Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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and assumed that the single-spin measure is
rotation invariant. Let us introduce the Fourier
transformation
^
xðpÞ¼
1
ffiffiffiffiffiffi
jj
p
X
‘2
x
‘
e
ið‘;pÞ
x
‘
¼
1
ffiffiffiffiffiffi
jj
p
X
p2
^
xðpÞe
ið‘; pÞ
½37
¼
p ¼ðp
1
; ...; p
d
Þjp
j
¼ þ
L
j
;
j
¼ 1; ...; 2L; j ¼ 1; ...; d
½38
Then we can set
b
K
ðkÞ
ðpÞ¼
^
x
ðkÞ
ðpÞ
^
x
ðkÞ
ðpÞ
hi
b
K
ðpÞ¼
X
N
k¼1
b
K
ðkÞ
ðpÞ
½39
Thereby, cf. [1], [2],
K
ð‘; ‘
0
Þ¼
def
½ðx
‘
; x
‘
0
Þ ¼
1
jj
X
p2
b
K
ðpÞe
iðp;‘‘
0
Þ
½40
By construction, for any ‘
0
2 ,
K
ð‘; ‘
0
Þ¼K
ð‘ þ ‘
0
;‘
0
þ ‘
0
Þ½41
where addition is componentwise modulo 2L. This
means that K
(‘, ‘
0
) is invariant with respect to the
translations on the corresponding torus. One can
show that K
(‘, ‘
0
) converges, as L !þ1,toK(‘, ‘
0
)
discussed in the Introduction. The corresponding
Gibbs state of the whole model is called the periodic
Gibbs state. By construction, it is translation
invariant. Set
EðpÞ¼
X
d
j¼1
½1 cos p
j
; p 2ð;
d
½42
Theorem 10 For all p 2
n {0},
b
K
ðpÞ
N
2JEðpÞ
½43
Proof Consider the function f () = Z
( h), 2 R,
where Z
(h) is defined by [25].ByTheorem 9 it has
a maximum at = 0; hence,
f
00
ð0Þ0 ½44
Obviously, f
00
(0) d epends on h = (h
‘‘
0
)
h‘, ‘
0
i2E
,
h
‘‘
0
2 R
N
. Let us choose h such that only the
first components h
(1)
‘‘
0
are nonzero. Then [4 4]
holds if
J
X
h‘
1
;‘
0
1
i2E
X
h‘
2
;‘
0
2
i2E
x
ð1Þ
‘
1
x
ð1Þ
‘
0
1
x
ð1Þ
‘
2
x
ð1Þ
‘
0
2
hi
h
ð1Þ
‘
1
‘
0
1
h
ð1Þ
‘
2
‘
0
2
X
h‘;‘
0
i2E
h
ð1Þ
‘‘
0
hi
2
½45
This means that the eigenvalues of the matrix of the
real quadratic form (with respect to h) defined by
the left-hand side of [45] do not exceed one. The
same ought to be true for the extension of this form
to the complex case. Let us show that the complex
eigenvectors h
(1)
‘‘
0
(p) of this matrix and the corre-
sponding eigenvalues (p) are
h
ð1Þ
‘‘
0
ðpÞ¼ðe
iðp;‘Þ
e
iðp;‘
0
Þ
Þ=
ffiffiffiffiffiffi
jj
p
ðpÞ¼2J EðpÞ
b
K
ð1Þ
ðpÞ
p 2
½46
For j = 1, ..., d, let
j
2 Z
d
be the unit vector with
the jth component equal to 1. Then for h‘, ‘
0
i2E,
there exists
j
such that ‘ ‘
0
=
j
. Since the edge
h‘, ‘
0
i is an unordered set, let us fix ‘
0
= ‘ þ
j
.
Thereby,
1
jj
1=2
X
h‘;‘
0
i2E
x
ð1Þ
‘
x
ð1Þ
‘
0
e
iðp;‘Þ
e
iðp;‘
0
Þ
¼
2
jj
1=2
X
‘2
X
d
j¼1
x
ð1Þ
‘
e
iðp;‘Þ
x
ð1Þ
‘
e
iðp;‘Þ
cosðp;
j
Þ
hi
¼ 2
^
x
ð1Þ
ðpÞEðpÞ
In view of [41], one has
½
^
x
ð1Þ
ðpÞ
^
x
ð1Þ
ðp
0
Þ ¼
0;pþp
0
b
K
ð1Þ
ðpÞ
Then employing the latter two facts and [37], we get
J
X
h‘
2
;‘
0
2
i2E
x
ð1Þ
‘
1
x
ð1Þ
‘
0
1
x
ð1Þ
‘
2
x
ð1Þ
‘
0
2
h
‘
2
‘
0
2
ðpÞ
h
¼ 2JEðpÞ
x
ð1Þ
‘
1
x
ð1Þ
‘
0
1
^
x
ð1Þ
ðpÞ
hi
¼ 2JEðpÞ
1
jj
1=2
X
p
0
2
^
x
ð1Þ
ðp
0
Þ
^
x
ð1Þ
ðpÞ
hi
e
iðp
0
;‘
1
Þ
e
iðp
0
;‘
0
1
Þ
¼ 2JEðpÞ
b
K
ð1Þ
ðpÞh
‘
1
‘
0
1
ðpÞ
which proves [46]. Then by [45]
b
K
(1)
(p) 1=2JE(p),
for p 6¼ 0. The same holds for
b
K
(k)
(p), k = 2, ..., N,
which by [39] yields [43].
