A) = F(A) for all x 2 Z
. Therefore, the
map F : A
L, sa
!C
(A
L, sa
,
!
) is not injective. This
expresses the physical phenomenon of coarse graining
and gives a mathematical signification of the fluctua-
tions being macroscopic observables.
In the above, we have constructed the new
macroscopic physical system of quantum fluctua-
tions for any microsystem with the property of
normal fluctuations (see Definition 2). The main
problem remains: when the microsystem does have
normal fluctuations. We end this section with the
formulation of a general sufficient clustering condi-
tion for the microstate ! in order that the micro-
system (B , !) has normal fluctuations.
Let ,
0
2D(Z
)and! a translation invariant
state, denote
!
ð;
0
Þ¼ sup
A 2A
;kAk¼1
B 2A
0
;kBk¼1
j!ðABÞ!ðAÞ!ðBÞj
The cluster function
!
N
(d ) is defined by
!
N
ðdÞ¼ sup
!
ð;
0
Þ: dð;
0
Þd and
f
maxðjj; j
0
jÞ N
g
where N, d 2 R
þ
and d(,
0
) is the Euclidean
distance between and
0
. It is obvious that
!
N
ðdÞ
!
N
ðd
0
Þ if d d
0
!
N
ðdÞ
!
N
0
ðdÞ if N N
0
The clustering condition is expressed by the follow-
ing scaling law:
9>0: lim
N !1
N
1=2
!
N
N
1=2
¼ 0 ½15
or, equivalently,
9>0: lim
N !1
N
þ
!
N
2ðþÞ
ðNÞ¼0 ½16
Note that this condition implies that
X
x 2Z
!
N
ðjxjÞ < 1
that is, that the function
!
N
( )isanL
1
(Z
)-
function for all N. In fact, this condition corre-
sponds to the uniform mi xing condition in the
commutative (classical) central-limit theore m (see,
e.g., Ibragimov and Linnick (1971)). This condition
can also be called the modulus of decoupling.
Product states, for example, equilibrium states of
mean-field systems are uniformly clustering with
!
(d ) = 0 for d > 0.
The normality of the fluctuations of the micro-
system (B, !) for product states is proved and
extensively studied in Goderis et al. (1989), and for
states satisfying the condition [15] or [16] in Goderis
and Vets (1989). In the latter case, the proofs are
very technical and based on a generalization of the
well-known Bernstein argument (Ibragimov and
Linnick 1971) of the classical central-limit theorem
to the noncommutative situation. A refinement of
these arguments can be found in Goderis et al.
(1990). For the sake of formal self-consistency we
formulate the theorem:
Theorem 3 (Central-limit theorem) Take the micro-
system (B, !) such that ! is lattice translation invariant
and satisfies the clustering condition [15]; then the
system has normal fluctuations for all elements of the
vector space of local observables A
L, sa
.
&
In Goldshtein (1982) a noncommutative central-
limit theorem is derived using similar techniques.
The main difference, however, is its strictly local
character, namely for one local operator separately.
The conditions depend on the spectral properties of
the operator. It excludes a global approach resulting
in a CCR algebra structure.
Even for quantum lattice systems, it is not
straightforward to check whether a state satisfies
Quantum Central-Limit Theorems 135
the degree of mixing as expressed in conditions
[15]–[16]. Clearly, one expects the condition to hold
for equilibrium states at high enough temperatures.
For quantum spin chains, a theorem analogous with
Theorem 3 under weaker conditions than [15] is
proved for example, in Matsui (2003).
So far we have reviewed the quantum central-limit
theorem for physical C
-spin systems (B, !) with
normal fluctuations.
Now we extend the physical system to a
C
-dynamical system (B, !,
t
)(Brattelli and Robinson
1979, 2002) and we investigate the properties of the
dynamics
t
under the central limit. As usual, the
dynamics is supposed to be of the short-range type in
order to guarantee the norm limit:
t
ðÞ ¼ n lim
e
itH
e
itH
and space homogeneous
t
x
=
x
t
, 8t 2 R, 8x 2
Z
. We suppose that the state ! is both space as
time translation invariant. Moreover, we assume
that the state ! satisfies the mixing condition [15]
for normal fluctuations.
In [10] we defined, for every local A 2A
L, sa
, the
local fluctuat ion F
(A) and obtained a clear meaning
of F(A) = lim
F
(A) from the central-limit theorem.
