Download Interacting Fock spaces: central limit theorems and quantum

UNIVERSITÀ DEGLI STUDI DI BARI
Dottorato di Ricerca in Matematica
XVII Ciclo – A.A. 2004–2005
Settore Scientifico-Disciplinare:
MAT/06 – Probabilità e statistica
Tesi di Dottorato
Interacting Fock spaces:
central limit theorems and
quantum stochastic calculus
Candidato:
Vitonofrio CRISMALE
Supervisore della tesi:
Prof. Yun Gang LU
Coordinatore del Dottorato di Ricerca:
Prof. Luciano LOPEZ
To the memory of my grandmother Teresa
Contents
Introduction
1
1 Preliminaries
1.1 From classical to quantum probability . . . . . . . . . . . . . .
1.2 The boson, fermion and free Fock spaces . . . . . . . . . . . . .
7
7
11
2 Interacting Fock space
2.1 Definition and first results . . . . . . . . . . . . . . . . . .
2.2 Standard interacting Fock spaces and module gaussianity
2.3 The 1-mode interacting Fock space . . . . . . . . . . . . .
2.4 1-mode IFS and orthogonal polynomials . . . . . . . . . .
25
26
29
43
45
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3 Universal Central Limit Theorem on interacting Fock space
3.1 Preservation operator on standard IFS and universal convolution
3.2 Moments of creators, annihilators and preservation operators
on IFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 1-mode type IFS . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . .
53
56
62
67
72
4 Boolean Central limit Theorem
81
4.1 Boolean independence and Boolean Fock space . . . . . . . . . 83
4.2 Moments of operators in discrete and Boolean case . . . . . . . 87
4.3 Boolean Central Limit Theorem . . . . . . . . . . . . . . . . . . 92
5 Quantum stochastic calculus on interacting
5.1 Simple adapted processes . . . . . . . . . .
5.2 Semi-martingale Inequalities . . . . . . . . .
5.3 Stochastic Integral . . . . . . . . . . . . . .
5.4 Quantum Stochastic Differential Equations
5.5 Quantum Ito Formula . . . . . . . . . . . .
Fock spaces
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95
97
101
118
126
130
vi
Index
5.6 Unitarity condition . . . . . . . . . . . . . . . . . . . . . . . . . 133
Bibliography
137
Introduction
This thesis is devoted to present some new results in Quantum Probability
obtained by means of a structure called Interacting Fock Space.
Quantum Probability is a recent mathematical theory based on the efforts
of several people to determine a unique field in which probability theory and
quantum mechanics can be discussed together. In fact it is well known that,
with the birth of quantum mechanics during the 1920’s, a new way of thinking to all physical objects emerged. From the beginning human experience
suggests that all things have a definite place and properties such as speed,
color,... But, the introduction of Heisenberg’s uncertainty principle in 1926,
according to which it is impossible to know exactly at the same time both the
position and the speed of a particle, broke these ideas, favoring the development of the Quantum Theory, a physical theory where it is not possible to
predict the result of an experiment with certainty, but yields randomness for
the outcomes.
During the 1930’s, Kolmogorov’s axioms brought to the birth of Probability
Theory. In fact he stated the structure of any probability space as a triple
(Ω, F, P) consisting of a measurable space (Ω, F) with a unit mass measure
P. As a consequence, Probability Theory inherited all the results on finite
measure theory, but was also enriched by new features, such as stochastic
processes, martingales, Markov chains,...
On the other hand the 1930’s were also the period in which a mathematical
model for quantum mechanics was formulated, mainly in the works by J. von
Neumann [63] and P. M. A. Dirac. The features of this new model were partly
imported from Functional Analysis (operators in Hilbert spaces, Banach algebras), and partly new (C ∗ -algebras, von Neumann algebras,...).The latter ones
represented the starting point for the diffusion of a new field in mathematics,
namely Operator Algebras Theory.
Probability Theory and Operator Algebras Theory ran separately for a
long time. It was during the 1970’s and 1980’s that people such as Ac-
2
Introduction
cardi, Belavkin, Hudson, Lewis, Parthasarathy, von Waldenfels,... developed
a framework, called Quantum Probability, in which it is possible to join these
theories into a unified picture.
A quantum (or algebraic) probability space is a couple (A, φ), where A is
a unital ∗-algebra and φ is a state on A, i.e. a linear map φ : A → C that
is positive ( for any a ∈ A φ (a∗ a) ≥ 0) and normalized (φ (1) = 1). The
relation between a classical probability space (Ω, F, P) and such a space is
intuitively given by the fact that the triple (Ω, F, P) uniquely determines the
space L∞ (Ω, F, P) of all bounded complex valued measurable functions. This
space has a natural structure of unital ∗-algebra. If we put A := L∞ (Ω, F, P),
it is possible to give a state on A by
Z
φ (f ) :=
f dP for any f ∈ A
Ω
The space (A, φ) so built is a commutative algebraic probability space. When
we drop commutativity, we find a general quantum probability space. In this
context becomes clear the name ”non-commutative Probability” sometimes
used to indicate Quantum Probability.
One of the most fertile fields of research in Quantum Probability during
the 1980’s and 1990’s has been made by the stochastic limit of quantum theory (see [10] for a complete description of the matter). On a certain stage
of its development it was shown in [8] that the stochastic limit of quantum
electrodynamics without dipole approximation led to the emergence of a new
mathematical feature: interacting Fock space. Such new structure has been
successively studied in many papers, such as [44], [45], [46],... All these results
have been joint together by Accardi, Lu and Volovich in [9], that represents
the first systematic exposition of interacting Fock spaces.
This thesis is related to the presentation of some new aspects linked to
such a structure emerged in the last years. It contains five chapters.
In the first chapter we present some already known notions and results
that are a useful tool for the future chapters.
In particular, after introducing how to pass from classical to Quantum
Probability, we present the boson, fermion and free Fock spaces, standing
out how they represent a mathematical model for physical particles and their
intimate Hilbert space structure. In fact boson, fermion and free Fock spaces
are defined as the symmetric, exterior and tensor algebra respectively, on a
given Hilbert space. They are also the framework in which one describes the
processes of ”birth” and ”death” of particles. The mathematical formulation
of such processes is given by creation and annihilation operators acting on an
Introduction
3
appropriate domain in these spaces. Hence, after introducing these operators,
some important properties are exposed. In particular we present the rules
of commutations, namely canonical commutation relations (CCR), canonical
anticommutation relations (CAR) and free commutation relations of these
operators respectively in boson, fermion and free case.
In the second chapter interacting Fock spaces are presented. Roughly
speaking an interacting Fock space over a pre-Hilbert space H is a space which,
as a vector space coincides with the free Fock space, but, for any n ∈ N, any nparticle space Hn has its own scalar product – not necessarily to be the tensor
product one. We give a general outline on this space, focusing our attention on
an important and wide class of them, the standard interacting Fock spaces. In
this case we introduce the basic operators of creation and annihilation, showing
that, as in the usual Fock spaces, they are mutually adjoint. Moreover, after
giving the field operator on interacting Fock space, we compute its moments,
i.e., the vacuum state of the powers of it. The reason of this computing is given
by the fact that such powers are exactly the moments of a distribution on the
real line, as a consequence of von Neumann’s spectral Theorem. These kinds
of calculi show one of the most important difference between interacting Free
Fock spaces and the usual ones, namely the deviation from gaussianity (see
Definition 2.2.11): in fact a more general concept of gaussianity arises, never
present in all the other previous cases: the so- called module gaussianity. The
chapter ends with the presentation of a special class of interacting Fock spaces:
the 1-mode interacting Fock spaces. In particular in Section 2.4, following [2],
a deep relation between this class and the L2 −space of a probability measure
µ on the real line is given. In fact it is shown that, under some conditions,
a 1-mode interacting Fock space and L2 (R, µ) are the same object, up to a
unitary isomorphism. A very important role in this Theorem is played by the
sequences of orthogonal polynomials and Jacobi coefficients associated to µ
(see Theorem 2.4.1). This result has been useful in Chapters 3 and 5.
In Chapter 3 we deal with a new result: a non-commutative Central Limit
Theorem. In particular we state that, given any probability measure µ on R
with finite moments of any order, there exists a suitable interacting Fock space
and a family of operator random variables {Qk }k in this space such that for
any m ∈ N
!m + Z
*Ã
N
1 X
√
= xm dµ
Qk
lim
N →∞
N k=1
It is worth of mention to illustrate two fundamental consequences of this result:
i) any one dimensional distribution with all finite moments is a central
4
Introduction
limit;
ii) we are able to construct the approximating sequence of random variables.
The main technical tool used to reach such a theorem is given by a special
class of interacting Fock spaces (IFS), namely the 1-mode type Free interacting
Fock spaces. More precisely, after introducing a new basic operator on the
standard IFS, that is called preservation operator and computing the mixed
moments of creation-annihilation-preservation operators in IFS, we show that
in 1-mode type free interacting Fock spaces, the moments of a non symmetric
field operator (i.e. an operator given by addiction of creation, annihilation
and suitable preservation) are uniquely determined by the norm of the test
function chosen. This result, which is fundamental in the proof of central limit
theorem, it is not true in a generic IFS, as shown by Lu in [46].
Chapter 4 is devoted to the presentation of a Boolean central limit theorem and it is based on [17]. In particular we firstly present the Boolean
Fock space, that can be seen as a particular IFS and then we recall the concept of boolean independence, introduced by von Waldenfels [64], Bożejko
[21] and successively in various fields of quantum probability. After we introduce the simplest Quantum Probability model (the quantum Bernoulli process) and construct a proper quantum stochastic process (namely creationannihilation-preservation processes) on Boolean Fock space and with discrete
time and compute the mixed moments of these operators with respect to the
vacuum state. Finally in the last section a Central Limit Theorem is given,
to get the creation-annihilation- preservation boolean independent family on
the Boolean Fock space starting from the boolean independent process with
discrete time. We underline the fact that a concrete construction is given both
for the approximating sequence of operators and for the limiting one.
In chapter 5 we develop a quantum stochastic calculus on standard interacting Fock spaces, as shown in [27]. Quantum stochastic calculus in Boson
Fock space appeared during the 1980’s thanks to the work by Hudson and
Parthasarathy [36]. Successively, a rich and deep collection of papers has been
devoted to this aim in Fock spaces different from the boson one (e.g. [15],
[35], [42], [19],...) and, consequently, many theories, depending on the representation chosen, of quantum stochastic calculus, arose. Accardi, Fagnola and
Quaegebeur in [4] presented a method to include all the quantum stochastic
calculi in a unified picture, free from the representation chosen, in analogy
with what happens in the classical case. Then Fagnola in [30], by means of a
suitable extension of these results, constructed a theory for the ”free” noise in-
Introduction
5
troduced in [58]. These motivations led us to extend and modify such a theory
on the IFS structure. We firstly introduce the ”semi-martingale inequalities”,
which are the main technical tool for the successive definition of stochastic
integral and computation of Ito table. The stochastic integral, and mainly its
properties there presented, allows us to prove the existence and uniqueness of
the solution for a class of quantum stochastic differential equations, whereas
the Ito formula in a ”weak” sense leads us to find a necessary and sufficient
condition for the unitarity of such a solution.
I wish to thank all people who gave me inspiration, encouragement and
love throughout this period.
My heartfelt gratitude goes to Prof. Yun Gang Lu who accepted to be my
advisor and introduced me into the charming world of Quantum Probability.
Without his constant help, patience and incredible disponibility, this work
would not have been completed.
I deeply thank Prof. Luigi Accardi for his recurring invitations to the
centre ”Vito Volterra” in Rome. His deep ideas have been a fertile source
of inspiration and his nice comments greatly improved my comprehension of
problems in Quantum Probability.
My thanks go to Profs. Michael Schürmann, Uwe Franz and Rolf Ghom
for the warm hospitality at the ”Ernst-Moritz-Arndt” University of Greifswald
during the spring 2004. I learnt very much from their lectures and seminars
as well as from joint discussions.
A special mention goes to Profs. Marek Bożejko, Franco Fagnola, Akihito Hora and Nobuaki Obata for fruitful suggestions and to Prof. Francesco
Fidaleo.
I like to thank the young researchers I met during these years for their
comments on my work and friendship and all my Ph.D. colleagues at the
Department of Mathematics in Bari.
A particular thank goes to Dr. Anis Ben Ghorbal, whose I enjoyed a
great number of exciting and improving mathematical discussions as well as a
beautiful friendship.
I am really indebted towards my parents and my family, who offered me
all the supports during these years as well as their understanding specially in
difficulties.
Finally no words can express all my feelings and gratitude to Iolanda.
Chapter 1
Preliminaries
The aim of this thesis is to study some important problems in Quantum Probability which are related with interacting Fock space structure. This first
chapter, which can be considered as an introduction, is mainly devoted to the
presentation of basic and fundamental definitions and results of those structures which will be used throughout the following chapters.
We firstly introduce the notions of quantum probability space, quantum
random variable and quantum stochastic process. Following an almost standard idea, in this setting we privilege the approach consisting of starting from
classical models, defining their non-commutative counterpart and then generalizing them into an abstract picture. Successively we introduce the most
famous Fock spaces, namely boson, fermion and free Fock spaces, whose importance in Quantum Probability will be stressed in Section 1.2 and present
some important results mainly used to investigate the most important properties of some operators acting on them, namely creation and annihilation
operators, whose rules of commutation in these spaces are stood out also.
1.1
From classical to quantum probability
During the 1930’s Kolmogorov formulated the axioms on which the modern
Probability Theory is based. The most important of them consists in the
statement of a ”probability space”, which is in fact taken as a measure space
(Ω, F, P), where Ω is a set (it represents the set of all possible outcomes of an
experiment submitted to law of chance), F is a σ-algebra on Ω (representing
8
Chapter 1. Preliminaries
”events”, i.e. statements on the outcomes of the experiment) and P is a
probability (i.e. finite and normalized) measure on (Ω, F) (representing the
probability that an event in F occurs).
As pointed out in the Introduction, the developments of Quantum Theory
during the last century, showed that the classical mathematical features were
not sufficient to describe all physical experiments and led to the emergence
of new mathematical models, namely noncommutative models. According to
a ”standard” common strategy in many fields of mathematics, also to make
probability spaces ”noncommutative”, one firstly transfers all the informations contained in the classical model into an algebra of functions and then
gives on this new structure some characterizing properties, among them, commutativity. Finally an abstract general structure is obtained by removing
commutativity.
Hence we are given a Kolmogorov’s triple (Ω, F, P) and we want to find an
algebra of functions
f :Ω→C
on which encode the informations on F and P. To this aim we consider all the
functions measurable with respect to F, i.e for all a, b ∈ R
{ω ∈ Ω : a ≤ Ref (ω) ≤ g} ∈ F
in order to consider them as random variables. Moreover for any such a
function we want to define its expectation with respect to P as a complex
valued linear functional
Z
φ (f ) :=
f dP
Ω
As a consequence we are looking for an algebra which is contained in L1 (Ω, F, P)
and then it is natural to get L∞ (Ω, F, P), which is furthermore closed with
respect to involution, i.e. it is a ∗-algebra. We also point out that the linear
map φ above introduced satisfies two conditions:
i) positivity ( for any f ∈ L∞ (Ω, F, P) φ (f ∗ f ) ≥ 0);
ii) normalization (φ (1) = 1)
i.e. it is a state on L∞ (Ω, F, P).
By the above discussion it is clear that (Ω, F, P) uniquely determines
L∞ (Ω, F, P). Moreover the following result holds.
9
1.1. From classical to quantum probability
Theorem 1.1.1. Let us given a probability space (Ω, F, P). Then the algebra
N := {Mf : f ∈ L∞ (Ω, F, P)}
is a commutative von Neumann algebra of operators on the Hilbert space H :=
L2 (Ω, F, P) and
Z
φ : Mf ∈ N 7→
f dP ∈ C
Ω
is a faithful normal state on Ω, where Mf is the multiplication operator by f
in H, i.e. for any g ∈ H Mf g := f g.
We omit the proof. General references for it as well as for von Neumann
algebras theory are, for example, [63], [37] or [25].
In order to generalize such a setting (namely drop the commutativity), one
can get a quantum probability space as a pair (N , φ), where N is an arbitrary
von Neumann algebra and φ a normal state on it. The advantage of such a
choice is moreover motivated by a theorem due to Gelfand (set in the more
general structure of C ∗ -algebras, see [16]) in which it is stated that, up to
an isomorphism, all the commutative von Neumann algebras endowed with a
faithful normal state are of the form L∞ (Ω, F, P) and the state φ is just the
expectation with respect to P. But in this thesis and in general one uses the
”noncommutative moments method” and, as argued in [53], we will deal with
not necessarily bounded operators. Then the following definition of quantum
probability space seems to be proper.
Definition 1.1.2. An algebraic probability space is a pair (A, φ), where A
is a unital ∗-algebra and φ is a state on A. If A is commutative we speak of a
classical probability space, if A is non-commutative, of a quantum probability
space.
In order to generalize into an algebraic (noncommutative) setting the notion of random variables and stochastic processes, one considers a classical
random variable, i.e. a measurable function
X : (Ω, F, P) → (E, E)
from a probability space (Ω, F, P) (sample space) to a measurable space (E, E)
(state space). We observe that such a random variable uniquely determines the
∗-algebra homomorphism (i.e. a linear map between two ∗-algebras h : A → B
such that for any a, b ∈ A h (a∗ ) = h (a)∗ , h (ab) = h (a) h (b))
j : L∞ (E, E) → L∞ (Ω, F, P)
10
Chapter 1. Preliminaries
such that for any f ∈ L∞ (E, E)
j (f ) := f ◦ X
Moreover any random variable X, up to stochastic equivalence, is determined
by such an isomorphism and we can identify these objects. The following
definition is now meaningful.
Definition 1.1.3. An algebraic random variable is a triple
{(A, φ) , B, j}
where (A, φ) is an algebraic probability space (sample space), B is a unital
∗-algebra (state space) and j : B → A is a ∗-algebra homomorphism.
Remark 1.1.4. If we restrict ourselves to von Neumann algebras, then the
∗-homomorphism above must be isometric.
Given an algebraic random variable j : B → (A, φ), we get the linear map
φj : B → C, such that φj := φ ◦ j. Then:
• for any b ∈ B φj (b∗ b) = φ (j (b)∗ j (b)) ≥ 0;
• φj (1B ) = φ (1A ) = 1
and hence φj is a state on B. In analogy with the classical case, we call it
the distribution of the random variable j.
Remark 1.1.5. As it is shown in [53], Section 1.2., given (H, π, Φ) the GNS
triple related to the pair (A, φ) and the algebraic probability space (H (H) , φΦ ),
where H (H) is the ∗-algebra given by the linear span of all hermitian operators in H and φΦ := h·, ·ΦiH is a state on H (H), then the algebraic random
variables j : B → (A, φ) and π ◦ j : B → (H (H) , φΦ ) have the same distribution.
We know that a classical stochastic process is a family (Xt )t∈T of random
variables indexed by a set T such that for any t ∈ T
Xt : (Ω, F, P) → (E, E)
Then, each Xt can be identified by the following homomorphism
jt : L∞ (E, E) → L∞ (Ω, F, P)
1.2. The boson, fermion and free Fock spaces
11
where for any f ∈ L∞ (E, E)
jt (f ) := f ◦ Xt
The passage to the quantum setting is achieved by the following definition,
due to Accardi, Frigerio and Lewis in [5] and based on the work of Nelson for
classical stochastic processes in [51].
Definition 1.1.6. An algebraic stochastic process is a triple
©
ª
(A, φ) , B, (jt )t∈T
where T is an index set and for any t ∈ T
{(A, φ) , B, jt }
is an algebraic random variable.
For any t ∈ T the state φt := φ ◦ jt is called the 1-dimensional distribution
of jt .
³ ´
(i)
Definition 1.1.7. Two algebraic stochastic processes jt
, i = 1, 2 over
t∈T
¡ (i) (i) ¢
two algebraic probability spaces A , φ
on the same ∗-algebra B are called
equivalent if their moments agree, i.e.
³
´
³
´
(1)
(1)
(2)
(2)
φ(1) jt1 (b1 ) · · · jtn (bn ) = φ(2) jt1 (b1 ) · · · jtn (bn )
for any n ∈ N, t1 , . . . tn ∈ T, b1 , . . . bn ∈ B.
1.2
The boson, fermion and free Fock spaces
In this section we recall some notions related to the free (or full) Fock space
and to its most important subspaces from a physical viewpoint: boson and
fermionic Fock spaces.
We follow the constructions given in [47] and [52], where also one can find
much more details either for the mathematics viewpoint or for the connections
with the quantum mechanics features.
Let us given H a complex separable Hilbert space and we denote by h·, ·i
its scalar product, with the convention that it is linear with respect to the
12
Chapter 1. Preliminaries
second component. For any n ∈ N we consider the sesquilinear map h·, ·i¯n :
H¯n × H¯n → C such that for any f1 , f2 , . . . , fn , g1 , g2 , . . . , gn ∈ H
hf1 ¯ f2 ¯ . . . ¯ fn , g1 ¯ g2 ¯ . . . ¯ gn i¯n :=
n
Y
hfh , gh i
(1.2.1)
h=1
where, as usual, ¯ denotes the algebraic tensor product. It is easy to see that
such a map is positive definite and non-degenerate, so H¯n is equipped with
a scalar product. From now on we will use the symbol H⊗n to denote the
Hilbert space given by completing H¯n with respect to the topology induced
by h·, ·i¯n and for any f1 , f2 , . . . , fn ∈ H, the tensor product f1 ¯ f2 ¯ . . . ¯ fn ,
regarded as an element of H⊗n , will be denoted by f1 ⊗f2 ⊗. . .⊗fn and will be
called scalar tensor product. From a physical hand, if H is the Hilbert space
which describes the state of one isolated particle, then H⊗n is the Hilbert
space describing a system of n identical particles and for this reason it is
called the n-particle space. Now we suppose that some changes in physical
characteristic occur on these n particles (e. g. some of them interchange
their positions) and it is not possible to detect such changes by any observable
means. This suggests the fact that H⊗n is too large and it is not well qualified
for the description of some dynamics concerning n identical particles. As a
consequence, it naturally arises the idea of restricting the Hilbert space H⊗n
to some suitable subspace, well qualified to describe the state of the particles.
To this aim one defines the two following products:
1 X
fσ(1) ⊗ fσ(2) ⊗ . . . ⊗ fσ(n)
n!
(1.2.2)
1 X
ε (σ) fσ(1) ⊗ fσ(2) ⊗ . . . ⊗ fσ(n)
n!
(1.2.3)
f1 ◦ f2 ◦ . . . ◦ fn :=
σ∈Sn
f1 ∧ f2 ∧ . . . ∧ fn :=
σ∈Sn
where f1 , f2 , . . . , fn are arbitrary elements of H, Sn is the group of all permutations σ of {1, . . . , n} and ε (σ) is the signature of σ, i. e. ε (σ) = ±1
according to whether the permutation σ is odd or even. As usual the symbols ◦ and ∧ are called respectively the symmetric product and the exterior
product. The closed subspaces of H⊗n
©
H◦n := f1 ⊗ f2 ⊗ . . . ⊗ fn ∈ H⊗n :
ª
f1 ⊗ f2 ⊗ . . . ⊗ fn = fσ(1) ⊗ fσ(2) ⊗ . . . ⊗ fσ(n) for all σ ∈ Sn
©
H∧n := f1 ⊗ f2 ⊗ . . . ⊗ fn ∈ H⊗n :
13
1.2. The boson, fermion and free Fock spaces
f1 ⊗ f2 ⊗ . . . ⊗ fn = ε (σ) fσ(1) ⊗ fσ(2) ⊗ . . . ⊗ fσ(n) for all σ ∈ Sn
ª
are called respectively the n-fold symmetric and the n-fold antisymmetric tensor product of H. We mention the fact that if the dynamics of n identical
particles is described in the space H◦n , then such a particle is called boson
(e. g. photon), whereas if it is described in the space H∧n , then it is called
fermion (e. g. electron).
Remark 1.2.1. From now on, if it is not strictly necessary, we will denote by
Hn indifferently H⊗n , H◦n or H∧n .
The n-fold symmetric and antisymmetric tensor product spaces, as closed
subspaces of the tensor product space, are endowed with a natural Hilbert
space structure; indeed it is possible to define new scalar products that are
equivalent to the original one, in the sense that the norms induced by them
are equal to the norm induced by (1.2.1). More precisely, for any n ∈ N, for
any f1 , f2 , . . . , fn , g1 , g2 , . . . , gn ∈ H
hf1 ◦ f2 ◦ . . . ◦ fn , g1 ◦ g2 ◦ . . . ◦ gn i◦
µ ¶2 X Y
n
­
®
1
fσ(i) , gτ (i)
=
n!
σ,τ ∈Sn i=1
and
hf1 ∧ f2 ∧ . . . ∧ fn , g1 ∧ g2 ∧ . . . ∧ gn i∧
µ ¶2 X
n
Y
®
­
1
fσ(i) , gτ (i)
=
ε (σ) ε (τ )
n!
σ,τ ∈Sn
i=1
define a scalar product respectively on H◦n and H∧n .
The above discussion discloses the idea that the tensor (respectively symmetric or antisymmetric) product of Hilbert spaces represents a good mathematical structure for the study of dynamics of a finite number of identical
particles. At this moment a natural question arises: there exist a ”suitable”
Hilbert space in which is possible to describe the state of an indefinite number
of identical particles in a system where, using Parthasarathy’s language, the
indefiniteness is due to the fact that such particles are ”created” and ”annihilated” according to some laws of chance? The answer is positive and it is
achieved by putting together all the finite tensor products into a direct sum.
More precisely we give the following definition.
14
Chapter 1. Preliminaries
Definition 1.2.2. Let us consider a Hilbert space H and for any n ∈ N let us
define H⊗n , H◦n and H∧n as above. Then the Hilbert spaces
Γ (H) :=
∞
M
H⊗n
n=0
Γs (H) :=
∞
M
H◦n
n=0
Γa (H) :=
∞
M
H∧n
n=0
with the conventions that
H⊗0 = H◦0 = H∧0 = C
and
H⊗1 = H◦1 = H∧1 = H
are called respectively the free (or full or Boltzmannian), the boson (or symmetric) and the fermion (or antisymmetric) Fock space over H. The n-th
direct summand in each case is called the n-th particle subspace and any element in this space is called an n-particle vector. Moreover the vector
Φ := 1 ⊕ 0 . . . ⊕ 0
is called the vacuum vector.
As a consequence of the definition above, any element F in a Fock space
(free, boson or fermion) can be seen in the following way:
F =
∞
X
fn , fn ∈ Hn , for any n ∈ N
n=0
We denote by Γ0 (H) , Γ0s (H) , Γ0a (H) the dense linear manifolds generated
by all n-particle vectors, n = 0, 1, 2, . . . in the corresponding Fock space and
call any element in them a finite particle vector.
The following definition introduces a very useful vector in boson Fock
space.
15
1.2. The boson, fermion and free Fock spaces
Definition 1.2.3. For any f ∈ H the element
E (f ) :=
∞
M
1
√ f ◦n
n!
n=0
where 0! = 1 and f ◦0 = 1, is called the exponential (or coherent) vector
associated with f .
Remark 1.2.4. For any f, g ∈ H , from Definition 1.2.3, it follows that
E (f ) ∈ Γs (H) ⊂ Γ (H)
and
hE (f ) , E (g)i = exp hf, gi
In fact
∞
∞
n=0
n=0
1 X ◦n ◦n
1 X ­ ⊗n ⊗n ®
hE (f ) , E (g)i =
hf , g i =
f ,g
n!
n!
=
∞
X
hf, gin
n=0
n!
= exp hf, gi
The identity above also explains the term ”exponential” used to indicate such
vectors.
We recall that any subset S ⊂ H is called total if the subspace generated
by S is dense in H.
Proposition 1.2.5. The set {E (f ) : f ∈ H} of all exponential vectors is linearly independent and total in Γs (H) .
Proof. See [52], Proposition 19.4.
¤
The subspace generated by the the set of exponential vectors is called
exponential domain and we denote it by E. As a consequence of Proposition
1.2.5, in boson Fock space it is possible to consider two dense sets: Γ0s (H) and
E.
In the Fock spaces there are important operators describing the ”birth” and
”death” processes of particles. We give their definition and some properties.
16
Chapter 1. Preliminaries
Definition 1.2.6. For any n ∈ N, for any f ∈ H the linear operator A+ (f ) :
H◦n → H◦n+1 such that for any f1 , . . . , fn ∈ H
A+ (f ) f1 ◦ . . . ◦ fn :=
√
n + 1f ◦ f1 ◦ . . . ◦ fn
and
A+ (f ) Φ = f
is called the creation operator on n-fold symmetric tensor product of H. Moreover the linear operator A (f ) : H◦n → H◦n−1 such that
n
1 X
A (f ) f1 ◦ . . . ◦ fn := √
hf, fi i f1 ◦ . . . ◦ fbi ◦ . . . ◦ fn
n
i=1
A (f ) Φ = 0
where the symbol b over fi indicates its omission, is called the annihilation
operator on n-fold symmetric tensor product of H.
Creation and annihilation operators in symmetric case can be extended by
linearity to the domain of all finite particle vectors and, by closure, to their
maximal domain and they are called boson creation and boson annihilation
operator respectively. It is possible to prove that the exponential domain E
is included in such a domain (see [52], Proposition 20.11), which is given by
all the vectors F in the Fock space such that A+ (F ) and A (F ) yet belong to
the Fock space.
Let K be an arbitrary Hilbert space and let us consider T, V two operators
acting on such space.The commutator operator of T and V on K is defined as
follows:
[T, V ] := T V − V T
Proposition 1.2.7. For any f ∈ H the following relations hold for boson
creation and annihilation operators:
∗
1. A (f ) = (A+ (f )) ;
2. A (f ) E (g) = hf, gi E (g) for any g ∈ H;
3. A+ (f ) E (g) =
d
dt E
(g + tf ) |t=0 for any g ∈ H;
17
1.2. The boson, fermion and free Fock spaces
4. for any g ∈ H
[A (f ) , A (g)] = 0
¤
A+ (f ) , A+ (g) = 0
£
¤
A (f ) , A+ (g) = hf, gi 1
£
(1.2.4)
(1.2.5)
(1.2.6)
Proof. In fact, for any n ∈ N, for any f1 , . . . , fn+1 , g1 , . . . , gn+1 ∈ H
­¡ +
¢∗
®
A (f1 ) g1 ◦ . . . ◦ gn+1 , f2 ◦ . . . ◦ fn+1 ◦
­
®
= g1 ◦ . . . ◦ gn+1 , A+ (f1 ) f2 ◦ . . . ◦ fn+1 ◦
√
n + 1 hg1 ◦ . . . ◦ gn+1 , f1 ◦ f2 ◦ . . . ◦ fn+1 i◦
=
µ
¶2 X n+1
Y­
√
®
1
=
n+1
fσ(i) , gτ (i)
(n + 1)!
σ,τ ∈Sn+1 i=1


hg1 , f1 i
· · · hg1 , fn+1 i
 hg2 , f1 i

√
1
· · · hg2 , fn+1 i

=
n+1
per 


···
··· ···
(n + 1)!
hgn+1 , f1 i · · · hgn+1 , fn+1 i

hg1 , f2 i
· · · hg1 , fn+1 i
 ···
··· ···

n+1
X
 hgi−1 , f2 i · · · hgi−1 , fn+1 i
√
1
hgi , f1 i per 
=
n+1
 hgi+1 , f2 i · · · hgi+1 , fn+1 i
(n + 1)!