&
The result ju st proved and the convergence of
K
(‘, ‘
0
) !K(‘, ‘
0
), as L !þ1, imply the infrared
bound [4]. It turns out that the estimate [43]
382 Reflection Positivity and Phase Transitions
may be used directly to prove the ph ase transi-
tion. Consider
P
¼
def
1
jj
2
X
‘
1
;‘
2
2
½ðx
‘
1
; x
‘
2
Þ
¼
1
jj
X
‘2
x
‘
2
0
@
1
A
0 ½47
where is the box [35].By[40] and [41 ], we have
P
¼
1
jj
b
K
ð0Þ½48
One can show that if P =
def
lim
L!þ1
P
is positive,
then there exist multiple Gibbs states. By [40], [41],
and [48], we get that for any ‘ 2 ,
K
ð‘; ‘Þ¼P
þ
1
jj
X
p2
nf0g
b
KðpÞ½49
Suppose that, cf. [5],
K
ð‘; ‘ÞK > 0 ½50
with K independent of and J. Employing in [49]
this estimate and [43], and passing to the limit
L !þ1, we get
P K IðdÞN=2J ½51
where
IðdÞ¼
def
1
ð2Þ
d
Z
ð;
d
dp
EðpÞ
½52
which is finite for d 3. Thereby, we have proved
the following:
Theorem 11 For the spin model [22], [23], there
exist multiple Gibbs states, and hence multiple
phases, if d 3 and J > I(d) N=2 K.
Finally, let us pay some attention to the estimate
[50], which is closely related with the properties of the
single-spin measure (note that played no role in
obtaining [26] and [43]). If it is the uniform measure
on the unit sphere S
N1
R
N
,thenK
(‘, ‘) = 1and
[50] is trivial. In general, one has to employ some
technique to obtain such an estimate.
Reflection Positivity and Phase
Transitions in Quantum Systems
As in the clas sical case, the way of proving the phase
transition for appropriate models leads from an
estimate like [17] to Gaussian domination and then
to the infrared bound. However, here this way is
much more complicated, so in the frames of this
article we can only sketch its main elements basing
on the original paper by Dyson et al. (1978), where
the interested reader can find the details. As above,
we start by studying reflection positive functionals.
Reflection Positivity in Nonabelian Case
Again we consider a finite set , jjbeing even. For every
‘ 2 , let a complex Hilbert space H
‘
be given. This is
the single-spin physical Hilbert space for our quantum
system. We suppose that all H
‘
, ‘ 2 , are the copies of a
certain finite-dimensional space H. The physical Hilbert
space H
corresponding to is the tensor product of
H
‘
, ‘ 2 .LetA
be the algebra of all linear operators
defined on H
. This is the algebra of observables in our
case; it is noncommutative (nonabelian) and contains the
unit element I – the identity operator. As above, splits
into two subsets
, which are the mirror images of each
other, that is, we are given a reflection : !,such
that (
þ
) =
.Thisallowsustointroducethe
corresponding subalgebras A
by setting the elements
of A
þ
to be of the form A I,whereA : H
þ
!H
þ
is a
linear operator and I is the identity operator on H
.
Respectively, the elements of A
aretobeoftheform
I A.Thenwedefinethemap# : A
þ
!A
as
#ðA IÞ¼I
A ½53
where A 7!
A is complex (not Hermitian) conjugation; it
may be realized as transposing and taking Hermitian
conjugation. For A
1
, ..., A
n
2A, one has
A
1
A
n
=
A
1
, A
n
. We also suppose that # possesses the
properties [8]. A linear functional : A
! R is called
RP (with respect to the pair , #)ifithastheproperty[9].
Definition 12 A functional is called generalized
reflection positive (GRP) if for any A
1
, ..., A
n
2A
þ
,
½A
1
#ðA
1
ÞA
n
#ðA
n
Þ 0 ½54
In principle, this notion differs from the reflection
positivity only in the nonabelian case. However, if
the algebras A
commute (they do commute in our
case), a functional is RP if and only if it is GRP.
Example 13 Let
ðAÞ¼traceðAÞ; A 2A
½55
Since the space H
is finite dimensional, this is
well defined. It is GRP. Indeed, as the algebras A
commute, we have
½A
1
I #ðA
1
IÞA
n
I #ðA
n
IÞ
¼ ½A
1
I A
n
I #ðA
1
IÞ#ðA
n
IÞ
¼ ½A
1
I A
n
I #ðA
1
I A
n
IÞ
¼ trace½A
1
A
n
trace½
A
1
A
n
¼ trace½A
1
A
n
jj
2
0
Reflection Positivity and Phase Transitions 383
The Cauchy–Schwarz inequality [13] obviously
holds also in the quantum case. By means of this
inequality and the Trotter product formula
expðA þ BÞ¼ lim
n!þ1
½expðA=nÞ expðB=nÞ
n
½56
one can prove that every RP functional obeys an
estimate like [17]. Thereby, we have the following
analog of Lemma 6:
Lemma 14 Let A, B, C
1
, ..., C
n
2A
þ
be any self-
adjoint operators posses sing real matrix representa-
tion and a
1
, ..., a
m
be any real numbers. Then
trace exp A þ #ðBÞ
X
m
n¼1
½C
n
#ðC
n
Þa
n
2
!()"#
2
trace exp A þ #ðAÞ
X
m
n¼1
½C
n
#ðC
n
Þ
2
!()
trace exp B þ #ðBÞ
X
m
n¼1
½C
n
#ðC
n
Þ
2
!()
½57
Gaussian Domination and Phase Transitions
To proceed further we need a concrete model with
finite-dimensional physical Hilbert spaces. As every
quantum model, it is defined by its Hamiltonian. Let
Z
d
be the box [35] and ( , E ) be the same
graph as in the sub section ‘‘Infrared bound.’’ The
periodic Hamiltonian of our model is
H
¼
X
‘2
Q
‘
þ
1
2
X
h‘;‘
0
i2E
jS
‘
S
‘
0
j
2
½58
where at each ‘ 2 we have the copies Q
‘
,
S
(1)
‘
, ..., S
(N)
‘
of N þ 1 basic operators, acting in the
Hilbert space H
‘
, and
jS
‘
S
‘
0
j
2
¼
X
N
k¼1
S
ðkÞ
‘
S
ðkÞ
‘
0
2
The only condition we impose so far is that all these
operators can simultaneously be chosen as real
matrices. For h = (h
‘‘
0
)
h‘,‘
0
i2E
2 R
NjEj
, we set
Z
ðhÞ¼trace
(
exp
X
‘2
Q
‘
2
X
h‘;‘
0
i2E
jS
‘
S
‘
0
h
‘‘
0
j
2
!)