Now we are interested in the dynamics of the
fluctuations F(A). Clearly, for all A 2A
L, sa
and all
finite :
t
F
ðAÞ¼F
ð
t
AÞ½17
and one is tempted to define the dynamics
~
t
of the
fluctuations in the -limit by the formula
~
t
FðAÞ¼Fð
t
AÞ½18
Note, however, that in general
t
A is not a local
element of A
L, sa
. It is unclear whether the central
limit of elements of the type
t
A, with A 2A
L, sa
exists or not and hence whether one can give a
meaning to F(
t
A). Moreover, if F(
t
A) exists, it
remains to prove that (
~
t
)
t
defines a weakly
continuous group of -automorphisms on the fluc-
tuation CCR algebra
~
M= C
(A
L, sa
,
!
)
00
(the von
Neumann algebra generated by the
˜
!-representation
of C
(A
L, sa
,
!
)). All this needs a proof. In Goderis
et al. (1990), one finds the proo f of the following
basic theorem about the dynamics.
Theorem 4 Under the conditions on the dynamics
t
and on the state ! expressed above, the limit
F(
t
A) = lim
F
(
t
A) exists as a central limit as in
Theorem 2, and the maps
~
t
defined by [18] extend
to a weakly continuous one-parameter group of
-automorphisms of the von Neumann algebra
~
M.
The quasifree state
˜
! is
~
t
-invariant (time invariant).
This theorem yields the existence of a dynamics
~
t
on the fluctuations algebra and shows that it is of
the quasifree type
~
t
FðAÞ¼Fð
t
AÞ
where F(A) is a representation of a Bose field in a
quasifree state
~
!, the noncommutative version of a
Gaussian distribu tion. In physical terms, it also
means that any microdynamics
t
induces a linear
process on the level of its fluctuations.
We can conclude that on the basis of the
Theorems 3 and 4 the quantum central-limit
theorem realized a map from the microdynamical
system (B, ! ,
t
) to a macrodynamical system
(C
(A
L, sa
,
!
),
˜
!,
~
t
) of the quantum fluctuations.
The latter system is a quasifree Boson system.
Note that, contrary to the central-lim it theorem,
the law of large numbers [4] maps local observables
to their averages forming a trivial commutative
algebra of macro-observables. The macrodynamics
is mapped to a trivial dynamics as well. Therefore,
the consideration of law of large numbers does not
allow one to observe genuine quantum phenomena.
On the other hand, on the level of the fluctuations,
macroscopic quantum phenomena are observable.
Abnormal Fluctuations
The results about normal fluctuations in the last
section contain two essential elements. On the one
hand, the central limit has to exist. The condition in
order that this occurs is the validity of the cluster
condition ([15] or [16]) guaranteeing the normality
of the fluctuations. On the other hand, there is the
reconstruction theorem, identifying the CCR algebra
representation of the fluctuation observables or
operators in the quasifree state, which is denoted
by
˜
!.
The cluster condition is in general not satisfied for
systems with long-range correlations, for example,
for equilibrium states at low temperatures with
phase transitions. It is a challenging question to also
study in this case the existence of fluctuations
operators and, if they exist, to study their mathe-
matical structure. Here we detect structures other
than the CCR structure, other states or distributions
different from quasifree states, etc.
Progress in the elucidation of all these questions
started with a detailed study of abnormal fluctua-
tions in the harmonic and anharmonic crystal
models (Verbeure and Zagrebnov 1992, Momont
et al. 1997). More general Lie algebras are obtai ned
than the Heisenberg Lie algebra of the CCR algebra,
and more general states
˜
! or quantum distributions
136 Quantum Central-Limit Theorems
are computed beyond quasifree states, which is the
case for normal fluctuations.
Abnormal fluctuations turn up, if one has an
ergodic state ! with long-range correlations. We
have in mind contin uous (second-order) phase
transitions, then typically, for example, the heat
capacity or some more general susceptibilities
diverge at critical points or lines. This means that
normally scaled (with the factor jj
1=2
) fluctuations
of some observables diverge. This is equivalent with
the divergence of sums of the type
X
x 2Z
ð!ðA
x
AÞ!ðAÞ
2
Þ
for some local observable A.