i=1
 ···
··· ···
hgn+1 , f2 i · · · hgn+1 , fn+1 i
√
n+1
=








n+1
X
1
n!
hgi , f1 i ×
(n + 1)!
i=1
× hg1 ◦ . . . ◦ gbi ◦ . . . ◦ gn+1 , f2 ◦ . . . ◦ fn+1 i◦
n+1
X
1
√
hgi , f1 i hg1 ◦ . . . ◦ gbi ◦ . . . ◦ gn+1 , f2 ◦ . . . ◦ fn+1 i◦
n + 1 i=1
=
On the other hand
hA (f1 ) g1 ◦ . . . ◦ gn+1 , f2 ◦ . . . ◦ fn+1 i◦
+
*n+1
X
1
= √
hf1 , gi i g1 ◦ . . . ◦ gbi ◦ . . . ◦ gn+1 , f2 ◦ . . . ◦ fn+1
n + 1 i=1
=
1
√
n+1
n+1
X
i=1
hgi , f1 i hg1 ◦ . . . ◦ gbi ◦ . . . ◦ gn+1 , f2 ◦ . . . ◦ fn+1 i◦
◦
18
Chapter 1. Preliminaries
Then 1. follows from the linearity of creation and annihilation operators and
the density of n−particle vectors. In order to prove 2. and 3. we consider an
arbitrary g ∈ H and we have
"
∞
X
1
√ g ◦n
A (f ) E (g) = A (f )
n!
n=0
#
∞
X
1 1
√ √ n hf, gi g ◦(n−1)
=
n! n
n=1
=
∞
X
n=1
p
1
(n − 1)!
hf, gi g ◦(n−1)
∞
X
1
√ g ◦n = hf, gi E (g)
= hf, gi
n!
n=0
Moreover, for any h ∈ H
®
­
E (h) , A+ (f ) E (g) = hA (f ) E (h) , E (g)i
= hh, f i hE (h) , E (g)i
On the other hand
¿
À
d
E (h) , E (g + tf ) |t=0
=
dt
d
hE (h) , E (g + tf )i |t=0
dt
d
= hE (h) , E (g)i hE (h) , E (tf )i |t=0
dt
h
i
= hE (h) , E (g)i hh, f i exphh,tf i |t=0
= hh, f i hE (h) , E (g)i
Then 3. follows from the totality of exponential vectors. In order to prove 4.,
we consider h ∈ H, hence
A (f ) A (g) E (h) − A (g) A (f ) E (h)
= hf, hi hg, hi E (h) − hf, hi hg, hi E (h) = 0
and (1.2.4) is proved as a consequence of the totality of exponential vectors.
19
1.2. The boson, fermion and free Fock spaces
(1.2.5) follows from (1.2.4) when one considers the adjoint operators. Finally
=
=
=
=
A (f ) A+ (g) E (h)
"∞
#
X 1 √
◦n
√
A (f )
n + 1g ◦ h
n!
n=0
√
∞
i
X
1
n+1h
√ √
hf, gi h◦n + n hf, hi g ◦ h◦(n−1)
n! n + 1
n=0
∞
i
X
1 h
√
hf, gi h◦n + n hf, hi g ◦ h◦(n−1)
n!
n=0
∞ r
X
n
hf, gi E (h) + hf, hi
g ◦ h◦(n−1)
(n − 1)!
n=1
On the other hand
Ã
!
∞
X
1
√ h◦n
A+ (g) A (f ) E (h) = hf, hi A+ (g)
n!
n=0
√
∞
X n+1
√
= hf, hi
g ◦ h◦n
n!
n=0
√
∞
X
n
p
g ◦ h◦(n−1)
= hf, hi
(n
−
1)!
n=1
(1.2.6) is another time given by the totality of exponential vectors.
¤
(1.2.4), (1.2.5), (1.2.6) are called canonical commutation (boson) relations.
Definition 1.2.8. For any n ∈ N, for any f ∈ H the linear operator A+ (f ) :
H∧n → H∧(n+1) such that for any f1 , . . . , fn ∈ H
A+ (f ) f1 ∧ . . . ∧ fn :=
√
n + 1f ∧ f1 ∧ . . . ∧ fn , n ≥ 1
and
A+ (f ) Φ = f
20
Chapter 1. Preliminaries
is called the creation operator on n-fold antisymmetric tensor product of H.
Moreover the linear operator A (f ) : H∧n → H∧n−1 such that
n
1 X
A (f ) f1 ∧ . . . ∧ fn := √
(−1)i−1 hf, fi i f1 ∧ . . . ∧ fbi ∧ . . . ∧ fn
n
i=1
A (f ) Φ = 0
is called the annihilation operator on n-fold antisymmetric tensor product of
H.
Remark 1.2.9. The creation and annihilation operators are bounded operators on n-fold antisymmetric tensor product of H.
Creation and annihilation operators in antisymmetric case can be extended
by linearity to the domain of all finite particle vectors in Γa (H) and, by their
boundedness, they can be uniquely extended to bounded operators on Γa (H)
(see [52], Proposition 23.3); they are called fermion creation and fermion
annihilation operator respectively. Moreover one can reassert the density of
the set of finite particle vectors on Γa (H) by the following result (see [52],
Proposition 23.4 for the proof).
Proposition 1.2.10. The set
©
ª
Φ, A+ (f1 ) , . . . A+ (fn ) Φ : fj ∈ H, j = 1, . . . n, n = 1, 2, . . .
is total in Γa (H) .
Let K be an arbitrary Hilbert space and let us consider T, V two operators
acting on such space.The anti-commutator operator of T and V is defined as
follows:
{T, V } := T V + V T
Proposition 1.2.11. For any f ∈ H the following relations hold for fermion
creation and annihilation operators (i.e. for A (f ) and A+ (f ) on Γa (H)):
∗
1. A (f ) = (A+ (f )) ;
21
1.2. The boson, fermion and free Fock spaces
2. for any g ∈ H
{A (f ) , A (g)} = 0
ª
A+ (f ) , A+ (g) = 0
©
ª
A (f ) , A+ (g) = hf, gi 1
©
(1.2.7)
(1.2.8)
(1.2.9)
Proof. 1. can be proved as 1. of Proposition 1.2.7 when one replaces the
symmetric tensor product by the antisymmetric one and the permanent of the
matrix (hgi , fj i)n+1
i,j=1 by the determinant of it. We turn to 2. By Proposition
1.2.10, it is enough to prove it on the n-particle vectors domain. Hence for
any n ∈ N, for any h1 , . . . , hn ∈ H
A+ (f ) A+ (g) Φ = f ∧ g
A+ (g) A+ (f ) Φ = g ∧ f
√
A+ (f ) A+ (g) h1 ∧ . . . ∧ hn =
n + 1A+ (f ) g ∧ h1 ∧ . . . ∧ hn
p
=
(n + 2) (n + 1) (f ∧ g) ∧ h1 ∧ . . . ∧ hn
√
n + 1A+ (g) f ∧ h1 ∧ . . . ∧ hn
p
=
(n + 2) (n + 1) (g ∧ f ) ∧ h1 ∧ . . . ∧ hn
A+ (g) A+ (f ) h1 ∧ . . . ∧ hn =
and (1.2.8) follows from the fact that (f ∧ g) + (g ∧ f ) = 0. By considering the
adjoint operators, one obtains (1.2.7). Finally
A (f ) A+ (g) Φ + A+ (g) A (f ) Φ = hf, gi Φ
A (f ) A+ (g) h1 ∧ . . . ∧ hn
√
=
n + 1A (f ) g ∧ h1 ∧ . . . ∧ hn
= hf, gi h1 ∧ . . . ∧ hn
n
X
+
(−1)i hf, hi i g ∧ h1 ∧ . . . ∧ hbi ∧ . . . ∧ hn
i=1
From another hand
A+ (g) A (f ) h1 ∧ . . . ∧ hn
Ã
!
n
1 X
i−1
+
(−1) hf, hi i h1 ∧ . . . ∧ hbi ∧ . . . ∧ hn
= A (g) √
n
i=1
= −
n
X
i=1
(−1)i hf, hi i g ∧ h1 ∧ . . . ∧ hbi ∧ . . . ∧ hn
22
Chapter 1. Preliminaries
and (1.2.9) is completely proved.
¤
The relations given by (1.2.7), (1.2.8) and (1.2.9) are called canonical anticommutation (fermion) relations.
Definition 1.2.12. For any n ∈ N, for any f ∈ H the linear operator A+ (f ) :
H⊗n → H⊗n+1 such that for any f1 , . . . , fn ∈ H
A+ (f ) f1 ⊗ . . . ⊗ fn := f ⊗ f1 ⊗ . . . ⊗ fn
and
A+ (f ) Φ = f
is called the creation operator on n-fold tensor product of H. Moreover the
linear operator A (f ) : H⊗n → H⊗n−1 such that
A (f ) f1 ⊗ . . . ⊗ fn := hf, f1 i f2 ⊗ . . . ⊗ fn
A (f ) Φ = 0
is called the annihilation operator on n-fold tensor product of H.
Remark 1.2.13. The creation and annihilation operators are bounded operators on n-fold tensor product of H.
As in the antisymmetric case, creation and annihilation operators in tensor
product case can be extended by linearity to the domain of all finite particle
vectors in Γ (H) and, by their boundedness, they can be uniquely extended to
bounded operators on Γ (H) (see [52], Proposition 20.24 and references therein)
and are called free creation and free annihilation operator respectively.
Proposition 1.2.14. For any f ∈ H the following relations hold for free
creation and annihilation operators (i.e. for A (f ) and A+ (f ) on Γ (H)):
∗
1. A (f ) = (A+ (f )) ;
2. for any g ∈ H
A (f ) A+ (g) = hf, gi 1
3.
°
°
kA (f )k = °A+ (f )° = kf k
1.2. The boson, fermion and free Fock spaces
23
Proof. For any n ∈ N, f1 , . . . , fn , g, g1 , . . . , gn ∈ H
­¡ +
¢∗
®
A (f ) g ⊗ g1 ⊗ . . . ⊗ gn , f1 ⊗ . . . ⊗ fn
= hg ⊗ g1 ⊗ . . . ⊗ gn , f ⊗ f1 ⊗ . . . ⊗ fn i
= hg, f i hg1 ⊗ . . . ⊗ gn , f1 ⊗ . . . ⊗ fn i
= hA (f ) g ⊗ g1 ⊗ . . . ⊗ gn , f1 ⊗ . . . ⊗ fn i
­¡ +
¢∗
® ­
®
A (f ) Φ, Φ = Φ, A+ (f ) Φ = 0
then 1. follows from the linearity of the operators and the density of all finite
particle vectors. We turn to prove 2. By definition
A (f ) A+ (g) Φ = hf, gi Φ
and
A (f ) A+ (g) f1 ⊗ . . . ⊗ fn
= A (f ) g ⊗ f1 ⊗ . . . ⊗ fn
= hf, gi f1 ⊗ . . . ⊗ fn
Furthermore, for any ψ ∈ Γ0 (H)
° +
°
­
®
°A (f ) ψ °2 = A+ (f ) ψ, A+ (f ) ψ
­
®
= ψ, A (f ) A+ (f ) ψ = kf k2 kψk2
Then, on Γ0 (H) and, consequently, by density of Γ0 (H) , on the free Fock
space,
° +
°
°A (f )° = kf k
The remaining part of 3. is a direct consequence of 1. and of a well known
property of adjoint operators on Hilbert spaces.
¤
Remark 1.2.15. The relations given at point 2. of the above proposition
are called the free (Cuntz) relations on the ∗−algebra A (Γ (H)) generated by
{A+ (f ) : f ∈ H}.
Remark 1.2.16. Introducing, for any fixed −1 ≤ q ≤ 1, the q-commutation
relations
A (f ) A+ (g) − qA+ (g) A (f ) = hf, gi 1
one recovers the free, boson and fermion commutation relations for q = 0,
q = 1, q = −1 respectively.
Chapter 2
Interacting Fock space
In this chapter we present the general structure of interacting Fock spaces.
This new feature emerged in the context of stochastic limit in quantum electrodynamics (see [9]). In particular the study of an electron coupled with an
electromagnetic field, led Accardi and Lu in [8] to compute the limit distribution of the field operators with respect to the vacuum state of the field and
some states on the system space of the electron. They found that it was possible to express them as the vacuum expectation of creators and annihilators in
an interacting Fock space (IFS). Here we follow more or less the construction
for IFS given in [9], emphasizing that they are more general Hilbert spaces
than the usual Fock spaces. But it is possible to achieve a relation between
such spaces and Hilbert and full Fock modules (see [55] for the definitions), as
presented by Accardi and Skeide in [14], where they show that, given a Fock
module, it is possible to associate an IFS and viceversa. Unfortunately, for
synthesis reasons, we will not treat such a beautiful approach in this chapter.
Roughly speaking an interacting Fock space over a given pre–Hilbert space
H is a space which, as a vector space, coincides with the usual Free Fock space
but in which, for any n, the n–particle space has its own scalar product h·, ·in .
The scalar products on different n–particle spaces are related by only three
conditions:
(i) it is possible to define in the usual way the (free) creation operator associated to any given vector f ;
(ii) linear combinations of the n–particle vectors are dense and the n–particle
spaces are mutually orthogonal;
26
Chapter 2. Interacting Fock space
(iii) the adjoint of the creation operator (annihilation) exists on the n–particle
vectors.
The Fock nature of the space is given by the fact that the annihilator still
kills the vacuum.
For any vector space H denote by L(H) the family of densely defined linear
operators on H and, as usual, an n–particle vector is any element of the tensor
n
product H⊗ of the form f1 ⊗ · · · ⊗ fn for any n ∈ N. From now on we adopt
the usual convention that
0
H⊗ = CΦ
(2.0.1)
where Φ is the vacuum vector as defined in the previous chapter.
2.1
Definition and first results
From now on a linear operator A on a pre–Hilbert space H is said to have an
adjoint if it has one on the Hilbert space completion of H.
Definition 2.1.1. Let H be a vector space. An interacting Fock space over
H is defined by the assignment, for each n ∈ N, of a pre-scalar product
n
(positive definite, possibly degenerate, sesquilinear form) ( · | · )n on H¯ with
the following properties:
i) For each n, the n–particle vectors are dense in each n–particle space for
the topology induced by the pre-scalar product ( · | · )n . Moreover
0
(zΦ|z 0 Φ)0 = zz 0 , H⊗ := CΦ
(2.1.1)
ii) The creation operator acts, on the n–particle vectors, as the usual creator,
i.e. for any f, f1 , . . . , fn ∈ H
³ n+1
´
¢
¡ ⊗n
⊗
A+
(f
)
:
H
,
(
·
|
·
)
→
H
,
(
·
|
·
)
n
n+1
n
such that
A+
0 (f )Φ = f
(2.1.2)
A+
n (f ) (f1 ⊗ ... ⊗ fn ) := f ⊗ f1 ⊗ ... ⊗ fn
(2.1.3)
27
2.1. Definition and first results
iii) On the domain of the n–particle vectors each creation operator A+ (f ) has
an adjoint operator, denoted by A(f ) and called annihilation operator
satisfying the condition
A0 (f )Φ = 0
(2.1.4)
Given the sequence (( · | · )n )n as above, a pre-scalar product on the tensor
algebra
M
n
0
H⊗
;
H⊗ := CΦ
(2.1.5)
n≥0
is uniquely determined by the requirement that the direct sum in (2.1.5) is
orthogonal.
Remark 2.1.2. Condition ii) above, i.e. that the creator is well defined, is
not trivial. It implies that, for the concrete construction of interacting Fock
spaces, one has to verify the implication:
(Fn | Fn )n = 0 =⇒ (f ⊗ Fn | f ⊗ Fn )n+1 = 0
(2.1.6)
n
for any n ∈ N, Fn ∈ H⊗ , f ∈ H.
Remark 2.1.3. When H is a Hilbert space, with scalar product h·, ·i and
for each n, the n–th scalar product h·, ·in coincides with the usual one
hf1 ⊗ · · · ⊗ fn , g1 ⊗ · · · ⊗ gn in =
n
Y
hfh , gh i
h=1
one recovers the free Fock space. The symmetrized and antisymmetrized scalar
products give respectively the Boson and Fermion Fock space.
The following theorem characterizes the interacting Fock spaces by means
of a sesquilinear map over the space H.
n
Theorem 2.1.4. Let be given an interacting Fock space {H⊗ , (·|·)n }n∈N over
a vector space H. Then, for any n ∈ N, there exists a sesquilinear map Tn :
n
n
n
H ×H → L(H⊗ ), where L(H⊗ ) is the set of all the linear operators on H⊗ ,
with the following properties:
28
Chapter 2. Interacting Fock space
i) for any f, g ∈ H, the map Tn (f, g) is well defined on the n–particle vectors
n
on the Hilbert space {H⊗ , (·|·)n } (this means that a zero (·|·)n –norm
vector is mapped into a zero (·|·)n –norm vector) and, for all n ∈ N and
for all f, f1 , . . . , fn , g, g1 , . . . , gn ∈ H, it satisfies the identity
(f ⊗ f1 ⊗ · · · ⊗ fn |g ⊗ g1 ⊗ · · · ⊗ gn )n+1
= (f1 ⊗ · · · ⊗ fn |Tn (f, g)g1 ⊗ · · · ⊗ gn )n
(2.1.7)
ii) The map Tn (·, ·) is (·|·)n –positive, i.e. one has
n
X
(Fi |Tn (fi , fj )Fj )n ≥ 0
(2.1.8)
i,j=1
n
for all n ∈ N, f1 , . . . , fn ∈ H, F1 , . . . , Fn ∈ H⊗ .
Conversely, given a sequence (Tn (·, ·))n of linear maps from H × H to
n
L(H⊗ ), conditions i), ii) ensure the possibility of defining inductively, using
(2.1.1) and (2.1.7), an interacting Fock space over H.
n
Proof. If the sequence {H⊗ , (·|·)n }n is an interacting Fock space over H, for
any n and for any f, g ∈ H, we define
Tn (f, g) := A(f )A+ (g)|H⊗n
(2.1.9)
and, because of the well definiteness of the creation operator, this map is well
n
defined on the n–particle vectors on the Hilbert space {H⊗ , (·|·)n }. Now, for
all n ∈ N and for all f, f1 , . . . , fn , g, g1 , . . . , gn ∈ H, we have
(f ⊗ f1 ⊗ . . . ⊗ fn |g ⊗ g1 ⊗ . . . ⊗ gn )n+1
= (A+ (f ) f1 ⊗ . . . ⊗ fn |A+ (g) g1 ⊗ . . . ⊗ gn )n+1
= (f1 ⊗ . . . ⊗ fn |A (f ) A+ (g) g1 ⊗ . . . ⊗ gn )n
= (f1 ⊗ . . . ⊗ fn |Tn (f, g)g1 ⊗ . . . ⊗ gn )n
and condition i) is verified. Condition ii) follows from (2.1.9) and the positive
n
definiteness of (·|·)n . In fact for all n ∈ N, f1 , . . . , fn ∈ H, F1 , . . . , Fn ∈ H⊗
n
X
(Fi |Tn (fi , fj )Fj )n =
i,j=1
n
X
(fi ⊗ Fi |fj ⊗ Fj )n+1
i,j=1
29
2.2. Standard interacting Fock spaces and module gaussianity
Conversely, given a sequence (Tn (·, ·)) as in the statement of the lemma and
defining inductively, using (2.1.1) and (2.1.7), the sesquilinear maps (· | ·)n ,
condition ii) implies that these sesquilinear maps are in fact pre–scalar prodn
ucts. Moreover for any Gn ∈ H⊗ and for any f, f1 , . . . , fn+1 ∈ H we define
(A+ (f )Gn |f1 ⊗ . . . ⊗ fn+1 )n+1 := (Gn |Tn (f, f1 )f1 ⊗ · · · ⊗ fn+1 )n
also obtaining the following inequality
| (A+ (f )Gn |f1 ⊗ · · · ⊗ fn+1 )n+1 |≤k Gn kn · k Tn (f, f1 )f2 ⊗ · · · ⊗ fn+1 ) kn
from which we deduce that:
i) A+ (f ) is defined on the dense n-particle vectors space and consequently
has an adjoint;
ii) whenever an n–particle vector Gn has zero (·|·)n –norm, then A+ (f )Gn
is equal to zero and its adjoint satisfies condition iii) of Definition 2.1.1.
¤
Theorem 2.1.4 shows that the creation and annihilation operators on an
interacting Fock space satisfy the commutation relation
A(f )A+ (g) (f2 ⊗ · · · ⊗ fn+1 ) = Tn (f, g)f2 ⊗ · · · ⊗ fn+1
n
(2.1.10)
n
for each n ∈ N. Introducing the number operator N : H⊗ → H⊗ for any
n ∈ N such that
N (f1 ⊗ f2 ⊗ · · · ⊗ fn ) = n(f1 ⊗ f2 ⊗ · · · ⊗ fn )
(2.1.11)
for any f1 , . . . , fn ∈ H, one can write (2.1.9) as the purely algebraic relation
A(f )A+ (g) = TN (f, g)
(2.1.12)
which is an operator generalization of the free commutation relations.
Remark 2.1.5. The above construction is functorial with the operator valued
inner product TN (f, g) replacing the usual scalar valued inner product.
2.2
Standard interacting Fock spaces and module
gaussianity
In this section we study a very important and useful class of interacting Fock
spaces over an Hilbert space H with scalar product h·, ·i.
30
Chapter 2. Interacting Fock space
n
Looking H as a linear space, we consider the algebraic tensor product H¯
for each n ∈ N. By introducing the usual tensor product Hilbert structure as
n
in Section 1.2, we have the usual tensor product Hilbert space H⊗ and the
tensor algebra
∞
M
n
Γ(H) :=
H⊗
n=0
is nothing else but the usual free Fock space over H.
n
Now let be given, for each n ∈ N, a positive self–adjoint operator on H⊗ ,
say Wn , and we define the pre-scalar product:
n
(Fn |Gn )n := hFn , Wn Gn i ; Fn , Gn ∈ H¯
(2.2.1)
¡ ¯n
¢
We denote by Hn := H , (·|·)n /(·|·)n the pre–Hilbert space obtained by
n
H¯ after the quotienting and call the (pre–)Hilbert space
ΓI (H) :=
∞
M
Hn
n=0
the interacting free Fock space over H determined by the sequence {Wn }n . As
usual, the 0–th space H0 is defined as the complex field, the vector
Φ := 1 ⊕ 0 ⊕ 0 ⊕ · · ·
is called the vacuum vector and for any f ∈ H we denote by Af the adjoint
+
of A+
f in the interacting Fock space. We want to recall that the action of Af
is the same in the free and interacting Fock spaces.
Definition 2.2.1. An interacting Fock space over an Hilbert space H, satisfying, for any n ∈ N, (2.2.1) and the following condition: for each n ∈ N,
k ≤ n, f1 , . . . , fn ∈ H, the operators
+
Afn−k · · · · · Afn A+
fn · · · · · Afn−k
(2.2.2)
are self–adjoint on Hn , shall be called standard .
Let H := L2 (X, X , µ), where X is a measurable space and µ is a σ–finite
measure on X and let be given a sequence of measurable, positive functions
{λn (xn , · · · , x2 , x1 )}∞
n=1
λn : (X n , X n ) 7−→ R+
(2.2.3)
31
2.2. Standard interacting Fock spaces and module gaussianity
with the property that for any measurable function Fn : (X n , X n ) → C if
Z
|Fn (xn , . . . , x1 )|2 λn (xn , . . . , x1 ) µ (dxn ) · · · µ (dx1 ) = 0
then for any measurable function f : (X, X ) → C,
Z
|f (x)|2 |Fn (xn , . . . , x1 )|2 λn+1 (x, xn , . . . , x1 ) µ (dx) µ (dxn ) · · · µ (dx1 ) = 0
(2.2.4)
Xn
We define, for each n ∈ N the measure µn on
by
Z
µn (E) :=
λn (xn , . . . , x1 ) µ (dxn ) · · · µ (dx1 )
E
for any E ∈ X n . On the space of linear combinations of the functions f0 ⊗
fn ⊗ · · · ⊗ f1 which are square integrable for the measure µn+1 , we define, for
each n, the pre-scalar product
hf0 ⊗ fn ⊗ · · · ⊗ f1 , g0 ⊗ gn ⊗ · · · ⊗ g1 in+1 :=
Z
µ(dx0 ) . . . µ(dx1 )λn+1 (x0 , . . . , x1 )f¯0 (x0 ) · · · f¯1 (x1 )g0 (x0 ) · · · g1 (x1 )
:=
(2.2.5)
Condition (2.2.4) implies that the creator is well defined and one has
hf0 ⊗ fn ⊗ · · · ⊗ f1 , A+
g0 gn ⊗ · · · ⊗ g1 in+1
Z
=
λn+1 (x0 , . . . , x1 )f¯0 (x0 ) . . . f¯1 (x1 )g0 (x0 ) . . . g1 (x1 )µ(dx0 ) . . . µ(dx1 )
(2.2.6)
Moreover condition (2.2.4) implies also that, if λn (xn , . . . , x1 ) = 0, then the
function x0 7−→ λn+1 (x0 , , xn . . . , x1 ) is equal to zero µ−a.s. Hence we can
divide λn+1 by λn almost everywhere and the expression (2.2.6) becomes
Z
µ(dxn ) · · · µ(dx1 )λn (xn , . . . , x1 )f¯n (xn ) · · · f¯1 (x1 )gn (xn ) · · · g1 (x1 )×
Z
×
µ (dx0 )
λn+1 (x0 , . . . , x1 ) ¯
f0 (x0 )g0 (x0 )
λn (xn , . . . , x1 )
From one hand (2.2.6) is nothing else than
hAg0 f0 ⊗ fn ⊗ · · · ⊗ f1 , gn ⊗ · · · ⊗ g1 in
(2.2.7)
32
Chapter 2. Interacting Fock space
From another hand (2.2.7) can be written as
µZ
¶
λn+1 (x0 , . . . , x1 )
g (x0 )f0 (x0 ) fn ⊗ · · · ⊗ f1 , gn ⊗ · · · ⊗ g1 in
h
µ (dx0 )
λn (xn , . . . , x1 ) 0
so the action of the annihilator Ag0 could be explicitly expressed by the following formula:
Ag0 f0 ⊗ · · · ⊗ f1 (xn , . . . , x1 )
·Z
¸
λn+1 (x0 , . . . , x1 )
=
µ(dx0 )
ḡ0 (x0 )f0 (x0 ) fn (xn ) · · · f1 (x1 )
(2.2.8)
λn (xn , . . . , x1 )
We define
Hn := L2 (X n , µn ) ∀n ≥ 2
H0 := C;
H1 := H
and notice that, for any n ∈ N, the n-particle vectors are dense in Hn . Hence,
for any n ∈ N, the creation operator with test function f ∈ H is such that
A+
f : Hn → Hn+1 and Af Φ = 0. As a consequence, by Definition 2.1.1 and
Definition 2.2.1, the sequence of pre-scalar products h·, ·in defines, by quotienting and completing, an interacting Fock space of standard type. It will be
called the interacting Fock space over H = L2 (X, dµ) with interacting functions (λn )n and denoted by
ΓI (H) := ΓI (L2 (X, dµ), (λn )n )
(2.2.9)
Remark 2.2.2. With the choice λn = 1 for any n = 1, 2, . . . , one recovers the
usual Boltzmannian Fock space over H.
The following Lemma, contained in [9], gives us some conditions for the
boundedness of the creation operator in the interacting Fock space above introduced.
Lemma 2.2.3. Let us suppose that there exists a sequence of functions
(λn (xn , · · · , x1 )) with λn : X n −→ R+ and two sequences of positive numbers
{bn , dn }∞
n=1 , such that, for any n ∈ N,
0 < λn ≤ bn ,
λn+1 (xn+1 , xn , · · · , x2 , x1 )
≤ dn
λ1 (xn+1 ) · λn (xn , · · · , x1 )
(2.2.10)
33
2.2. Standard interacting Fock spaces and module gaussianity
Then for any g ∈ H and for any n ∈ N, the creation operator A+ (g) is a
bounded operator from Hn into Hn+1 . Moreover
√
||A+ (g)||n→n+1 ≤ dn · ||g||1
(2.2.11)
and in this case the operator Tn (f, g) introduced by Lemma 2.1.4 has the form
Z
λn+1 (x, xn , · · · , x1 )
Tn (f, g) :=
µ(dx)λ1 (x)f (x)g(x) ·
(2.2.12)
λ1 (x)λn (xn , · · · , x1 )
X
Proof. By definition, for any n ∈ N, g1 , . . . , gn ∈ H
° +
°
°A (g) [gn ⊗ · · · ⊗ g1 ]°2 = ||g ⊗ gn ⊗ · · · ⊗ g1 ||2
= hg ⊗ gn ⊗ · · · ⊗ g1 , g ⊗ gn ⊗ · · · ⊗ g1 i
Z
=
X n+1
µ(dx)µ(dxn ) · · · µ(dx1 )λn+1 (x, xn , · · · , x1 )|g(x)|2 ·
n
Y
|gh |2 (xh )
h=1
2
= ||gn ⊗ · · · ⊗ g1 || ×
·Z
λn+1 (x, xn , · · · , x1 )
µ(dx)λ1 (x)|g(x)| ·
λ1 (x)λn (xn , · · · , x1 )
¸
2
×
(2.2.13)
By assumption (2.2.10), the quantity in square brackets on the right hand side
of (2.2.13) is less than or equal to dn · ||g||21 . Moreover the right hand side of
(2.2.13) and (2.2.6) also give the explicit form of the operator Tn (f, g). In fact
for any f, f1 , . . . , fn , g, g1 , . . . , gn ∈ H
hf ⊗ fn ⊗ · · · ⊗ f1 , g ⊗ gn ⊗ · · · ⊗ g1 i
Z
=
Xn
µ(dxn ) · · · µ(dx1 )λn (xn , · · · , x1 )
·Z
(2.2.14)
n
Y
¡
¢
f h gh (xh )×
h=1
¸
¡ ¢
λn+1 (x, xn , · · · , x1 )
×
µ(dx)λ1 (x) f g (x) ·
λ1 (x)λn (xn , · · · , x1 )
X
¸
À
¿
·Z
¡ ¢
λn+1 (x, xn , · · · , x1 )
gn ⊗ · · · ⊗ g1
= fn ⊗ · · · ⊗ f1 ,
µ(dx)λ1 (x) f g (x) ·
λ1 (x)λn (xn , · · · , x1 )
X
But, on the other hand, by (2.1.9), (2.2.14) is equal to
hfn ⊗ · · · ⊗ f1 , Tn (f, g) g ⊗ gn ⊗ · · · ⊗ g1 i
¤
As a direct consequence of the above Lemma, we want to present the
following corollary, whose proof is clear.
34
Chapter 2. Interacting Fock space
Corollary 2.2.4. In the assumptions of Lemma 2.2.3, for any g ∈ H:
i) the creation operator A+ (g) is a densely defined operator on ΓI (H) and
the set of linear combinations of finite particle vectors
)
(N
X
Γ0 :=
cn Gn : N ∈ N, Gn ∈ Hn , cn ∈ C, n = 0, 1, 2, · · ·
n=0
(2.2.15)
is included in the domain D(A+ (g)) of A+ (g).
ii) the annihilation operator A(g) is bounded from Hn to Hn−1 ; densely defined on ΓI (H) and Γ0 ⊂ D(A(g)).
√
iii) If sup dn < ∞, then the creation operator A+ (g) is bounded on ΓI (H)
n
and
√
||A+ (g)|| ≤ sup dn · ||g||
n
(2.2.16)
Motivated by concrete examples of interacting Fock spaces (see for example
[43], [45], [46]), from now on we shall assume that the boundedness condition
(2.2.11) is satisfied. Thus any annihilator Ag is bounded on each n–particle
space and its action on the n–particle space is given by
(2.2.17)
Ag Gn+1 (xn+1 , . . . , x1 )
Z
λn+1 (xn+1 , . . . , x1 )
ḡ(xn+1 )Gn+1 (xn+1 , . . . , xn )
=
µ(dxn+1 )
λn (xn , . . . , x1 )
where xn+1 , . . . , x1 ∈ X, Gn+1 ∈ Hn+1 for any n. In the following, we consider
the creation and annihilation operators only on Γ0 , i.e. we do not distinguish
the operators themselves from their restriction on the domain Γ0 where they
are adjoint each other (according to Definition 2.2.1).
After defining the interacting Fock space and some of the basic objects related to it, like the vacuum vector and the creation and annihilation operators,
we begin to investigate the joint distribution of such operators with respect to
the ”vacuum” state, i.e. the state given by hΦ, ·Φi . This is due to the fact that
important and deep problems concerning interacting Fock spaces (e.g. finding
the distribution of symmetric field operator Q (f ) = A (f ) + A+ (f ) for any
f ∈ H), can be reduced to the explicit computation of the joint correlators of
35
2.2. Standard interacting Fock spaces and module gaussianity
the creation and annihilation operators with respect to the vacuum state, i.e.
of expressions of the form
D
E
Φ, Aε(n) (fn ) · · · Aε(1) (f1 )Φ
(2.2.18)
where n ∈ N, ε ∈ {0, 1}n , f1 , · · · , fn ∈ H and, for ε ∈ {0, 1}, we adopt the
convention
½
A, if ε = 0
ε
A :=
(2.2.19)
A+ if ε = 1
Since A(f )Φ = 0, it is clear that the quantity (2.2.18) differs from zero only if
ε(n) = 0,
ε(1) = 1
(2.2.20)
Every fixed sequence ε = (ε(1), . . . , ε(n)) ∈ {0, 1}n with the property (2.2.20)
determines, in a unique way, a number 1 ≤ m ≤ n/2 and a sequence 1 ≤ k1 <
k2 < · · · < k2m−1 < n, such that
1 = ε(k2j + 1) = ε(k2j + 2) = · · · = ε(k2j+1 )
(creators)
0 = ε(k2j+1 + 1) = ε(k2j+1 + 2) = · · · = ε(k2j+2 )
(2.2.21)
(annihilators) (2.2.22)
for each j = 0, 1, · · · , m−1, where we have introduced the convention k2m := n
and k0 := 0. In these notations we rewrite the correlator (2.2.18) as
*
Φ,
kY
2m
h=k2m−1 +1
k2m−1
A(gh ) ·
Y
h=k2m−2 +1
A+ (gh ) · · ·
k2
Y
A(gh ) ·
h=k1 +1
+
k1
Y
A+ (gh )Φ
h=k0 +1
(2.2.23)
h=1 ah shall
Qr
where hereinafter in this chapter a product of the form
always be interpreted as the ordered product a1 · ... · ar .
The following lemma makes explicit the action of a finite family of annihilators on any n−particle vector.
Lemma 2.2.5. For any m, n ∈ N, f1 , · · · , fm ∈ H and Gn ∈ Hn , the vector
A(fm ) · · · A(f1 ) Gn ∈ Hn−m
is equal to zero if m > n; to
hf1 ⊗ · · · ⊗ fn , Gn i
(2.2.24)
36
Chapter 2. Interacting Fock space
if m = n and, if m < n, to
A(fm ) · · · A(f1 ) Gn (xn−m , . . . , x1 )
Z
=
m
Y
λn (xn , . . . , x1 )
µ(dxn−m ) · · · µ(dx1 )
f¯h (xh ) · Gn (xn , . . . , x1 )
λn−m (xn−m , . . . , x1 )
h=1
(2.2.25)
Proof. From (2.2.17) we know that
Z
λn (xn , . . . , x1 ) ¯
A(f1 )Gn (xn−1 , . . . , x1 ) = µ(dxn )
f1 (xn )Gn (xn , . . . , x1 )
λn−1 (xn−1 , . . . , x1 )
Therefore
Z
=
A(f2 )A(f1 )Gn (xn−2 , . . . , x1 ) = [A(f2 ) [A(f1 )Gn ]] (xn−2 , . . . , x1 ) =
Z
λn−1 (xn−1 , . . . , x1 ) ¯
f2 (xn−1 )×
= µ(dxn−1 )
λn−2 (xn−2 , . . . , x1 )
Z
λn (xn , . . . , x1 ) ¯
f1 (xn ) · Gn (xn , xn−1 , . . . , x1 )
× µ(dxn )
λn−1 (xn−1 , . . . , x1 )
µ(dxn )µ(dxn−1 )
λn (xn , . . . , x1 ) ¯
f1 (xn )f¯2 (xn−1 )Gn (xn , xn−1 , . . . , x1 )
λn−2 (xn−2 , . . . , x1 )
The proof follows by induction using the fact that the action of an annihilator
to the vacuum vector gives zero.
¤
One notices that the correlator (2.2.23) differs from zero only if the number of annihilators equals the number of creators and, if one fix an arbitrary
operator of the sequence, the number of annihilators on its right can not be
larger than the number of creators, as the following lemma suggests.
Lemma 2.2.6. The correlator (2.2.23) differs from zero only if the following
two conditions hold:
i) for any j = 1, . . . , 2n,
|{ε (k) = 1 : k = 1, . . . , j}| ≥ |{ε (k) = 0 : k = 1, . . . , j}|
(2.2.26)
ii) for j = 2n the equality in (2.2.26) holds, equivalently
|{ε (k) = 1 : k = 1, . . . , 2n}| = |{ε (k) = 1 : k = 1, . . . , 2n}|
(2.2.27)
37
2.2. Standard interacting Fock spaces and module gaussianity
Proof. The right hand side of (2.2.26) is nothing else than the total number
of annihilators up to the j–th step and the left hand side is the total number of
creators at the same step. If for some j = 1, 2, · · · , 2n, condition (2.2.26) is not
true, then at some step less than or equal to j, one shall have an annihilator
acting on a multiple of the vacuum, which gives zero. Moreover, if the total
number of creators is strictly larger than the total number of annihilators,
then the vector
kY
2m
h=k2m−1 +1
k2m−1
A(gh ) ·
Y
+
A (gh ) · · ·
h=k2m−2 +1
k2
Y
k1
Y
A(gh ) ·
h=k1 +1
A+ (gh )Φ (2.2.28)
h=k0 +1
belongs to some p–particle space for some p > 1. Therefore it shall be orthogonal to the vacuum.
¤
Lemma 2.2.6 shows that the only non zero correlators of the form (2.2.23)
are those for which conditions (2.2.26) and (2.2.27) are satisfied.
As a consequence the number of operators in the sequence (2.2.18) must
be even and we shall use the notations:
2n
{0, 1}2n
: (2.2.26) and (2.2.27) hold }
+ := {ε ∈ {0, 1}
2n
{0, 1}2n
\ {0, 1}2n
− := {0, 1}
+
(2.2.29)
(2.2.30)
For each ε ∈ {0, 1}2n , there are a natural integer m ≤ 2n and an ordered set
1 ≤ l1 < l2 < · · · lm ≤ 2n, such that
{l1 , . . . , lm } = {k : ε(k) = 0}
(2.2.31)
Clearly both m and the ordered set {l1 , . . . , lm } are uniquely determined by ε
and that, conversely, the ordered set {l1 , . . . , lm } uniquely determines ε. Conditions (2.2.20), (2.2.26) and (2.2.27) can be restated, in terms of {l1 , . . . , lm },
m
as follows: ε ∈ {0, 1}2n
+ if and only if the set {lh }h=1 is such that:
m = n,
lm = 2n,
l1 > 1
(2.2.32)
and, for any k = 1, 2, . . . , 2n
|{1, 2, . . . , k} ∩ {lh }nh=1 }| ≤
≤ |{1, 2, . . . , k} ∩ ({1, 2, · · · , 2n} \ {lh }nh=1 )|
(2.2.33)
38
Chapter 2. Interacting Fock space
In the following any subset {l1 , . . . , lm } of the set {1, 2, · · · , 2n}, satisfying
conditions (2.2.32) and (2.2.33) shall be called a left–index set (or annihilator–
index set) and the set
{r1 , . . . , rn } := {1, . . . , 2n} \ {l1 , . . . , lm }
(2.2.34)
shall be called the associated right–index set (or creator–index set). By construction any ε ∈ {0, 1}2n
+ uniquely determines its left and right index sets and
conversely and such a pair of sets uniquely determines a ε ∈ {0, 1}2n
+ .
Let us now recall some basic properties about non crossing pair partitions
of the set {1, 2, · · · , 2n}.
Definition 2.2.7. A partition of {1, 2, · · · , 2n} of the form {lh , rh } (h =
1, . . . , n) is called a pair partition. We shall always assume that the pairs
{lh , rh }nh=1 are ordered so that
lh > rh , ∀h = 1, 2, · · · , n
(2.2.35)
A pair partition {(lh , rh ) : h = 1, . . . , n} is called non–crossing if, for each
pair of indices k, h, if lh > rk , then also rh > rk .
For a given ε ∈ {0, 1}2n
+ we can consider all the pair partitions
{l1 , l10 , . . . , ln , ln0 } whose left–index set {l1 , . . . , ln } coincides with the left–index
set of ε, defined by (2.2.31). Clearly the set
{l10 , . . . , ln0 } := {1, 2, · · · , 2n} \ {l1 , . . . , ln }
must satisfy the following conditions:
(i) for any h = 1, 2, · · · , n, lh0 < lh ;
(ii) for any h, k = 1, 2, · · · , n, lh0 differs from lk0 . If n > 1, for each fixed
left–index set {l1 , . . . , ln } (or equivalently, for each fixed ε ∈ {0, 1}2n
+ ),
there are many partitions and their number depends on the left–index
set. But among them only one is a non-crossing pair partition. More
precisely the following result can be stated:
2.2. Standard interacting Fock spaces and module gaussianity
39
Lemma 2.2.8. For each fixed ε ∈ {0, 1}2n
+ (or equivalently, for each fixed
left–index set {l1 , . . . , ln }) there is exactly one non–crossing pair partition
{(lh , rh ) : h = 1, . . . , n}. Moreover the indices rh are determined by the
following rule: for any p = 0, 1, · · · , n − 1
rp = max {k ∈ {1, 2, . . . , lp − 1} \ {l1 , · · · , lp−1 , r1 , . . . , rp−1 }}
(2.2.36)
Proof. The statement is a consequence of the following two observations:
i) {(lh , rh ) : h = 1, . . . , n} is a non–crossing pair partition only if r1 =
l1 − 1.
ii) {(lh , rh ) : h = 2, . . . , n} is a non–crossing pair partition of {1, 2, · · · , 2n}\
{l1 , r1 } if and only if {(lh , rh ) : h = 1, . . . , n} is a non–crossing pair partition
of {1, 2, · · · , 2n}.
¤
Given a left–index set {lh }nh=1 , the set {rh }nh=1 such that {lh , rh }nh=1 is the
unique non–crossing pair partition determined by the lh ’s, shall be called the
right–index set of {lh }nh=1 . Notice that, while the lh are in increasing order,
the associated rh in general are not ordered. In the above notations, the main
result of the present section can be stated as follows:
Theorem 2.2.9. The correlator (2.2.18) is non zero only if n is an even
number and ε ∈ {0, 1}n+ . In this case, replacing n by 2n one has
D
E
Φ, Aε(2n) (f2n ) · · · Aε(1) (f1 )Φ
Z
=
µ(dyn ) · · · µ(dy1 )
n
Y
(f lh (yh )frh (yh ))·K({lh , rh }nh=1 ; yn , . . . , y1 ) (2.2.37)
h=1
{lh , rh }nh=1
where
is the left–right–index set of ε and K({lh , rh }nh=1 ; y1 , . . . , yn )
is a function of n variables uniquely determined by ε and by the first n interacting functions λ1 , λ2 , . . . , λn through the following rule: defining on the
space X 2 = X × X the σ–finite measure
µ2 (dx, dy) := µ(dx)µ(dy) δ(x − y)
where δ is the Dirac measure, the integral (2.2.37) is equal to
Z
Z
n
Y
µ2 (dxln , dxrn ) · · ·
µ2 (dxl1 , dxr1 )
(f lh (yh )frh (yh ))
X2
X2
h=1
40
Chapter 2. Interacting Fock space
¡
¢
λl(h+1) −2h−1 {xlh −1 , . . . , x1 } \ {xlp , xrp }hp=0
³
´
h
λ
{x
,
·
·
·
,
x
}
\
({x
,
x
}
∪
{x
})
1
rp p=0
rh
lh −1
lp
h=0 l(h+1) −2h−2
n−1
Y
(2.2.38)
where hereinafter for any N ∈ N, m ≤ N , any set {ph }m
h=1 ⊂ {1, 2, · · · , N }
with cardinality m and any function F of N − m variables, we adopt the
notation:
F ({x1 , x2 , · · · , xN } \ {xph }m
c
d
p1 , · · · , x
pm , · · · , xN )
h=1 ) := F (x1 , · · · , x
Proof. We refer to [44] for a direct proof.
¤
Remark 2.2.10. Formula (2.2.38) is useful to get an intuitive understanding
of the new features introduced by the interacting Fock space with respect to
the usual Fock space. These new features manifest themselves in a deviation
from Gaussianity in the sense specified below.
It is well known that the powers of vacuum expectation of operators given
by addiction of creators and annihilators in Fock spaces (free, fermion, boson,
interacting,...) uniquely determine a distribution. More precisely the moments
of the field operator (symmetric position operator)
Q (f ) := A (f ) + A+ (f )
with respect to the vacuum state are the same of a distribution. This result
is given by von Neumann’s spectral theorem (see [52], Th. 12.4 and [63]) by
which self-adjoint operators on Hilbert spaces are put in one-to-one correspondence to spectral measures on the real line. Hence Q (f ) and, more generally
the set {A (f ) , A+ (f )} , where f ∈ H, is called a quantum random variable.
Roughly speaking we call gaussian any field operator whose moments,
with respect to the vacuum, are determined by the moments of second order,
in analogy to what happens in the classical case. More precisely, following [10],
we say that gaussianity in the non-commutative sense consists in the fact that
expectation values of products of random variables (correlators), are expressed
as weighted sums, over a certain subset of pair partitions, of the products of
pair correlation functions over all the pairs of a single partition. By varying
the weights and the subsets of pair partitions, one obtains the various notion
of gaussianity. The following definition sums up all these informations.
41
2.2. Standard interacting Fock spaces and module gaussianity
Definition 2.2.11. The set {A (f ) , A+ (f )} is called Gaussian with respect
to the vacuum state if for each n ∈ N, for each f1 , . . . , fn ∈ H and each
ε ∈ {0, 1}n one has:
D
E
Φ, Aε(n) (fn ) · · · Aε(1) (f1 )Φ = 0 if n is odd
(2.2.39)
D
E
Φ, Aε(2p) (f2p ) · · · Aε(1) (f1 )Φ
=
X
{lh ,rh }ph=1 ∈P.P.(2p)
p
Y
¡
­
®
p ¢
² {lh , rh }h=1
Φ, A(flh )A+ (frh )Φ if n (2.2.40)
= 2p
h=1
where P.P. (2p) denotes the set of all pair partitions
of {1,
¡
¢ 2, · · · , 2p}, the sum
is extended to all such pair partitions and ² {lh , rh }ph=1 is a complex number
depending on the partition {lh , rh }ph=1 .
If one considers the usual (boson, fermion, free, q–deformed (see [23]) for
details) Fock spaces over a given one–particle Hilbert space H, the vacuum
expectations of products of the form
Aε(n) (gn ) · · · Aε(1) (g1 )
(2.2.41)
with n ∈ N, g1 , · · · , gn ∈ H and ε ∈ {0, 1}n , vanish when n is odd or when
n = 2N for some N ∈ N but ε ∈ {0, 1}2N
− and, when n is even (we write 2n)
n
0 n
and ε ∈ {0, 1}2n
with
left–index
set
{l
}
h h=1 and right–index set {lh }h=1 , they
+
have the form
n D
E
X
Y
f ({lh , lh0 }nh=1 )
Φ, A(glh )A+ (glh0 )Φ
(2.2.42)
0 }n
{lh ,lh
h=1 ∈P.P.(2n)
h=1
The combinatorial factor f ({lh , lh0 }nh=1 ) depends on the structure of the n–
particle spaces and in fact mainly on the type of product chosen to define this
space (usual tensor product in the boson case, Z2 –graded tensor product in
the fermion case, Schürmann tensor product in the q–deformed case (see [53]),
free product in the boltzmannian case, . . .). More precisely:
(i) f ({lh , lh0 }nh=1 ) = 1 in the boson case;
(ii) f ({lh , lh0 }nh=1 ) = (−1)ε(σ) in the fermion case, where ε (σ) is the signature
of the permutation
σ : {1, 2, 3, 4, · · · , 2n − 1, 2n} −→ {l1 , l10 , l2 , l20 , · · · , ln , ln0 }
42
Chapter 2. Interacting Fock space
(iii) in the free Fock case
½
f ({lh , lh0 }nh=1 )
=
1, if {lh , lh0 }nh=1 ∈ N.C.P.P. (2n)
0, otherwise
where and hereinafter, by N.C.P.P. (2n) we denote the totality of all
non–crossing pair partitions of {1, 2, · · · , 2n}.
(iv) a power of q in the q–deformed case.
By introducing the physical terminology of calling any correlator made by
n elements a n point function, we can say that in all cases above any 2n − 1
correlator is equal to zero and any 2n points function equals a sum of products
of two points functions weighted with a combinatorial factor which depends
only on the pair partition. This property is the manifestation of the fact that
in all these cases we are dealing with gaussian operators, in the sense given by
Definition 2.2.11.
On the contrary, in the interacting Free Fock space, formula (2.2.38) shows
that the even correlators of creation and annihilation operators involves a
weight factor which depends not only on the pair partition but also on the
space–time (or momentum–time) variables (i.e. the xj –variables in (2.2.38)).
This property is characteristic of the interaction. Formula (2.2.38) also shows
that some characteristics of gaussianity still survive in the interacting case,
namely:
i) the odd correlators are still equal to zero
ii) If we consider an inner product with values in the algebra generated
by the multiplication operators by functions of the variables xj (module (or
operator) inner product), as shown in [9], Section 17, and then an integral
over these variables, then one can see that, with respect to this module inner
product, the combinatorial weight factor again depends only on the pair partition. In this sense we can claim that the generalization of gaussianity that
arises in the interacting Fock space can be considered as a module (or operator
deformed) gaussianity.
That this notion of gaussianity cannot be reduced to the usual one is
evident already at the level of the four point function (i.e. n = 2 in formula
(2.2.38)), in fact in this case the expectation value
­
®
Φ, A(g4 )A(g3 )A+ (g2 )A+ (g1 )Φ
43
2.3. The 1-mode interacting Fock space
is equal to
Z
µ(dy)µ(dx) (ḡ3 · g2 ) (x) (ḡ4 · g1 ) (y)λ2 (y, x)
and which in general cannot be represented by a form like
Z
X
¡
¢
¡
¢
c(σ) µ(dy)µ(dx) ḡσ(3) · gσ(2) (x) ḡσ(4) · gσ(1) (y)
σ∈S4
where, as usual, Sn denotes the n−permutation group.
This shows that the notion of module Gaussianity is essentially different
from usual gaussianity. From an algebraic point of view, the emergence of
module gaussianity is due to the fact that relations (2.1.12) reply the usual
free commutation relations.
2.3
The 1-mode interacting Fock space
In order to illustrate examples and applications of interacting Fock space, let
us consider the case in which K is just the set of complex numbers which,
in physical language, corresponds to a 1–particle space in zero space–time
n
dimensions. In this case, for each n ∈ N also K⊗ is 1–dimensional, so we
identify it to the multiple of a vector denoted by a+n Φ, where Φ is the vacuum
vector. The pre-scalar products (·|·)n can only have the form
(z|w)n := λn (z, w) z̄w
;
z, w ∈ C
(2.3.1)
where the λn are strictly positive numbers.
The 1-mode interacting Fock space ΓI (C, {λn }) is given by taking quotient
and completing the orthogonal sum
∞
M
¢
¡ ⊗n
K , (·|·)n
n=0
with the convention that λ0 = λ1 = 1. Moreover we require the sequence
{λn }n satisfies the condition
λn = 0 =⇒ λm = 0 ∀m ≥ n
(2.3.2)
Under the assumption (2.3.2), the creation operator
a+ : a+n Φ 7→ a+(n+1) Φ
(2.3.3)
44
Chapter 2. Interacting Fock space
n
where for any n ∈ N a+n Φ represents an element of K⊗ , is well defined.
Moreover if we use the convention that 0/0 = 0, the annihilator a is given by:
a : a+(n+1) Φ 7→
Hence for any n ∈ N
λn+1 +n
a Φ
λn
(2.3.4)
¡
¢ λn+1 +n
aa+ a+n Φ =
a Φ
λn
¡
¢
λn +n
a+ a a+n Φ =
a Φ
λn−1
and the following commutation relations arise:
aa+ =
λN +1
,
λN
a+ a =
λN
λN −1
(2.3.5)
where N is the number operator and the right hand side of (2.3.5) is uniquely
determined by the spectral theorem. These relations, together with the condition
aΦ = 0
characterize the interacting Fock space structure.
Remark 2.3.1. The 1-mode interacting Fock space is finite dimensional if
and only if λn = 0 for some n ∈ N.
Remark 2.3.2. The commutation relations (2.3.5) can be regarded as generalizations of the usual commutation relations for usual Fock spaces over the
complex field C. In fact
i) canonical commutation (boson) relation
aa+ − a+ a = 1
is obtained by taking λn = n! in (2.3.5);
ii) canonical anticommutation (fermion) relation
aa+ + a+ a = 1
is obtained by taking λ0 = λ1 = 1 and λn = 0 for any n ≥ 2 in (2.3.5);
iii) free commutation relation
aa+ = 1
45
2.4. 1-mode IFS and orthogonal polynomials
is obtained by taking λn = 1 for any n in (2.3.5);
iv) if one considers the q−deformed Fock space as in [23], the CCR, CAR
and free commutation relations are special cases of q−commutation relations
aa+ − qa+ a = 1,
for − 1 ≤ q ≤ 1
(2.3.6)
Yet in this case the relation (2.3.6) above can be obtained by (2.3.5) by taking
λn = [n]q ! := [n]q [n − 1]q · · · [1]q
where [n]q := 1 + q + q 2 + . . . + q n−1 for any n ∈ N.
2.4
1-mode IFS and orthogonal polynomials
One of the main interests driven by the introduction of 1-mode interacting
Fock space is given by its relation with the system of orthogonal polynomials
of a given probability measure on the real line. The results we shall present
give an outline of this intimate connection. We follow [2] and [33] and we
remind to these papers for more details.
Theorem 2.4.1. Let µ be a probability measure on R with finite
any order. There exist two sequences of real numbers (αn )n , (ωn )n
for any n such that, denoting (Pn )n the sequence of orthogonal
asoociated to µ, normalized so that P−1 = 0, P0 = 1, P1 (x) =
following relations hold for any n ∈ N
xPn (x) = Pn+1 (x) + αn+1 Pn (x) + ωn Pn−1 (x)
Z
n
Pn (x) Pm (x) dµ (x) = δm
ω1 · · · ωn
moments of
with ωn ≥ 0
polynomials
x − α1 , the
(2.4.1)
(2.4.2)
n is the Kronecker symbol. Moreover µ is symmetric if and only if
where δm
αn = 0 for each n and µ is supported in a finite number of points if and only
if ωn = 0 for some n ∈ N.
Remark 2.4.2. The above sequences (αn )n and (ωn )n are called the Jacobi
parameters of the measure µ.
n
Let us consider the number operator over the complex field K N : K⊗ →
n
n
K , then the operator αN : K⊗ → K⊗ such that
⊗n
αN (xn ⊗ . . . ⊗ x1 ) := αn (xn ⊗ . . . ⊗ x1 )
(2.4.3)
46
Chapter 2. Interacting Fock space
for any x1 , . . . , xn ∈ K, is uniquely determined by the spectral theorem. Now
we present a theorem by Accardi and Bożejko, which states, under some hypothesis, that, up to unitary isomorphism, 1-mode interacting Fock spaces and
L2 (R, µ) (with µ as in the previous theorem) are equivalent.
Theorem 2.4.3. (Accardi-Bożejko) Let µ be a probability measure on R
with finite moments of any order, let (Pn )n be the sequence of its orthogonal
polynomials in L2 (R, µ) with Jacobi parameters (αn )n , (ωn )n . Then there
exists a unique 1-mode interacting Fock space ΓI (C, {λn }) and an isometry
U : ΓI (C, {λn }) → L2 (R, µ)
such that
U Φ = P0
+
U a U ∗ Pn = Pn+1
¡
¢
U a + a+ + αN U ∗ = Q
where Q is the multiplication operator by x densely defined in L2 (R, µ) and
αN is defined in (2.4.3).
Proof. See [2], Theorem. 5.2. In this setting we only want to underline the
relationship between the families (ωn ) and (λn ), which, for each n ∈ N, is
given by
λn
ωn =
λn−1
¤
At this step a natural question arises: is the operator U of the above
theorem unitary? An affirmative answer is given whenever the polynomials
span a dense subset in L2 (R, µ). This circumstance is verified if the moment
problem for the measure µ is determined (see [29]). A sufficient condition
P∞ tn zfor
n
the determinancy is the analiticy in a neighbour of 0 of the function n=0 n! ,
where {tn } is the sequence of the moments of µ. Hence we give the following
corollary.
Corollary 2.4.4. Under the same hypothesis of Theorem 2.4.3, let
P∞{tn }tn zbe
n
the sequence of the moments of the measure µ. If the function
n=0 n!
is analytic in a neighbour of 0, the operator U above introduced is unitary.
Furthermore, given a 1-mode interacting Fock space, the vacuum distribution
µ of any self-adjoint extension of a + a+ + αN is such that the interacting Fock
space associated to µ by the operator U is the original one.
47
2.4. 1-mode IFS and orthogonal polynomials
Remark 2.4.5. As a consequence of Theorem 2.4.3, one has that for any
m∈N
Z
­ ¡
¢m ®
xm dµ = Φ, a + a+ + αN Φ Γ (C,{λn })
I
R
In fact for any m ∈ N
­ ¡
¢m ®
Φ, a + a+ + αN Φ Γ
I (C,{λn })
­
¡
¢m ®
= U Φ, U a + a+ + αN Φ L2 (R,µ)
®
­
¡
¢m
= U Φ, U a + a+ + αN U ∗ P0 L2 (R,µ)
Z
m
= hP0 , Q P0 iL2 (R,µ) =
xm dµ
R
The above remark suggests us the idea to construct the so called quantum
decomposition of a classical random variable. In fact, let X be a classical
random variable with distribution µ having moments of any order. Define an
algebraic probability space (A, φ) by taking A the ∗−algebra generated by Q
and
φ (a) := hP0 , aP0 iL2 (R,µ) , a ∈ A
Then X and Q are stochastically equivalent:
Z
m
E (X ) =
xm dµ = φ (Qm ) ,
m = 0, 1, 2, . . .
(2.4.4)
R
On the other hand, let B the ∗−algebra generated by {a, a+ , αN } in ΓI (C, {λn })
and we consider a state ψ on B given by
ψ (b) := hΦ, bΦiΓI (C,{λn }) ,
b∈B
By Remark 2.4.5 and (2.4.4) we have that X and a+a+ +αN are stochastically
equivalent. In this sense we write
X = a + a + + αN
which is an example of quantum decomposition of a classical random variable. It is worth to mention that now the quantum components are not
commuting each other, i.e. the algebraic probability space (B, ψ) is noncommutative, whereas (A, φ) is commutative.
48
Chapter 2. Interacting Fock space
The result by Accardi and Bożejko has been extended to several dimensions
in [13], by the help of multi-mode interacting Fock spaces.
A very deep and interesting application of such results has been recently
given by Accardi, Kuo and Stan in [11] and [12].
They have introduced three types of operators on the L2 −space of a probability measure µ on Rd , d ≥ 1, namely
© 0 ª
© + ª
© − ª
a (i) 1≤i≤d ,
a (i) 1≤i≤d ,
a (i) 1≤i≤d
respectively called preservation, creation and annihilation operators and showed
that for any vector-valued random variable X = (X1 , . . . Xd ) , d ≥ 1, one has
the following multi-dimensional quantum decomposition
Xj = a+ (j) + a− (j) + a0 (j) ,
j = 1, . . . , d
This fact suggests other problems to analize:
1) is it possible to code the whole informations of a probability measure
µ into a set of relations on creation, annihilation and preservation operators
canonically associated (in the sense above explained) to it?
2) Let be given V the vector space of all polynomials on Rd and let ai , aj,k
be linear operators on V , for 1 ≤ i, j, k ≤ d , what are the conditions on these
operators such that there exists a probability measure µ on Rd satisfying
£
¤
ai = a0 (i) , aj,k = a− (j) , a+ (k) , 1 ≤ i, j, k ≤ d
The work is still in progress. In fact it has been possible to characterize some
properties of µ in terms of creation, preservation and annihilation operators,
whereas the second problem has been solved in one-dimensional case.
Moreover, by means of recursion formula (2.4.1) and by the knowledge of
orthogonal polynomials related to a given probability measure µ on the real
line, it is easy to compute the Jacobi parameters of µ. By combining such a
result with Theorem 2.4.3 it is possible to know the λn ’s appearing on the 1mode interacting Fock space canonically associated to µ. As a consequence the
Jacobi parameters can be used to know explicitly the creation, annihilation
and preservation operators canonically associated to µ. All these informations
are summed up into the following tables, for the most useful distributions,
which are given for any n ≥ 0. They appear in [11], and we present them in
the following pages almost without any change.
49
2.4. 1-mode IFS and orthogonal polynomials
−
a0µ (Pn ) := αn Pn , a+
µ (Pn ) := Pn+1 , aµ (Pn ) := ωn Pn−1
Measure
(2.4.5)
Polynomial Pn
Hermite
Hn (x; σ 2 )
2
2
2
2
= (−σ 2 )n ex /2σ ∂xn e−x /2σ
Gaussian
N (0, σ 2 )
Charlier
Cn (x; a) =
Poisson
Poi(a)
(−1)n a−x Γ(x + 1)∆n
x+
h
ax
Γ(x−n+1)
i
∆x+ f (x) = f (x + 1) − f (x)
Gamma Γ(α),
1
xα−1 e−x ,
Γ(α)
Laguerre
(α−1)
(x)
n
= (−1)n x−α+1 ex ∂xn [xn+α−1 e−x ]
(α > 0)
x>0
Legendre
e n (x) = n 1
L
∂ n [(x2 − 1)n ]
2 (2n−1)!! x
Uniform on [−1, 1]
Arcsine
√1
,
π
1−x2
Chebyshev (1st kind)
Te0 (x) = 1
1
Ten (x) = 2n−1
cos(n cos−1 x), n ≥ 1
|x| < 1
Semicircle
√
2
1 − x2 ,
π
Γ(β+1)
√1
(1
π Γ(β+ 1 )
2
Chebyshev (2nd kind)
cos−1 x]
en (x) = 1n sin[(n+1) −1
U
2
sin(cos
x)
|x| < 1
2 β− 1
2
−x )
,
|x| < 1
β > − 12 ,
β 6= 0, 1
Negative binomial
r > 0, 0 < p < ¡1 ¢
P (X = x) = pr −r
(−1)x (1 − p)x ,
x
x ∈ ∪{0}
Gegenbauer
(β)
2 1 −β
e (β)
G
u(x),
n = Cn (1 − x ) 2
1
u(x) = ∂xn [(1 − x2 )n+β− 2 ]
n n
(β)
2 Γ(2β+n)
Cn = (−1)Γ(2β+2n)
Meixner
(r,p)
Mn (x) = (−1)n p1n
u(x) =
h
(1 − p)−x ∆n
x+
Γ(x+1)
u(x),
Γ(x+r)
Γ(x+r)
(1
Γ(x−n+1)
− p)x
i
,
50
Chapter 2. Interacting Fock space
Measure
Szegö-Jacobi parameters
Gaussian
N (0, σ 2 )
αn
ωn
(λn
=
=
=
0
σ 2 (n + 1)
σ 2n n!)
Poisson
Poi(a)
αn
ωn
(λn
=
=
=
n+a
a(n + 1)
an n!)
αn
ωn
(λn
=
=
=
2n + α
(n + 1)(n + α)
n!(n + α − 1) · · · α)
αn
ωn
=
=
0
(λn
=
αn
=
ωn
=
(λn
=
αn
ωn
(λn
=
=
=
Gamma Γ(α),
1
xα−1 e−x ,
Γ(α)
(α > 0)
x>0
Uniform on [−1, 1]
Arcsine
√1
,
π
1−x2
|x| < 1
Semicircle
√
2
1 − x2 ,
π
Γ(β+1)
√1
(1
π Γ(β+ 1 )
2
|x| < 1
1
− x2 )β− 2 ,
|x| < 1
− 12 ,
β 6= 0, 1
β>
Negative binomial
r > 0, 0 < p < ¡1 ¢
P (X = x) = pr −r
(−1)x (1 − p)x ,
x
x ∈ N ∪ {0}
αn
ωn
αn
ωn
(λn
=
=
=
=
=
(n+1)2
(2n+3)(2n+1)
(n!)2
)
[(2n−1)!!]2 (2n+1)
0
½
1
,
2
1
,
4
1
22n−1
n=0
n≥1
)
0
1
4
1
)
4n
0
(n+1)(n+2β)
4(n+β+1)(n+β)
(2−p)n+r(1−p)
p
(n+1)(n+r)(1−p)
p2
n!(1−p)n (n+r−1)···r
)
p2n
51
2.4. 1-mode IFS and orthogonal polynomials
+
[a−
µ , aµ ]Pn
Measure
a0µ Pn
Gaussian
N (0, σ 2 )
σ 2 Pn
0
Poisson
Poi(a)
aPn
(a + n) Pn
Gamma Γ(α),
1
xα−1 e−x ,
Γ(α)
(α > 0)
x>0
(2n + α)Pn
1
− (2n+3)(2n+1)(2n−1)
Pn
Uniform on [−1, 1]
Arcsine
√1
,
π
1−x2