½59
where >0 is the inverse temperature.
Theorem 15 For the model [58] and any
h = (h
‘‘
0
)
h‘, ‘
0
i2E
2 R
NjEj
,
Z
ðhÞZ
ð0Þ½60
The proof is performed by means of Lemma 14.
The periodic local Gibbs state of the model [58] at
the inverse temperature , analogous to the state [31],is
ðAÞ¼tracefA expðH
Þg=Z
ð0Þ; A 2A
½61
As in the classical case, one can define the parameter
[47]. However, now the fact that lim
L!þ1
P
> 0
does not yet imply the phase transition. One has to
prove a more general fact
lim
L
0
!þ1
lim
L!þ1
1
j
0
j
X
‘2
0
S
‘
2
0
@
1
A
8
<
:
9
=
;
> 0 ½62
where
0
is the box [35] of side 2L
0
. Furthermore, in
the quantum case the Gaussian domination [60]
does not lead directly to the estimate [43], which
yields [51]. Instead, one can get a bound like [43]
but for the Duhamel two-point function (DTF).
Given A, B 2A
, their DTF is
ðA; BÞ¼
Z
1
0
ðAe
H
Be
H
Þd ½63
By means of [56] one can show that
ðA;BÞ¼
1
Z
ð0Þ
@
2
@@
trace½expðA þB H
Þ
¼¼0
½64
Let
^
S(p)=(
^
S
(1)
(p),...,
^
S
(N)
(p)),p 2
, be the Fourier
image of S
‘
, defined by [37], [38]. Then
^
SðpÞ;
^
SðpÞ
¼
X
N
k¼1
^
S
ðkÞ
ðpÞ;
^
S
ðkÞ
ðpÞ
Theorem 16 For all p 2
n{0}, it follows that
^
SðpÞ;
^
SðpÞ
N
2EðpÞ
½65
To prove this statement one has to use the
Gaussian bound [60] exactly as in the case of
Theorem 10. The second derivative with respect to
gives the corresponding DTF (see [64]).
Now let us indicate how the infrared bound [65]
leads to the phase transition. To this end we use the
simplest quantum spin model with the Hamiltonian
[58], for which Q
‘
= 0, N = 2, and S
(k)
‘
, k = 1, 2,
being the copies of the Pauli matrices
S
ð1Þ
¼
01
10
; S
ð2Þ
¼
10
0 1
Then
K
ðkÞ
ð‘; ‘Þ¼
S
ðkÞ
‘
S
ðkÞ
‘
¼ 1
for all ‘ 2 ; k ¼ 1; 2 ½66
384 Reflection Positivity and Phase Transitions
which gives the bound K (see [50]). For A, B 2A
,
by [A, B] we denote the commutator AB BA. Set
ðkÞ
ðpÞ¼
^
S
ðkÞ
ðpÞ; H
;
^
S
ðkÞ
ðpÞ
hihi
k ¼ 1; 2 ½67
The phase transition in the model we consider can
be established by means of the following statement
(see Dyson 1978, Theorem 5.1).
Proposition 17 Suppose there exist
(k)
(p), k = 1, 2,
p 2 (, ]
d
such that, for all L 2 N,
ðkÞ
ðpÞ
ðkÞ
ðpÞ; k ¼ 1; 2; p 2
½68
Then the model undergo es a phase transition at a
certain finite if d 3 and
1
ð2Þ
d
Z
ð;
d
ðkÞ
ðpÞ
8EðpÞ
1=2
dp < 1 ½69
for a certain, and hence for both, k = 1, 2.
Thus to prove the phase transition we have to
estimate
(k)
(p), k = 1, 2. By means of the Cauchy–
Schwarz inequality, the estimate [69] may be
transformed into the following:
1
ð2Þ
d
Z
ð;
d
ð1Þ
ðpÞþ
ð2Þ
ðpÞ
hi
dp < 16=IðdÞ
where I(d) is the same as in [52]. The integral on
the left-hand side can be estimated from above by
8
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d( d þ 1)
p
; hence, the latter inequality holds if
IðdÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dðd þ 1Þ
q
< 2
which holds for all d 3. In particular, I(3) 0.505.
Bibliographic Notes
As the original sources on the RP method in the
theory of phase transitions we mention the papers
Fro¨ hli ch et al. (1976) (classical case), Dyson et al.
(1978) (quantum case), and Fro¨ hlich and Lieb (1978)
(both cases). In a unified way and with many examples,
this method is described in Fro¨ hlich et al. (1978,
1980). A detailed analysis of the method, especially in
its applications to classical models with unbounded
spins, was given in Shlosman (1986). The techniques
based on the chessboard and contour estimates are
described in Fro¨ hlich and Lieb (1978) and Shlosman
(1986). As was mentioned above, the quantum case is
much more complicated; it gets even more compli-
cated if one deals with quantum models employing
infinite-dimensional physical Hilbert spaces and
unbounded operators, such as quantum crystals.
The adaptation of the RP method to such models
was made in Driessler et al. (1979), Pastur and
Khoruzhenko (1987), Barbulyak and Kondratiev
(1992),andKondratiev (1994). In the latter two
papers a general version of the quantum crysta l was
studied in the framework of the Euclidean approach,
based on functional integrals (see Albeverio et al.
(2002)). In this approach the quantum crystal is
represented as a lattice spin model with unbounded
infinite-dimensional spins. Like in the case of classical
models with unbounded spins, here establishing the
estimate [5] becomes a highly nontrivial task. In
particular cases, for example, for
4
-models, one
applies special tools like the Bogoliubov inequalities
(see Driessler et al. (1979) and Pastur and Khoruz-
henko (1987)). In the general case quasiclassical
asymptotics allow us to get the lower bound [5] (see
Barbulyak and Kondratiev (1992) and Kondra tiev
(1994)). There is one more technique based on
reflection positivity (see Lieb (1989)). It employs
reflections in spin spaces, whereas the properties of
the index sets (lattices) play no role. This technique
proved to be useful in the theory of strongly correlated
electron systems, see Tian (2004).Finally,wemention
the books of Georgii (1988), Prum (1986),andSinai
(1982) where different aspects of the RP method are
described. In Georgii (1988), one can also find
extended bibliographical and historical comments on
this subject.