In order to deal with these situations, we rescale
the local fluctuations. One determines a scaling
index
A
2 (1=2, 1=2), depending on the observa-
ble A, such that the abnormally scaled local
fluctuations
F
A
¼jj
A
F
ðAÞ
with F
(A)asin[10], yield a nontrivial character-
istic function: 8t 2 R,
lim
!
ðe
itF
A
ðAÞ
Þ
A
ðtÞ½19
where we limit our discussion to states !
local
Gibbs states. The index
A
is a measure for the
abnormality of the fluctuation of A. Note that
A
= 1=2 yields a triviality and that
A
= 1=2
would lead to a law of large numbers (theory of
averages). Observe also that in general the char-
acteristic function
A
or the corresponding state
˜
!
need not be Gaussian or quasifree.
In the physics literature, one describes the long-
range order by means of the asymptotic form of the
connected two-point function in terms of the critical
exponent
!
ða
x
AÞ!
ðAÞ
2
’ 0
1
jxj
2þ
!
; jxj!1 ½20
Our scaling index
A
is related to the critical
exponent by the straightforward relation
¼ 2 2
A
As stated above, the index
A
is determined by the
existence of the central limit and exp licitly com-
puted in several model calculations, for example,
Verbeure and Zagrebnov (1992), and for equili-
brium states. Apart from the strong model depen-
dence, the indices also depend strongly on the
chosen boundary conditions. This fact draws a new
light on the universality of the critical exponents.
Suppose now that the indices
A
are determined
by the existence of the central limit [19]. The next
problem is to find out whether also in these cases a
reconstruction theorem, comparable to, for exam-
ple, Theorem 2, can be proved giving again a
mathematical meaning to the limits
lim
F
A
ðAÞF
A
ðAÞ½21
as operators, in general unbounded, on a Hilbert space.
Here we develop a proof of the Lie algebra
character of the abnormal fluctuations under the
conditions: (1) the -indices are determined by the
existence of the variances (second moments), and
(2) the existence of the third moments (for more
details see, e.g., Momont et al. (1997 )).
Consider a local algebra, namely an n-dimensional
vector space G with basis {v
i
}
i = 1,..., n
and product
v
j
v
k
½v
j
; v
k
¼
X
n
‘¼1
c
‘
jk
v
‘
½22
with structure constants c
‘
jk
satisfying
c
‘
jk
þ c
‘
kj
¼ 0
X
r
ðc
r
ij
c
s
rk
þ c
r
jk
c
s
ri
þ c
r
ki
c
s
rj
Þ¼0
Consider the concrete Lie algebra basis of operators
in A
{0}
fL
0
¼ i1; L
1
; ...; L
m
g; m < 1
such that L
j
= L
j
, j = 0, 1, ..., m and !(L
j
) =
lim
!
(L
j
) = 0 for j > 0. Clearly, !
(L
0
) = i for all
,andthe{L
i
}satisfyeqn [22]. Because of the
special choices of L
0
one has c
‘
ok
= c
‘
ko
= 0and
c
o
jk
= i lim
!
([L
j
, L
k
]). We consider now the
fluctuations of these generators and we are looking
for a characterization of the Lie algebra of the
fluctuations if any.
For a translation-invariant local state !
, Z
,
such that ! = lim
!
is mixing, define the local
fluctuations, for j = 1, ..., m,
F
j
j
; ¼
1
jj
1=2þ
j
X
x 2
x
L
j
!
ðL
j
Þ
½23
and for notational convenience, take
F
0;
¼ i1
Now we formulate the conditions for our purposes.
Condition A We assume that the parameters
j
are
determined by the existence of the finite a nd
nontrivial variances: for all j = 1, ..., m,
0 < lim
!
ðF
j
j;
Þ
2
< 1½24
Quantum Central-Limit Theorems 137
After reordering, take 1=2 >
1
2
m
>
1=2.
Condition B Assume that all third moments are
finite, that is,
lim
!
F
j
j;
F
k
k;
F
‘
‘;
< 1
We have in mind, that the !
’s are Gibbs states
for some local Hamiltonians with some specific
boundary conditions. The limit !Z
may depend
very strongly on these boundary conditions, in the
sense that they are visible in t he values of the
indices
j
(see, e.g., Verbeure and Zagrebnov
(1992)). If for some j 1, the corresponding
j
= 0
then the ope rator L
j
has a normal fluctuation
operator
F
j
j
¼ lim
F
j
j;
½25
where the limit is understood in the sense of
Condition A, namely a finite nontrivial variance. If,
for some j 1, the corresponding
j
6¼ 0, then the
fluctuation [25] is called an abnormal fluctuation
operator. In order to satisfy Condition A, it happens
sometimes that
j
has to be chosen negative (see,
e.g., Verbeure and Zagrebnov (1992)). In this case,
it is reasonable to limit our discussion to the
situation that all
j
> 1=2.