|x| < 1
Semicircle
√
2
1 − x2 ,
π
Γ(β+1)
√1
(1
π Γ(β+ 1 )
2

½
1
P ,
2 0
− 14 P1 ,
0,
1
P ,
4 0
0,
|x| < 1
1
− x2 )β− 2 ,
|x| < 1
− 12 ,
β 6= 0, 1
β>
(α + 2n) Pn
Negative binomial
r > 0, 0 < p < ¡1 ¢
(−1)x (1 − p)x ,
P (X = x) = pr −r
x
x ∈ N ∪ {0}
n=0
n=1
n≥2
n=0
n≥1
β 2 −β
P
2(n+1+β)(n+β)(n−1+β) n
(2n+r)(1−p)
Pn
p2
0
0
0
0
(2−p)n+r(1−p)
Pn
p
Chapter 3
Universal Central Limit
Theorem on interacting Fock
space
The present chapter is based on the paper [3] and is devoted to present a
general central limit theorem in non-commutative case. One knows that in
classical probability one essential notion underlying all the types of central
limit theorems is independence, so anyone should suppose that the same take
place in the quantum case too. But when one leaves the commutative frame,
such a notion looses its determination and unicity and, as a consequence, after
stating ”what” is independence in quantum probability, it is possible to introduce various, inequivalent notions of it. Roughly speaking, ”independence”
in non-commutative contest, can be seen as a property of algebraic random
variables that allows to compute the mixed moments of them with respect
to a given state. Generally, given an algebraic probability space (A, φ) and a
family of ∗-subalgebras (Ai )i∈I of A, where I is an arbitrary index set, we say
that (Ai )i∈I satisfies a certain condition of independence with respect to φ if
the φ-state of any finite sequence of elements belonging to the family (Ai )i∈I
can be factorized into a product of states of the type
¢
¡
φi ai1 · · · aini
where aij ∈ Ai , φi = φ ¹Ai for any i ∈ I, for any ni ∈ N, for any j = 1, . . . ni .
On the other hand, the functoriality of the construction of the IFS suggested, since the beginning of the theory, that it might be a natural tool to
construct examples of inequivalent notions of stochastic independence.
54
Chapter 3. Universal Central Limit Theorem on IFS
The proliferation of notions of stochastic independence has motivated the
development of different points of view concerning these notions. In particular
we mention:
(i) the axiomatic point of view, based on various notions of coproduct and
developed by Schürmann [54], Speicher [59], Ben Ghorbal [18], Muraki
[50], according to which there exist exactly five different types of independence;
(ii) the reductionistic point of view, developed by Lenczewski, which reduces
all notions of independence to tensor independence [40], see also [32];
(iii) the individualistic point of view, which concentrates on a specific notion
of independence and extensively develops the corresponding probability theory in analogy with the classical one. This has been followed
by Voiculescu [62], Bożejko and Speicher [24], Speicher [58], Bercovici
[20], for free probability; by Lu [44, 46] and especially Muraki [49], for
monotone probability; by Speicher and Woroudi [61], Ben Ghorbal and
Schürmann [19] for boolean probability;
(iv) the constructive approach, which emphasizes concrete and explicit representations of the random variables involved. Examples of this approach can be found in the paper [22] by Bozejko, Kümmerer, Speicher
on q–Gaussian processes and in several other papers of the Polish QP–
school; in the papers by Accardi, Hashimoto, Obata [6], Hashimoto [34],
Hashimoto, Hora and Obata [33], Lu [46], which use the IFS as a basic
tool.
From the Lenczewski approach [40] it begun to emerge the idea that the
various notions of quantum independence are in fact masked forms of classical
dependencies. This point of view received two independent confirmations: one
in the result, proved by Cabanal Duvillard and Ionescu [26], that any symmetric probability measure on R, with moments of any order can be obtained
as the central limit distribution, in the sense of convergence in moments of
a sequence of symmetrically distributed algebraic random variables satisfying
a notion of stochastic independence. Another one was obtained by Accardi
and Bożejko [2] who introduced a universal convolution and proved that every symmetric probability law on R with moments of any order is infinitely
divisible with respect to this convolution. Then, from a certain viewpoint, the
result in [26] proves that really happens what, according to [2] can happen.
55
The result of [2] was based on interacting Fock space techniques while
the result of Cabanal–Duvillard and Ionescu exploited the combinatorial techniques of [31] combined with an extension of the notion of (ϕ, ψ)–independence,
due to Bożejko and Speicher [24]. This construction was then generalized by
Mlotkowski [48]. Between the results obtained in [2, 26, 48] and the central
limit theorem proved in this chapter, there are three main differences:
1. we prove that any mean–zero distribution with moments of all orders,
not only the symmetric ones, can be obtained as central limits, in the
sense of convergence of moments, of self-adjoint random variables;
2. we explicitly realize both the approximating random variables and their
limits as sums of creation, annihilation and number operators in suitable
IFS;
3. with respect to the reference state, the random variables considered here
do not satisfy, in the case of non symmetric distribution, the symmetrically distributed condition, used in [26].
The present approach also suggests a general notion of independence, naturally abstracting the properties of the special class of interacting Fock spaces
which is used here (cf. Section 3.3), and different from the notion of weak
independence used by Cabanal–Duvillard and Ionescu. This development is
now still in progress.
It was known from [2], Theorem 5.1, that the mixed moments of any probability measure with moments of any order can be expressed in terms of singletons and (non crossing) pair partitions (Gaussianization) and that the symmetric case is characterized by the absence of singletons. The functoriality of
the IFS construction provides us a natural and easy way to extend here this
construction from single random variables to processes.
A nontrivial difference between the symmetric and the non symmetric central limit theorem is that we are unable to prove the latter with identically
distributed random variables. We are able to reduce this non homogeneity to
a simple multiplication by a constant (cf. Section 3.4 formula (3.4.4)), but not
to eliminate it completely. It is not known whether this is a limitation of our
method or the manifestation of an intrinsic difference between the two cases.
The analogy with the classical case and the recent paper [38] suggests the
conjecture that the latter hypothesis is true. In fact Krystek and Wojakowski
in [38] showed that the convolution arising from the addition of field operators
in suitable IFS is exactly the universal one by Accardi and Bożejko. By means
56
Chapter 3. Universal Central Limit Theorem on IFS
of this result they proved the central limit theorem of [3] in a different way,
where yet the same multiplication by a constant is necessary.
This chapter is organized as follows in Section 3.1 we introduce a basic
operator on standard interacting free Fock space, namely the preservation
operator, in addition to creation and annihilation operators already presented
in Chapter 2. In Section 3.2 one gives a formula for the moments of a quantum
stochastic process of the form of A(f ) + A+ (f ) + Λα . In Section 3.3, we
introduce the notion of “1–mode–type IFS” (1–MT–IFS) and show that on
any 1–MT-IFS, the vacuum distribution of A(f ) + A+ (f ) + Λα depends only
on the module of f but not on f itself. This is the main difference between
IFS and 1–MT-IFS, for more details see [46]. Section 3.4 is devoted to the
proof of our main result i.e. Theorem 3.4.5. Firstly we prove an estimate on
the mixed moments (Lemma 3.4.4) which allows us to apply Lemma 2.4 of
[6]. This, combined with a quantum extension of the moment formula proved
in Theorem 5.1 of [2], allows to conclude the proof of our Theorem 3.4.5 in
the symmetric case. Up to this point we work with identically distributed
random variables. Then we extend the proof to the non symmetric case trying
to emphasize the steps of the proof that prevent us from using only identically
distributed random variables (cf. formula (3.4.12)). Corollary 3.4.6 specializes
the above result to the associated classical process and corresponds to the non
symmetric extension of the Cabanal–Duvillard and Ionescu result.
3.1
Preservation operator on standard IFS and universal convolution
In this and following sections we give definitions and some properties of the
standard IFS, that are necessary to formulate the main results of this chapter.
Definition 3.1.1. Let (X, X , µ) be a measure space and let
ΓI (H, {λn }n ) :=
∞
M
Hn
(3.1.1)
n=0
be the standard interacting Fock space over the Hilbert space H = L2 (X, µ),
as presented in Chapter 2. If the {λn }∞
n=1 are constant, such space is called a
1–mode type free interacting Fock space (1–MT-IFS in short).
57
3.1. Preservation operator on standard IFS and universal convolution
Definition 3.1.2. For any X ∈ B (H) , for any α = {αn }∞
n=0 ⊂ R with
α0 := 0, for any n ∈ N and for any f1 , . . . , fn ∈ H, one defines
Λα (X) (f1 ⊗ f2 ⊗ · · · ⊗ fn ) := αn (Xf1 ) ⊗ f2 ⊗ · · · ⊗ fn
the preservation operator with intensity ({αn }∞
n=0 , X).
For simplicity we shall adopt throughout this paper the following conventions:
• if X is the identity operator, Λα (X)will be denoted simply
Λα := Λα (1)
(3.1.2)
Λα (f ) := Λα (Mf )
(3.1.3)
• for any f ∈ H we write
where Mf is the multiplication by f , i.e. Mf g := f g, for any g ∈ H.
¡ ¢
For any ε ∈ {−1, 0, 1} , f ∈ H α ∈ `∞ RN we will denote
 +
 A (f ) , if ε = 1
ε
ε
Λα (f ) , if ε = 0
A (f ) = Af :=