See also: Phase Transition Dynamics; Phase Transitions
in Continuous Systems; Quantum Spin Systems;
Renormalization: Statistical Mechanics and Condensed
Matter.
Further Reading
Albeverio S, Kondratiev YG, Kozitsky Y, and Ro¨ ckner M (2002)
Euclidean Gibbs states of quantum lattice systems. Reviews in
Mathematical Physics 14: 1–67.
Barbulyak VS and Kondratiev YG (1992) The quasiclassical limit
for the Schro¨dinger operator and phase transitions in quantum
statistical physics. Functional Analysis and its Applications
26(2): 61–64.
Driessler W, Landau L, and Fernando-Perez J (1979) Estimates of
critical lengths and critical temperatures for classical and quantum
lattice systems. Journal of Statistical Physics 20: 123–162.
Dyson FJ, Lieb EH, and Simon B (1978) Phase transitions in
quantum spin systems with isotropic and nonisotropic inter-
actions. Journal of Statistical Physics 18: 335–383.
Fro¨hlich J, Israel R, Lieb EH, and Simon B (1978) Phase transitions
and reflection positivity, I. General theory and long range lattice
models. Communications in Mathematical Physics 62: 1–34.
Fro¨ hlich J, Israel R, Lieb EH, and Simon B (1980) Phase
transitions and reflection positivity, II. Lattice systems with
short-range and Coulomb interactions. Journal of Statistical
Physics 22: 297–347.
Fro¨ hlich J and Lieb EH (1978) Phase transitions in anisotropic
lattice spin systems. Communications in Mathematical Physics
60: 233–267.
Reflection Positivity and Phase Transitions 385
Fro¨ hlich J, Simon B, and Spenser T (1976) Infrared bounds, phase
transitions and continuous symmetry breaking. Communica-
tions in Mathematical Physics 50: 79–85.
Georgii H-O (1988) Gibbs Measures and Phase Transitions.
Studies in Mathematics, vol.9. Berlin: Walter de Gruyter.
Kondratiev JG (1994) Phase transitions in quantum models of
ferroelectrics. In: Stochastic Processes, Physics and Geometry,
vol. II, pp. 465–475. Singapure: World Scientific.
Lieb EH (1989) Two Theorems on the Hubbard Model. Physical
Review Letters 62: 1201–1204.
Pastur LA and Khoruzhenko BA (1987) Phase transitions in
quantum models of rotators and ferroelectrics. Theoretical
and Mathematical Physics 73: 111–124.
Prum P (1986) Processus sur un Re´seau et Mesures de Gibbs.
Applications. Techniques Stochastiques. Paris: Masson.
Prum B and Fort J-C (1991) Stochastic Processes on a Lattice
and Gibbs Measures (translated from the French by Bertram
Eugene Schwarzbach and revised by the authors). Mathema-
tical Physics Studies, vol.11. Dordrecht: Kluwer Academic
Publishers Group.
Shlosman SB (1986) The method of reflection positivity in the
mathematical theory of first-order phase transitions. Russian
Mathematical Surveys 41: 83–134.
Sinai Ya (1982) Theory of Phase Transitions. Rigorous Results.
Oxford: Pergamon.
Tian G-S (2004) Lieb’s spin-reflection-positivity method and its
applications to strongly correlated electron systems. Journal of
Statistical Physics 116: 629–680.
Regularization for Dynamical -Functions
V Baladi, Institut Mathe
´
matique de Jussieu,
Paris, France
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
If A is a finite, say N N, matrix with
complex coefficients, the following easy equality
gives an expres sion for the polynomial
Q
N
k = 1
(1 z
k
) = det (Id zA):
detðId zAÞ¼exp
X
1
n¼1
z
n
n
tr A
n
!
½1
(here, Id denotes the identity matrix and tr is the
trace of a matrix). Even in this trivial f inite-
dimensional case, the z-radius of convergence of
the logarithm of the right-hand side only gives
information about the spectral radius (the modulus
of the largest eigenvalue) of A. The zeros of the
left-hand side (i.e., the inverses z = 1=
k
of the
nonzero eigenvalues of A) can only be located
after extending holomorphically the right-hand
side. The purpose of this article is to discuss
some dynamical situations in which A is replaced
by a linear bounded operator L,actingonan
infinite-dimensional space, and for which a dyna-
mical determinant (or dynamical -function), con-
structed from periodic orbits, takes the part of the
right-hand side. In the examples presented, L will
be a transfer operator associated to a weighted
discrete-time dynamical system: given a transfor-
mation f : M ! M on a compact manifold M and
a function g : M ! C,weset
L’ ¼ g ’ f
1
½2
(If f is not inversible, i t is understood, e.g., that f
has at most finitely many inverse branches, and
that the right-hand side of [2] is the sum over
these inverse branches, see the next section.) We
let L act on a Banach space of functions or
distributions ’ on M. For suitable g (in particular
g = jdet Tf
1
j when this Jacobian makes sense), the
spectrum of L is related to the fine statistical
properties of the dynamics f:existenceand
uniqueness of equilibrium states (related t o the
maximal eigenvector of L), decay of correlations
(related to the spe ctral gap), lim it l aws, entro-
pies, etc: see, for example, Baladi (1998) or
Cvitanovic
´
et al. (2005). The operator L is not
always trace-class, indeed, it sometimes is not
compact on any reasonable space. Even worse, its
essential spectral radius may coincide with its
spectral radius. (Recall that the essential spectral
radius of a bounded linear operator L acting on a
Banach space is the infimum of those >0, such
that the spectrum of L outside of the disk of
radius is a finite set of eigenvalues of finite
algebraic multiplicity.) However, various techni-
ques allow us to prove that a suitable dynamically
defined replacement for the right-hand side of [1]
extends holomorphically to a disk in which its
zeros describe at least part of the spectrum of L.