On the basis of Condition A, the limit set
{F
j
j
}
j = 0,..., m
of fluctuation operators generates a
Hilbert space H with scalar product
F
j
j
; F
k
k
¼ lim
!
ðF
j
j;
Þ
F
k
k;
½26
On the basis of Condition B, the fluctuation
operators are defined as multiplication operat ors of
the Hilbert space H. Note that the Conditions A and
B are not sufficient to obtain a characteristic
function. However, they are sufficient to obtain the
notion of fluctuation operator. Now we proceed to
clarify the Lie algebra character of these fluctuation
operators on H.
Consider the Lie product of two local fluctuations
for a finite , one gets
F
j
j;
; F
k
k;
hi
¼
X
m
‘¼0
c
‘
jk
ðÞF
‘
‘;
½27
with
c
‘
jk
ðÞ¼
c
‘
jk
jj
1=2þ
j
þ
k
‘
;‘¼ 1; ...; m
c
0
jk
ðÞ¼jj
j
k
X
m
‘¼0
c
‘
jk
!
ðF
‘;
Þ
It is an easy exercise to check that the {c
‘
jk
()} are the
structure coefficients of a Lie algebra G(). Hence,
by considering local fluctuations, one constructs a
map from the Lie algebra G onto the Lie algebra
G() by a nontrivial change of the structure
constants. When the transformed structure constants
approach a well-defined limit, a new nonisomorphic
Lie algebra might appear. The limit algebra G(Z
),
called the contracted one of the original one G is
always nonsemisimple. This contraction is a typical
Ino¨nu¨ –Wigner contraction (Ino¨nu¨ and Wigner
1953). About the limit algebra G(Z
), the following
results are obtained (see Momont et al. (1997)):
lim
c
‘
jk
ðÞ¼
0if
1
2
þ
j
þ
k
‘
> 0
c
‘
jk
if ............... ¼ 0
0if...............< 0
8
<
:
½28
It is interesting to distinguish a number of special
cases:
1. If all fluctuations are normal, one recovers the
Heisenberg algebra of the canonical commuta-
tion relations with the right symplectic form
!
.
2. If 1=2 þ
j
þ
k
‘
> 0 for all j, k, ‘ one obtains
an abelian Lie algebra of fluctuations.
3. One gets the richest structure if 1=2 þ
j
þ
k
‘
= 0 for all j, k, ‘ or for some of them. One
notes a phenomenon of scale invariance, the
c
‘
jk
() are -independent. Algebras different from
the CCR algebra are observed. A particularly
interesting case turns up if
j
=
k
6¼ 0, that is,
one of the indices is negative, for example,
j
< 0,
the corresponding fluctuation F
j
j
shows a prop-
erty of space squeezing, and then
k
> 0, the
fluctuation F
k
k
expresses the property of space
dilation. These phenomena are observed and
computed in several models (see, e.g., Verbeure
and Zagrebnov (1992)). This yields in particular
a microscopic explanation of the phenomenon of
squeezing (squeezed states and all that) in
quantum optics. We refer also to the section
‘‘Spont aneous symm etry breaking’’ for this phe-
nomenon as being the basis of the construction of
the Goldstone normal modes of the Goldstone
particle appearing in systems showing sponta-
neous symmetry breakdown.
Some Applications
The notion of fluctuation operator as presented
above, and the mathematical structure of the algebra
of fluctuations have been tested in several soluble
models. Many applications of this theory of quan-
tum fluctuations can be found in the list of
references. Here we are not entering into the details
138 Quantum Central-Limit Theorems
of any model, but we limit ourselves to mention
three applications which are of a general nature and
totally model independent.
Conservation of the KMS Property under
the Transition from Micro to Macro
Suppose that we start with a micro-dynamical
system (B, !,
t
) with normal fluctuations, that is,
we are in the situation as treated in the section
‘‘No rmal fluctuat ions.’’ Hence, we know that the
quantum central-limit theorem maps the system
(B, !,
t
) onto the macrodynamical system
(C
(A
L, sa
,
!