A (f ) , if ε = −1
(3.1.4)
The properties of standard IFS imply that, for any n ∈ N, ε (1) , . . . , ε (n) ∈
{−1, 1}, f1 , . . . , fn ∈ H, for any sequence (αn ) with α0 = 0, and for any
X ∈ B (H), the following identity holds (in the sense that both sides are well
defined and the identity holds):
Λα (X) Aε(n) (fn ) · · · Aε(1) (f1 ) Φ = αm Aε(n) (gn ) · · · Aε(1) (g1 ) Φ
(3.1.5)
where
• among the vectors g1 , . . . , gn , there is exactly one index i such that
gi = Xfi ,
gj = fj ∀j 6= i
• also the index m is uniquely determined by the family {ε (1) , . . . , ε (n)}.
58
Chapter 3. Universal Central Limit Theorem on IFS
As a consequence any product of creation, annihilation and number operators applied to the vacuum can be always reduced to a multiple of a product
of only creation and annihilation operators applied to the vacuum.
The Φ–statistics of the operator stochastic process {A (f ) , A+ (g) , Λα (X) :
f, g ∈ H, X ∈ B (H)} is coded into its mixed moments
D
E
D
E
Aε(n) (fn ) · · · Aε(1) (f1 ) := Φ, Aε(n) (fn ) · · · Aε(1) (f1 ) Φ
through the algebraic rules (3.1.5) and:
­
®
A (f ) Φ = 0, A (f ) A+ (g) = hf, gi
(3.1.6)
In the introduction to this chapter we mentioned the universal convolution
given by Accardi and Bożejko in [2] and in particular we stood out the property
that any symmetric probability measure on the real line is infinitely divisible
with respect to this convolution. As a consequence any such measure can be
the central limit of a sequence of algebraic random variables. Here we give its
definition and some properties.
Let µ1 , µ2 be probability measures on R with finite moments³of any
´ order.
(k)
For any k = 1, 2 and n = 0, 1, 2, . . . we consider the sequence Pn
of the
orthogonal polynomials associated to µk , as introduced in Section 2.4 and, for
(k)
(k)
each n ∈ N, let αn ∈ R, ωn ∈ R+ be the corresponding Jacobi parameters.
We know that
(k)
(k)
(k)
xPn(k) (x) = Pn+1 (x) + αn+1 Pn(k) (x) + ωn(k) Pn−1 (x)
n³
´ ³
´o
(k)
(k)
and, viceversa, each sequence
αn , ωn
gives a unique state on the
n
∗-algebra C [x] of polynomials in one real variable.
Definition 3.1.3. (Accardi-Bożejko) The universal convolution of µ1 and
µ2 is the unique state µ = µ1 × µ2 on C [x] whose Jacobi parameters αn (µ) ,
ωn (µ) for any n ∈ N are characterized by
αn (µ) = αn(1) + αn(2)
ωn (µ) = ωn(1) + ωn(2)
(3.1.7)
Theorem 3.1.4. The universal convolution has the following properties:
i) associativity;
ii) commutativity;
3.1. Preservation operator on standard IFS and universal convolution
59
iii) positivity, i.e. µ1 × µ2 is a positive
¡ −1 ¢functional on C [x];
iv) for a dilation Dλ µ (E) := µ λ E , E any Borel subset of R, λ > 0,
one has:
Dλ (µ1 × µ2 ) = Dλ (µ1 ) × Dλ (µ2 )
v) for each symmetric measure µ and N = 1, 2, . . .
µ
¶
µ
¶
Sn (µ) := D √1 µ × . . . × D √1 µ = µ
N
N
|
{z
}
N −times
if δx denotes the Dirac measure centered in x ∈ R, then (δx ) × µ = (δx ) ∗ µ,
where ∗ denotes the classical convolution.
Proof. i) and ii) clearly follows from (3.1.7), in fact, for any µ1 , µ2 , µ3 probability measures on the real line with finite moments of any order, the identity
(µ1 × µ2 ) × µ3 = µ1 × (µ2 × µ3 )
is determined by the fact that for any n ∈ N
³
´
³
´
αn(1) + αn(2) + αn(3) = αn(1) + αn(2) + αn(3)
and
³
´
³
´
ωn(1) + ωn(2) + ωn(3) = ωn(1) + ωn(2) + ωn(3)
whereas the identity
µ1 × µ2 = µ2 × µ1
is determined by the fact that for any n ∈ N
αn(1) + αn(2) = αn(2) + αn(1)
and
ωn(1) + ωn(2) = ωn(2) + ωn(1)
iii) is verified because any state on a ∗-algebra is a positive map by definition.
Let us turn to prove iv). In fact, from (2.4.2), we obtain, for any n ∈ N,
Z
Pn2 (x) dµ (x) = ω1 ω2 · · · ωn
(3.1.8)
Then we want to prove the following relations on µ = µ1 × µ2 :
ωn (Dλ µ) = λ2 ωn (µ)
60
Chapter 3. Universal Central Limit Theorem on IFS
αn (Dλ µ) = λαn (µ)
(3.1.9)
for any n ∈ N, for any λ > 0. We use an induction procedure on n.
If n = 1
Z
Z
2
2
2
λ ω1 (µ) = λ
(x − α1 ) dµ (x) = (λx − λα1 )2 dµ (x)
If we put x := λx, then dµ (x) = d (Dλ µ) (x) and the above expression becomes
Z
(x − λα1 )2 d (Dλ µ) (x) = ω1 (Dλ µ)
when α1 (Dλ µ) = λα1 (µ) . Now let us suppose the relations (3.1.9) are true
for any k ≤ n and prove that
ωn+1 (Dλ µ) = λ2 ωn+1 (µ) , αn+1 (Dλ µ) = λαn+1 (µ)
In fact, from (3.1.8), (2.4.1) and the orthogonality of the system (Pn ) ,
Z
Z
2
ωn+1 (µ) Pn2 (x) dµ (x) = Pn+1
(x) dµ (x)
Z
=
Z
2
(x − αn+1 )
Pn2 (x) dµ (x)
−
ωn2 (µ)
then
Z
2
Pn−1
(x) dµ (x)
Z
2
(x − αn+1 )
Hence
Pn2 (x) dµ (x)
= [ωn+1 (µ) − ωn (µ)]
Pn2 (x) dµ (x)
(3.1.10)
Z ³
´2
[ωn+1 (Dλ µ) − ωn (Dλ µ)]
Pn(λ) (x) d (Dλ µ) (x)
Z
³
´2
= (x − αn+1 (Dλ µ))2 Pn(λ) (x) d (Dλ µ) (x)
³
´
(λ)
where Pn
denotes the orthogonal polynomials sequence related to the
measure Dλ µ. By the hypothesis of induction, and putting λx := x in the
integral on the right side above, the last equality is equivalent to
Z
£
¤ ³ (λ) ´2
2
ωn+1 (Dλ µ) − λ ωn (µ)
Pn
(x) d (Dλ µ) (x)
¶
Z µ
αn+1 (Dλ µ) 2 2
Pn (λx) dµ (x)
=λ
x−
λ
2
61
3.1. Preservation operator on standard IFS and universal convolution
From (3.1.10) we obtain:
£
¤
ωn+1 (Dλ µ) − λ2 ωn (µ)
Z ³
´2
Pn(λ) (x) d (Dλ µ) (x)
Z
2
= λ [ωn+1 (µ) − ωn (µ)] Pn2 (λx) dµ (x)
Z ³
´2
= λ2 [ωn+1 (µ) − ωn (µ)]
Pn(λ) (x) d (Dλ µ) (x)
where the last equality is derived by putting x := λx. From this it follows that
ωn+1 (Dλ µ) = λ2 ωn+1 (µ) , αn+1 (Dλ µ) = λαn+1 (µ)
As a consequence, for any n ∈ N
ωn (Dλ (µ1 × µ2 )) = λ2 (ωn (µ1 ) + ωn (µ2 ))
= ωn (Dλ µ1 ) + ωn (Dλ µ2 ) = ωn (Dλ µ1 × Dλ µ2 )
Analogously
αn (Dλ (µ1 × µ2 )) = αn (Dλ µ1 × Dλ µ2 )
and iv) is proved. Finally we prove v). By definition we know that for any
n ∈ N, N ≥ 1, µ symmetric measure
µµ
¶
µ
¶¶
µ
¶
ωn (Sn (µ)) = ωn
D √1 µ × . . . × D √1 µ
= N ωn D √1 µ
N
N
N
hence, by (3.1.9)
ωn (Sn (µ)) = ωn (µ)
and, since αn (µ) = 0 for any n ∈ N by Theorem 2.4.1, we have Sn (µ) = µ
because the Jacobi parameters uniquely determine a measure.
¤
The extension of the Cabanal-Duvillard and Ionescu theorem, mentioned
in the introduction, can be formulated as follows (cf. Corollary (3.4.6)): for
any mean–zero probability measure µ on R with moments of any order, there
exists an IFS ΓI (H, {λn }n ) and a family of operator random variables {Qk }∞
k=1
such that, for any m ∈ N
*Ã
!m + Z
N
1 X
√
Qk
= xm dµ
lim
N →∞
N k=1
62
Chapter 3. Universal Central Limit Theorem on IFS
Moreover, both the family {λn }n and the construction of {Qk }∞
k=1 are determined by the Jacobi parameters of the measure µ. This result will be obtained
as a corollary of a more general quantum central limit theorem (cf. Theorem
(3.4.5)).
3.2
Moments of creators, annihilators and preservation operators on IFS
In this section we present some properties of joint expectation of the three
basic operators on interacting Fock spaces which are useful tools for our future
discussion on 1-mode type IFS and central limit theorem.
Lemma 3.2.1. For any n ∈ N and ε belonging to the set
{−1, 0, 1}n := {ε = (ε (n) , · · · , ε (1)) : ε (i) ∈ {−1, 0, 1} , ∀i = 1, . . . , n }
©
ª
i) if among Aε(n) (fn ) , · · · , Aε(1) (f1 ) there are the same number of annihilators and creators, then there exists a constant c such that
Aε(n) (fn ) · · · Aε(1) (f1 ) Φ = cΦ
(3.2.1)
©
ª
ii) if among Aε(n) (fn ) , · · · , Aε(1) (f1 ) there are more annihilators than
creators, then
Aε(n) (fn ) · · · Aε(1) (f1 ) Φ = 0
iii) if the cardinality of the set {i : ε (i) = ±1} is odd , then
Aε(n) (fn ) Aε(n−1) (fn−1 ) · · · Aε(1) (f1 ) Φ = 0
iv) if either ε (1) ∈ {0, −1} or ε (n) ∈ {1, 0} , the scalar product
D
E
Aε(n) (fn ) Aε(n−1) (fn−1 ) · · · Aε(1) (f1 )
is equal to zero.
3.2. Moments of creators, annihilators and preservation operators on IFS
63
Proof. Indeed, from the properties of preservation operator, we know that
any sequence of creators, annihilation and preservation operators applied to
the vacuum vector, can be reduced to a sequence of creation and annihilation
operators. Hence Lemma 2.2.5 and Lemma 2.2.6 imply i)-iii). Moreover, by
(3.1.6), we know that, if ε (1) = −1, the scalar product
D
E
Aε(n) (fn ) Aε(n−1) (fn−1 ) · · · Aε(1) (f1 )
is equal to zero. The same is verified whenever ε (1) = −0, as a consequence
of Definition 3.1.2. A similar argument can be used for ε (n) after passing to
the adjoint operator. Hence iv) follows.
¤
Definition 3.2.2. For n ∈ N and ε = (ε (n) , · · · , ε (1)) ∈ {−1, 0, 1}n we
define the depth function (of the string ε) dε : {1, ..., n} → {0, ±1, ..., ±n} by
dε (j) =
j
X
ε (k)
(3.2.2)
k=1
= |{ε (k) : ε (k) = 1; k < j}| − |{ε (k) : ε (k) = −1; k < j}|
Thus dε (j) gives the relative number of creators (annihilators, if negative)
in the product Aε(n) · · · Aε(1) which are on the right side of Aε(j) or, equivalently, the number of pairs which contain j in their “interior”.
Definition 3.2.3. {−1, 0, 1}n+ is defined as the totality of all {−1, 0, 1}n satisfying the following conditions:
i)
Pn
k=1 ε (k)
= dε (n) = 0;
ii) ε (1) = 1 and ε (n) = −1;
iii) for all i = 1, . . . , n, dε (i) ≥ 0.
Lemma 3.2.4. For any n ∈ N, {fk }nk=1 ⊂ H and ε ∈ {−1, 0, 1}n , the scalar
product
D
E
Aε(n) (fn ) Aε(n−1) (fn−1 ) · · · Aε(1) (f1 )
(3.2.3)
can be nonzero only if ε ∈ {−1, 0, 1}n+ .
64
Chapter 3. Universal Central Limit Theorem on IFS
Proof. We must check that, if conditions i)-iii) of Definition 3.2.3 do not
hold, the scalar product (3.2.3) is equal to zero. In fact, if i) is not satisfied,
then there is a different number of creators and annihilators in the sequence
(3.2.3). In particular, if the number of annihilators is greater than the number
of creators, then, by Lemma 3.2.1, condition ii), (3.2.3) is equal to zero. The
same occurs in the other case, after passing to the adjoint operators. Also if
condition ii) is not verified, Lemma 3.2.1 implies that (3.2.3) vanishes. Finally,
if there exist an index i = 1, . . . , n such that dε (i) < 0, then, because of the
reduction to creation and annihilation operators, from Lemma 2.2.6, condition
i), the scalar product is equal to zero.
¤
Remark 3.2.5. If we restrict our consideration only to creation and annihilation operators, the analogue of the set {−1, 0, 1}n+ , is denoted by {−1, 1}n+ .
Condition i) of Definition 3.2.3, which can be realized only if n is even,
means that within the set
n
o
Aε(n) (fn ) , Aε(n−1) (fn−1 ) , . . . , Aε(1) (f1 )
the number of creators is equal to ©the number of annihilators. Condition
ª
iii) means that for any i, in the set Aε(i) (fi ) , Aε(i−1) (fi−1 ) , · · · , Aε(1) (f1 )
the number of creators is bigger or equal to the number of annihilators. As
a consequence, ε (1) must be equal to 1. Moreover, since one must verify
contemporarily both
|{h : h ≤ n − 1, ε (h) = 1}| ≥ |{h : h ≤ n − 1, ε (h) = −1}|
and
n
X
ε (k) = 0,
k=1
ε (n) has to be equal to −1.
When one considers not only creation and annihilation, but also number
operators, then the cardinality of number operators is arbitrary, so that n is
not necessarily even.
Definition 3.2.6. For any pair partition {(ip , jp )}np=1 of the set {1, 2, · · · , 2n} ,
©¡
¢ªn
aip , ajp p=1 will be called a pair partition of the set {a1 , a2 , · · · , a2n }. More©¡
¢ªn
over aip , ajp p=1 is called non-crossing if {(ip , jp )}np=1 is non-crossing.
65
3.2. Moments of creators, annihilators and preservation operators on IFS
We shall ©¡
adopt the¢ªconvention that, for any pair partition (non-crossing
n
or crossing) aip , ajp p=1 , we have in < · · · < i1 and ip > jp for all p =
1, 2, · · · , n.
We have shown that for any ε ∈ {−1, 0, 1}n+ , the cardinality of the set
{i : ε (i) = ±1} must be even and we shall denote this set by {i1 , i2 , · · · , i2N }
with the order i2N < · · · < i1 . Define
ε0 (h) := ε (ih ) , ∀h = 1, 2, · · · , 2N
0
then ε0 ∈ {−1, 1}2N
+ and, as proved in [7], ε determines a unique non-crossing
pair partition of the set {1, 2, · · · , 2N } , hence a non-crossing pair partition of
{i1 , i2 , · · · , i2N }. In the following this pair partition will be called the noncrossing pair partition determined by ε. It is clear that any ε ∈ {−1, 0, 1}n+
determines exactly one ε0 and therefore exactly one pair partition.
The following lemma allows us to write the mixed moments of non symmetric position operator A(f ) + A+ (f ) + Λα as sums of scalar products of
creation - annihilation operators weighted by some numbers depending on the
cardinality and position of preservation operators.
Lemma 3.2.7. In the notation (3.1.4), one has:
D
E
X
¢m ®
­ ¡
Φ, A(f ) + A+ (f ) + Λα Φ =
Φ, Aε(m) (f ) · · · Aε(1) (f )Φ
ε∈{−1,0,1}m
+
=
m−1
X
X
X
Ã
i1 <...<ij
ε(is )=0 ∀ s=1,...,j
*
× Φ,
Y
1≤h≤m
αdε (is ) ×
s=1
0≤j≤m−2 ij ,··· ,i1 =2 ε∈{−1,0,1}m,j
+
m−j∈2N
j
Y

+


Aε(h) (f ) Φ 

(3.2.4)
j
h∈{i
/ s }s=1
where, for any j ∈ {h : h = 0, 1, · · · , m − 2} such that m − j is even and for
any s = 1, . . . , j, we define
©
ª
m
{−1, 0, 1}m,j
+ := ε ∈ {−1, 0, 1}+ : |{h : h = 1, 2, · · · , m, ε (h) = 0}| = j
m
Proof. Expanding the power (A(f ) + A+ (f ) + Λα ) , we find that
D
E
X
­ ¡
¢m ®
Φ, A(f ) + A+ (f ) + Λα Φ =
Φ, Aε(m) · · · Aε(1) Φ
ε∈{−1,0,1}m
66
Chapter 3. Universal Central Limit Theorem on IFS
By Lemma 3.2.4, for any ε ∈ {−1, 0, 1}m \ {−1, 0, 1}m
+ , the scalar product
­
®
ε(m)
ε(1)
Φ, A
· · · A Φ is equal to zero and so one gets the first identity in (3.2.4).
For any ε ∈ {−1, 0, 1}m , denoting
j := |{h : h = 1, 2, · · · , m, ε (h) = 0}|
we see that j can take values in the set {0, 1, 2, · · · , m} , that is,
X
D
ε(m)
Φ, A
ε(1)
···A
E
Φ =
m
X
ε∈{−1,0,1}m
D
E
Φ, Aε(m) · · · Aε(1) Φ
X
j=0 ε∈{−1,0,1}m,j
+
If j = m, i.e if ε (i) = 0 for any i = 1, 2, · · · , m,
the definition
/
© by
ª ε ∈
ε(m) , · · · , Aε(1) there
{−1, 0, 1}m
.
If
j
=
m
−
1,
then
among
all
operators
A
+
is
exactly
one creator or annihilator and all others are number operator, so
Pm
/ {−1, 0, 1}m
k=1 ε (k) = ±1 and, by the definition, ε ∈
+ . In conclusion, as ε
m
runs over {−1, 0, 1}+ , the index j should run over the set {0, 1, 2, · · · , m − 2}.
©
ª
Moreover, since in the set Aε(1) , · · · , Aε(m) , the number of creation and
­
®
annihilation operators is m − j , the scalar product Φ, Aε(m) · · · Aε(1) Φ is
equal to zero if m − j is odd. In other words,
D
E
D
E
X
X
X
Φ, Aε(m) · · · Aε(1) Φ =
Φ, Aε(m) · · · Aε(1) Φ
ε∈{−1,0,1}m
0≤j≤m−2 ε∈{−1,0,1}m,j
+
m−j∈2N
(3.2.5)
For any j = 0, 1, · · · , m−2 such that m−j is even and for any ε ∈ {−1, 0, 1}m,j
+ ,
we denote {i1 , · · · , ij } = {h : ε (h) = 0} with the order ij > · · · > i1 . By the
definition of {−1, 0, 1}m
/ {i1 , · · · , ij } and m ∈
/ {i1 , · · · , ij }, i.e. 2 ≤ i1 <
+ , 1 ∈
· · · < ij ≤ m − 1. So
D
E
X
X
Φ, Aε(m) · · · Aε(1) Φ
0≤j≤m−2 ε∈{−1,0,1}m,j
m−j∈2N
=
X
+
X
X
D
Aε(m) · · · Aε(1)
E
0≤j≤m−2 i1 <...<ij ∈{2,...,m−1} ε∈{−1,0,1}m,j :ε(is )=0 ∀s=1,2,··· ,j
m−j∈2N
+
where h·i = hΦ, ·Φi on the right hand side above.
For any j = 0, 1, · · · , m − 2 such that m − j is even, for any 2 ≤ i1 < · · · <
ij ≤ m − 1 and for any ε ∈ {−1, 0, 1}m,j
such that ε (is ) = 0 ∀s = 1, 2, · · · , j,
­ +ε(m)
®
by definition, the scalar product Φ, A
· · · Aε(1) Φ must have the form
D
Φ, Aε(m) (f ) · · · Aε(ij +1) (f ) Λα Aε(ij −1) (f ) · · ·
(3.2.6)
67
3.3. 1-mode type IFS
· · · Aε(i1 +1) (f ) Λα Aε(i1 −1) (f ) · · · Aε(1) (f ) Φ
E
By definition, the vector Λα Aε(i1 −1) (f ) · · · Aε(1) (f ) Φ is a certain αn multiplied by Aε(i1 −1) (f ) · · · Aε(1) (f ) Φ, where the number n must be equal to the
difference between the number of creators among
n
o
Aε(i1 −1) (f ) , . . . , Aε(1) (f )
and that of annihilators, i.e. dε (i1 ) . So we have that
Λα Aε(i1 −1) (f ) · · · Aε(1) (f ) Φ = αdε (i1 ) Aε(i1 −1) (f ) · · · Aε(1) (f ) Φ
By induction we prove that the scalar product (3.2.6) is equal to
j
Y
D
αdε (is ) Φ, Aε(m) (f ) · · · Aε(ij +1) (f ) Aε(ij −1) (f ) · · ·
s=1
E
· · · Aε(i1 +1) (f ) Aε(i1 −1) (f ) · · · Aε(1) (f ) Φ
and this proves the second identity in (3.2.4).
¤
Remark 3.2.8. In the notations of Lemma 3.2.7, if instead of the operator
Λα = Λα (I) one considers Λα (χ) , where χ is the indicator of any measurable
set including the support of f , the identity (3.2.4) remains true.
Remark 3.2.9. A similar version of the above result is given in [2], Theorem
3.4.5, where a different notation is used.
3.3
1-mode type IFS
The simplest class of standard IFS is that for which the functions (λn )n in
Definition 3.1.1 are constants. The sequence {αn }∞
n=1 ⊂ R is arbitrary while
condition (ii) of Definition 3.1.1 becomes in this case: {λn }∞
n=1 ⊂ R+ and
λn = 0 ⇒ λn+1 = 0 for any n.
In our notation this class is characterized by the following relations:
Af A+
g = ωΛ+1 hf, gi for any f, g ∈ H
(3.3.1)
68
Chapter 3. Universal Central Limit Theorem on IFS
Af Φ = 0 for any f ∈ H
ωΛ =
∞
X
ωn Pn
(3.3.2)
(3.3.3)
n=0
where Pn denotes the projection on the n−particle space in the decomposition
(3.1.1) and
λn
ωn :=
(3.3.4)
λn−1
This means that all the mixed moments of Aεf (ε ∈ {−1, 0, 1}) are uniquely
determined by f and by the basic relation
αΛ Aε(n) (fn ) · · · Aε(1) (f1 ) Φ = αdε (n) Aε(n) (fn ) · · · Aε(1) (f1 ) Φ
(3.3.5)
where dε is the depth function defined in (3.2.2).
The next results, set in 1-mode type IFS, are some of the main tools in
order to prove our central limit theorem.
Lemma 3.3.1. If the functions {λn }∞
n=1 are constants, then for any m ∈ N
and for any ε ∈ {−1, 1}m , the scalar product
D
E
Φ, Aε(m) (f ) · · · Aε(1) (f ) Φ
has the form
kf km · C (ε; {λn }∞
n=1 )
where k · k is the norm in L2 (X, µ), C (ε; {λn }∞
n=1 ) is a constant uniquely
determined by ε and {λn }m
.
Moreover
it
is
equal
to zero if either m is odd
n=1
m
m
or m is even and ε ∈ {−1, 1} \ {−1, 1}+ .
Proof. If m is odd, the conclusion is trivial and so we consider only the
case
m = 2N for some N ∈
­
® N. As proved in Lemma 3.2.4, one knows that
Φ, Aε(2N ) (f ) · · · Aε(1) (f ) Φ = 0 if ε ∈ {−1, 1}2N \ {−1, 1}2N
+ . In the case of
2
N = 1 and ε ∈ {−1, 1}+
D
E ­
®
Φ, Aε(2) (f ) Aε(1) (f ) Φ = Φ, A (f ) A+ (f ) Φ = λ1 kf k2
We suppose the statement is true for any k ≤ N and prove it for k = N + 1.
2(N +1)
For any ε ∈ {−1, 1}+
, by denoting
h := min {k ∈ {1, ..., 2N + 2} , ε (k) = −1}
69
3.3. 1-mode type IFS
we know that, just by definition, h 6= 1 and ε (h − 1) = · · · = ε (1) = 1.
Moreover since
A (f ) A+ (fn ) · · · A+ (f1 ) Φ = ωn hf, fn i A+ (fn−1 ) · · · A+ (f1 ) Φ
we have that
D
(3.3.6)
E
Φ, Aε(2(N +1)) (f ) · · · Aε(1) (f ) Φ
D
E
= hf, f i ωdε (h) Φ, Aε(2(N +1)) (f ) · · · Aε(h+1) (f ) Aε(h−2) (f ) · · · Aε(1) (f ) Φ
The proof follows now by induction.
¤
Corollary 3.3.2. In the same hypothesis of the previous lemma, for any f, g ∈
H the moments of A(f ) + A+ (f ) and of A(g) + A+ (g) are the same if and
only if kf k = kgk.
Proof. Our conclusion is easy to see as follows
­ ¡
¢m ®
Φ, A(f ) + A+ (f ) Φ =
D
E
Φ, Aε(m) (f ) . . . Aε(1) (f ) Φ
X
ε∈{−1,1}m
+
= kf km
X
C (ε; {λn }∞
n=1 )
ε∈{−1,1}m
+
= kgkm
X
­ ¡
¢m ®
+
C (ε; {λn }∞
Φ
n=1 ) = Φ, A(g) + A (g)
ε∈{−1,1}m
+
¤
Corollary 3.3.3. For any f, g ∈ H and χf , χg two indicators such that χf
(resp. χg ) is the indicator of a measurable set including the support of f (resp.
g), the moments of A(f ) + A+ (f ) + Λα (χf ) and A(g) + A+ (g) + Λα (χg ) are
the same if and only if kf k = kgk .
70
Chapter 3. Universal Central Limit Theorem on IFS
Proof. For any m ∈ N
­ ¡
¢m ®
Φ, A(f ) + A+ (f ) + Λα (χf ) Φ
à j
m−1
X
X
X
Y
=
αdε (is ) ×
0≤j≤m−2 ij ,··· ,i1 =2 ε∈{−1,0,1}m,j
+
m−j∈2N
*
i1 <...<ij
Y
× Φ,
s=1
ε(is )=0 ∀ s=1,...,j

+
ε(h)
A
1≤h≤m


(f ) Φ 

j
h∈{i
/ s }s=1
and
­ ¡
¢m ®
Φ, A(g) + A+ (g) + Λα (χg ) Φ
à j
m−1
X
X
X
Y
=
αdε (is ) ×
0≤j≤m−2 ij ,··· ,i1 =2 ε∈{−1,0,1}m,j
+
m−j∈2N
*
i1 <...<ij
Y
× Φ,
1≤h≤m
s=1
ε(is )=0 ∀ s=1,...,j