Some of these techniques have a ‘‘regularization’’
flavor, and we shall concentrate on them.
In the following section, we present the simplest
case: analytic expanding or hyperbolic dynamics,
for which no regularization is necessary and the
Grothendieck–Fredholm theory can be applied.
Next, we consider analytic situations where
finitely many neutral periodic orbits introduce
branch cuts in the dynamical determinant, and
see how to ‘‘regularize’’ them. Finally, we discuss a
386 Regularization for Dynamical -Functions
kneading operator regularization approach,
inspired by the work of Milnor and Thurston,
and applicable to dynamical systems with finite
smoothness.
Despite the terminology, none of the regulariza-
tion techniques discussed below match the following
‘‘-regularization’’ formula:
Y
1
k¼1
a
k
¼ exp
d
ds
X
1
k¼1
a
s
k
j
s¼0
!
½3
(For information about the above -regularization
and its applications to physics, we refer, e.g., to
Elizalde 1995. See also Voros (1987) and Fried
(1986) for more geometrical approaches and further
references, e.g., to the work of Ray and Singer.)
We do not cover all aspects of dynamical
-functions here. For more information and refer-
ences, we re fer to our surve y Baladi (1998),tothe
more recent surveys by Pollicott (2001) and Ruelle
(2002), and also to the exhaustive account by
Cvitanovic
´
et al. (2005), which contains a rich
array of physical applications.
The Grothendieck–Fredholm Case
Let M be a real analytic compact manifold (e.g., the
circle or the d-torus), and let f : M ! M be real
analytic and g : M !C be analytic.
First suppose that f is uniformly expanding, that
is, there is >1 so that kTf (v)kkvk. (For
example, f (z) = z
2
on the unit circle, or a small
analytic perturbation thereof.) Consider
L
f ; g
’ðxÞ¼
X
y: f ðyÞ¼x
gðyÞ’ðyÞ½4
(For example, with g(y) = 1=jdet Tf (y)j or
1=jdet Tf (y)j
s
.) Ruelle (1976) proved that an
operator L
0
, which is essentially the same as L
f , g
(the difference, if any, arises from the use of Markov
partitions, especially in higher dimensions), acting
on a Banach space of holomorphic and bounded
functions, is not only compact, but is in fact a
nuclear operator in the sense of Grothendieck. In
particular, the traces of all its powers are well
defined, and the Grothendieck–Fredholm (Gohberg
et al. 2000) determinant
d
0
ðzÞ¼exp
X
1
n¼1
z
n
n
tr L
n
0
!
½5
extends to an entire function of finite order, the
zeros of which are exactly the inverses of the
nonzero eigenvalues of L
0
. (The order of the zero
coincides with the algebraic multiplicity of the
eigenvalue.) Ruelle also proved that the traces can
be written as sums over periodic orb its:
tr L
n
0
¼
X
x: f
n
ðxÞ¼x
Q
n1
k¼0
gðf
k
xÞ
jdetðId Tf
n
x
Þj
where
P
means that the fixed points of f
n
lying in
the intersection of two or more elements of the
Markov partition must be counted two or more
times. (Note that if f
n
(x) = x, then this closed orbit
gives a natural inverse branch for f
n
.) Taking into
account the periodic orbits on the boundaries of the
Markov partition, Ruelle expresses the following
‘‘dynamical determinant’’:
d
f; g
ðzÞ
¼ exp
X
1
n¼1
z
n
n
X
x: f
n
ðxÞ¼x
Q
n1
k¼0
gðf
k
xÞ
jdetðId Tf
n
x
Þj
2
4
3
5
½6
as an alternated product of determinants d
0
(z)asin[5].
The expression [6] is sometimes also called a
‘‘dynamical -function,’’ but we prefer to reserve this
terminology for the following power series:
f; g
ðzÞ¼exp þ
X
1
n¼1
z
n
n
X
x: f
n
ðxÞ¼x
Y
n1
k¼0
gðf
k
xÞ
2
4
3
5
½7
It is not difficult to write
f , g
(z)as(Baladi 1998)an
alternated product of determinants d
f , g
i
, for
i = 0, ..., d, and appropriate weights g
i
.
In fact, the results just described hold in more
generality, for example, for piecewise bijective and
analytic interval maps. Such maps, f, appear
naturally, for example, when considering Schottky
subgroups of PSL(2, Z). We mention the recent
work of Guillope´–Lin–Zworski (200 4), who let the
transfer operator associated to such f and weights
g
s
(y) = 1=jf
0
(y)j
s
act (as trace-class operators) on
suitable Hilbert spaces of holomorphic functions.
This allows them to obtain precise estimates for the
number of zeros of s 7!d
f , g
s
[1] in the complex
plane: these zeros are the resonances (in the sense of
the spectrum of the Laplacian).
Note that the nuclearity properties extend also to
the Gauss map f ( x) = {1=x}, which has infinitely
many inverse branches, if the weight g has summa-
bility properties over the branches (e.g.,
g
s
(y) = j1=f
0
(y)j
s
, where s is a complex parameter,
with <s > 1=2). The dynamical determinant d
f , g
s
(z)
for the transfer operator of the Gauss map is related
to the Selberg -function (see e.g., Chang and Mayer
(2001) and references therein).
Next, assume that M and g are as before, but f is a
uniformly hyperbolic real analytic diffeomorphism.