),
˜
!,
~
t
) of quantum fluctuations.
If the microstate ! is
t
-time invariant (!
t
= !
for all t 2 R), then it also follows readily that the
macrostate
˜
! is
~
t
-time invariant (see Theorem 4,
i.e.,
˜
!
~
t
=
˜
! for all t 2 R).
A less trivial question to pose is: suppose that the
microstate ! is an equilibrium state for the micro-
dynamics
t
, is then the macrostate
˜
! also an
equilibrium state for the macrodynamics
~
t
of the
fluctuations? In Goderis et al. (1990) this question is
answered positively in the following more technical
sense: if ! is an
t
-KMS state of B at inverse
temperature , then
˜
! is an
~
t
-KMS state at the
same temperature.
This property proves that the notion of equili-
brium is preserved under the operation of coarse
graining induced by the central-limit theorem. This
statement constitutes a proof of one of the
basic assumptions of the phenomenological theory
of Onsager about small oscillations around
equilibrium.
This result also yields a contribution to the
discussion whether or not quantum systems should
be described at a macroscopic level by classical
observables. The result above states that the macro-
scopic fluctuation observables behave clas sically if
and only if they are time invariant. In other words, it
can only be expected a priori that conserved
quantities behave classically. In principle, other
observables follow a quantum dynamics.
Linear Response Theory
In particular, in the study of equilibrium states
(KMS states) a standard procedure is to perturb the
system and to study the response of the system as a
function of the perturbation. The response eluci-
dates many, if not all, of the properties of the
equilibrium state.
Technically, one considers a perturbation of the
dynamics by adding a term to the Hamiltonian. One
expands the perturbed dynamics in terms of the
perturbation and the unperturbed dynamics. It is
often argued that when the perturbation is small,
one can limit the study of the response to the first-
order term in the perturbation in the corresponding
Dyson expansion. This is the basis of what is called
the ‘‘linear response theory of Kubo.’’
A long-term debate is going on about the validity
of the linear response theory. The question is how to
understand from a microscopic point of view the
validity of the response theory being linear or not.
One must realize that the linear response theory
actually observed in macroscopic systems seems to
have a significant range of validity beyond the
criticism being expressed about it.
Here we discuss the main result of the paper
(Goderis et al. 1991) in which contours are sketched
for the exactness of the response being linear.
We assume:
1. that the microdynamics
t
is the norm-limit of
the local dynamics
t
= e
itH
e
itH
, where H
contains only standard finite-range interactions
(as in the sect ion ‘‘No rmal fluc tuations’’);
2. that the !
are states such that ! = lim
!
is a
state which is time and space translation invar-
iant; and
3. that ! satisfies the cluster condition [15] or [16].
From the time invariance of the state, one has a
Hamiltonian GNS representation of the dynamics:
t
= e
itH
e
itH
. On the basis of Theorem 4, one has
the dynamics
~
t
of the fluctuation algebra
C
(A
L, sa
,
!
) in the state
˜
!. This GNS representation
yields a Hamiltonian representation for
~
t
:
~
t
¼ e
it
~
H
e
it
~
H
Now take any local perturbation P 2A
L, sa
of
t
,
namely
P
t;
¼ e
itðHþF
ðPÞÞ
e
itðHþF
ðPÞÞ
where F
(P) is the local fluctuation of P in !. Then
one proves the following central-limit theorem
(Goderis et al. 1991): for all A and B in A
L, sa
, one
has the perturbed dynamics
~
P
t
¼ e
itð
~
HþFðPÞÞ
e
itð
~
HþFðPÞÞ
of the fluctuation algebra in the sense of [18] :
~
P
t
FðAÞ¼lim
Fð
P
t;
ðAÞÞ
This proves the existence and the explicit form of
the perturbed dynamics lifted to the level of the
fluctuations. In particular, one has
lim
!
P
t;
ðF
ðAÞÞ
¼
~
!ð
~
P
t
FðAÞÞ
Quantum Central-Limit Theorems 139
This is nothing but the existence of the relaxati on
function of Kubo but lifted to the level of the
fluctuations and instead of dealing with strictly local
observables here one considers fluctuations.