+


Aε(h) (g) Φ 

j
h∈{i
/ s }s=1
For any 0 ≤ j ≤ m − 2 such that m − j ∈ 2N, for any 2 ≤ i1 < . . . < ij ≤ m − 1
and for any ε ∈ {−1, 0, 1}m,j
+ such that ε (is ) = 0 ∀s = 1, 2, · · · , j,from Lemma
3.3.1, it follows that
*
+ *
+
Y
Y
ε(h)
ε(h)
Φ,
A
(f ) Φ = Φ,
A
(g) Φ
1≤h≤m
j
h∈{i
/ s }s=1
1≤h≤m
j
h∈{i
/ s }s=1
if and only if kf k = kgk. Hence the thesis is achieved.
¤
Remark 3.3.4. Both Corollary 3.3.2 and Corollary 3.3.3 strongly depend on
the fact that the λn ’s are constant. There are many standard IFS in which
the distribution of A(f ) + A+ (f ) + Λα (χf ) (even A(f ) + A+ (f ) ) depends not
only on kf k but also on f itself. For example, if we take H := L2 ([0, 1]) and
71
3.3. 1-mode type IFS
λn (x1 , . . . , xn ) := x2 x23 · · · xn−1
for any n, then the distribution of A(χ[0,1] ) +
n
√
√
+
A (χ[0,1] ) is the arcsine law but A( 2χ[0,1/2] ) + A+ ( 2χ[0,1/2] ) has a different
distribution. For more example see [9, 44, 46] and references therein.
Corollary 3.3.5. Let kf k = 1. Then the polynomial distribution of A+
f +Af +
αN is the (polynomially unique) probability measure with Jacobi parameters
(ωn ), (αn ).
Proof. Let {a± , H1 , Φ1 } denote the 1–mode interacting Fock space with Jacobi parameters (ωn ), (αn ), whose construction is given in Theorem 2.4.3. The
identities (3.3.6) and (3.3.4) imply that
Af A+
f =
λN
=ωN
λN −1
(3.3.7)
Since Af Φ = 0, it follows that the norm preserving map
a+n Φ1 7→ A+n
f Φ
extends to a unitary isomorphism U , from H1 to the closed subspace generated
+n = A+ |
by the vectors {A+n
Hn for any n ∈ N. By
f Φ : n ∈ N}, where A
construction U satisfies
U a+ = A+
fU
U Φ1 = Φ
and this implies the statement.
¤
Lemma 3.3.6. If the {λn }∞
n=1 are constant, then for any N ∈ N, for any
ε ∈ {−1, 1}2N
and
for
any
{f
1 , · · · , f2N } ⊂ H, with the convention λ0 := 1,
+
one has:
D
N
E Y
hflk , frk i ωdε (r
Φ, Aε(2N ) (f2N ) · · · Aε(1) (f1 ) Φ =
k=1
k)
(3.3.8)
where {lk , rk }N
k=1 is the unique non–crossing pair partition determined by ε ∈
2N
{−1, 1}+ .
Proof. If N = 1, (3.3.8) is clear. Suppose that it holds for N = n. Then, for
N = n + 1 we have:
D
E
Φ, Aε(2n+2) (f2n+2 ) · · · Aε(1) (f1 ) Φ
72
Chapter 3. Universal Central Limit Theorem on IFS
D
E
¡
¢
¡
¢
= Φ, Aε(2n+2) (f2n+2 ) · · · Aε(ln+1 ) fln+1 · · · Aε(rn+1 ) frn+1 · · · Aε(1) (f1 ) Φ
where ln+1 is the index of the first annihilator from the right in the sequence
above. By the non-crossing principle rn+1 = ln+1 − 1 and ε (h) = 1 for any
h ≤ rn+1 . So the quantity above is equal to
D
¡
¢ ¡
¢
Aε(2n+2) (f2n+2 ) · · · Aε(ln+1 )+1 fln+1 +1 A fln+1 ×
¡
¢
¡
¢
®
×A+ frn+1 A+ frn+1 −1 · · · A+ (f1 )
­
®
= fln+1 , frn+1 ωdε (ln+1 ) ×
D
E
¡
¢
¡
¢
× Aε(2n+2) (f2n+2 ) · · · Aε(ln+1 +1) fln+1 +1 A+ frn+1 −1 · · · A+ (f1 )
­
®
= fln+1 , frn+1 ωdε (ln+1 ) ×
D
E
¡
¢
¡
¢
× Aε(2n+2) (f2n+2 ) · · · Aε(ln+1 +1) fln+1 +1 Aε(rn+1 −1) frn+1 −1 · · · A+ (f1 )
then (3.3.8) now follows from the induction assumption.
¤
3.4
Central Limit Theorem
The main goal in this section is to show the central limit theorem. For
∞
any pair of Jacobi sequences {αn }∞
n=1 ⊂ R, {ωn }n=1 ⊂ R+ , in the notations
¡ (3.1.2), (3.1.3)¢ we consider the following operators on the 1–MT-IFS
Γ L2 (R+ ) , {λn }∞
n=1 :
+
Ak := A(fk ) ; A+
k := A (fk ), Λk := Λα , k = 0, 1, . . .
(3.4.1)
2
where {fk }∞
k=0 is an othonormal set in L (R+ ) and for any bounded sequence
∞
{ck }k=1 ⊂ R we consider
Qk = Ak + A+
k + ck Λk ,
k = 0, 1, 2, . . .
(3.4.2)
and c0 := 1.
Remark 3.4.1. It is worth to mention that in [3] the authors considered the
following operators of creation and annihilation
+
Ak := A(χ[k,k+1) ) ; A+
k := A (χ[k,k+1) ), k = 0, 1, . . .
Later, Krystek and Wojakowski, in [38], generalized such a choice to the test
functions appearing in (3.4.1).
73
3.4. Central Limit Theorem
We point out that the proof of the main theorem is divided into two parts:
one related to the symmetric measures case and the other to nonsymmetric
measures. The symmetric case needs some technical conditions which are
guaranteed by the following results. Firstly we introduce the definition of
singleton condition, necessary for the successive lemma.
³ ´∞
(1)
Definition 3.4.2. Let (A, φ) be an algebraic probability space and bn
,
n=1
³ ´∞
³ ´
(2)
(j)
bn
, . . . , sequences of elements of A such that φ bn
= 0 for any j.
n=1
We say that the set of such sequences satisfies the singleton condition with
respect to φ if, for any k ≥ 1, j1 , . . . , jk , n1 , . . . nk ∈ N
³
´
(jk )
1)
φ b(j
=0
n1 · · · bnk
holds whenever there exists ns 6= nt for any s 6= t.
©
ª∞
Lemma 3.4.3. The family Ak , A+
k k=1 satisfies the singleton condition with
respect to the state hΦ, ·Φi.
Proof. It follows from (3.3.8) and from the orthogonality of the set {fk }∞
k=0 .
¤
Lemma 3.4.4. (Uniform boundedness of the mixed moments). For any N ∈
N, for any {k1 , . . . , km } ⊂ N and for any ε ∈ {−1, 1}2N
+
¯D
E¯
¯ ε(2N )
ε(1) ¯
2N
¯ Ak2N · · · Ak1 ¯ ≤ [λ (N )]
(3.4.3)
with the conventions: λ0 := 1 and, for any m ∈ N:
¾
½
λ3
λm
λ2
, λm .
λ (m) := max 1, λ1 , , λ2 , , λ3 , · · · ,
λ1
λ2
λm−1
D
E
ε(2) ε(1)
Proof. If N = 1, Ak2 Ak1
is different from zero only if ε (2) = −1 and
ε (1) = 1. Moreover
¯D
E¯ ¯D
E¯
¯ ε(2) ε(1) ¯ ¯
¯
¯ Ak2 Ak1 ¯ = ¯ Ak2 A+
k1 ¯ = λ1 hf2 , f1 i ≤ λ1 ≤ λ (1)
2(N +1)
Let us suppose the result is true for N. For any ε ∈ {−1, 1}+
, we denote
+1
{lh , rh }N
the
non–crossing
pair
partition
determined
by
ε.
We
know,
by the
h=1
74
Chapter 3. Universal Central Limit Theorem on IFS
non-crossing principle, that rN +1 = lN +1 − 1, ε (h) = 1 for any h ≤ rN +1 , and
¯D
E¯
¯ ε(2N +2)
ε(1) ¯
¯ Ak2N +2 · · · Ak1 ¯
¯D
E¯
¯ ε(2N +2)
ε(l
+1)
+
+ ¯
= ¯ Ak2N +2 · · · Akl N +1+1 AklN +1 A+
A
·
·
·
A
krN +1 krN +1 −1
k1 ¯
N +1
¯­
®
= ωdε (l ) ¯ flN +1 , frN +1 ×
E¯
D N +1
ε(l
+1)
ε(2N +2)
+ ¯
× Ak2N +2 · · · Akl N +1+1 A+
·
·
·
A
krN +1 −1
k1 ¯
N +1
¯D
E¯
¯ ε(2N +2)
−1)
ε(l
+1) ε(r
ε(1) ¯
≤ ωdε (l ) ¯ Ak2N +2 · · · Akl N +1+1 Akr N +1−1 · · · Ak1 ¯
N +1
N +1
N +1
By the induction assumption one has that
¯D
E¯
¯ ε(2N +2)
ε(lN +1 +1) ε(rN +1 −1)
ε(1) ¯
2N
A
·
·
·
A
A
·
·
·
A
¯ k2N +2
¯ ≤ [λ (N )]
kl
kr
k1
−1
+1
N +1
N +1
Therefore
¯D
E¯
¯ ε(2N +2)
ε(1) ¯
2N
¯ Ak2N +2 · · · Ak1 ¯ ≤ [λ (N )] · ωdε (l
N +1 )
Moreover dε (lN +1 ) = dε (rn+1 ) + 1 and, since ε (h) = 1 for any h ≤ rN +1 and
|{h : ε (h) = 1}| = N + 1, one gets that dε (lN +1 ) = dε (rN +1 ) + 1 ≤ N + 1.
Consequently
ωdε (l ) ≤ λ (N + 1)
N +1
By definition, the sequence {λ (N )}∞
N =1 is increasing, hence
¯D
E¯
¯ ε(2N +2)
ε(1) ¯
2N
2(N +1)
¯ Ak2N +2 · · · Ak1 ¯ ≤ [λ (N )] · ωdε (l ) ≤ λ (N + 1)
N +1
¤
Theorem 3.4.5. Let µ be a mean–zero probability measure on (R, B) with
moments ¡of any order and
¢ with Jacobi coefficients given by {ωnλ}nn , {αn }n .
2
Denote Γ L (R+ ) , {λn }n the 1–mode type free IFS with ωn = λn−1 for any
©
ª∞
n and consider the operators Ak , A+
defined by (3.4.1), (3.4.2) on
,
c
Λ
k
k
k
k=0
¡
¢
Γ L2 (R+ ) , {λn }n , where {ck }∞
⊂
R
is
a
bounded
sequence which satisfies
k=1
the condition
N
1 X
√
ck → 1
(3.4.4)
N k=1
75
3.4. Central Limit Theorem
√
√
(e.g. ck = k − k − 1 for any k) if the sequence (αn ) does not vanish
identically.
Then
i) for any k = 0, 1, 2 · · · , the distribution of Ak + A+
k + Λk with respect to
the state hΦ, ·Φi is exactly the measure µ;
ii) the operator quantum stochastic process
(
)∞
N
N
N
1 X
1 X + 1 X
√
Ak , √
A ,√
ck Λk
N k=1
N k=1 k
N k=1
k=1
converges in the sense of the mixed moments for N → ∞, to
©
ª
A0 , A+
0 , Λ0
Proof. i) follows from Corollary 3.3.3, Corollary 3.3.5 and [2]. To prove ii)
we first deal with the symmetric case i.e. when the sequence (αn ), hence
(Λ
vanishes
and the quantum operator process is reduced to
n k ),√identically
o
PN
±
(1/ N ) k=1 Ak . Our goal consists in showing that for any m ≥ 1 and
for any
ε = (ε (1) , · · · , ε (m)) ∈ {−1, +1}m
D
E D
E
X
1
ε(m)
ε(1)
ε(m)
ε(1)
Aβm · · · Aβ1
= A0 · · · A0
(3.4.5)
lim
m
N →∞ N 2
1≤β ,...,βm ≤N
1
©
ª∞
The operator process Ak , A+
k k=0 satisfies the singleton condition and the
boundedness of the mixed moments as shown in Lemma 3.4.3 and Lemma
3.4.4 respectively. Hence from [6], Lemma 2.4 it follows that the limit (3.4.5)
is nonzero only if m = 2p and, in this case, the left hand side of (3.4.5) is equal
to:
D
E
X
1
ε(2p)
ε(1)
lim
A
·
·
·
A
(3.4.6)
βm
β1
N →∞ N p
β:{1,...,2p}→{1,...N }
|Range(β)|=p
|β −1 (β(j))|=2
∀j=1,...,2p
where |·| denotes cardinality. Each map β in the above summation induces a
pair partition of the set {1, 2, . . . , 2p}. We denote by P.P. (2p) the set of all
these partitions and use the notations
β −1 (β (j)) =: {lj , rj } , lj > rj , j = 1, . . . , p
Moreover we know, from Lemma 3.2.1, that a term in the sum (3.4.6) can be
nonzero only if ε ∈ {−1, +1}2p
+ . In particular only non crossing pair partitions
76
Chapter 3. Universal Central Limit Theorem on IFS
may give nonzero contributions. Hence the quantity (3.4.6) can be rewritten
as
D
E
X
1
ε(lj )
ε(rj )
+
lim
A
·
·
·
A
·
·
·
A
·
·
·
A
(3.4.7)
β2p
βlj
βrj
β1 ,
N →∞ N p
p
{lj ,rj }j=1 ∈N.C.P.P.(2p)
where N.C.P.P. (2p) denotes the set of all non crossing pair partitions of
{1, 2, . . . , 2p} and the first operator on the left is an annihilator (and corresponds to an lj -index) while the first operator on the right is a creator (and
corresponds to an rj -index). Let βl1 be the index of the first annihilator from
the right in (3.4.7). Hence the operator with index βl1 − 1 is a creator. If
βl1 − 1 6= βr1 , then one has
Aβl1 A+
βl
1
−1
=0
because we are dealing with orthogonal functions. Therefore the nonzero contributions can come only from those pairs satisfying βl1 − 1 = βr1 . Using
(3.3.7) and the depth function (3.2.2), (3.4.7) becomes
¿
À
X
1
\
ε(l1 ) \
ε(r1 )
+
Aβ2(p−1) · · · Aβl Aβr · · · Aβ1
ωdε (l1 ) lim
1
1
N →∞ N p
p
{lj ,rj }j=1 ∈N.C.P.P.(2p)
Iterating the same procedure for any lj (j = 2, . . . , p), (3.4.7) takes the form
p
Y
j=1
1
N →∞ N p
X
ωdε (lj ) lim
(3.4.8)
p
∈N.C.P.P.(2p)
j=1
{lj ,rj }
Now we observe that, as a set:
{1, . . . , 2p} = {l1 , r1 , . . . , lp , rp }
so any map of the sum in (3.4.8) can be written in the following way:
β : {l1 , r1 , . . . , lp , rp } → {1, . . . , N } , |Range β| = p,
β −1 (β (j)) = {lj , rj } , j = 1, . . . , p
Moreover, since any ε ∈ {−1, +1}2p
+ determines exactly one non-crossing pair
partition of {1, . . . , 2p} and lp < · · · < l1 , then, for any j = 1, . . . , p, βrj is
77
3.4. Central Limit Theorem
uniquely determined once one knows βlj . So assigning β is equivalent to assign
an injective map
β : {l1 , . . . , lp } → {1, . . . , N }
Therefore (3.4.8) is equal to
p
Y
j=1
ωdε (lj ) lim
N →∞
1
|{β : {l1 , . . . , lp } → {1, . . . , N } : β injective}|
Np
(3.4.9)
where | · | denotes cardinality. Since
|{β : {l1 , . . . , lp } → {1, . . . , N } : β injective}| =
N!
(N − p)!
in the limit (3.4.9) we have
p
p
Y
Y
1
ωdε (lj )
ω
=
N
(N
−
1)
·
·
·
(N
−
(p
+
1))
dε (lj )
N →∞ N p
lim
j=1
j=1
On the other hand, in the above notations,
D
E D
E
ε(2p)
ε(1)
ε(l ) ε(r )
A0
· · · A0
= A0 · · · A0 1 A0 1 · · · A+
0
(3.4.10)
¿
À
\
ε(l ) \
ε(r )
= ωdε (l1 ) A0 · · · A0 1 A0 1 · · · A+
0
and iterating for any j, one obtains
p
D
E Y
ε(2p)
ε(1)
A0
· · · A0
=
ωdε (lj )
(3.4.11)
j=1
Now let us prove (3.4.5) in the non-symmetric case.
ε(j)
In this case ε = (ε (1) , · · · , ε (m)) ∈ {−1, 0, +1}m
+ and Aβj = cβj Λ. The
proof is by induction on m ≥ 1. If m = 1, then (3.4.5) is verified because each
side is identically equal to zero. Now we suppose that (3.4.5) is verified for
any h ≤ m − 1 and prove it for h = m. We denote by IZ the set of indices in
ε corresponding to number operators, i.e.
IZ := {j ∈ {1, ..., m} : ε (j) = 0}
Denote s = |IZ | the number of singletons, then 0 ≤ s ≤ m − 2, because
1, m ∈
/ IZ and m − s = 2p, because of iii) of Lemma 3.2.1. If IZ = ∅, then the
78
Chapter 3. Universal Central Limit Theorem on IFS
thesis follows because we find the symmetric case. If IZ 6= ∅, then there exist
k = 1, ...m such that ε (k) = 0. Denoting z1 = min {k ∈ {1, ..., m} : k ∈ IZ } ,
the right hand side of (3.4.5) can be written as
¿
À
D
E
\
ε(m)
ε(z1 )
ε(1)
ε(m)
ε(z1 )
ε(1)
A0 · · · A0
· · · A0
= αdε (z1 ) A0 · · · A0
· · · A0
The left hand side of (3.4.5) can be written as
E
D
X
1
ε(1)
ε(z1 )
ε(m)
lim
A
·
·
·
A
·
·
·
A
m
β1
βz1
βm
N →∞ N 2
1≤β ,...,βm ≤N
1

1


×


X
1
= lim αdε (z1 ) √
N →∞
N βz ∈F
X
1
N
m−1
2
cβz1  ×
N,z1

¿
À
\
ε(m)
ε(z )
ε(1) 

Aβm · · · Aβz 1 · · · Aβ1 

1
1≤β1 ,...,βm ≤N
(3.4.12)
βk 6=βz1
where FN,z1 is the subset of {1, . . . , N } in which βz1 can be chosen. FN,z1 =
{1, . . . , N } \ {βm , β1 } (because 1, m ∈
/ IZ ). Hence |FN,z1 | = N − 2. As a
N
P
P
cβz1 has the same asymptotic of √1N
ck for N → ∞
consequence √1N
z1 ∈FN,z1
k=1
because {ck } is a bounded sequence in R. Moreover we notice that
¿
À
X
\
ε(m)
ε(z1 )
ε(1)
Aβm · · · Aβz · · · Aβ1
1
1≤β1 ,...,βm ≤N
βk 6=βz1
in (3.4.12) can be written as
X
0
1≤β10 ,...,βm−1
≤N
where ε0 =
¿
À
ε0 (m−1)
ε0 (1)
A 0
···A 0
βm−1
β1
³
´
ε (1) , ..., ε[
(z1 ), ..., ε (m) and β 0 : {1, . . . , m − 1} → {1, . . . , N }
such that βk0 = βk if k = 1, . . . , z1 − 1 and βk0 = βk+1 if k = z1 , . . . , m − 1.
By hypothesis of induction we know that
¿
À D
E
X
1
ε0 (m−1)
ε0 (1)
ε0 (m−1)
ε0 (1)
lim
A
·
·
·
A
=
A
·
·
·
A
0
0
m−1
0
0
βm−1
β1
N →∞ N 2
1≤β 0 ,...,β 0
≤N
1
m−1
79
3.4. Central Limit Theorem
and the right hand side above is
¿
À
\
ε(m)
ε(z1 )
ε(1)
A0 · · · A0
· · · A0
As a consequence (3.4.12) is equal to
¿
À
\
ε(m)
ε(z1 )
ε(1)
αdε (z1 ) A0 · · · A0
· · · A0
¤
Corollary 3.4.6. In the same notations of Theorem 3.4.5 we have that for
any m ≥ 0
*Ã
!m + Z
N
1 X
√
lim
Qk
= xm dµ
N →∞
N k=1
where, for any k ∈ N, the operator stochastic process Qk is defined by:
Qk = Ak + A+
k + ck Λk
Proof. For any m ≥ 1
!m +
*Ã
N
1
1 X
√
= m
Qk
N k=1
N2
Z
xm dµ =
X
D
X
1≤k1 ,...,km ≤N
ε∈{−1,0,1}m
+
¢m ®
­¡
=
A0 + A+
0 + Λ0
X
ε∈{−1,0,1}m
+
ε(m)
ε(1)
E
Akm · · · Ak1
D
E
ε(m)
ε(1)
A0 · · · A0
P
Since the sum ε∈{−1,0,1}m involves only a finite number of terms, the thesis
+
follows by applying Theorem 3.4.5 to each element of the sum.
¤
Chapter 4
Boolean Central limit
Theorem
As observed in the previous chapter, in quantum probability the notion of
independence is not unique and any notion of non-commutative independence
generates a central limit theorem. As for independencies, where an axiomatic
point of view has been developed during the past years (see [59], [18], [50]), for
quantum central limits also a unified abstract approach has been made by Accardi, Hashimoto and Obata in [6] and by Speicher and von Waldenfels in [60],
where non-commutative general central limits are proved. More recently the
interacting Fock space structure has suggested an idea to obtain constructive
central limit theorems and, as shown in Chapter 3, any mean zero measure on
the real line with finite moments of any order is the central limit in the sense
of moments of a sequence of processes, satisfying rather weak conditions, on
1-mode type interacting Fock space.
Furthermore Accardi and Bach showed in [1] that creation, annihilation
and number operators in Boson Fock space without test function, can be
obtained as central limit of operators in toy Fock spaces. This is possible
also for Fermionic, free and monotone Fock space (see [43], [28], [41]).
One wish naturally to find a similar central limit theorem also for the
boolean independence and Boolean Fock space and here, following [17], we
present such a result. In particular we point out that the Boolean Fock space
can be seen as a 1-mode type interacting Fock space and then some useful
techniques on this spaces, presented in the previous chapter, can be used to
reach this goal.
82
Chapter 4. Boolean Central limit Theorem
The notion of boolean independence was introduced in [64] by W. von
Waldenfels (namely interval partition) and by Bozejko in [21] in the theory
of the free groups and successively Speicher and Woroudi in [61] defined the
Boolean convolution. This kind of non-commutative independence has been
used also by Lenczewski in [39] and by Skeide in [56] for the theory of quantum
stochastic calculus in the framework of Hilbert modules. More recently a
quantum stochastic calculus on Boolean Fock space has been developed by
Ben Ghorbal and Schürmann in [19].
The chapter is organized as follows: in Section 4.1 we define the boolean
independence in an algebraic probability space, the boolean Fock space, the
simplest quantum probability model – quantum Bernoulli process – and proper
constructive quantum stochastic processes (namely creation – annihilation –
preservation processes) on the boolean Fock space and with discrete time. In
Section 4.2 we give results on the moments of such operator processes with
respect to the ”vacuum” state. In section 4.3 we present our main result:
a quantum central limit argument is used to get the creation – annihilation
– number processes on the Boolean Fock space starting from discrete time
processes. It is worth to mention the fact that the family of processes used
in our central limit theorem satisfies the singleton condition with respect to
the vacuum state and the boundedness of the mixed moments (see [6] and
Chapter 3). Then, as pointed out in the previous chapter, as a consequence
if [6], Lemma 2.4, only pair partitions survive in the limit. Our constructive
approach shows that discrete - time processes give exactly one pair partition,
i.e. the interval partition from one hand and the explicit formulation either
for the approximating sequences of random variables as operators in discrete
time, or for the central limit in the Boolean Fock space from another one.
Finally we want to point out the differencies between [60] and our approach:
(i) we give a concrete construction rather than an abstract and general
approach of existence of a central limit;
(ii) in [60] the *-algebra and the state are the same either for the convergent sequence of the quantum random variables or their resulting limit,
¡ ¢⊗N
whereas our sequence is made of operators on R2
, but in the limit
one finds operators on the Boolean Fock space;
(iii) in [60] the GNS representation for the limit state has a Fock like structure
on which the elements act as creation and annihilation operators, instead
4.1. Boolean independence and Boolean Fock space
83
in our case the constructive approach allows to find also the number
operator either in the convergent sequence or in the limit.
4.1
Boolean independence and Boolean Fock space
In this section we firstly introduce the definition of boolean independence in
a purely algebraic contest.
Definition 4.1.1. Let (A, φ) be an algebraic probability space. Let us given
an index set I and a family (Ai )i∈I of ∗-subalgebras of A. We say that such a
family is boolean independent with respect to the state φ if for any n ≥ 1,
for any ε1 , . . . , εn ∈ I, where εi = εj for i 6= j is not being excluded, for any
ε
aj j ∈ Aεj , j = 1, . . . , n
φ (aε11 · · · aεnn ) =
n
Y
³ ´
ε
φεj aj j
j=1
holds whenever εj 6= εj+1 for any j = 1, . . . , n − 1 and φεj := φ |Aεj for any
j = 1, . . . , n.
For the construction of Boolean Fock space, we start from an Hilbert space
H. The Hilbert space
Γ (H) := C ⊕ H
is called the Boolean Fock space over H. We denote by Φ the vacuum vector
Φ := 1 ⊕ 0
and, as usual, the vacuum expectation h·i is the state
h·i := hΦ, ·Φi
Remark 4.1.2. The Boolean Fock space is a particular 1MT-IFS, where the
sequence of interacting constants is given by
λ0 = λ1 = 1, λn = 0 ∀n ≥ 2
We introduce the three basic operators of creation, annihilation and preservation on Γ (H) as follows.
84
Chapter 4. Boolean Central limit Theorem
Definition 4.1.3. For any f ∈ H and for any α ∈ C, g ∈ H the creation
operator
A+ (f ) (α ⊕ g) := 0 ⊕ αf
is defined as a linear operator A+ (f ) : Γ (H) → Γ (H) and has an adjoint
A (f ) : Γ (H) → Γ (H) called the annihilation operator.
Remark 4.1.4. For any f ∈ H and for any α ∈ C, g ∈ H
A (f ) (α ⊕ g) = hf, gi ⊕ 0
In fact, for any β ∈ C, h ∈ H one has
hβ ⊕ h, A (f ) (α ⊕ g)i =
­ +
®
A (f ) (β ⊕ h) , α ⊕ g
= h0 ⊕ βf, α ⊕ gi = β hf, gi
= hβ ⊕ h, hf, gi ⊕ 0i
Remark 4.1.5. For any f, g ∈ H
­
®
A (f ) A+ (g) = hf, gi
Definition 4.1.6. For any B ∈ B (H) and for any α ∈ C, g ∈ H we define
the preservation operator Λ (B) : Γ (H) → Γ (H) such that
Λ (B) (α ⊕ g) := 0 ⊕ B (g)
For any ε ∈ {−1, 0, 1} , for any B ∈ B (H) and f ∈ H we will denote
 +
 A (f ) if ε = 1
ε
Λ (B)
if ε = 0
A (f, B) :=

A (f )
if ε = −1
In order to state our main result we need some further definitions and
notations starting
Bernoulli process as in [28], [47]. Let us
µ ¶from theµquantum
¶
1
0
denote e1 :=
, e2 :=
. Clearly {e1 , e2 } gives the usual base of R2 .
0
1
For any N ≥ 1, 0 ≤ n ≤ N and 1 ≤ k1 < k2 < . . . < kn ≤ N , we define

(N )
⊗N

if n = 0
 δ∅ := e1
(N )
⊗(k
−1)
⊗(k2 −k1 −1)
⊗(N −kn )
1
δ(k1 ,...,kn ) :=
e1
⊗ e2 ⊗ e1
⊗ e2 ⊗ . . . ⊗ e2 ⊗ e1


if 1 ≤ n ≤ N
(4.1.1)
85
4.1. Boolean independence and Boolean Fock space
where hereinafter, when n = 0, (k1 , . . . , kn ) is understood as ∅. We notice that
the family
n
(N )
(N )
δ(k1 ,...,kn ) ⊗ δ(h1 ,...,hm ) : 0 ≤ n, m ≤ N, 1 ≤ k1 < . . . < kn ≤ N,
1 ≤ h1 < . . . < hm ≤ N }
2
2
is a base of the space M⊗N
2 , where M2 := R ⊗R is the space of 2×2-matrices.
The definition (4.1.1) can be generalized by inserting suitable ”test functions”. Hence we consider the space of all complex Riemann integrable functions L ([0, 1]) and for 1 ≤ n ≤ N, 1 ≤ k1 < . . . < kn ≤ N, f1 , . . . , fn ∈
L ([0, 1]) , define
µ
(N )
δ(k1 ,...,kn ) (f1 , . . . , fn ) := fn
kn
N
#
¶ "n−1
Y µ kh ¶ (N )
δ(k1 ,...,kn )
fh
kh+1
h=1
For any k = 1, . . . , N, for any f ∈ L ([0, 1]) , for any 0 ≤ n ≤ N and 1 ≤
¡ 2 ¢⊗N
¡ ¢⊗N
k1 < . . . < kn ≤ N we define a linear operator T+
→ R2
N (f, k) : R
as follows
(
(N )
h
i
δ(k) (f ) if n = 0
(N
)
T+
(f,
k)
δ
:=
N
(k1 ,...,kn )
0
if n ≥ 1
and we denote by TN (f, k) the adjoint of T+
N (f, k) . The following result
allows us to compute the value of TN (f, k) .
Lemma 4.1.7. For any k = 1, . . . , N, for any f ∈ L ([0, 1]) , for any 0 ≤ n ≤
N and 1 ≤ k1 < . . . < kn ≤ N,
(
h
i
(N ) ¡ ¢
δ∅
f
if n = 1, k = k1
(N )
TN (f, k) δ(k1 ,...,kn ) =
0
otherwise
(N ) ¡
where δ∅
¢
¡ ¢ (N )
f := f Nk · δ∅ .
Proof. We firstly observe that for any n, m ∈ N, 1 ≤ k1 < . . . < kn ≤ N,
1 ≤ h1 < . . . < hm ≤ N