Regularization for Dynamical -Functions 387
For example, M is the 2-torus and f is a small real
analytic perturbation of the linear automorphism
21
11
More generally, we may assume that f is a real
analytic Anosov diffeomorphism, that is, there are
C 1and>1 such that the tangent bundle
decomposes as TM = E
u
E
s
, whe re th e dynam ical
bundles E
u
and E
s
are Tf-invariant, with kTf
n
j
E
s
k
C
n
and kTf
n
j
E
u
kC
n
for all n 2 Z
þ
.In
general, the smoothness of x 7!E
u
(x)andE
s
(x)is
only Ho¨ lder. Under the very strong additional
assumption that E
u
(x)andE
s
(x) are real analytic,
Ruelle (1976) (see also Fried (1986)) showed that
the power series d
f , g
(z) can again be written as a
finite alternated product (this product being again
an artifact of the Markov partition) of entire
functions of finite order. For this, he constructed
auxiliary transfer operators associated to the
expanding (and analytic!) quotiented dynamics
acting on holomorphic functions on disks. The
analyticity assumption on the dynamical bundles
was later lifted by Rugh (1996) (see also Fried
(1995)), who let their transfer operators act on
Banach topological t ensor products of spaces of
holomorphic functions on a disk with the dual of
such a space. In all these cases, the transfer
operator is a nuclear operator in the sense of
Grothendieck and no regularization is needed.
(More recent work of Kitaev (1999), when applied
to this analytic setting, shows that the ‘‘mero-
morphic’’ function d
f , g
(z) in fact does not have
poles.)
Regularization and Intermittency
Consider the interval M = [0, 1], and f defined on M
by f (x) = f
1
(x) = x=(1 x) on [0, 1/2], and f (x) =
f
2
(x) = (1 x)=x on [1/2, 1]. (This is the Farey
map, which appears naturally when consideri ng
continued fractions.) Each of the two branches is
an analytic bijection onto [0, 1]. The second branch
is expanding, but the first one, f
1
, has a (parabolic)
neutral fixed point at x = 0 (the expansion is
f (x) = x þ x
2
þ x
3
þ). Let g = g
s
be an analytic
weight of the form g(y) = 1=jf
0
(y)j
s
for <s 1=2. We
are interested in the spectrum of the operator L
f , g
associated with the pair (f , g)by[4]. Clearly, the
expression [6] is not a good candidate for an analog
of the Fredholm determinant of L
f , g
. Rugh (1996)
introduced a Banach space B of functions in a
complex neighborhood of M, having a controlled
singularity at 0, and such that the spectral radius of
L
f , g
on B is equal to 1, and such that the following
regularized determinant
d
f; g
ðzÞ
¼ exp
X
1
n¼1
z
n
n
X
x2ð0;1: f
n
ðxÞ¼x
Q
n1
k¼0
g
s
ðf
k
xÞ
1 Tf
n
x
2
4
3
5
½8
is a holomorphic function in the cut complex plane
{z 2 C jz 62 [1, 1)}. Furthermore, its zeros z in this
cut plane are in bijection with the spectrum of L
f , g
j
B
outside of the unit interval [0, 1], and this spectrum
consists of eigenvalues 1/z of finite multiplicities.
Finally, these eige nvalues can only accumulate at 0
or 1, although each point in the unit interval belongs
to the spectrum of L
f , g
. In particular, the essential
spectral radius of L
f , g
on B coincides with its
spectral radius.
Let us define the Banach space B and explain the
key ideas in the proof of the above result (Rugh’s
claim is in fact more general than the statement
above and applies to a class of maps f with neutral
fixed points). The starting point is the decomposition
L
f ; g
¼L
1
þL
2
where L
i
’ = ’ f
1
i
j(f
1
i
)
0
j
s
. The operator L
2
is of
the type discussed in the previous section, and it is
nuclear when acting, for example, on bounded
holomorphic functions in a complex neighborhood
of M. Since f
1
is not expanding (because of the
parabolic fixed point at 0), other ideas must be used
to handle the operator L
1
. The change of coordinates
(this idea goes back to Fatou) w = 1=x replaces the
weak contraction f
1
1
by the translation w 7!w þ 1in
a suitable domain containing a half-plane <w > w
0
.
In order to take into account the weight g
s
,itis
convenient to use the change of variab les
(w) = ’(1=w) w
2s
. Indeed, in the new coordinates
the operator L
1
reads as
M
1
ðwÞ¼ðw þ 1Þ
The next step consists in letting M
1
act on the
Banach space B
w
of Laplac e transforms of
L
1
(R
þ
, Lebesgue), that is, functions
ðwÞ¼
Z
1
0
e
ðww
0
Þt
ðt Þdt
with the induced norm kk
B
w
=
R
j (t)jdt.SinceM
1
maps to e
t
(t), it is not difficult to see that the
spectrum of M
1
on B
w
(and thus of L
1
on the pullback
B of B
w
by , which consists of functions in a complex
neighborhood of [0,1], holomorphic in a sector at 0,
and with a possible, but controlled, singularity at 0) is
the closed unit interval. One can check that L
2
is
nuclear on B. Composing a bounded operator with a
388 Regularization for Dynamical -Functions
nuclear operator gives a nuclear operator. If 1=z 62
[0, 1], the resolvent (1 zL
1
)
1
is a bounded operator,
and therefore, for such z, the operator
PðzÞ :¼ zL
2
ð1 zL
1
Þ
1
½9
is nuclear on B . We view P(z) as a ‘‘regularized’’
version of L
f , g
= L
1
þL
2
. Now, since
ð1 zL
f ; g
Þ
1
¼ð1 zðL
1
þL
2
ÞÞ
1
¼ð1 zL
1
Þ
1
1 zL
2
ð1 zL
1
Þ
1
1
it is not surprising that one can prove (Rugh 1996)
that the Fredholm determinant
u 7!det 1 L
2
ðu L
1
Þ
1
(which is holomorphic in u 62 [0, 1]) has as its zero set
sp(L
f , g
j
B
) n [0, 1], and that this set consists in isolated
eigenvalues of finite multiplicity (equal to the order of
the corresponding zero) for L
f , g
. Formally,
ð1 zL
1
Þ
1
¼
X
1
k¼0
z
k
L
k
1
½10
so that the regularization we just described can be
viewed as mirroring an induction (or renormaliza-
tion) procedure, where the dynamics f is replaced by
the first-return map to the ‘‘chaotic’’ part of the
phase space [0, 1/2]. (For the Farey map, the induced
map is just the Gauss map.) The formal equality [10]
is also behind the fact that (Rugh 1996)
tr PðzÞ
n
¼
X
x6¼0: f
n
ðxÞ¼x
Q
n1
k¼0
g
s
ðf
k
xÞ
1 Tf
n
x
An extension of this theory to the two-dimensional
setting has been obtained by Baladi, Pujals, and
Sambarino.