Assume, furthermore, that the state ! is an (
t
, )-
KMS state; then one derives readily Kubo’s famous
formula of his linear response theory:
d
dt
~
!ð
~
P
t
FðAÞÞ ¼ i
~
! ½FðPÞ;
~
t
FðAÞðÞ
which shows full linearity in the perturbation
observable P. Kubo’s formula arises as the central
limit of the microscopic response to the dynamics
perturbed by a fluctuation observable. We remark
that if ! is an equilibrium state, then the right-hand
side of the formula above can be expressed in terms
of the Duhamel two-point function, which is the
common way of doing in linear response theory.
Spontaneous Symmetry Breaking
SSB is one of the basic phenomena accompanying
collective phenomena, such as phase transitions in
statistical mechanics, or specific ground states in
field theory. SSB goes back to the Goldstone
theorem. There are many different situations to
consider, for example, in the case of short-range
interactions, it is typical that SSB yields a
dynamics w hich remains symmetric, whereas for
long-range interactions SSB also breaks the sym-
metry of t he dynamics. However, in all cases the
physics li teratu re predicts the appearance of a
particular particle, namely the Goldstone boson, to
appear as a result of SSB. The theory of fluctua-
tion operators allows the construction of the
canonical coordinates of this particle. The most
general result can be found in Michoel and
Verbeure (2001). We sketch the essentials in two
cases, namely for systems of long-range interac-
tions (mean fields) and for systems with short-
range interactions.
Long-range (mean-field) interactions Here we give
explicitly the example of the strong-coupling BCS
model in one dimension ( = 1). The microscopic
algebra of observables is B=
i
(M
2
)
i
, where M
2
is
the algebra of 2 2 complex matrices. The local
Hamiltonian of the models is given by
H
N
¼
X
N
i¼N
z
i
1
2N þ 1
X
N
i;j¼N
þ
i
j
0 <<
1
2
where
z
,
are the usual 2 2 Pauli matrices. In
the thermodynamic limit, the KMS equation has the
following product state solutions: !
=
i
tr
,
where
¼
e
h
tr e
h
;¼ tr
¼!
ð
Þ
h
¼
z
þ
Note that = tr
is a nonlinear equation for
whose solutions determine the density matrix
.
This equation always has the solution = 0,
describing the so-called normal phase. For >
c
,
with th
c
= 2, one has a solution 6¼ 0, describing
the superconducting phase. Remark that if is a
solution, then also e
i
for all is a solution as
well. It is clear that H
N
is invariant under the
continuous gauge transformation automorphism
group G= {
’
j’ 2 [0, 2 ]} of B:
’
ð
þ
i
Þ¼e
i’
þ
i
Hence G is a symmetry group. On the other hand:
!
(
’
(
þ
i
)) = e
i’
!
(
þ
i
) 6¼ !
(
þ
i
). The gauge group
G is spontaneously broken. Remark also that the
gauge transformations are implemented locally by
the charges
Q
N
¼
X
N
j¼N
z
i
; i:e:;
’
ð
þ
i
Þ¼e
i’Q
N
þ
i
e
i’Q
N
and
z
is the symmetry generator density. As the
states !
are product states, all fluctuations are
normal (see the section ‘‘N ormal fluc tuations’’). One
considers the local operators
Q ¼
jj
2
2
z
þ
2
ð
þ
þ
Þ
P ¼
i
ð
þ
Þ
where = (
2
þjj
2
)
1=2
. Note that P is essentially
the order parameter operator, that is, the operator P
is breaking the symmetry:
d
d’
!
ð
’
ðAÞÞ 6¼ 0;!
ðAÞ¼0
On the other hand, Q is essentially the generator of
the symmetry
z
normalized to zero, that is,
!
(Q) = 0.
Michoel and Verbeure (2001) proved in detail
that the fluctuations F(Q)andF(P) form a
canonical pair
½FðQÞ; FðPÞ ¼ i
4jj
2
and that they behave, under the time evolution, as
harmonic oscillator coordinates oscillating with a
140 Quantum Central-Limit Theorems
frequency equal to 2. This frequency is called a
plasmon frequency. Moreover, the variances are
~
!
ðFðQÞ
2
Þ¼
jj
2
2
¼
~
!
ðFðPÞ
2
Þ
This means that these coordinates vanish or dis-
appear if = 0. The coordinates F(Q)andF(P) are
the canonical coordinates of a particle appearing
only if there is spontaneous symmetry breakdown.