n
D
E
Y
k
(N )
(N )
n 
δ(h1 ,...,hm ) , δ(k1 ,...,kn ) = δm
δhjj 
j=1
86
Chapter 4. Boolean Central limit Theorem
n is the Kronecker symbol. Hence we have that
where δm
D
h
iE D
h
i
E
(N )
(N )
(N )
(N )
δ(h1 ,...,hm ) , TN (f, k) δ(k1 ,...,kn ) = T+
(f,
k)
δ
,
δ
N
(h1 ,...,hm )
(k1 ,...,kn )
E
¡ k ¢ D (N ) (N )
δ
,
δ
N
(k)
(k1 ,...,kn )
(
f
=
½
=
0
f
0
otherwise
¡k¢
if m = 0, n = 1, k = k1
otherwise
N
From another hand
E
( D
(N )
(N ) ¡ ¢
δ(h1 ,...,hm ) , δ∅
f
(
f
=
½
=
0
0
f
0
¡ k ¢D
N
¡k¢
N
if m = 0
(N )
(N )
δ(h1 ,...,hm ) , δ∅
if n = 1, k = k1
otherwise
E
if n = 1, k = k1
otherwise
if m = 0, n = 1, k = k1
otherwise
¤
¡
¢
For any B ∈ B L2 ([0, 1] , dx) , we define a linear
¡ 2 ¢⊗N
¡ ¢⊗N
R
→ R2
such that for any 0 ≤ n ≤ N, for any
and 1 ≤ k1 < . . . < kn ≤ N
(
(N )
i
h
δ(k1 ) (Bf1 )
(N )
ΛN (B) δ(k1 ,...,kn ) (f1 , . . . fn ) :=
0
operator ΛN (B) :
f1 , . . . fn ∈ L ([0, 1])
if n = 1
otherwise
One notices that if I is the identity operator on L2 ([0, 1] , dx), then ΛN (I) is
¡ ¢⊗N
the identity on R2
.
Remark 4.1.8. The operators T+
N (f, k) , TN (f, k) and ΛN (B) above defined
can be seen as creation, annihilation and preservation operators respectively,
¡ ¢⊗N
on R2
.
4.2. Moments of operators in discrete and Boolean case
87
¡
¢
Proposition 4.1.9. For any B ∈ B L2 ([0, 1] , dx) , for any h, k = 1, . . . , N,
for any f, g ∈ L ([0, 1]) , for any 0 ≤ n ≤ N and 1 ≤ k1 < . . . < kn ≤ N,
h
i ¡ ¢µ k ¶h
i
(N )
(N )
(g,
h)
δ
=
TN (f, k) T+
f
g
δ
δkh
(4.1.2)
N
∅
∅
N
h
i
i
h
(N )
(N )
+
(Bf,
k)
δ
(4.1.3)
=
T
ΛN (B) T+
(f,
k)
δ
N
N
(k1 ,...,kn )
(k1 ,...,kn )
¡
¢
Proof. Let us fixed B ∈ B L2 ([0, 1] , dx) , h, k = 1, . . . , N, f, g ∈ L ([0, 1]) ,
0 ≤ n ≤ N, 1 ≤ k1 < . . . < kn ≤ N. Then
· µ ¶
¸
h
i
h
(N )
(N )
+
TN (f, k) TN (g, h) δ∅
= TN (f, k) g
δ(h)
N
µ ¶h
i
¡ ¢ k
(N )
δ∅
δkh
= fg
N
Moreover
h
i
h
i
(N )
(N )
(N )
ΛN (B) T+
(f,
k)
δ
=
Λ
(B)
δ
(f
)
= δ(k) (Bf )
N
N
(k1 ,...,kn )
(k)
h
i
(N )
= T+
(Bf,
k)
δ
N
(k1 ,...,kn ) .
¤
As a consequence of the previous results, one has that any product of creation - annihilation - preservation operators can be ever reduced to a product
of only¡ creation - annihilation
operators. For any ε ∈ {−1, 0, 1} , for any
¢
2
B ∈ B L ([0, 1] , dx) , for any 1 ≤ k ≤ N, and f ∈ L ([0, 1]) we will denote
 +
 TN (f, k) if ε = 1
ε
Λ (B)
if ε = 0
TN (f, k, B) :=
 N
TN (f, k) if ε = −1
4.2
Moments of operators in discrete and Boolean
case
In our main result we need to know the value of the following joint expectations
D
E
(N )
(N )
δ∅ , TεNn (fn , kn , Bn ) · · · TεN1 (f1 , k1 , B1 ) δ∅
88
Chapter 4. Boolean Central limit Theorem
hAεn (fn , Bn ) · · · Aε1 (f1 , B1 )i
in the notations introduced in the previous section. To this goal we present
the following results for the discrete case. We observe that they hold in the
Boolean case too. Their validity can be obtained just by replying all the cases
(N )
H to L ([0, 1]) , Aε to TεN and Φ to δ∅ or, for some of them, consequence of
the results obtained in Chapter 3, Section 3.2.
Lemma 4.2.1. For any n ∈ N and ε belonging to the set
{−1, 0, 1}n := {ε = (ε (n) , . . . , ε (1)) : ε (i) ∈ {−1, 0, 1} , ∀i = 1, . . . , n}
©
ª
i) if among TεNn (fn , kn , Bn ) , . . . , TεN1 (f1 , k1 , B1 ) there are more annihilators than creators, then
(N )
TεNn (fn , kn , Bn ) · · · TεN1 (f1 , k1 , B1 ) δ∅
=0
ii) if the cardinality of the set {i : ε (i) = ±1} is odd, then
D
E
(N )
(N )
δ∅ , TεNn (fn , kn , Bn ) · · · TεN1 (f1 , k1 , B1 ) δ∅
=0
iii) if either ε (n) ∈ {0, 1} or ε (1) ∈ {−1, 0}, the scalar product
D
E
(N )
(N )
δ∅ , TεNn (fn , kn , Bn ) · · · TεN1 (f1 , k1 , B1 ) δ∅
is equal to zero.
©
ª
iv) if among TεNn (fn , kn , Bn ) , . . . , TεN1 (f1 , k1 , B1 ) there are same number
of annihilators and creators, then there exists a constant c such that
(N )
TεNn (fn , kn , Bn ) · · · TεN1 (f1 , k1 , B1 ) δ∅
(N )
= cδ∅
(4.2.1)
Proof. The proof follows by using the same arguments used in Lemma 3.2.1.
¤
As in the previous chapter, we consider the depth function of a given
sequence ε ∈ {−1, 0, 1}n , whose definition is recalled here for a clearer reading.
4.2. Moments of operators in discrete and Boolean case
89
Definition 4.2.2. For n ∈ N and ε = (ε (n) , · · · , ε (1)) ∈ {−1, 0, 1}n we
define the depth function (of the string ε) dε : {1, ..., n} → {0, ±1, ..., ±n} by
dε (j) =
j
X
ε (k)
k=1
= |{ε (k) : ε (k) = 1; k < j}| − |{ε (k) : ε (k) = −1; k < j}|
Definition 4.2.3. {−1, 0, 1}n+ is defined as the totality of all {−1, 0, 1}n satisfying the following conditions:
i)
Pn
k=1 ε (k)
= dε (n) = 0;
ii) ε (1) = 1 and ε (n) = −1;
iii) for all i = 1, . . . , n, dε (i) ≥ 0;
We denote by {−1, 0, 1}n+,B the set {−1, 0, 1}n+ such that for any j =
1, . . . , n − 1, if ε (j) = 1 (respectively ε (j) = −1) then ε (j + 1) 6= 1 (respectively ε (j + 1) 6= −1).
Lemma 4.2.4. For any n ∈ N, and ε ∈ {−1, 0, 1}n , the scalar products
E
D
(N )
(N )
(4.2.2)
δ∅ , TεNn (fn , kn , Bn ) · · · TεN1 (f1 , k1 , B1 ) δ∅
hAεn (fn , Bn ) · · · Aε1 (f1 , B1 )i
can be nonzero only if ε ∈
{−1, 0, 1}n+,B
(4.2.3)
.
Proof. We prove only (4.2.2). The conditions fulfilling i)-iii) of Definition
4.2.3 can be obtained by using the same arguments developed in the proof
of Lemma 3.2.4. In order to prove that (4.2.1) does not vanish only if ε ∈
{−1, 0, 1}n+,B , we firstly observe that if ε (2) = 1, then from the definition of
¡ ¢⊗N
the creation operator on R2
, we have that
D
E
(N )
(N )
+
δ∅ , TεNn (fn , kn , Bn ) · · · T+
(f
,
k
)
T
(f
,
k
)
δ
2
2
1
1
N
N
∅
D
E
(N )
(N )
= δ∅ , TεNn (fn , kn , Bn ) · · · T+
(f
,
k
)
δ
(f
)
=0
2 2 (k)
N
90
Chapter 4. Boolean Central limit Theorem
By induction we suppose that such condition holds for all h ≤ j − 1 and prove
it for h = j. If ε (j + 1) = 1, we have to compute
D
(N )
+
δ∅ , TεNn (fn , kn , Bn ) · · · T+
N (fj+1 , kj+1 ) TN (fj , kj ) · · ·
(N )
· · · TN (f2 , k2 ) T+
N (f1 , k1 ) δ∅
E
As observed at the end of the previous section, we reduce the product of
creation-annihilation-preservation operators on the right side of T+
N (fj , kj ) to
one involving only creators and annihilators. From our hypothesis of induction
in such case there is the same number of creators and annihilators. Hence,
from (4.2.1) it follows that there exists a constant c such that the quantity
above becomes equal to
D
E
(N )
(N )
+
c δ∅ , TεNn (fn , kn , Bn ) · · · T+
(f
,
k
)
T
(f
,
k
)
δ
j+1
j+1
j
j
N
N
∅
and this is clearly equal to zero.
¤
As a consequence of Lemma 4.2.1, the scalar products (4.2.2) and (4.2.3)
are different from zero only if the number of creation operators and annihilation operators is the same. Hence hereinafter, if m is the cardinality of the
operators in (4.2.2), then m = 2n + j, where n is the cardinality of creators
(equivalently annihilators) and j the cardinality of preservation operators.
Furthermore we denote
V := {i = 1, . . . , m : ε (i) = 0}
(so |V | = j, where as usual |·| is the cardinality of the set), put V =: {i1 , . . . ij }
and {h1 , . . . h2n } := {1, . . . , m} \ {i1 , . . . ij } .
The following result states that any scalar product of the form
D
E
(N )
(N )
δ∅ , TεNm (fm , km , Bm ) · · · TεN1 (f1 , k1 , B1 ) δ∅
or
hAεm (fm , Bm ) · · · Aε1 (f1 , B1 )i
can be reduced to a product of factors, each of them formed by a scalar product
containing exactly one annihilator and its following creator, i.e. our families
of operators satisfy the Boolean condition of independence.
91
4.2. Moments of operators in discrete and Boolean case
Lemma 4.2.5. For any m ∈ N, for any ε ∈ {−1, 0, 1}m
, for any f1 , . . . , fm ∈
+,B
¡ 2
¢
L ([0, 1]), 1 ≤ k1 < . . . < km ≤ N and B1 , . . . , Bm ∈ B L ([0, 1] , dx)
D
E
(N )
(N )
δ∅ , TεNm (fm , km , Bm ) · · · TεN1 (f1 , k1 , B1 ) δ∅
=
2n
D
Y
´
E
³
¡
¢
(N )
(N )
f
,
k
δ
δ∅ , TN fhp+1 , khp+1 T+
B
V
h
h
p
p
hp
N
∅
(4.2.4)
p=1
p∈2N+1
hAεm (fm , Bm ) · · · Aε1 (f1 , B1 )i
2n
D ¡
³
´E
Y
¢
=
A fhp+1 A+ BVhp fhp
(4.2.5)
p=1
p∈2N+1
where for any p = 1, . . . , 2n, p ∈ 2N + 1,
Vhp := {ir , . . . , iq ∈ V : hp+1 < ir < . . . < iq < hp }
and BVhp := Bir · · · Biq , ir , . . . , iq ∈ Vhp .
Proof. In the notations introduced above, let h1 := 1, let hl and i1 be the
indices relative respectively to the first annihilator and to the first preservation
operator from the right in the left hand side of (4.2.4). From Lemma 4.2.4 we
have that the left hand side of (4.2.4) is equal to
D
¢
εh +1 ¡
(N )
δ∅ , TN (fm , km ) · · · TN l
fhl +1 , khl +1 , Bhl+1 TN (fhl , khl ) · · ·
(N )
· · · ΛN (Bi1 ) T+
N (f1 , k1 ) δ∅
E
Proposition 4.1.9 implies that the quantity above can be reduced to
D
εh +1
(N )
δ∅ , TN (fm , km ) · · · TN l (fhl +1 , khl +1 , Bhl +1 ) ×
(N )
×TN (fhl , khl ) · · · T+
N (Bi1 f1 , k1 ) δ∅
E
and, by using repeatedly Lemma 4.2.4 and Proposition 4.1.9, we have
D
εh +1
(N )
δ∅ , TN (fm , km ) · · · TN l (fhl +1 , khl +1 , Bhl +1 ) TN (fhl , khl ) ΛN (Bir ) · · ·
92
Chapter 4. Boolean Central limit Theorem
(N )
· · · ΛN (Bi2 ) T+
N (Bi1 f1 , k1 ) δ∅
E
D
εh +1
(N )
= δ∅ , TN (fm , km ) · · · TN l (fhl +1 , khl +1 , Bhl +1 ) ×
E
(N )
×TN (fhl , khl ) T+
(B
·
·
·
B
B
f
,
k
)
δ
ir
i2 i1 1 1 ∅
N
Thus, necessarily l = 2 and consequently Bir · · · Bi2 Bi1 = BVh1 . The above
scalar product can be rewritten as
D
E
(N )
(N )
δ∅ , TN (fm , km ) · · · T+
(f
×
,
k
)
δ
h3 h3 ∅
N
D
´
³
E
(N )
(N )
× δ∅ , TN (fh2 , kh2 ) T+
B
f
,
k
δ
Vh1 1 1
N
∅
Iterating the same procedure for any p = 3, . . . , 2n, p ∈ 2N + 1, the thesis
follows.
¤
4.3
Boolean Central Limit Theorem
In this section we present the main result of the chapter. From now on H :=
L2 ([0, 1] , dx) . In order to prove our central limit theorem, as in the previous
chapter, we consider for any ε ∈ {−1, 0, 1} , for any B ∈ B (H) , for any
1 ≤ k ≤ N, and f ∈ L ([0, 1]) the ”normalized family”
 +
if ε = 1
 TN (f, k)
ε
e
c Λ (B)
if ε = 0
TN (f, k, B) :=
 k N
TN (f, k)
if ε = −1
where {ck } is a bounded sequence in R satisfying
N
1 X
ck = 1
lim √
N →∞ N
k=1
For any ε ∈ {−1, 0, 1} , we introduce the centered sum
N
ε
SN
1 Xeε
(f, B) := √
TN (f, k, B)
N k=1
and state the following central limit theorem.
(4.3.1)
93
4.3. Boolean Central Limit Theorem
Theorem 4.3.1. For any m ∈ N, for any ε ∈ {−1, 0, 1}m , for any f1 , . . . , fm ∈
L ([0, 1]), and B1 , . . . , Bm ∈ B (H) , the limit, for N → ∞, of
D
E
(N )
(N )
ε1
εm
δ∅ , SN
(fm , Bm ) · · · SN
(f1 , B1 ) δ∅
(4.3.2)
m
is equal to zero whenever ε ∈ {−1, 0, 1}m \ {−1, 0, 1}m
+,B and if ε ∈ {−1, 0, 1}+,B
it is equal to
hAεm (fm , Bm ) · · · Aε1 (f1 , B1 )i
Proof. In fact (4.3.2) can be rewritten as
N
D
E
X
1
(N ) e εm
(N )
ε1
e
δ∅ , T
(f
,
k
,
B
)
·
·
·
T
(f
,
k
,
B
)
δ
m
m m
m
1 1
1 ∅
N
N
N 2 k ,...,km =1
1
hence obtaining the first part of the thesis from Lemma 4.2.4. If ε ∈ {−1, 0, 1}m
+,B ,
then, from Lemma 4.2.5, this quantity is equal to



j
N
Y
X
1
 j

ckiq  ×
2
N q=1 kiq =1


2n
 1 Y
×
Nn
N D
X
p=1 khp =1
p∈2N+1
³
´
E
¢
¡
(N )
(N ) 

B
δ∅ , TN fhp+1 , khp+1 T+
f
,
k
δ
Vhp hp hp
N
∅

where m = 2n + j and, as defined above, 2n and j are respectively the cardinality of annihilator - creators and preservation operators. By (4.1.2), we
obtain





¶
µ
µ
¶¶
µ
j
N
2n
N
X
Y
X

 1 Y
khp
khp

 1j

BVhp fhp
ckiq  · 
f hp+1


n
N
N
N
2
N
q=1
p=1 khp =1
p∈2N+1
kiq =1
 


µ
¶
µ
¶
j
N
N
Y
X
X
1
1
kh1
kh1 

= j
ckiq  · 
f h2
BVh1 fh1
×
N
N
N
N 2 q=1 kiq =1
kh1 =1


µ
¶
µ
¶
N
X
k
k
1
h2n−1
h2n−1 
×... × 
f h2n
BVh2n−1 fh2n−1
N
N
N

kh1 =1
94
Chapter 4. Boolean Central limit Theorem
Taking the limit for N → ∞, the first factor converges to 1, thanks to (4.3.1),
while, for the remaining others, we recognize in each of them, a RiemannLebesgue sum. Hence we have
µZ
2n
Y
p=1
p∈2N+1
0
1
¶
f hp+1 (x) BVhp fhp (x) dx =
2n
D
Y
E
fhp+1 , BVhp fhp
p=1
p∈2N+1
H
By definition of annihilations, creation and preservation operators in the Boolean
Fock space, the quantity above is equal to
2n
D ¡
´
Y
¢ ³
¡ ¢E
A fhp+1 Λ BVhp A+ fhp =
p=1
p∈2N+1
2n
D ¡
³
´E
Y
¢
A fhp+1 A+ BVhp fhp
p=1
p∈2N+1
= hAεm (fm , Bm ) · · · Aε1 (f1 , B1)i
where the last equality is given by (4.2.5).
¤
Chapter 5
Quantum stochastic calculus
on interacting Fock spaces
Quantum stochastic calculus was initiated by Hudson and Parthasarathy in
[36] for Boson Fock space. After this pioneer work a great number of papers
was devoted to develop a theory in non Boson cases (see e.g. [15] for the
Fermion case, [35] for universal invariant case, [58] for free [42] for general
quasi-free, [19] for Boolean, [56] for full Fock module, [57] for q-deformed Fock
case). Accardi, Fagnola and Quaegebeur in [4] reached a double result: developing a theory independent on the particular representation chosen (as in the
classical case) from one hand and including all the quantum stochastic calculi
already appeared (boson and fermion) into a unifying picture from another
one. Successively Fagnola in [30] showed that a suitable extension of this theory allowed to construct a quantum stochastic integral for the ”free” noise case
introduced in [58] and moreover to give necessary and sufficient conditions for
the unitarity of the solution of quantum stochastic differential equations. On
the other hand, in the 90’s interacting Fock spaces appeared and later Accardi
and Bożejko in [2] (see also Chapter 2, Th. 2.4.3) proved that for 1-mode
interacting Fock spaces the interacting factors are related to the orthogonal
polynomials of probability measures by a unitary isomorphism between such
IFS and the L2 -space associated with the probability distribution chosen. This
result allows to express the interacting factors by means of the Jacobi parameters of the probability measure. Such an approach was later generalized to
finite dimensions bigger than one in [13] and to infinite dimensions in [12]. As
a consequence it emerges the need of developing a quantum stochastic calculus
for a class of standard interacting Fock spaces containing the 1-mode IFS and
96
Chapter 5. Quantum stochastic calculus on IFS
this is the aim of the present chapter. Since the free Fock space is a particular
interacting Fock space, it has been natural for us the idea of extending the
results contained in [30] to a much more general structure in which the presence of interacting functions deforms in a modular sense the free commutation
relations. In Section 5.1, after introducing some ”constraints” to get creation
and annihilation on IFS as bounded operators and to give the proof of the
main technical tool used (the semi-martingale estimates), we define stochastic processes as a family of operators whose domains contain a subset of the
number vectors. Moreover we introduce a family of ∗-subalgebras of the algebra of bounded operators on IFS, which plays the role of filtration in classical
stochastic calculus. Finally, yet in analogy with the classical case, we define
either simple adapted processes or stochastic integrals of them with respect
to the basic processes of creation and annihilation. Section 5.2 is devoted to
the proof of semi-martingale inequalities for simple adapted processes, which
allows to majorize stochastic integrals of simple adapted processes by ordinary
ones and to reach the successive results. In particular we present two different proofs either for the case of non constant interacting functions or for the
1-mode type interacting Fock spaces (1-MT IFS). In Section 5.3 we introduce
a quantum stochastic integral for a class of processes wider than the simple
adapted ones. Namely this class consists of limit operators of simple adapted
processes with respect to the topologies of strong ∗-convergence and the one
induced by a family of semi-norms on suitable domains. The necessity of the
latter topology reveals in order to prove existence and uniqueness of solutions
for a class of quantum stochastic differential equations (QSDE). In fact in
Section 5.4 we study the problem of giving existence and uniqueness for the
solution of a class of quantum stochastic differential equations (QSDE) with
respect to the basic processes in standard interacting Fock spaces. Moreover
we investigate the necessary and sufficient condition on the ”terms” of our
QSDE, under which this solution is a unitary operator. To this aim, thanks to
the semi-martingale estimates developed in 5.2, in Section 5.5 we construct a
Ito table and a quantum Ito formula in the weak sense, which is the technical
tool in order to prove the the unitarity condition, as presented in Section 5.6.
It is worth to mention that the Ito table here is, as usual, trivial in all cases
but one, namely dA (t) dA+ (t). In such a case we find an operator valued
deformation, due to the presence of interacting functions, which generalizes in
a modular sense the relation achieved in the free Fock case. We notice that all
the results contained in the last three sections are set in standard interacting
Fock space with non constant interacting functions; the 1-mode type interact-
97
5.1. Simple adapted processes
ing Fock space case can be similarly treated by using slight modifications.
5.1
Simple adapted processes
As presented in the introduction, our purpose is to set stochastic calculus
theory over standard interacting free Fock space, hence we firstly introduce
some basic definitions and properties on such a structure. From now on we
suppose there exist a sequence of positive numbers (Bn )n≥1 and a constant
M > 0 such that for any n ≥ 1
λn (xn , ..., x1 ) ≤ Bn ,
λn (xn , ..., x1 )
λn+1 (x0 , xn , ..., x1 )
≤ M;
≤ M for any k = 0, 1, ..., n
λn+1 (x0 , xn , ..., x1 )
λn (x0 , ..., x
ck , ..., x1 )
(5.1.1)
for almost all (x0 , xn , ..., x1 ) ∈ X n+1 . The conditions above are introduced
for technical reasons, in particular they will used to satisfy some boundedness
conditions in the proof of the ”semi-martingale inequalities”, but they do not
seem very natural in the context of 1-mode type interacting Fock spaces. In
fact, as a consequence of the unitary isomorphism theorem due to Accardi and
Bożejko (see [2], Th. 3.1), Accardi, Kuo and Stan in [12] computed the weights
λn for the most important distributions on the real line uniquely determined
by their moments, finding that for Gauss, Poisson and Gamma distributions
for any n ∈ N
λn
=: ωn ∼ n
λn−1
Then in such a case we need to replace the second line of (5.1.1) and namely
we request there exists a sequence of increasing positive numbers (Mn )n≥1
such that for any n ≥ 1
λn
λn
≤ Mn ;
≤ Mn
λn+1
λn−1
(5.1.2)
From now on in the paper we will stand out the differences between the general
case and the 1-MT IFS.
Moreover as a consequence of (5.1.1) or (5.1.2) respectively, both creator
and annihilator can be extended to a common invariant dense subset of ΓI (H)
where there are still mutually adjoint.
98
Chapter 5. Quantum stochastic calculus on IFS
From now on we shall take¡ the measure
space (X, dµ) as (R+ , dx), i.e.
¢
2
2
H := L (R+ ), denote FI := ΓI L (R+ ) and denote
D := {Φ, u1 ⊗ ... ⊗ uk : k ∈ N,
ª
uj ∈ L2 (R+ , dx) ∩ L1 (R+ , dx) , kuj kL2 ≤ 1 and kuj kL1 ≤ 1
then LispanD ⊆Dom(A+ (f )) ∩Dom(A (f )) . Furthermore by the symbol L (D, FI )
we denote the vector space of all linear operators A with domain containing D
such that their (essential) adjoint operator A+ also contains D in its domain.
From the above discussion, it follows that this set is not empty.
The following definition introduces the notion of stochastic process.
Definition 5.1.1. A family (X (t))t≥0 of elements of L (D, FI ) is called a
stochastic process in FI if for any ξ ∈ D, the map t 7→ X (t) ξ is strongly
measurable.
The notations and definitions below will be used throughout the paper:
• for any s, t ∈ R+ such that s ≤ t
¡
¢
A (t) := A χ[0,t) ,
¡
¢
A (s, t) ; = A χ[s,t) ,
¡
¢
A+ (t) := A+ χ[0,t)
¡
¢
A+ (s, t) := A+ χ[s,t)
and
M 01 := (A (t))t≥0 ;
¡
¢
M 10 := A+ (t) t≥0 ;
M 00 := (t1)t≥0
are called the annihilation, creation and deterministic processes respectively and they are stochastic processes in the sense of Definition 5.1.1.
• for any t ∈ [0, +∞] one denotes by At] the linear span of
n
Bt := Aε(1) (g1 ) Aε(2) (g2 ) · · · Aε(n) (gn ) : n ∈ N∪ {0} ,
ª
ε ∈ {−1, 1}n , gk ∈ L2 (0, t) for any k = 1, 2, · · · , n
(5.1.3)
where Aε(1) (g1 ) Aε(2) (g2 ) · · · Aε(n) (gn ) is understood as the identity if
n = 0 and
½
A (g)
if ε = −1
ε
A (g) :=
(5.1.4)
+
A (g) if ε = 1
It is clear that At] is a ∗-subalgebra of B (FI ) for any t.
99
5.1. Simple adapted processes
Remark 5.1.2. In 1-MT IFS the annihilation operator is such that for any
n ∈ N, for any f, fn , . . . , f1 ∈ H
A (f ) fn ⊗ fn−1 ⊗ · · · ⊗ f1 =
λn
hf, fn i fn−1 ⊗ · · · ⊗ f1
λn−1
then each element of At] can be written as a finite sum of operators of the
form
c (t, f, g) A+ (fh ) · · · A+ (f1 ) A (gl ) · · · A (g1 )
(5.1.5)
¡
¢
¡
¢
f ∈ L2 (0, t) , Cd , g ∈ L2 (0, t) , Cl , d, l ∈ N, c (t, f, g) ∈ C.
A stochastic process is called simple adapted if it can be written as
n
X
F (tk ) χ[tk ,tk+1 )
(5.1.6)
k=1
where
n ∈ N, 0 ≤ t1 < t2 < ... < tn+1 < +∞, F (tk ) ∈ Atk ] , ∀ k = 1, ..., n.
Let S be the vector space of simple adapted processes. In order to prove
the ”semi-martingale” inequalities for the case of non constant interacting
functions, in the notations above we introduce the set
©
¡ ¢
Ptk := A (g) A+ g 0 | g, g 0 ∈ L2 (0, tk ) , ∀n ∈ N, ∀ G ∈ Hn :
o
¡ 0¢
­ 0®
A (g) A g G = K g, g L2 (0,t ) G, K ∈ R+
+
k
The reason we consider only the general case for the definition above is due to
the fact that in 1-MT IFS for any g, g 0 ∈ L2 (0, tk ) A (g) A+ (g 0 ) belongs to Ptk ,
hence the definition becomes meaningless, hence from now on any property
related to Ptk will be understood only for the case of non constant interacting
functions.
Let us fix 0 ≤ t < +∞ and³let´us take G, G0 ∈ Bt such that there exist
¡ ¢
g1 , . . . , gn ∈ L2 (0, t) , A g j A+ g 0j ∈ Pt , j = 1, . . . , m, m ≤ n, such that
¡ ¢
¡ ¢
G (t) = Aε(1) (g1 ) · · · A (g 1 ) A+ g 01 · · · A (g m ) A+ g 0m · · · Aε(n) (gn )
+ (g 0 ) · · · A
+ (g 0 ) · · · Aε(n) (g )
\
\
G0 (t) = Aε(1) (g1 ) · · · A
(g 1 )A\
(g m )A\
n
m
1
(5.1.7)
100
Chapter 5. Quantum stochastic calculus on IFS
or
+ (g 0 ) · · · A
+ (g 0 ) · · · Aε(n) (g )
\
\
G (t) = Aε(1) (g1 ) · · · A
(g 1 )A\
(g m )A\
n
m
1
¡
¢
¡ 0 ¢
0
ε(1)
+
0
+
ε(n)
G (t) = A
(g1 ) · · · A (g 1 ) A g 1 · · · A (g m ) A g m · · · A
(gn ) (5.1.8)
Definition 5.1.3. Let G and G0 be defined as above. We say that G and G0
are related and we will denote it by G (t) Rt G0 (t) , whenever one of the two
conditions (5.1.7) or (5.1.8) holds.
Remark 5.1.4. For any t ≥ 0 clearly Rt realizes an equivalence relation on
Bt whose quotient will be denoted by Bt0 . Without loss of generality, we can
consider, for any G (t) in Bt , [G (t)]Rt as the element in Bt0 which does not
contain any operator in Pt . From now on such a choice will be assumed and
we denote the equivalence class simply by G (t) .
As a consequence of the remark above, we can give the following definition
Definition 5.1.5. For any t ≥ 0, for any G (t) in Bt0 we define the order of
G (t) and denote it by ordG (t), the cardinality of the set of operators (creators
and annihilators) forming G (t) .
Remark 5.1.6. In 1-MT IFS we know that for any t ≥ 0 Pt is not defined,
hence the definition of order is given as above for any element in Bt .
Example 5.1.7. Let us take G (t) in Bt0 , then G (t) = Aε(1) (g1 ) · · · Aε(m) (gm ) ,
ε ∈ {0, 1}m , gj ∈ L2 (0, t). In this case ordG (t) = m.
We notice that for any t ≥ 0, any element F (t) of the ∗−subalgebra At]
can be written as
m
X
αh Fh (t)
F (t) =
k=1
Bt0
αh ∈ C, Fh (t) ∈
(Bt in 1-MT IFS) h = 1, . . . , m and we denote by S 0 the
vector space of simple adapted processes F such that
F =
n
X
F (tk ) χ[tk ,tk+1 ) , F (tk ) =
k=1
m
X
αh Fh (tk ) , Fh (tk ) ∈ Bt0 k
h=1
Clearly in 1-MT IFS S 0 = S . Moreover by definition ordFh (tk ) < +∞ for any
tk , then we set
(F )
(F )
Ntk := max ordFh (tk ) , N (F ) := max Ntk
1≤h≤m
1≤k≤n
101
5.2. Semi-martingale Inequalities
Right and left stochastic integrals of simple adapted processes can be defined
as usual
Z t
n
X
F (s) dA (s) :=
F (tk ) A (tk ∧ t, tk+1 ∧ t)
0
Z
k=1
t
dA (s) F (s) :=
0
Z
t
A (tk ∧ t, tk+1 ∧ t) F (tk )
k=1
+
F (s) dA (s) :=
0
Z
n
X
n
X
F (tk ) A+ (tk ∧ t, tk+1 ∧ t)
k=1
t
dA+ (s) F (s) :=
0
n
X
A+ (tk ∧ t, tk+1 ∧ t) F (tk )
k=1
where tk ∧ t := min {tk , t} .
5.2
Semi-martingale Inequalities
This section is devoted to the proof of the ”semi-martingale” estimates for left
and right stochastic integral with respect to simple adapted processes. They
were firstly introduced in [4] and successively used in [30] and are the main
technical tool in order to extend the stochastic integral to a wider class of
processes from one hand and, as shown in [?], to prove a quantum Ito formula
from another one.
Two different main propositions will be presented: for the general (standard) case and for the 1-MT IFS,which reflect the different choices of the
boundedness conditions (5.1.1) or (5.1.2)respectively. We will firstly treat the
case of general IFS, giving some preliminary results, which can be considered
as technical tools for the main result.
Lemma 5.2.1. For any interacting Fock space FI , with interacting functions
(λn ) satisfying (5.1.1), the following inequalities hold:
i) for any d, k ∈ N, for any x1 , . . . , xd , y1 , . . . , yk+1 ∈ R+
λd+k+1 (yk+1 , . . . , y1, xd , . . . , x1 )
λd+k (yk+1 , . . . , ybj , . . . , y1, xd , . . . , x1 )
≤M
λd (xd , . . . , x1 )
λd (xd , . . . , x1 )
for all j = 1, . . . , k + 1.
102
Chapter 5. Quantum stochastic calculus on IFS
ii) for any d, k ∈ N, for any x1 , . . . , xd , y1 , . . . , yk , z ∈ R+
λd+k (yk , . . . , y1, xd , . . . , x1 )
λd+k+1 (z, yk , . . . , y1, xd , . . . , x1 )
≤M
λd (xd , . . . , x1 )
λd (xd , . . . , x1 )
iii) for any d, k, m ∈ N, for any x1 , . . . , xd , y1 , . . . , yk , t1 , . . . , tm , z ∈ R+
λd+k (yk , . . . , y1, xd , . . . , x1 )
λd+k+1 (z, yk , . . . , y1, xd , . . . , x1 )
≤ M2
λd+m+1 (z, tm , . . . , t1, xd , . . . , x1 )
λd+m (tm , . . . , t1, xd , . . . , x1 )
Proof. i). In fact using the the notations of the statement
λd+k+1 (yk+1 , . . . , y1, xd , . . . , x1 )
λd (xd , . . . , x1 )
λd+k+1 (yk+1 , . . . , y1, xd , . . . , x1 )
λd+k (yk+1 , . . . , ybj , . . . , y1, xd , . . . , x1 )
=
·
λd+k (yk+1 , . . . , ybj , . . . , y1, xd , . . . , x1 )
λd (xd , . . . , x1 )
λd+k (yk+1 , . . . , ybj , . . . , y1, xd , . . . , x1 )
≤M
λd (xd , . . . , x1 )
ii) can be obtained in the same way. Let us turn to prove iii).
λd+k+1 (z, yk , . . . , y1, xd , . . . , x1 )
λd+m+1 (z, tm , . . . , t1, xd , . . . , x1 )
λd+k (yk , . . . , y1, xd , . . . , x1 )
λd+m (tm , . . . , t1, xd , . . . , x1 )
=
×
λd+m (tm , . . . , t1, xd , . . . , x1 ) λd+m+1 (z, tm , . . . , t1, xd , . . . , x1 )
λd+k+1 (z, yk , . . . , y1, xd , . . . , x1 )
×
λd+k (yk , . . . , y1, xd , . . . , x1 )
λd+k (yk , . . . , y1, xd , . . . , x1 )
≤ M2
λd+m (tm , . . . , t1, xd , . . . , x1 )
¤
Lemma 5.2.2. Let us take d ∈ N. For all f, g, h, k, u1 , . . . , ud ∈ L2 (R+ ) ,
under the conditions (5.1.1), we have:
A (f ) A (g) A+ (h) A+ (k) (ud ⊗ · · · ⊗ u1 )
≤ M hf, kiL2 (R+ ) A (g) A+ (h) (ud ⊗ · · · ⊗ u1 )
103
Proof. In the same notations of the statement, we consider x1 , . . . , xd ∈ R+
and
A (f ) A (g) A+ (h) A+ (k) (ud ⊗ · · · ⊗ u1 ) (xd , . . . , x1 )
·Z
¸
d
Y
λd+2 (z, y, xd , . . . , x1 )
= A (f )
dz
g (z) h (z) k (y)
uj (xj )
λd+1 (y, xd , . . . , x1 )
R+
j=1
Z Z
λd+1 (y, xd , . . . , x1 )
dydz
=
f (y) k (y)
λd (xd , . . . , x1 )
R+ R+
d
Y
λd+2 (z, y, xd , . . . , x1 )
g (z) h (z)
uj (xj )
×
λd+1 (y, xd , . . . , x1 )
j=1
By i) of Lemma 5.2.1, the quantity above is less than or equal to
Z
d
Y
λd+1 (z, xd , . . . , x1 )
dz
g (z) h (z)
uj (xj )
λd (xd , . . . , x1 )
R+
M hf, kiL2 (R+ )
j=1
+
= M hf, kiL2 (R+ ) A (g) A (h) (ud ⊗ · · · ⊗ u1 ) (xd , . . . , x1 )
¤
The following proposition is the first part of ”semi-martingale” estimates
for standard IFS with non constant interacting functions.
Proposition 5.2.3. Let F ∈ S 0 , d ∈ N . For all ξ = gd ⊗ · · · ⊗ g1 ∈ D and
h = 1, ..., d, let us denote ηh := gd ⊗ · · · ⊗ gh+1 ⊗ gh−1 ⊗ · · · ⊗ g1 ∈ D (with the
convention that ηh := 0 if ξ = Φ ; ηh := Φ if d = 1 ). Then, for all t ∈ R+
we have
°Z t
°2
Z t
°
°
° F (s) dA (s) ξ ° ≤ M 2
kF (s) ηd k2 ds
(5.2.1)
°
°
0
0
°Z t
°2
Z t
°
°
° dA+ (s) F (s) ξ ° ≤ M
kF (s) ξk2 ds
°
°
(5.2.2)
°Z t
°2
d−1 Z t
X
°
°
° dA (s) F (s) ξ ° ≤ M 2
kF (s) ηd−h k2 ds
°
°
(5.2.3)
°2
°Z t
Z t
°
°
° F (s) dsξ ° ≤ t
kF (s) ξk2 ds
°
°
(5.2.4)
0
0
0
h=0 0
0
0
104
Chapter 5. Quantum stochastic calculus on IFS
Proof. We begin by showing (5.2.1).
°Z t
°2
°
°
° F (s) dA (s) ξ °
°
°
0
°
°2
n
°X
°
°
°
= ° F (tk ) A (tk ∧ t, tk+1 ∧ t) ξ °
°
°
k=1
°
°2
n Z tk+1 ∧t
°X
°
λd (xd , . . . , x1 )
°
°
F (tk ) dxd
=°
gd (xd ) ηd °
°
°
λd−1 (xd−1 , . . . , x1 )
k=1 tk ∧t
°
°
n Z tk+1 ∧t
°X
°2
°
2°
≤M °
dxd gd (xd ) F (tk ) ηd °
°
°
k=1 tk ∧t
°Z t
°2
Z t
°
°
2°
2
°
= M ° gd (s) F (s) ηd ds° ≤ M
kF (s) ηd k2 ds
0
0
where the last inequality is obtained by using Cauchy-Schwarz inequality.
Now we turn to prove (5.2.2).
°Z t
°2
°
°
° dA+ (s) F (s) ξ °
°
°
0
=
=
n
X
­ +
®
A (tk ∧ t, tk+1 ∧ t) F (tk ) ξ, A+ (th ∧ t, th+1 ∧ t) F (th ) ξ
k,h=1
n
X
­
®
F (tk ) ξ, A (tk ∧ t, tk+1 ∧ t) A+ (th ∧ t, th+1 ∧ t) F (th ) ξ
(5.2.5)
k,h=1
We observe that, by definition, the quantity above vanishes when h 6= k. This
implies that only the diagonal elements of the sum above survive, i.e. (5.2.5)
is equal to
n
X
­
®
F (tk ) ξ, A (tk ∧ t, tk+1 ∧ t) A+ (tk ∧ t, tk+1 ∧ t) F (tk ) ξ
k=1
=
m
n X
X
­
®
αl αr Fl (tk ) ξ, A (tk ∧ t, tk+1 ∧ t) A+ (tk ∧ t, tk+1 ∧ t) Fr (tk ) ξ
k=1 l,r=1
(5.2.6)
¡
¢
where, for any r = 1, . . . , m, Fr (tk ) = Aε(1) (gr1 ) · · · Aε(p) grp , grj ∈ L2 (0, tk ) ,
ε (j) ∈ {−1, 1} for any j = 1, . . . , p. Let us consider
c := |{j = 1, . . . , p : ε (j) = 1}| − |{j = 1, . . . , p : ε (j) = −1}|
105
where, as usual, |·| denotes the cardinality. Notice that c must be non negative,
in fact if in the sequence of operators of Fr (tk ) we have more annihilators than
creators, the scalar product vanishes. Hence (5.2.6) is equal to
n X
m
X
¿
Z
αl αr Fl (tk ) ξ,
tk+1 ∧t
λd+c+1 (x, xc , . . . , x1 , yd , . . . y1 )
Fr (tk ) ξ
λd+c (xc , . . . , x1 , yd , . . . y1 )
t
∧t
k
k=1 l,r=1
Z t
n
X
≤M
h(F (tk )) ξ, (tk+1 ∧ t − tk ∧ t) F (tk ) ξi = M
kF (s) ξk2 ds
À
dx
0
k=1
For (5.2.3) we consider
°2
°Z t
°2 °
n X
m
°X
°
°
°
°
°
° dA (s) F (s) ξ ° = °
α
A
(t
∧
t,
t
∧
t)
F
(t
)
ξ
°
r
r
k
k+1
k
°
°
°
°
0
(5.2.7)
k=1 r=1
Let us denote, for any r = 1, . . . , m, p := ordFr (tk ) and, in the same notations
as the previous case, we call c0 = −c. By the adaptness of the process, the
quantity above can be different from zero only if 0 ≤ c0 < d, ε (p) = −1.
As a consequence there exist some annihilators in Fr (tk ) (whose number is
c0 ) acting on the vector ξ, whereas the remaining annihilators act on the
creators of itself. Hence p = c0 + 2q. Then, if {lh , ∇h }qh=1 is the non crossing
pair partition determined by ε ∈ {−1, 1}2q (see [9] for details) and, for any
j = 1, . . . , c0 , g (r,j) denotes the test function if any annihilator in Fr (tk ) not
coupled with any creator therein and for any h = 1, . . . , q g (r,lh ) g (r,∇h ) one
obtains
A (tk ∧ t, tk+1 ∧ t) Fr (tk ) gd ⊗ · · · ⊗ g1 (xd−c0 −1 , . . . , x1 )
ÃZ
Z t Z tk+1 ∧t
t
λd−c0 (xd−c0 , . . . , x1 )
gd−c0 (xd−c0 ) ×
=
···
λd−c0 −1 (xd−c0 −1 , . . . , x1 )
0
0
tk ∧t
0
c
´
Y
λd−j+1 (xd−j+1 , . . . , x1 ) ³ (r,j)
×
g
gd−j+1 (xd−j+1 ) ×
λd−j (xd−j , . . . , x1 )
j=1
!
q
³
´
Y
×
Λ · g (r,lh ) g (r,∇h ) (x∇h ) dx∇h dxd−j+1 dxd−c0 ×
h=0
× gd−c0 −1 ⊗ · · · ⊗ g1 (xd−c0 −1 , . . . , x1 )
where Λ is a product of a certain number of fractions of λn ’s. The module of
106
Chapter 5. Quantum stochastic calculus on IFS
the quantity can be majorized by
µZ t
Z t Z tk+1 ∧t
M
···
gd−c0 (xd−c0 ) ×
0
0
tk ∧t
c0
Y
´
λd−j+1 (xd−j+1 , . . . , x1 ) ³ (r,j)
g
gd−j+1 (xd−j+1 ) ×
λd−j (xd−j , . . . , x1 )
j=1
!
q
´
³
Y
(r,lh ) (r,∇h )
g
(x∇h ) dx∇h dxd−j+1 dxd−c0 ×
×
Λ· g
×
h=0
× gd−c0 −1 ⊗ · · · ⊗ g1 (xd−c0 −1 , . . . , x1 )
Since c0 depends on the choice of Fr (tk ), it can takes all values among 0 and
d − 1, hence (5.2.7) is less than or equal to to
° n d−1 Z
°2
°X X tk+1 ∧t
°
°
2°
M °
gd−h (r) F (tk ) ηd−h dr°
°
°
k=1 h=0 tk ∧t
°2
d−1 °Z t
X
°
°
2
°
°
≤M
° gd−h (s) F (s) ηd−h ds°
h=0
0
and (5.2.3) follows from the Cauchy-Schwarz inequality, that also gives directly
(5.2.4).
¤
Rt
The proof of a semi-martingale estimate for 0 F (s) dA+ (s) ξ can not be
achieved by direct computations, but needs of some preliminary notations and
results.
0
Let us consider F ∈ S
P,nthen there exist n ∈ N, 0 ≤ t1 < t2 < . . . < tn+1 <
+∞ such that F (s) = k=1 F (tk ) χ[tk ,tk+1 ) (s), F (tk ) ∈ Atk ] . By using the
same notations and arguments developed in the case (5.2.2) of the previous
proposition, we have:
°Z t
°2
°
°
° F (s) dA+ (s) ξ °
°
°
0
=
n
X
­
®
ξ, A (tk ∧ t, tk+1 ∧ t) (F (tk ))∗ F (tk ) A+ (tk ∧ t, tk+1 ∧ t) ξ
k=1
By definition, the right hand side above can be written as
n X
m
X
k=1 l,r=1
­
®
αl αr ξ, A (tk ∧ t, tk+1 ∧ t) (Fl (tk ))∗ Fr (tk ) A+ (tk ∧ t, tk+1 ∧ t) ξ
107
The adaptness of the Fl (tk )’s and Fr (tk )’s implies that the possible non zero
contributions to the scalar products above can be obtained when in any element of the sums above A (tk ∧ t, tk+1 ∧ t) acts on A+ (tk ∧ t, tk+1 ∧ t) . This
circumstance, together with the non-crossing principle for interacting Fock
spaces, gives us some conditions on (Fl (tk ))∗ Fr (tk ) for any l, r = 1, . . . , m,
which we present in the following lemmata.
Firstly we denote, for any l, r = 1, . . . , m
³
´
³
´
(l,r)
(l,r)
(Fl (tk ))∗ Fr (tk ) := Aε(l,r) (p) gp,tk · · · Aε(l,r) (1) g1,tk
¡
¢
(l,r)
where p ∈ N, ε(l,r) (p) , . . . , ε(l,r) (1) ∈ {−1, 1}p , gj,tk ∈ L2 (0, tk ) , j =
1, . . . , p.
The successive result is similar to Lemma 2.2.6, but we prefer to put here
again for the sake of completeness of the chapter.
Lemma 5.2.4. In the same notations as above
³
´
³
´
(l,r)
(l,r)
A (tk ∧ t, tk+1 ∧ t) Aε(l,r) (p) gp,tk · · · Aε(l,r) (1) g1,tk A+ (tk ∧ t, tk+1 ∧ t) ξ
is different
from zero only if the following
¯©
ª¯ ¯© conditions are satisfied: ª¯
i) ¯ ε(l,r) (j) = 1 : j = 1, . . . , p ¯ = ¯ ε(l,r) (j) = −1 : j = 1, . . . , p ¯ ;
ii) ε(l,r) (1) = 1, ε(l,r) (p) = −1;
iii) for any j = 2, . . . , p
¯©
ª¯ ¯©
ª¯
¯ ε(l,r) (k) = 1 : 1 ≤ k < j ¯ ≥ ¯ ε(l,r) (k) = −1 : 1 ≤ k < j ¯
Proof. In fact
not hold, namely take the
¯© , let us firstly suppose that
ª¯ i) ¯does
©
ª¯
case in which ¯ ε(l,r) (j) = 1 : j = 1, . . . , p ¯ < ¯ ε(l,r) (j) = −1 : j = 1, . . . , p ¯ .
By the non-crossing principle, there exists an annihilator in the sequence
∗
+
(F
¯© l (tk )) Fr (tk ) coupled with
ª¯ A ¯©(tk ∧ t, tk+1 ∧ t) , thus givingª¯zero. The case
¯ ε(l,r) (j) = 1 : j = 1, . . . , p ¯ > ¯ ε(l,r) (j) = −1 : j = 1, . . . , p ¯ is similar.
³
´
(l,r)
Let us turn to prove ii). If ε(l,r) (1) = −1, then A g1,tk is coupled with
A+ (tk ∧ t, tk+1 ∧ t) , whereas if ε(l,r) (p) = 1, then A (tk ∧ t, tk+1 ∧ t) is coupled
³
´
(l,r)
with A+ gp,tk , giving zero in both cases.
Finally, if there exists j = 1, . . . , p such that iii) is not verified,
³
´ then, by the
(l,r)
ε(l,r) (j)
non-crossing principle, on the right hand side of A
gj,tk there exists
an annihilator coupled with A+ (tk ∧ t, tk+1 ∧ t) thus giving zero.
¤
108
Chapter 5. Quantum stochastic calculus on IFS
¡
¢
As already done, we introduce the notation ε(l,r) = ε(l,r) (p) , . . . , ε(l,r) (1) ∈
{−1, 1}2N
ε(l,r) realizes the conditions i),..., iii)
+ to express that the partition
n³
´
o
of the Lemma above, denote by
lN(l,r) , rN(l,r) , . . . , (l1 , r1 ) the unique non©
ª
crossing pair partition on the set 1, . . . , 2N(l,r) and assume that it is increasingly ordered with respect to the left indices lj ’s.
The following definition is given in order to prove a very useful result for
the semi-martingale estimate we will present later.
Definition 5.2.5. A non-crossing pair partition {lj , rj }nj=1 of {1, 2, . . . , 2n}
such that l1 < l2 < . . . < ln is called connected if for any k = 1, . . . , n one has
{lk , rk } ⊆ {ln , rn } (i.e. ln > lk > rk > rn ). A subset {li , ri }i∈I , I ⊆ {1, . . . , n} ,
|I| ∈ 2N, of {lj , rj }nj=1 is called a connected component of {lj , rj }nj=1 if it is a
connected non-crossing pair partition. A non-crossing pair partition {lj , rj }nj=1
is called interval partition if for any j = 1, . . . , n lj = rj + 1.
Remark 5.2.6. Any connected component of an interval partition is given
by a pair of two consecutive ”left-right” elements of the set {1, 2, . . . , 2n} .
Lemma 5.2.7. In the same notations of Lemma 5.2.4, let us suppose that the
pair partition {lj , rj }nj=1 induced by ε ∈ {−1, 1}2n
+ is such that 1 = rn < . . . <
r1 < l1 < . . . < ln = 2n. Then, for any gl1 , . . . , gln , gr1 , . . . , grn ∈ H = L2 (R+ ) ,
any d ∈ N, ξ ∈ Hd , any yln , . . . , yl1 , xd , . . . , x1 ∈ R+ one has:
h
i
Aε(ln ) (gln ) · · · A (gl1 ) A+ (gr1 ) · · · Aε(rn ) (grn ) ξ (xd , . . . , x1 )
"Z
λd+dε (l1 ) (yln , . . . , yl1 , xd , . . . , x1 )
×
=
n
λd (xd , xd−1 , . . . , x1 )
R+