Regularization and Kneading
Determinants
Up to now we have only discussed analytic dynamical
systems, for which hyperbolicity (or uniform expan-
sion) guaranteed that the transfer operator (or a
regularized version thereof) was compact, even
nuclear, on a natural Banach space. When considering
hyperbolic invertible (or expanding noninvertible)
maps f, and weights g with ‘‘finite smoothness,’’ say
C
r
for some finite r > 1, the transfer operator defined
by [2] or [4] is usually not compact on any infinite-
dimensional space. However, one can often prove a
‘‘Lasota–Yorke’’ type inequality (see e.g., Baladi
(1998)) which ensures that the essential spectral radius
ess
(L
f , g
), defined in the ‘‘Introduction,’’ is strictly
smaller than the spectral radius. Then, the goal is to
prove that the dynamical determinant [6] defines a
holomorphic function in the disk of radius 1=
ess
,and
that its zeros in this disk are exactly the inverses of the
eigenvalues of L
f , g
. For uniformly expanding C
r
maps
f on compact manifolds, and C
r
weights, denoting by
>1 the expansion coefficient as in the section ‘‘The
Grothendieck–Fredholm case,’’ this goal was essen-
tially attained by Ruelle (1990).ForL
f , g
acting on the
Banach space of C
r
functions on M, Ruelle proved
ess
(L
f , g
)
r
and was able to extend d
f , g
(z)(and
interpret its zeros) in the disk of radius
r
.
For C
r
Anosov diffeomorphisms f,andC
r
weights g,
Pollicott, Ruelle, Haydn, and others obtained important
results using the symbolic dynamics description (for
which the maximal smoothness which can be used is
r 1, because of the metric-space model). Later, Kitaev
(1999) was able to show that d
f , g
(z) extends to a
holomorphic function in the disk of radius
r=2
,
but did not give any spectral interpretation of the
zeros of d
f , g
(z). More recently, Liverani (2005) was able
to give such an interpretation, in a smaller disk however.
All the works mentioned in the previous paragraph
are based on some approximation scheme (Taylor
expansion style). In the early 1990s, a new approach,
with a regularization flavor, was launched (see e.g.,
Baladi and Ruelle (1996)), initially for piecewise
monotone interval maps. We present it next.
Consider a finite set of local homeomorphisms
!
: U
!
!
!
(U
!
), where each U
!
is a bounded
open interval of R, and of associated weight functions
g
!
which are continuous, of bounded variation, and
have support inside U
!
. For example, the
!
can be the
inverse branches of a single piecewise monotone
interval map f,andg
!
can be g
!
for a single g.
(No contraction assumption is required on the
!
:
their graph can even coincide with the diagonal on a
segment.) The transfer operator is now
M’ ¼
X
!
g
!
ð’
!
Þ
Ruelle obtained an estimate, noted
b
R, for the essential
spectral radius of M acting on the Banach space BV of
functions of bounded variation. The main result of
Baladi and Ruelle (1996) links the eigenvalues of
M:BV ! BV outside of the disk of radius
b
R,with
the zeros of the following ‘‘sharp determinant’’:
det
#
ðId zMÞ ¼ exp
X
1
n¼1
z
n
n
tr
#
M
n
!
½11
where (with the understanding that y=jyj= 0ify = 0)
tr
#
M¼
X
!
Z
1
2
!
ðxÞx
j
!
ðxÞxj
dg
!
ðxÞ
Regularization for Dynamical -Functions 389
If the
!
are strict contra ctions which form the set
of inverse branches of a piecewise monotone interval
map f, and g
!
= g
!
, then integration by parts
together with the key property that
d
x
2jxj
¼ ; the Dirac delta at the origin of R
show that det
#
(Id zM) = 1=
f , g
(z) (recall [7]). If
one assum es instead only that the graph of each
admissible composition
n
w
of n successive
!
’s (with
n 1) intersects the diagonal transversally, then
det
#
ðId zMÞ
¼ exp
X
1
n¼1
z
n
n
X
admissible
n
w
X
x:
n
w
ðxÞ¼x
Lx;
n
w
2
4
Y
n1
k¼0
g
!
k
k
w
ðxÞ
#
½12
where L(x, ) 2 {1, 1} is the Lefschetz number of a
transversal fixed point x = (x)(if is C
1
this is just
sgn (1
0
(x))). Therefore, we call the sharp determi-
nant det
#
(Id zM) a Ruelle–Lefschetz (dynamical)
determinant. For a class of ‘‘unimodal’’ interval maps f
and constant weight g = 1, the expression [12] with
Lefschetz numbers, coming from the additional
transversality assumption, gives that det
#
(Id zM)
is just 1=
(z), where the ‘‘negative -function’ ’
ðzÞ¼exp þ
X
1
n¼1
z
n
n
2#Fix
ðf
n
Þ1ðÞ
"#
½13
is defined by counting (twice) the sets
Fix
ðf
n
Þ¼fxjf
n
ðxÞ¼x; f strictly decreasing
in a neighborhood of xg
of ‘‘negative fixed points.’’ This negative -function
was studied by Milnor and Thurston, who proved
the remarkable identity
ð
ðzÞÞ
1
¼ detð1 þ
b
DðzÞÞ
where
b
D(z)isa11 ‘‘matrix,’’ which is just a
power series in z with coefficients in {1, 0, þ1},
given by the signed itinerary of the im age of the
turning point (the so-called ‘‘kneading’’ data).
Returning now to the general setup
!
, g
!