They are the canonical coordinates of the Goldstone
boson, which arise if SSB occurs.
Short-range interactions An analogous result, as
for long-range interactions, can be derived for
systems with short-range interactions. However, in
this case we have equilibrium states with poor
cluster properties. We are now in the situation as
descr ibed in the ‘‘Abno rmal Fluct uations’’ sect ion.
Also in this case we have the phenomenon of SSB,
which shows the appearance of a Goldstone particle.
Also in this case one is able to construct its
canonical coordinates. The details of this construc-
tion can be found in Michoel and Verbeure (2001).
Here we give a heuristic picture of this construction.
Consider again a microsystem (B, !,
t
) and let
s
be a strongly continuous one-parameter symmetry
group of
t
which is locally generated by
Q
=
P
x2
q
x
. SSB amounts to find an equilibrium
(KMS) or ground state ! which breaks the symme-
try, that is, there exists a local observable A 2A
L, sa
such that for s 6¼ 0 holds: !(
s
(A)) 6¼ !(A) and
t
s
=
s
t
. This is equivalent to
d
ds
!ð
s
ðAÞÞ
s¼0
¼ lim
!ð½Q
; AÞ ¼ c 6¼ 0
with c a constant.
Now we turn this equation into a relation for
fluctuations. Using space translation invariance of
the state, one gets
lim
1
jj
!
X
x 2
ðq
x
!ðqÞÞ
X
y 2
ð
x
A !ðAÞÞ
"# !
¼ c
We now use another consequence of the Gold-
stone theorem, namely that SSB implies poor
clustering properties for the order parameter A,
that is, in the line of what is done in the last
section, we assume that the lack of clustering is
expressed by the existence of a positive index
such that
lim
!
1
jj
1þ2
X
x 2
ð
x
A !ðAÞÞ
!
2
0
@
1
A
is nontrivial and finite. This means that the fluctua-
tion F
(A) exists. Then we get
lim
!
1
jj
1=2
X
x 2
ðq
x
!ðqÞÞ ;
"
1
jj
1=2þ
X
y 2
ð
x
A !ðAÞÞ
#!
¼ c
Hence
~
! F
ðqÞ; F
ðAÞ
¼ c
which for equilibrium states !, turns into the
operator equation for fluctuations
½F
ðqÞ; F
ðAÞ¼ c1
In other words, one obtains a canonical pair
(F
(q), F
(A)) of normal coordinates of the collec-
tive Goldstone mode.
Note that the long-range correlation of the
order-parameter operator (positive )isexactly
compensated by a squeezing, described by the
negative index , for the fluctuation operator of
the local generator of the broken symmetry. This
result can also be expressed as typical for SSB,
namely that the symmetry is not completely
broken, but only partially. More detailed informa-
tion about all this is found in Michoel and
Verbeure (2001).
See also: Algebraic Approach to Quantum Field Theory;
Large Deviations in Equilibrium Statistical Mechanics;
Macroscopic Fluctuations and Thermodynamic
Functionals; Quantum Phase Transitions; Quantum
Spin Systems; Symmetry Breaking in Field Theory;
Tomita–Takesaki Modular Theory.
Further Reading
Brattelli O and Robinson DW (1979) Operator Algebras and
Quantum Statistical Mechanics, vol. I. New York–Heidelberg–
Berlin: Springer.
Brattelli O and Robinson D (2002) Operator Algebras and
Quantum Statistical Mechanics, vol. II. New York–Heidelberg–
Berlin: Springer.
Cushen CD and Hudson RL (1971) A quantum mechanical central
limit theorem. Journal of Applied Probability 8: 454–469.
Fannes M and Quaegebeur J (1983) Central limits of product
mappings between CAR-algebras. Publications of the Research
Institute for Mathematical Studies Kyoto 19: 469–491.
Giri N and von Waldenfels W (1978) An algebraic version of the
central limit theorem. Zeitschrift fu¨ r Wahrscheinlichkeitstheorie
und Verwandte gebiete 42: 129–134.
Goderis D, Verbeure A, and Vets P (1989) Non-commutative
central limits. Probability and Related Fields 82: 527–544.
Goderis D, Verbeure A, and Vets P (1990) Dynamics of
fluctuations for quantum lattice systems. Communications in
Mathematical Physics 128: 533–549.
Quantum Central-Limit Theorems 141