n ³
´¡ ¢
Y
×
g lj grj ylj dylj  ξ (xd , . . . , x1 )
j=1
where dε is the depth function introduced in Definition 3.2.2.
109
Proof. In fact by definition, for any yln , . . . , yl1 , xd , . . . , x1 ∈ R+
h
i
Aε(ln ) (gln ) · · · A (gl1 ) A+ (gr1 ) · · · Aε(rn ) (grn ) ξ (xd , . . . , x1 )
·Z
¸
¢
λd+dε (l1 ) (yln , . . . , yl1 , xd , . . . , x1 ) ¡
=
dyl1
g gr (yl1 ) ×
λd+dε (l1 )−1 (yln , . . . , yl2 , xd , . . . , x1 ) l1 1
R+
¸
·Z
¢
λd+dε (l2 ) (yln , . . . , yl2 , xd , . . . , x1 ) ¡
×
dyl2
g gr (yl2 ) ×
λd+dε (l2 )−1 (yln , . . . , yl3 , xd , . . . , x1 ) l2 2
R+
·Z
¸
¢
λd+dε (ln ) (yln , xd , . . . , x1 ) ¡
× ··· ×
dyl2
g ln grn (yln ) ξ (xd , . . . , x1 )
λd (xd , . . . , x1 )
R+
The thesis follows by recalling that for any j = 1, . . . , n − 1, dε (lj ) − 1 =
dε (lj+1 ) .
¤
As a consequence of the lemma above, any sequence of operators giving
a pair partition ε ∈ {−1, 1}2n
+ of the set {1, 2, . . . , 2n} such that, in our
notations, dε (l1 ) = n, once applied to a d particle vector of the space, give
only one fraction of the λn ’s in analogy with the case of a pair of operators.
The difference between the two partitions is that in the latter case the λn ’s
in the fraction are consecutive, that is not true in the former general case;
anyway in both cases the index of the numerator indicates the depth of the
partition. Now we can investigate what happens for a sequence of annihilators
and creators, inducing a connected pair partition, which acts on a certain
vector. We firstly deal with a particular case.
Lemma 5.2.8. Let us given a non-crossing pair partition {lj , rj }nj=1 such
that for any j = 1, . . . , n − 1, lj = rj + 1, ln = 2n, rn = 1. Then, for
any gl1 , . . . , gln , gr1 , . . . , grn ∈ L2 (R+ ) , any d ∈ N, ξ ∈ Hd , any yln , . . . , yl1 ,
xd , . . . , x1 ∈ R+ one has:
h
i
Aε(ln ) (gln ) · · · A (gl1 ) A+ (gr1 ) Aε(rn ) (grn ) ξ (xd , . . . , x1 )
Z
¢
¡
¢
λd+2 (yln , yl1 , xd , . . . , x1 ) ¡
=
dyl1 dyln
g ln grn (yln ) g l1 gr1 (yl1 ) ×
λd (xd , . . . , x1 )
Rn
 +

¢
¡
n−1
´¡ ¢
Y λd+2 yln , ylj , xd , . . . , x1 ³
×
g lj grj ylj dylj  ξ (xd , . . . , x1 )
λd+1 (yln , xd , . . . , x1 )
j=2
110
Chapter 5. Quantum stochastic calculus on IFS
Proof. In fact
h
i
Aε(ln ) (gln ) · · · A (gl1 ) A+ (gr1 ) Aε(rn ) (grn ) ξ (xd , . . . , x1 )
·Z
¢
λd+2 (yln , yl1 , xd , . . . , x1 ) ¡
=
dyl1
g l1 gr1 (yl1 ) ×
λd+1 (yln , xd , . . . , x1 )
R+
Z
¢
λd+2 (yln , yl2 , xd , . . . , x1 ) ¡
×
dyl2
g l2 gr2 (yl2 ) ×
λd+1 (yln , xd , . . . , x1 )
R+
¡
¢
Z
´¡
¢
λd+2 yln , yln−1 , xd , . . . , x1 ³
× ··· ×
dyln−1
g ln−1 grn−1 yln−1 ×
λd+1 (yln , xd , . . . , x1 )
R+
¸
Z
¢
λd+1 (yln , xd , . . . , x1 ) ¡
g ln grn (yln ) ξ (xd , . . . , x1 )
×
dyln
λd (xd , . . . , x1 )
R+
from which the thesis follows.
¤
Hence, for any connected pair partition satisfying the assumptions of the
above lemma, after the action on a vector ξ, the number of fractions of the
λn ’s is exactly given by the number of the indices consecutive pairs. This
result, together with Lemma 5.2.7, allows us to state that the computation
of the number of fractions of λn ’s appearing after the action of a sequence
of annihilators and creators inducing a connected pair partition, is given by
the number of indices consecutive pairs. This is the content of the following
proposition.
Proposition 5.2.9. In the same notations as above, let us denote
k := |{{lj , rj } : rj = lj − 1, j = 1, . . . , n}|
Then, after computing
¡ ¢
¡ ¢
Aε(ln ) (gln ) · · · A glj · · · A+ grj · · · Aε(rn ) (grn ) ξ
it appears exactly k fractions of the λn ’s.
Proof. The thesis can be obtained by iteration. In fact, let us fix the first pair
of consecutive left-right indices from the right in the sequence, say {lk1 , rk1 } .
If either lk1 − 1 or rk1 + 1 is a left index, we turn to the successive indices
consecutive pair. On the contrary, if lk1 − 1 is a left index and rk1 + 1 is a
right index, by Lemma 5.2.7, in the computation of the number of fractions,
we reduce it to a single pair. After removing we repeat the same reasoning
111
and iterate it for all the pairs of consecutive left-right indices. Finally we find
the same partition described in Lemma 5.2.8.
¤
Now let us consider two elements Fl (tk ) , Fr (tk ) ∈ Bt0 k such that
(Fl (tk ))∗ Fr (tk ) = Aε(2n) (g2n ) · · · Aε(1) (g1 ) , ε ∈ {−1, 1}2n
+ , let us denote
(l,r)
N tk
:= |{{lj , rj } : rj = lj − 1, j = 1, . . . , n}|
and give the following
Lemma 5.2.10. For any d ∈ N, for any ξ = gd ⊗ . . . ⊗ g1 ∈ D
¯­
®¯
¯ ξ, A (tk ) (Fl (tk ))∗ Fr (tk ) A+ (tk ) ξ ¯
¯¿ Z t ∧t
À¯
k+1
(l,r)
¯
¯
2N t −1 ¯
∗
¯
k
≤M
ξ,
dy
(F
(t
))
F
(t
)
ξ
r
l
k
k
¯
¯
(5.2.8)
tk ∧t
Proof. We firstly notice that the sequence A (tk ) (Fl (tk ))∗ Fr (tk ) A+ (tk ) de(l,r)
termines a connected pair partition with N tk indices consecutive pairs. Then,
(l,r)
by Proposition 5.2.9, its action on ξ gives N tk
fractions of the λn ’s. We
(l,r)
prove (5.2.8) by induction. In fact, let us suppose that N tk
ε0 ∈ {−1, 1}2(n+1) such that
 0
 ε (1) = 1
ε0 (j) = ε (j − 1) , j = 2, . . . , n + 1
 0
ε (2n + 2) = −1
= 1 and consider
(5.2.9)
on+1
n
the non-crossing pair partition determined by ε0 ,
Let us denote by lj0 , rj0
j=1
112
Chapter 5. Quantum stochastic calculus on IFS
0
0
hence rn+1
< rn0 < . . . < r10 < l10 < . . . < ln0 < ln+1
. By Lemma 5.2.7, we have
¯­
®¯
¯ ξ, A (tk ) (Fl (tk ))∗ Fr (tk ) A+ (tk ) ξ ¯
´
³
¯

¯*
Z
Z
0
0
λ
,
x
,
.
.
.
,
x
y
,
.
.
.
,
y
1
¯
d+d 0 0
d
ln+1
l1
t
∧t
ε (l1 )
¯
 k+1
0
= ¯ ξ, 
×
dyln+1
¯
λd (xd , . . . , x1 )
tk ∧t
Rn
¯
 +¯
¯
n ³
´³ ´
Y
¯
×
g lj0 grj0 ylj0 dylj0  ξ ¯¯
¯
j=1
´
³
¯

¯*
Z tk+1 ∧t
Z λd+d 0 0 −1 yl 0 , . . . , y[
0
,
x
,
.
.
.
,
x
1
¯
d
l
1
n+1
ε (l1 )
¯

0
×
≤ M ¯ ξ, 
dyln+1
¯
λd (xd , . . . , x1 )
n
t
∧t
R
k
¯
 +¯
¯
n ³
´³ ´
Y
¯

×
g lj0 grj0 ylj0 dylj0 ξ ¯¯
¯
j=1
where the last inequality is given by Lemma 5.2.1. We notice that dε0 (l 0 ) =
j
dε(lj ) + 1 for any j = 1, . . . , n, then the quantity above is equal to
¯* " Z
Z λ
¯
tk+1 ∧t
d+dε(l1 ) (yl1 , . . . , yln , xd , . . . , x1 )
¯
0
M ¯ ξ,
×
dyln+1
¯
λd (xd , . . . , x1 )
tk ∧t
Rn
 +¯
¯
n ³
´¡ ¢
Y
¯

×
g lj grj ylj dylj ξ ¯¯
¯
j=1
¯¿ Z t ∧t
À¯
k+1
¯
¯
∗
¯
= M ¯ ξ,
dy (Fl (tk )) Fr (tk ) ξ ¯¯
tk ∧t
(l,r)
(l,r)
Let us suppose the result holds for any N tk ≤ N and prove it for N tk =
¶
µ
0
0
N + 1. We denote by l (l,r) , r (l,r) the first index consecutive pairs moving
Nt
k ,1
Nt
k ,1
from the right in the sequence where ε0 is as in (5.2.9), define
(
Yd




ε0 l 0
(l,r) 
N t ,1
k
:=
³
´
(y 1 , . . . , y k ) ⊆ yr10 , . . . , yrn0 : y j ≤ yl 0
(l,r)
N t ,1
k
)
, j = 1, . . . , k
113
Hence, denoting by y the variable relative to the operator A (tk ):
¯­
®¯
¯ ξ, A (tk ) (Fl (tk ))∗ Fr (tk ) A+ (tk ) ξ ¯
¯*
Ã
!
Ã
!
+¯
¯
¯
¯
¯
ε(2n)
+
ε(1)
+
0
0
= ¯ ξ, A (tk ) A
(g2n ) · · · A gl
A
gr
···A
(g1 ) A (tk ) ξ ¯
(l,r)
(l,r)
¯
¯
N t ,1
N t ,1
k
k
¯D
¯
= ¯ ξ, A (tk ) Aε(2n) (g2n ) · · · ×







 +1 Yd 
 , y, xd , . . . , x1 
λd+d 







Z
εl 0
ε0 l 0


(l,r)
(l,r)
 t
N t ,1
N t ,1

k
k
×


×
 0
(
)





 , y, xd , . . . , x1 

 Yd 
\
0
λd+d 
y
l



(l,r)

N
 0


0 0
εl

(l,r)
N t ,1
k
Ã
×
!Ã
gl 0
(l,r)
N t ,1
k
gr 0
!
yl 0
(l,r)
N t ,1
k
tk ,1
ε l

(l,r)
N t ,1
k
(l,r)
N t ,1
k
#
ε(1)
dyl 0
(l,r)
N t ,1
k
···A
+¯
¯
¯
(g1 ) A (tk ) ξ ¯
¯
+
Moreover, by Lemma 5.2.1, we obtain the quantity above is less than or equal
to
¯D
¯
M 2 ¯ ξ, A (tk ) Aε(2n) (g2n ) · · · ×



Z
 t

×
 0



λd+d

λd+d




εl 0

(l,r)
N t ,1
k

Yd


 −1



εl 0

(l,r)
N t ,1
k

Yd

gl 0
(l,r)
N t ,1
k
gr 0
(l,r)
N t ,1
k




ε0 l 0

(l,r)
N t ,1
k


(l,r)
N t ,1
k


ε0 l 0

(l,r)
N t ,1
k
(l,r)
N t ,1
k
#
dyl 0
)
\ yl 0
!
yl 0

, yb, xd , . . . , x1 

(
!Ã
Ã
×



(l,r)
N t ,1
k

×


 , yb, xd , . . . , x1 


+¯
¯
¯
· · · Aε(1) (g1 ) A+ (tk ) ξ ¯
¯
Now in the sequence of operators on the right hand side of the scalar product
above, we have exactly N index consecutive pairs, hence it is possible to use
the induction hypothesis, thus obtaining the quantity above is less than or
114
Chapter 5. Quantum stochastic calculus on IFS
equal to
2
M M
¯¿ Z
¯
¯ ξ,
2N −1 ¯
tk+1 ∧t
tk ∧t
À¯
¯
ds (Fl (tk )) Fr (tk ) ξ ¯¯
∗
and the thesis follows.
¤
Before proving the last semi-martingale inequality, we introduce the following useful notation:
(l,r)
N tk := max N tk , N := max N tk
1≤l,r≤m
k=1,...,n
Proposition 5.2.11. Using the same notations as above, one has
°Z t
°2
Z t
°
°
° F (s) dA+ (s) ξ ° ≤ M 2N −1
kF (s) ξk2 ds
°
°
0
(5.2.10)
0
Proof. In fact, using the adaptness arguments, the left hand side of (5.2.10)
is equal to
n X
m
X
¯­
®¯
αl αr ¯ ξ, A (tk ) (Fl (tk ))∗ Fr (tk ) A+ (tk ) ξ ¯
k=1 l,r=1
By Lemma 5.2.10, this is less than or equal to
¯¿ Z t ∧t
À¯
m
n X
X
k+1
(l,r)
¯
¯
2N t −1 ¯
∗
α l αr M k
dy (Fl (tk )) Fr (tk ) ξ ¯¯
¯ ξ,
tk ∧t
k=1 l,r=1
n ¯¿ Z
X
¯
2N −1
¯ ξ,
≤M
¯
≤ M 2N −1
k=1
Z t
tk+1 ∧t
tk ∧t
À¯
¯
dy (F (tk ))∗ F (tk ) ξ ¯¯
kF (s) ξk2 ds
0
where the last majorization follows from Hölder’s inequality.
¤
For the 1MT-IFS the results obtained can be modified by using (5.1.2). In
0
fact, given a simple adapted process F ∈ S (= S ) , denote, for any d ∈ N,
ωN (F ) +d+1 := max ωordFh (tk )+d+1
tk
1≤h≤m
ωN (F ) +d+1 := max ωN (F ) +d+1
1≤k≤n
tk
115
Moreover we define
Hk ³
´
³
´
³
´
X
(h,k)
(h,k)
c tk , f (h,k) A+ fh
F+ (tk ) :=
· · · A+ f1
F− (tk ) :=
h=0
Hk
X
³
´ ³
´
³
´
(h,k)
(h,k)
c tk , f (h,k) A fh
· · · A f1
h=0
³
´
¡
¢
(h,k)
(h,k)
f (k,h) := fh , . . . f1
∈ L2 (0, t) , Ch . Hence for left and right stochastic integrals of simple adapted processes, we find the following estimates.
Proposition 5.2.12. Under the same notations of Proposition 5.2.3, for
1MT-IFS one has:
°Z t
°2
Z t
°
°
2
° F (s) dA (s) ξ ° ≤ M
kF (s) ηd k2 ds
(5.2.11)
d
°
°
0
0
°Z t
°2
k−1 Z t
X
°
°
° dA (s) F (s) ξ ° ≤ M 2
kF− (s) ηd−h k2 ds
d
°
°
0
(5.2.12)
h=0 0
°Z t
°2
Z t
°
°
° dA+ (s) F (s) ξ ° ≤ M (F )
kF (s) ξk2 ds
N
+d+1
°
°
0
0
°Z t
°2
Z t
°
°
° F (s) dA+ (s) ξ ° ≤ M (F )
kF+ (s) ξk2 ds
N
+d+1
°
°
0
(5.2.13)
(5.2.14)
0
Proof. In fact
°2
°Z t
°2 °
n
°X
°
°
°
°
° F (s) dA (s) ξ ° = °
F
(t
)
A
(t
∧
t,
t
∧
t)
ξ
°
°
k
k
k+1
°
°
°
°
0
k=1
°2
° n
°
°X Z tk+1 ∧t
°
°
=°
dxd gd (xd ) F (tk ) ηd °
ωd
°
°
tk ∧t
k=1
°Z t
°2
°
°
°
≤ Md2 °
° dsgd (s) F (s) ηd °
0
and (5.2.11) follows from Cauchy-Schwarz inequality. Let us prove (5.2.12):
°2
°Z t
°2 °
n
°X
°
°
°
°
° dA (s) F (s) ξ ° = °
(5.2.15)
A
(t
∧
t,
t
∧
t)
F
(t
)
ξ
°
°
− k
k
k+1
°
°
°
°
0
k=1
116
Chapter 5. Quantum stochastic calculus on IFS
where the equality above follows from (5.1.5) and the adaptness of F (tk ) .
(h)
Denoting, as in [30], by F− the simple adapted process
(h)
F− (r) =
n
´ ³
³
´
³
´
X
(h,k)
(h,k)
c tk , f (h,k) A fh
· · · A f1
χ[tk ,tk+1 ) (r)
k=1
we have that, for any k = 1, . . . , n
A (tk ∧ t, tk+1 ∧ t) F− (tk ) gd ⊗ · · · ⊗ g1
Hk ∧(d−1)
X
=
h
³
´Y
D
E
(h,k)
ωd−j+1 fj
c tk , f (h,k)
, gd−j+1 ×
j=1
h=0
µZ
×
tk+1 ∧t
tk ∧t
Hk ∧(d−1)
X
=
×
ωd−h gd−h (r) dr gd−h−1 ⊗ . . . ⊗ g1
h D
E
³
´ λ Y
(h,k)
d
fj
, gd−j+1 ×
c tk , f (h,k)
λd−h
j=1
h=0
µZ
¶
tk+1 ∧t
tk ∧t
¶
ωd−h gd−h (r) dr gd−h−1 ⊗ . . . ⊗ g1
Z
Hk ∧(d−1)
=
X
ωd−h
h=0
tk+1 ∧t
tk ∧t
(h)
gd−h (r) F− (r) ηd−h dr
(h)
(h0 )
As a consequence of orthogonality of F− (r) ηd−h and F−
h 6= h0 , the right hand side of (5.2.15) is equal to:
Hk ∧(d−1)
X
h=0
≤ Md2
(r) ηd−h0 when
°Z t
°2
°
°
(h)
°
ωd−h °
° gd−h (r) F− (r) ηd−h dr°
0
d−1 Z
X
h=0 0
t
kF− (r) ηd−h k2 dr
where in the last estimate we used the fact that the Mn ’s are increasing and
the Cauchy-Schwarz inequality. For (5.2.13) we have
°2
°Z t
°
°
+
° dA (s) F (s) ξ °
°
°
0
117
=
n
X
­
®
F (tk ) ξ, A (tk ∧ t, tk+1 ∧ t) A+ (tk ∧ t, tk+1 ∧ t) F (tk ) ξ
k=1
=
n
X
m
X
­
®
αl αr Fl (tk ) ξ, A (tk ∧ t, tk+1 ∧ t) A+ (tk ∧ t, tk+1 ∧ t) Fr (tk ) ξ
k=1 l,r=1
≤
n X
m
X
αl αr ωordFh (tk )+d+1 (tk+1 ∧ t − tk ∧ t) hFl (tk ) ξ, Fr (tk ) ξi
k=1 l,r=1
≤
n
X
ωN (F ) +d+1 (tk+1 ∧ t − tk ∧ t) kF (tk ) ξk2
tk
k=1
Z
≤ ωN (F ) +d+1
t
kF (s) ξk2 ds
0
Hence (5.2.13) follows. Finally, as a consequence of adaptness of F (tk ), we
have
°2
°Z t
°2 °
n
°X
°
°
°
°
°
+
+
° F (s) dA (s) ξ ° = °
F
(t
)
dA
(t
∧
t,
t
∧
t)
ξ
°
+ k
k
k+1
°
°
°
°
0
k=1
which, by using usual arguments, is equal to
n
X
­
®
ξ, A (tk ∧ t, tk+1 ∧ t) F− (tk ) F+ (tk ) A+ (tk ∧ t, tk+1 ∧ t) ξ
k=1
Again by adaptness we have that for any k = 1, . . . , n the unique non zero
contributions to the scalar product above can be obtained if A (tk ∧ t, tk+1 ∧ t)
acts on A+ (tk ∧ t, tk+1 ∧ t) . It is easy to prove that the quantity above is equal
to
n λ (F )
X
Nt +d+1
k
k=1
≤
n
X
k=1
λd
·
λd
(tk+1 ∧ t − tk ∧ t) hξ, F− (tk ) F+ (tk ) ξi
λN (F ) +d
tk
MN (F ) +d+1 (tk+1 ∧ t − tk ∧ t) hξ, F− (tk ) F+ (tk ) ξi
tk
≤ MN (F ) +d+1
Z
0
t
kF+ (s) ξk2 ds
¤
118
Chapter 5. Quantum stochastic calculus on IFS
5.3
Stochastic Integral
In this section we define the stochastic integral in the vector space of processes
that can be approximated by elements of S 0 (or S in 1-MT IFS) by means
of sequences. We will follow the methods of [4] and [30] in order to set a
definition of a stochastic integral satisfying our semi-martingale inequalities.
Let us take ξ = ud ⊗. . .⊗u1 ∈ D and the set J (ξ) ⊂ D whose elements are
Φ and uσ(h) ⊗ . . . ⊗ uσ(1) , h ∈ {1, ..., d} , σ : {1, ..., h} → {1, ..., d} increasing.
As in [4] we want to establish a τ − semi-martingale inequality with respect
to a topology τ induced by a family of semi–norms and we recall that in [4],
topology τ is induced by semi–norms
Z t
kF k2ξ,t,µ :=
kF (s) ξk2 dµ (s) ,
0
where F is a simple adapted process, ξ ∈ D, t ∈ R+ are arbitrarily chosen.
In our case, for any F ∈ S 0 , ξ ∈ D , t ∈ R+ and for any N ≥ 1, the
topology τ is determined by the seminorms
n
o
qξ,t,N (F ) := max M 2 , M 2N −1 , t ×
×
X
(µZ
η∈J(ξ)
or
×
X
t
¶ 21 µZ t
¶ 12 )
kF (s) ηk2 ds
kF ∗ (s) ηk2 ds
+
0
(5.3.1)
0
n
o
2
×
qξ,t,N (F ) (F ) = max Md4 , MN
(F ) +d+1 , t
(µZ
¶ 21 µZ t
¶ 12 )
t
X
2
2
∗
kFε (s) ηk ds
+
kFε (s) ηk ds
(5.3.2)
η∈J(ξ) ε∈{−1,0,1}
0
0
according to we consider the general case with non constant interacting functions or the 1-MT IFS, where F−1 = F− , F0 = F, F1 = F+ . From now on
we will consider only the general case, as the 1MT-IFS can be obtained just
by replying (5.3.1) with (5.3.2). Denote by αβ an arbitrary element of the set
{01, 10}. As a consequence of Proposition 5.2.3 and Proposition 5.2.11, we
have that the maps
Z t
F ∈ S 0 7→
F (s) dM αβ (s) ∈ L (D, FI )
0
119
5.3. Stochastic Integral
Z
0
F ∈ S 7→
0
t
dM αβ (s) F (s) ∈ L (D, FI )
are continuous with respect to the topology on S 0 induced by seminorms
(5.3.1) and the topology of strong ∗-convergence on D. Hence, if we call τ
and τ 0 respectively the topologies we are dealing with, we can say that the
basic processes are (τ − τ 0 ) -semimartingales, according to [4], Definition 2.1.
Let
vector space of processes F such that there exists a sequence
¡ (n)
¢ S be the
0
F
in S for which the following property holds:
n≥0
¡
¢
(∗) for any t ∈ R+ F (n) n≥0 converges to F (t) in the topology τ 0 of
strong ∗-convergence on D.
The following definition gives us the class of integrable processes.
Definition
A process F ∈ S is said to be integrable if there exists a
¡ 5.3.1.
¢
sequence F (n) n≥0 in S 0 satisfying the condition (∗) and such that for any
ξ ∈ D, t ∈ R+
³
´
(5.3.3)
lim qξ,t,N F (n) − F = 0.
n→+∞
We denote by I the class of all integrable processes and notice that for any
t ∈ R+ , any F ∈ I, and any (F n )n≥0 satisfying Definition 5.3.1, the sequences
of stochastic integrals:
µZ t
¶
(n)
αβ
F (s) dM (s)
0
µZ
n≥0
¶
t
dM αβ (s) F (n) (s)
0
n≥0
are convergent in the topology of strong ∗-convergence on D. In fact, for any
ξ ∈ D, by Proposition 5.2.3
°·Z t
¸ °2
Z t
°
°
(n)
°
0≤°
F (s) dA (s) −
F (s) dA (s) ξ °
°
0
0
Z t °³
´ °2
° (n)
°
≤ M2
° F (s) − F (s) ηd ° ds
0
Because of condition (∗), the last integral above converges to zero. The other
cases can be similarly treated. Hence we can define the left and right stochastic
integrals of elements of I with respect to the basic processes in the following
way
Z
Z
t
0
F (s) dM αβ (s) ξ := lim
n→+∞ 0
t
F (n) (s) dM αβ (s) ξ
120
Chapter 5. Quantum stochastic calculus on IFS
Z
Z
t
dM
αβ
(s) F (s) ξ := lim
n→+∞ 0
0
t
dM αβ (s) F (n) (s) ξ,
for any ξ ∈ D, where the limit is understood in the topology of strong ∗convergence on FI .
By Definition 5.3.1, Proposition 5.2.3 and Proposition 5.2.11, the following
estimates hold
°Z t
°
°Z t
°
°
°
°
°
αβ
° F (s) dM αβ (s) ξ ° ≤ q
°
dM (s) F (s) ξ °
ξ,t,N (F ) , °
°
°
° ≤ qξ,t,N (F )
0
0
(5.3.4)
for any F ∈ I and ξ ∈ D.
¡
¢
In fact let us consider a sequence F (n) n≥0 in S 0 satisfying conditions (∗)
and (5.3.3) with respect to F . Then, by (5.2.1) for any n ∈ N
°Z t
°2
Z t°
°2
°
°
° (n)
°
(n)
° F (s) dA (s) ξ ° ≤ M 2
F
(s)
η
°
d ° ds
°
°
0
0
According to the definition of the family of semi-norms, this implies that
°Z t
°
³
´
°
°
(n)
° F (n) (s) dA (s) ξ ° ≤ q
F
ξ,t,N
°
°
0
By taking the limit for n → ∞ on each side above, one of the estimates in
(5.3.4) is given. The others can be obtained by using the same arguments.
Proposition 5.3.2. Let F ∈ I. Then the following propositions hold
i) The maps
Z t
(s, t) 7→
F (r) dM αβ (r)
s
Z
(s, t) 7→
t
dM αβ (r) F (r)
s
are strongly continuous on D.
ii) I is a vector space and left and right stochastic integrals are linear in
F .
Proof. The above discussion implies that i) and ii) hold for all elements in
S 0 . Then the thesis follows by approximation.
¤
121
5.3. Stochastic Integral
The successive section of this chapter will be devoted to the study of a
class of quantum stochastic differential equations (QSDE) with respect to the
basic processes of creation and annihilation in interacting Fock spaces. In
order to prove an existence and uniqueness theorem for QSDE one wants that
stochastic integrals of integrable processes are yet integrable processes and
moreover that the qξ,t,N - seminorm of a stochastic integral can be majorized
by an ordinary integral. These conditions are ensured by the following result.
Theorem 5.3.3. Let F ∈ I. Then the left and right stochastic integral of F
with respect to the basic processes is an element of I. Moreover we have that
·
µZ
qξ,t,N
t
F (s) dM
0
αβ
¶¸2
X Z t
(s)
≤ c (M, t) |J (ξ)|
(qη,s (F ))2 ds
η∈J(ξ) 0
(5.3.5)
where c is a constant depending on M and t.
Proof.
M αβ simply by M . Let us consider
¡ (n) ¢ We denote any process of the form
0
F
a sequence of elements of S satisfying the conditions of Definition
n≥0
5.3.1 with respect to F and suppose that every F (n) is defined by means of a
partition 0 = sn0 < sn1 < ... < snkn +1 < +∞ such that
lim
sup
n→+∞j∈{0,...,k }
n
¯ n
¯
¯sj+1 − snj ¯ = 0
For any n ∈ N we consider the simple adapted process:
I (n) (s) :=
kn
X
i=1