, the
crucial step in the proof of the spectral interpreta-
tion of the zeros of this Ruelle–Lefschetz determi-
nant consists in establishing the following
continuous version of the Milnor–Thurston identity:
det
#
ðId zMÞ ¼ det
ðId þ
b
DðzÞÞ ½14
where the ‘‘kneading operator’’
b
D(z) replaces (for-
mally) the finite kneading matrix of Milnor and
Thurston. In a suitable z-disk, one proves that this
operator
b
D(z) is a Hilbert–Schmidt operator on an
L
2
space (its kernel is bounded and compactly
supported), thus allowing the use of regularized
determinants of order 2 (see e.g., Gohberg et al.
(2000)). By definition, det
(Id þ
b
D(z)) is the product of
this regularized determinant with the exponential of the
average of the kernel of
b
D(z) along the diagonal, which
is well defined. Another kneading operator, D(z), is
essential. If 1=z is not in the spectrum of M (on BV),
then D(z) is also Hilbert–Schmidt, and one can show
det
(Id þ
b
D(z)) = det
(Id þD(z))
1
. The initial defini-
tions of
b
D(z)andD(z) were technical and we shall not
give them here. However, a more conceptual definition
of the D(z) was later implemented:
DðzÞ¼NðId zMÞ
1
S½15
where N is an auxiliary transfer operator and S is
the convolution
S’ðxÞ¼
Z
1
2
x y
jx yj
’ðyÞd
where is an auxiliary non-negative finite measure.
From [15], it becomes clear that the kneading
operator is a regularized (through the convolution
S) object which describes the inverse spectrum of the
transfer operator: the resolvent (Id zM)
1
in [15]
means that poles can only appear if 1=z is an
eigenvalue. Since det
(Id þ
b
D(z)) = det
(Id þD(z))
1
,
this can be translated into a statement for zeros of
det
(Id þ
b
D(z)). The Milnor–Thurston identity [14]
then implies that any zero of det
#
(Id zM)isan
inverse eigenvalue of M.
The one-dimensional kneading regularization we
just presented is well understood. The higher-
dimensional theory is not as developed yet. Let
U
!
be now finitely many bounded open subsets of R
d
,
!
: U
!
!
!
(U
!
)belocalC
r
homeomorphisms or
diffeomorphisms, while g
!
: U
!
! C are compactly
supported C
r
functions, for r 1.
In 1995, A Kitaev wrote a two-page sketch proving a
higher-dimensional Milnor–Thurston formula, under
an additional transversality assumption. This assump-
tion guarantees that the set of fixed points of each fixed
period m is finite, so that the Ruelle–Lefschetz
determinant det
#
(Id zM) can be defined through
[12]. Inspired by Kitaev’s unpublished note, Baillif
(2004) proved the following Milnor–Thurston formula:
det
#
ðId zMÞ ¼
Y
d1
k¼0
det
[
ðId þD
k
ðzÞÞ
ð1Þ
kþ1
½16
Here, the D
k
(z) are kernel operators acting on (k þ 1)-
forms, constructed with the resolvent (Id zM
k
)
1
,
together with a convolution operator S
k
,mapping
390 Regularization for Dynamical -Functions
(k þ 1)-forms to k-forms and which satisfies the
homotopy equation dSþSd = 1. The kernel
k
(x, y)
of S
k
has singularities of the form (x y)=k x yk
d
.
The transversality assumption allows Baillif to interpret
the determinant obtained by integrating the kernels
along the diagonal as a flat determinant in the sense of
Atiyah and Bott, whence the notation det
[
in the right-
hand side of [16].
Baillif (2004) did not give a spectral interpretation
of zeros or poles of the sharp determinant [16], but
he noticed that for jzj very small, suitably high
iterates of the D
k
(z) are trace-class on L
2
(R
d
),
showing that the corresponding regul arized determi-
nant has a nonzero radius of convergence under
weak assumptions. The spectral interpretation of the
sharp determinant [12] in arbitrary dimension, but
under additional assumptions, was subsequently
carried out by Baillif and the author of the present
article, giving a new proof of some of the results in
Ruelle (1990).
See also: Chaos and Attractors; Dynamical Systems and
Thermodynamics; Ergodic Theory; Hyperbolic Dynamical
Systems; Number Theory in Physics; Quantum
Ergodicity and Mixing of Eigenfunctions; Quillen
Determinant; Semi-Classical Spectra and Closed Orbits;
Spectral Theory for Linear Operators.
Further Reading
Baillif M (2004) Kneading operators, sharp determinants, and
weighted Lefschetz zeta functions in higher dimensions. Duke
Mathematical Journal 124: 145–175.
Baladi V (1998) Pe riodic Orbits and Dynamical Spectr a,
Ergodic Theory Dynam. Systems, vol. 18, pp. 255–292
(with an addendum by Dolgopyat D and Pollicott M, pp.
293–301.)
Baladi V and Ruelle D (1996) Sharp determinants. Inventiones
Mathematicae 123: 553–574.
Chang CH and Mayer DH (2001) An extension of the
thermodynamic formalism approach to Selberg’s zeta function
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Relativistic Wave Equations Including Higher Spin Fields
R Illge and V Wu¨ nsch, Friedrich-Schiller-Universita
¨
t
Jena, Jena, Germany
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
The description of phenomena at high energies
requires the investigation of relativistic wave equa-
tions, that is, equations which are invariant under
Lorentz transformations. Our discussion will be given
classically (i.e., nonquantum). A classification of the
wave equations may be based on the spin of the
particles (or physical fields), which was discovered
for the electron by Goudsmith and Uhlenbeck in
1925. For the greater part of physics, the three spin
numbers s = 0, 1=2, and 1 are sufficient; the respec-
tive equations named after their discoverers Klein–
Gordon, Dirac, and Proca for massive fields and
D’Alembert, Weyl, and Maxwell for massless fields,
respectively (see the following section).
In their original form, these equations look rather
different. However, their translation into spinor form
shows that the wave equations for bosons and fermions
Relativistic Wave Equations Including Higher Spin Fields 391