X
¡
¢
¡
¢
 F (n) snj M snj , snj+1  χ n n (s) .
[si ,si+1 )
j<i
(5.3.6)
122
Chapter 5. Quantum stochastic calculus on IFS
For any ξ ∈ D, t ∈ R+ we have:
°µ
¶ °2
Z s
° (n)
°
°
F (r) dM (r) ξ °
sup ° I (s) −
°
s≤t
0
°·
Z s
° (n)
= sup °
F (r) dM (r)
° (I (s) −
s≤t
0
¸ °2
°
+
F (r) dM (r) −
F (r) dM (r)) ξ °
°
0
0
°µZ s ³
°
¶
´
°
°2
(n)
°
≤ 2sup °
F
(r)
−
F
(r)
dM
(r)
ξ
°
°
Z
s
Z
s
(n)
s≤t
0
°µ
Z
° (n)
+ 2sup °
I
(s)
−
°
s≤t
s
F
(n)
0
(n)
¶ °2
°
(r) dM (r) ξ °
°
Now, because of (5.3.4) the first term can be majorized by
h
³
´i2
2 qξ,t,N F (n) − F
it goes to zero for n going to infinity. The second term can be majorized by
°Z
°2
° s
°
°
°
(n)
F (r) dM (r) ξ °
2 sup
sup °
°
°
n
sj
j∈{k1 ,...kn }s∈[sn ,sn ]
j
j+1
By Proposition 5.2.3, we obtain that it is less than or equal to
"Z
°2
o
n
s°
° (n)
°
sup
sup
max M 2 , M 2N −1
°F (r) ξ ° dr
sn
j∈{k1 ,...kn }s∈[sn ,sn ]
j
j j+1
#
Z s°
°2
° (n)∗
°
+
(r) ξ ° dr
°F
sn
j
n
o X
≤ max M 2 , M 2N −1
Z
sup
sn
j+1
n
η∈J(ξ)j∈{k1 ,...kn } sj
µ°
°2 °
°2 ¶
° (n)
°
° (n)∗
°
F
(s)
η
+
F
(s)
η
°
°
°
° ds
(5.3.7)
Following the method developed in [30], Th.4.4., one can show that for any
η ∈ D, for any t ∈ R+ , for any n ≥ 0 the sequence
µ°
°2 °
°2 ¶
³
´
° (n)
°
° (n)+
°
(n)
φ
:= °F (·) η ° + °F
(·) η °
n≥0
n≥0
123
5.3. Stochastic Integral
converges in L1 (0, t) and it is uniformly integrable. In fact for any n, m ≥ 0
°
°
° (n)
(m) °
°φ − φ °
L1
¯ ¯
¯
Z t ¯³
¯ (n) ´ 21 ³ (m) ´ 12 ¯ ¯³ (n) ´ 12 ³ (m) ´ 12 ¯
¯
¯
¯
¯ ds
=
+ φ
− φ
¯ φ
¯·¯ φ
¯
0
1
ÃZ ¯
!
2
°
°1
´1 ³
´ 1 ¯¯2
t ¯³
2
2
° (n) ° 2
¯ φ(n) − φ(m) ¯ ds
≤ 2 sup °φ ° 1
¯
¯
L
n
0
where we used the Cauchy-Schwarz inequality and the fact that for any a, b ∈
³√
√ ´2
R+
a+ b
≤ 2 (a + b). Moreover, recalling that for any a, b ∈ R+
¯√
¯
√ ¯2
¯
¯ a − b¯ ≤ |a − b|, we get that the quantity above is less than or equal
to
°
° 1 µZ t ¯³
´ ³
´¯ ¶ 12
° (n) ° 2
¯ (n)
(m) ¯
2 sup °φ ° 1
− φ
¯ φ
¯ ds
L
n
0
°2
°
° 1 µZ t ¯¯°
°2 °
° (n) ° 2
°
° (n)∗
°
(n)
¯°
= 2 sup °φ ° 1
F
(·)
η
+
F
(·)
η
°
°
°
°
¯
n
L
n
L
n
L
0
µ°
°2 °
°2 ¶¯¯ ¶ 12
° (m)
°
° (m)∗
°
− °F
(·) η ° + °F
(·) η ° ¯¯ ds
°
° 1 µZ t µ°
°2 °
°2
° (n) ° 2
° (n)
°
°
°
≤ 2 sup °φ ° 1
°F (·) η ° + °F (n)∗ (·) η °
0
°
°2 °
°2 ¶ ¶ 21
° (m)
°
° (m)∗
°
− °F
(·) η ° − °F
(·) η ° ds
°
° 1 µZ t µ°³
´ °2
° (n) ° 2
° (n)
°
(m)
≤ 4 sup °φ ° 1
(·) η °
° F (·) − F
0
°³
´ °2 ¶ ¶ 12
° (n)+
°
(m)∗
+° F
(·) − F
(·) η ° ds
¡
¢
Because L1 (0, t) is complete, φ(n) n≥0 is uniformly convergent in such space.
This property, together with (5.3.6), gives that (5.3.7) is infinitesimal as n
going to infinity. As a consequence we have that ∀ ξ ∈ D, ∀ t ∈ R+ :
°¡
¢ °2
Rs
lim sup ° I (n) (s) − 0 F (r) dM (r) ξ ° = 0
n→+∞ s≤t
°¡
¢ °2
Rs
lim sup ° I (n)∗ (s) − 0 dM ∗ (r) F ∗ (r) ξ ° = 0,
n→+∞ s≤t
(5.3.8)
124
Chapter 5. Quantum stochastic calculus on IFS
where the second limit can be evaluated in the same way as the first one. It
follows that left and right stochastic integrals are in I. Now we prove (5.3.5).
Denote
Z s
Z s
I (s) :=
F (r) dM (r) ;
I ∗ (s) :=
dM ∗ (r) F ∗ (r) ;
0
therefore
0
h
i2
©
ª
qξ,t,N (I) = max M 4 , M 4N −2 , t2 ×
×

 X µZ

0
η∈J(ξ)
t

¶ 12 µZ t
¶ 21 2
kI (s) ηk2 ds
+
kI ∗ (s) ηk2 ds

0
µ
Recalling that for all n ∈ N, x = (x1 , ...xn ) ,
is less than or equal to

à Z
 X Z t °
°
°
×
°

η∈J(ξ) 0
0
Since

à Z
 X Z t °
°
°
°

s
η∈J(ξ) 0
is majorized by
0
s
n
P
¶2
xi
i=1
≤n
n
P
i2
h
x2i , qξ,t,N (I)
i=1
n
o
4
4N −2 2
max M , M
, t |J (ξ)| ×
°2 ! 12 ðZ
°
°
+ °
F (r) dM (r) η °
°
°
0
°2 ! 12 ðZ
°
°
F (r) dM (r) η °
+ °
°
°
0

2
X Z
η∈J(ξ) 0
t
s
s

°2 ! 12 2
°
ds
dM ∗ (r) F ∗ (r) η °
°


°2 ! 21 2
°
dM ∗ (r) F ∗ (r) η °
ds
°

2
qη,s,N (F ) ds
(5.3.5) follows.
¤
Remark 5.3.4. We observe that the stochastic ¡integral
¢ I has the following
approximation property: there exists a sequence I (n) n≥0 of elements of S 0
such that for any ξ ∈ D, for any t ∈ R+
°³
°³
´ °
´ °
°
°
°
°
lim sup ° I (n) (s) − I (s) ξ ° = lim sup ° I (n)∗ (s) − I ∗ (s) ξ ° = 0.
n→+∞ s≤t
n→+∞ s≤t
(5.3.9)
In fact this property follows from (5.3.8).
125
5.3. Stochastic Integral
Let Ic be the vector space of processes F ∈ S with the above uniform
approximation property by elements of S 0 , strongly ∗-continuous on D. As a
consequence of Proposition 5.3.2 and Theorem 5.3.3, we have the following
Corollary 5.3.5. The right and left stochastic integral of any member F ∈ I
with respect to M αβ is an element of Ic .
Proof. In fact from (5.3.8) the stochastic integral of an integrable process
has the property (5.3.9). The strong ∗-continuity follows from i) and ii) of
Proposition 5.3.2.
¤
We present a useful result in the discussion of existence and uniqueness
of quantum stochastic differential equations. In the following by the upper
symbol # we denote indifferently either the operator which is applied or its
adjoint.
¡
¢
Proposition 5.3.6. Let U (n) n≥0 be a sequence of elements of Ic such that,
for all ξ ∈ D, and t ∈ R+ :
°³
´ °
° (n)#
°
(m)#
(s) − U
(s) ξ ° = 0.
(5.3.10)
lim sup ° U
m,n→+∞ s≤t
Then there exists an element U ∈ Ic such that for any αβ ∈ {01, 10}
°³
´ °
°
°
lim sup ° U (n)# (s) − U # (s) ξ ° = 0
(5.3.11)
n→+∞ s≤t
Z
lim
t
n→+∞ 0
Z t
lim
n→+∞ 0
Z
U (n) (s) dM αβ (s) ξ =
dM αβ (s) U (n)+ (s) ξ =
t
0
Z t
U (s) dM αβ (s) ξ
(5.3.12)
dM αβ (s) U + (s) ξ.
(5.3.13)
0
Proof. Let U be the process defined in the following way
U # (s) ξ := lim U
n→∞
(n) #
(s) ξ
for any ξ ∈ D, s ∈ R+ . Firstly we prove (5.3.11). In fact
°³
´ °
°
°
lim sup ° U (n)# (s) − U # (s) ξ °
n→+∞ s≤t
°³
´ °
° (n)#
°
(m)#
= lim sup lim ° U
(s) − U
(s) ξ °
n→+∞ s≤t m→+∞
°³
´ °
° (n)#
°
(m)#
≤ lim sup ° U
(s) − U
(s) ξ ° = 0
m,n→+∞ s≤t
126
Chapter 5. Quantum stochastic calculus on IFS
where the last equality is given by (5.3.10). Moreover we show that U ∈ Ic ,
i.e. the approximation property (5.3.11) is strongly continuous on D. In fact
e (n) be an element of S 0 such that for any ξ ∈ D
for any n ∈ N, let U
°³
´ ° µ 1 ¶n
° e (n)#
°
(n)#
sup ° U
(s) − U
(s) ξ ° <
2
s≤t
Then
°³
´ °
° e (n)#
°
sup ° U
(s) − U # (s) ξ °
s≤t
°³
°³
´ °
´ °
° e (n)#
°
°
°
≤ sup ° U
(s) − U (n)# (s) ξ ° + sup ° U (n)# (s) − U # (s) ξ °
s≤t
s≤t
µ ¶n
°³
´ °
1
°
°
+ sup ° U (n)# (s) − U # (s) ξ °
<
2
s≤t
By taking the limit in both sides above, by (5.3.11) we obtain
°³
´ °
° e (n)#
°
#
sup ° U
(s) − U (s) ξ ° = 0
s≤t
hence U ∈ Ic . We also notice that (5.3.11) implies
³
´
lim qξ,t U (n) − U = 0
n→∞
and, according to Definition 5.3.1, U is integrable, then (5.3.12), (5.3.13) follow.
¤
5.4
Quantum Stochastic Differential Equations
The main result of this section consists in providing a result for the existence
and uniqueness of the solution for a class of quantum stochastic differential
equations (QSDE) with respect to the basic processes of creation and annihilation on IFS. We firstly observe that all the results obtained in Section 5.3 hold
even if we consider an initial Hilbert space Hs coupled with the interacting
Fock space FI . From now on we will consider such a situation and denote by
xξ := x ⊗ ξ, x ∈ Hs , ξ ∈ FI .
The inequality 5.3.5 is fundamental to prove the following result.
127
5.4. Quantum Stochastic Differential Equations
Theorem 5.4.1. Let Γ be a finite set of indices and for all γ ∈ Γ let M γ be
a basic process. If Fγ , Gγ , X0 ∈ B (Hs ) , then there exists a unique solution
of the quantum stochastic differential equation
½
dX (t) = X (t) Fγ dM γ (t) + dM γ (t) Gγ X (t)
(5.4.1)
X (0) = X0
where (5.4.1) is understood as the integral equation
Z
X (t) = X0 +
0
t
Z
X (s) Fγ dM γ (s) +
t
0
dM γ (s) Gγ X (s)
(5.4.2)
Remark 5.4.2. This equation has the same form of that presented in [30],
Theorem 5.1, because our basic processes do not satisfy any %− commutation
relation in the sense defined in [4], Definition 6.2.
Proof. For the existence, without loss of generality we can not consider in
our discussion the last integral on the right side in (5.4.2), in fact the results
obtained can be clearly extended to the general case.
Recalling that the stochastic integral of an integrable process is integrable
itself, define by induction for any t ∈ R+
Z
X (0) (t) := X0 ,
X (n+1) (t) :=
0
t
X (n) (s) Fγ dM γ (s)
The sequence is well defined, because X0 is an adapted process strongly
continuous on D and, by means of Corollary 5.3.5, the left and right stochastic
integral of an element of I is an element of Ic . Then X (n) ∈ Ic for all n ∈ N.
We fix n ∈ N, for any ξ ∈ D, and any t ∈ R+ , by 5.3.4
°2
Z
°
°2 °
° t (n−1)
°
° (n)
°
γ
(t1 ) Fγ dM (t1 ) ξ °
°X (t) ξ ° = °
° X
°
0
³
´i2
h
≤ qξ,t,N X (n−1) Fγ
h
n
oi2
2N −1
2
2
×
= max MX
(n−1) , M (n−1) , t
X


°2
X Z t°
° (n−1)
°
×
(t1 ) Fγ η1 ° dt1 
°X
η1 ∈J(ξ) 0
128
Chapter 5. Quantum stochastic calculus on IFS
Put C := max
j=1,...,n
h
n
oi2
2N −1 2 2
2
max MX
,
M
,
t
,
t
then the quantity above is
(j)
j
X (j)
less than or equal to
°Z t1
°2
°
°
(n−2)
γ
°
C |J (ξ)|
dt1 °
X
(t2 ) Fγ dM (t2 ) η1 °
°
0
0
h
n
oi2
2N −1
2
2
≤ C max MX
|J (ξ)| ×
(n−2) , M (n−2) , t1
X
Z
Z
°2
t
t1 °
X
° (n−2)
°
×
dt1
(t2 ) Fγ η2 ° dt2
°X
Z
t
η2 ∈J(ξ) 0
0
An n - fold iteration of this procedure gives
Z t
Z tn−2
X µZ
n−1
n
C |J (ξ)|
dt1 · · ·
dtn−1
0
0
ηn ∈J(ξ)
C n |J (ξ)|n max kηk2 kFγ k2n kX0 k2 tn
η∈J(ξ)
tn−1
0
°
°
°X0 Fγn ηn °2 dtn
¶
1
n!
P
(n) converges in the strong topology on D and,
Therefore the series ∞
n=0 X
because of Proposition 5.3.6, its sum, denoted by X, is strongly continuous
on D.
By the next step we show that X is a solution of (5.4.1). In fact, thanks
to (5.3.4), for any ξ ∈ D
°Z
°2
Z tX
n
° t
°
°
°
(k)
γ
γ
X (s) Fγ dM (s) ξ °
° X (s) Fγ dM (s) ξ −
° 0
°
0 k=0
°Z
°2 "
!#2
à ∞
∞
° t X
°
X
°
°
(k)
(k)
γ
X (s) Fγ
=°
X (s) Fγ dM (s) ξ ° ≤ qξ,t,N
° 0
°
k=n+1
k=n+1


Z
∞
°2
oi2
h
n
t X °
X
° (k)
°
2N −1 2
2

≤ max MX
°X (s) Fγ η ° ds
(k) , M (k) , t
X
η∈J(ξ) 0 k=n+1
(5.4.3)
Following the same arguments that led us to have (??), we obtain that for any
s ∈ [0, t) , ξ ∈ D, k ∈ N
°2
°
1
° (k)
°
2
2k
2
k
°X (s) Fγ ξ ° ≤ max kηk kFγ k kX0 k C k |J (ξ)| tk
k!
η∈J(ξ)
129
5.4. Quantum Stochastic Differential Equations
∞ °
°
P
°X (k) (s) Fγ ξ °2 converges in the strong topology on D.
Hence the series
k=0
Thanks to Lebesgue’s Theorem, the right hand side of (5.4.3) is infinitesimal
for n going to infinity, hence
lim
Z tX
n
n→∞ 0
k=0
Z
X
(k)
γ
(s) Fγ dM (s) =
t
0
X (s) Fγ dM γ (s)
strongly on D. Then X satisfies equation (5.4.1) as a consequence of the
identity
n+1
X
X
(k)
(t) ξ = X0 ξ +
k=0
Z tX
n
0 k=0
X (k) (s) Fγ dM γ (s) ξ
In fact
X (t) ξ = lim
n→∞
n
X
X (k) (t) ξ
k=0
Z tX
n
X (k) (s) Fγ dM γ (s) ξ
= X0 ξ + lim
n→∞ 0
k=0
Z t
= X0 ξ +
0
X (s) Fγ dM γ (s) ξ
For the uniqueness also, the estimate (5.3.5) is the basic tool. In fact we
consider X (t) , Y (t) solutions of (5.4.1). Therefore
Z
X (t) = X0 +
0
Z
γ
X (s) Fγ dM (s) +
0
Z
Y (t) = X0 +
t
t
Y (s) Fγ dM γ (s) +
t
0
Z t
0
dM γ (s) Gγ X (s)
dM γ (s) Gγ Y (s)
Arguing as before, without loss of generality it is possible to not consider the
integrals on the right sides above.
For any ξ ∈ D, for any t ∈ R+ , by using arguments already developed, one
130
Chapter 5. Quantum stochastic calculus on IFS
achieves the following majorization
h
i2
k(X (t) − Y (t)) ξk2 ≤ qξ,t,N (X − Y )
i2
X Z th
qη1 ,t1 ,N 1 (X − Y ) dt1
≤ c (M, t) |J (ξ)|
η1 ∈J(ξ) 0
≤ c (M, t) c (M1 , t1 ) |J (ξ)|2 ×
Z t1 h
i2
X Z t
×
dt1
qη2 ,t2 ,N 2 (X − Y ) dt2
η2 ∈J(ξ) 0
0
where the last inequality follows from 5.3.5. After applying such inequality
n − 2 times and recalling the definition of qξ,t,N (·), one finally obtain that the
quantity is less than or equal to
K n |J (ξ)|n kFγ k2 max kηk2 sup kX (s) − Y (s)k2 tn
η∈J(ξ)
where
½
K := max c (M, t) ,
s≤t
max
1
n!
¾
c (Mj , tj )
j=1,...,n−1
Since such a majorization holds for any n ∈ N, then for any t, (X (t) − Y (t)) ξ =
0.
¤
5.5
Quantum Ito Formula
By the above result we know that for a class of quantum stochastic differential
equations, the solution exists and it is unique. Our goal is now to investigate
when such a solution is a unitary operator. This question leads us to study
the problem of achieving a quantum Ito formula for the basic processes here
considered. Hence in this section, following the methods developed in [4] and
[30] we construct a Ito formula for the stochastic integrals in the interacting
Fock case. As usual we deal with the general standard IFS, but it is possible
to achieve similar results in the case of 1-MT IFS using the semi - martingale
estimates presented in Proposition 5.2.3.
Firstly we recall the o (dt) -notation used in [4]. Let
©
ª
R2≤ := (s, t) ∈ R2 | s ≤ t
131
5.5. Quantum Ito formula
and consider a function φ : R2≤ → C. We say that φ is of order o (dt) if for
every bounded interval (s, t) in R+
X
lim
φ (tj , tj+1 ) = 0
|P (s,t)|→0
s=t1 <t2 <...<tk =t
where |P (s, t)| denotes the length of the partition t1 < t2 < ... < tk in (s, t).
From now on by the upper symbol # we denote indifferently the operator
which is applied or its adjoint and define dA# (t) := A# (t, t + dt) for any
t ∈ R+ . In order to construct a quantum Ito table we need to consider four
cases.
1. dA (t) dA+ (t)
Let us consider
Ψ1 := A+ (fm ) · · · A+ (f1 ) Φ,
Ψ2 := A+ (gn ) · · · A+ (g1 ) Φ,
m ≥ 0, f1 , . . . , fm ∈ H
n ≥ 0, g1 , . . . , gn ∈ H
two number vectors and the correlator
®
­
Ψ1 , dA (t) dA+ (t) Ψ2
­
®
= A+ (fm ) · · · A+ (f1 ) Φ, dA (t) dA+ (t) A+ (gn ) · · · A+ (g1 ) Φ
­
®
= Φ, A (f1 ) · · · A (fm ) dA (t) dA+ (t) A+ (gn ) · · · A+ (g1 ) Φ
The quantity above can be different from zero only if n = m, then it
becomes
­
®
Φ, A (f1 ) · · · A (fn ) dA (t) dA+ (t) A+ (gn ) · · · A+ (g1 ) Φ
¡
¢ ®
­
= Ψ1 , Tn χ[t,t+dt) , χ[t,t+dt) Ψ2
where for any f, g ∈ H, Tn (f, g) = A (f ) A+ (g) |Hn as defined in 2.1.9.
Since number vectors are total in our space, we have the following relation
¡
¢
dA (t) dA+ (t) = TN χ[t,t+dt) , χ[t,t+dt) =: TN (dt)
(5.5.1)
132
Chapter 5. Quantum stochastic calculus on IFS
2. dA+ (t) dA (t).
We recall from 5.2.10 that for any T ∈ R+ , F ∈ S 0 , ξ ∈ D, for all
s1 , s2 ∈ [0, T ] , s1 < s2
kA (s1 , s2 ) F (s2 ) ξk ≤ c (F, ξ, T ) (s2 − s1 )
(5.5.2)
for each ξ ∈ D, F ∈ S 0 .
Notice that the constant c is not depending on s1 , s2 , so if we consider
Ψ1 and Ψ2 two number vectors as in the previous case, we have:
¯­
®¯
¯ Ψ1 , dA+ (t) dA (t) Ψ2 ¯
¯­
®¯
= ¯ dA (t) A+ (fn ) · · · A+ (f1 ) Φ, dA (t) A+ (gn ) · · · A+ (g1 ) Φ ¯
Applying the Hölder inequality and (5.5.2) we have that this quantity is
less than or equal to
°
°
° °
°A (t, t + dt) A+ (fn ) · · · A+ (f1 ) Φ° · °A (t, t + dt) A+ (gn ) · · · A+ (g1 ) Φ°
≤ c1 (Ψ1 , Φ, t) c2 (Ψ2 , Φ, t) (dt)2
Hence
dA+ (t) dA (t) = 0
(5.5.3)
3. dA+ (t) dA+ (t)
From 5.2.2 we have that for any T ∈ R+ , F ∈ S 0 , ξ ∈ D, for all s1 , s2 ∈
[0, T ] , s1 < s2
° +
°
°A (s1 , s2 ) F (s2 ) ξ ° ≤ k (F, ξ, T ) (s2 − s1 )
(5.5.4)
where the constant k is again not depending on s1 , s2 . Hence one
can follow the same arguments presented in 2. and, by using (5.5.2),
(5.5.4) and the Hölder inequality, find a majorization of order (dt)2 for
|ψ1 , dA+ (t) dA+ (t) ψ2 | and therefore
dA+ (t) dA+ (t) = 0
(5.5.5)
4. dA (t) dA (t) .
Following the same arguments used in the cases 2. and 3., one finds
133
5.6. Unitarity condition
dA (t) dA (t) = 0
(5.5.6)
As a consequence we can construct the module (module on the algebra
generated by the number operator) Ito table
dA (t)
dA+ (t)
dA (t)
0
0
dA+ (t)
T N (dt)
0
The following proposition is the basic tool to find the unitarity condition
for the solution of a quantum stochastic differential equation and can be proved
by using the Ito table above.
Proposition 5.5.1. For all F, G ∈ Ic , ξ, η ∈ D, x, y ∈ Hs the following
formulas hold:
D
E
dM αβ (t) G (t) yη, dM δγ (t) F (t) xξ
½
hG (t) yη, TN (dt) F (t) xξi if α = δ = 1
=
(5.5.7)
0
otherwise
(left weak Ito formula)
D
E
G (t) dM αβ (t) yη, F (t) dM δγ (t) xξ
½
=
hG (t) yΦ, F (t) xΦi hη, TN (dt) ξi if α = δ = 1
0
otherwise
(5.5.8)
(right weak Ito formula).
5.6
Unitarity condition
This section is devoted to find a necessary and sufficient condition for the
unitarity of the solution of a quantum stochastic differential equation. The
relation obtained contains a deformation in the sense of operators of a similar
condition found by Fagnola in [30], which can be seen as a manifestation of
134
Chapter 5. Quantum stochastic calculus on IFS
the interaction. For simplicity we will consider only the unidimensional noise;
in fact the result can be enlarged also to the more general case.
Let us take the quantum stochastic differential equation
½
dUt = Ut Fαβ dM αβ (t)
(5.6.1)
U (0) = 1,
where Fαβ ∈ B (Hs ) . Then we have the following theorem.
Theorem 5.6.1. If Fαβ ∈ B (Hs ) , the solution of (5.6.1) is unitary if and
only if the following condition holds
¡
¢
∗
∗
(F00 + F00
) t1 + F10
F10 TN χ[0,t) , χ[0,t) = 0 ∀ t ∈ R+
(5.6.2)
Proof. Let Ut be the solution of (5.6.1) and suppose it is unitary. By using
the right weak Ito formula (5.5.8) we have for all v, u ∈ Hs , η, ξ ∈ D
0 = d hUt vη, Ut uξi = hdUt vη, Ut uξi + hUt vη, dUt uξi + hdUt vη, dUt uξi
D
E D
E
= Ut Fαβ dM αβ (t) vη, Ut uξ + Ut vη, Ut Fαβ dM αβ uξ
D
E
+ Ut Fαβ dM αβ (t) vη, Ut Fδγ dM δγ (t) vη
D
E D
E
= Fαβ dM αβ (t) vη, uξ + vη, Fαβ dM αβ (t) uξ
+ hF1α vΦ, F1β uΦi hη, TN (dt) ξi
Therefore, choosing α = β = 0, η = ξ = Φ , ∀ t ∈ R+ we have:
¢
¡
∗
∗
(F00 + F00
) t1 + F10
F10 TN χ[0,t) , χ[0,t) = 0
Now we prove that condition (5.6.2) is sufficient.
For all v, u ∈ Hs , η, ξ ∈ D we have by the left weak Ito formula
d hUt∗ vη, Ut∗ uξi = hdUt∗ vη, Ut∗ uξi + hUt∗ vη, dUt∗ uξi + hdUt∗ vη, dUt∗ uξi
D
E
∗ ∗
α∗ β ∗
∗
= dM
(t) (Fαβ ) Ut vη, Ut uξ
D
E
∗ ∗
+ Ut∗ vη, dM α β (t) (Fαβ )∗ Ut∗ uξ
­
®
∗
+ Ut∗ vη, F0α (t) F0β
TN (dt) Ut∗ uξ
¡
¢∗
∗ ∗
where M α β (t) = M αβ (t) .
If α = β = 1, η = ξ = Φ
∗
∗
hUt∗ vΦ, ((F00 + F00
) dt + F01 F01
TN (dt)) Ut∗ uΦi
135
5.6. Unitarity condition
After noticing that
∗
F01 = −F10
∗
F10 = −F01
by (5.6.2) the scalar product above becomes:
∗
∗
hUt∗ vΦ, ((F00 + F00
) dt + F10
F10 TN (dt)) Ut∗ uΦi = 0
hence U is a coisometry. Now we prove that U is an isometric operator.
For all v, u ∈ Hs , η, ξ ∈ D , applying the right weak Ito formula, we
have:
D
E D
E
d hUt vη, Ut uξi = Ut Fαβ dM αβ (t) vη, Ut uξ + Ut vη, Ut Fαβ dM αβ uξ
+ hUt F1α vΦ, Ut F1β uΦi hη, TN (dt) ξi
(5.6.3)
Choosing α = β = 0, η = ξ = Φ, we obtain the ordinary integral equation:
Z t
hUt vΦ, Ut uΦi = hv, ui +
hU (s) F10 vΦ, U (s) TN dsF10 uΦi
0
+ hU (s) F00 vΦ, U (s) uΦi + hU (s) vΦ, U (s) F00 uΦi)ds
where
Z
0
t
hU (s) F10 vΦ, U (s) TN (ds) F10 uΦi
:= lim
n→∞
n D
X
U (tj ) F10 vΦ, U (tj ) TN
³³
´´
E
χ[tj ,tj+1 ) , χ[tj ,tj+1 ) F10 uΦ
j=0
for any n ∈ N, 0 = t0 < t1 < · · · < tn = t.
Let be V (t) := U ∗ (t) U (t) . A slight modification of the method developed
in [4], implies that this integral equation has a unique solution. From condition
(5.6.2) we have that
hvΦ, V (t) uΦi = hv, ui
is a solution, hence
hUt F1α vΦ, Ut F1β uΦi = hF1α v, F1β ui
From (5.6.3) we obtain that V = V (t) satisfies the quantum stochastic differential equation
½
∗ V (t) + F ∗ F T (dt)
dV (t) = V (t) Fαβ dM αβ (t) + dM αβ (t) Fαβ
1α 1β N
V (0) = 1.
(5.6.4)
136
Chapter 5. Quantum stochastic calculus on IFS
Now we take α = β = 0 and by (5.6.2) we have that for all t ∈ R+ , V (t) = 1
is a solution.Iin order to prove the isometry of U , it is sufficient to show that
V (t) = 1 is the unique solution of equation (5.6.4). To this aim we take V,
Z solutions of (5.6.4). Therefore X := V − Z is a solution of
½
∗ X (t)
dX (t) = X (t) Fαβ dM αβ (t) + dM αβ (t) Fαβ
X (0) = 0.
From Theorem 5.4.1 we know that X ≡ 0 is the unique solution of such a
QSDE, then V (t) ≡ Z (t) .
¤
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