Hilbert Module Realization of the Square of White
Noise and the Finite Difference Algebra
Luigi Accardi
Centro Vito Volterra
Università degli Studi di Roma “Tor Vergata”
Via di Tor Vergata, s.n.c.
00133 Roma, Italia
E-mail: accardi@volterra.mat.uniroma2.it
Homepage: http://volterra.mat.uniroma2.it
Michael Skeide∗
Lehrstuhl für Wahrscheinlichkeitstheorie und Statistik
Brandenburgische Technische Universität Cottbus
Postfach 10 13 44, D–03013 Cottbus, Germany
E-mail: skeide@math.tu-cottbus.de
Homepage: http://www.math.tu-cottbus.de/INSTITUT/lswas/ skeide.html
September 1999
Abstract
We present an approach to the representations theory of the square of white
noise based on Hilbert modules. We recover the unique Fock representation and
show that the representation space is a usual symmetric Fock space. However,
although we started with one degree of freedom we end up with countably many
degrees of freedom.
Surprisingly, our representation turns out to have a close relation to Feinsilver’s finite difference algebra. In fact, there exists a homomorpic image of the finite
difference algebra in the algebra of square of white noise. Our representation restricted to this image is Boukas’ representation on the finite difference Fock space.
Thus, we can say that we extend Boukas’ representation to a bigger algebra which
is generated by creators, annihilators, and number operators.
∗
MS is supported by Deutsche Forschungsgemeinschaft.
1
1
Introduction
Following [ALV99], by a white noise we understand operator-valued distributions b+
t and
bt (indexed by t ∈ R) which fulfill the CCR [bt , b+
s ] = δ(t − s). Therefore, formally the
2
square of white noise should be the operator-valued distributions Bt+ = b+
and Bt = b2t
t
fulfilling commutation relations which follow from the CCR.
Unfortunately, it turns out that the objects Bt+ , Bs are too singular. This becomes
manifest in the fact that their formal commutator has a factor δ 2 (t − s), which a priori
does not make sense. To overcome this trouble it was proposed in [ALV99] is to consider a
renormalization of the singular object δ 2 in which δ 2 is replaced by 2cδ where c > 0. This
choice is motivated by a regularization procedure where δ is approximated by functions δε
such that δε2 → 2cδ in a suitable sense (where c might be even complex).
After this renormalization the commutator [Bt , Bs+ ] = δ(t − s)(2c + 4b+
t bt ) has an
operator part on the right-hand side, namely, the number density Nt = b+
t bt . Since
[Nt , Bs+ ] = δ(t − s)2Bt+ , the Lie algebra is closed. Smearing out the densities by setting
R
R
Bf+ = f (t)Bt+ dt and Na+ = a(t)Nt+ dt, and computing the formal commutators, we
find the following relations.
[Bf , Bg+ ] = 2c Tr(f g) + 4Nf g
f, g ∈ L2 (R) ∩ L∞ (R)
(1.1a)
+
[Na , Bf+ ] = 2Baf
a ∈ L∞ (R), f ∈ L2 (R) ∩ L∞ (R)
(1.1b)
R
and [Bf+ , Bg+ ] = [Na , Na0 ] = 0 where we set Tr f = f (t) dt. Our goal is to find a
representation of the ∗–algebra generated by these relations.
In [ALV99] a representation has been constructed with the help of a Kolmogorov
decompostion for a certain positive definite kernel. It was not so difficult to define the
correct kernel, but, it was difficult to show that it is, indeed, positive definite. In [Sni99]
Sniady found an explicit form of the kernel from which positivity is apparent. Here we
proceed in a different way, motivated by the following observations. Relation (1.1a), looks
like the usual CCR, except that the inner product on the right-hand side takes values in
the algebra generated by the number operators Na . It is, therefore, natural to try a
realization on a Hilbert module over the algebra of number operators. Additionally, on a
Hilbert module we have a chance to realize also Relation (1.1b), by putting explicitly a
suitable left multiplication by number operators.
Since the number operators are unbounded, we cannot use the theory of Hilbert modules over C ∗ –algebras, but, we need the theory of pre-Hilbert modules over ∗–algebras as
described in [AS98]. In Section 2 we sum up the necessary notions. In Section 3 we show
how to define representations of the algebra of number operators just by fixing the values
of the representation on Na . This is essential for the definition of the left multiplication.
It allows to identify the algebra generated by Na not just as an abstract algebra generated
by relations, but, concretely as an algebra of number operators.
2
The main part is Section 4. Here we construct a two-sided pre-Hilbert module E,
and we show that it is possible to construct a symmetric Fock module Γ(E) over E. We
see that the natural creation operators a∗ (f ) on this Fock module and the natural left
multiplication by Na fulfill Relation (1.1a) up to an additive term and Relation (1.1b). By
a tensor product construction we obtain a pre-Hilbert space where Relations (1.1a,1.1b)
are realized. This representation coincides with the one constructed in [ALV99].
In Section 5 we show that our representation space is isomorphic to the usual symmetric Fock space over L2 (R, `2 ). In the final Section 6 we show that our representation
may be considered as an extension of Boukas’ representation of Feinsilver’s finite difference algebra [Fei87] on the finite difference Fock space. The calculus based on the square
of white noise should generalize Boukas’ calculus in [Bou91b]. In [PS91] Parthasarathy
and Sinha realized the finite difference algebra by operators on a symmetric Fock space.
They do, however, not consider the question, whether this representation is equivalent to
Boukas’ representation. It is likely that the square of white noise allows for a similar representation. Of course, it would also be interesting to know, whether our representation
of the square of white noise is faithful. We leave such questions to future work.
2
Hilbert modules over ∗–algebras
For our goals we need a notion of Hilbert modules over ∗–algebras of unbounded operators.
This makes the definition of positivity somewhat tricky. In a C ∗ –algebra there are many
equivalent ways to define positive elements and positive linear functionals. In a general
∗–algebra different definitions give rise to different notions of positivity. For instance, the
algebraic definition where positive elements are those in the convex cone generated by all
elements of the form b∗ b does not contain enough positive elements for our purposes. On
the other hand, a weak definition where we say an element b is positive, if ϕ(b) ≥ 0 for
all positive functionals is uncontrollable. It does, for instance, not allow to show that the
inner product on a tensor product of Hilbert modules is again positive.
Here we follow the approach of [AS98] where we put by hand certain elements to
be positive and consider a certain cone generated by these elements. This definition of
positivity remains controllable, because, in principle, it is algebraic. It makes, however,
necessary to involve directly the left multiplication on a two-sided Hilbert module. Therefore, we do not give a definition of a right Hilbert module, but immediately of a two-sided
Hilbert module.
2.1 Definition. Let B be a unital ∗–algebra and let S be a distinguished subset of selfadjoint elements in B containing 1. By P (S) we denote the convex B–cone generated by
S (i.e. the set of all sums of elements of the form a∗ ba with b ∈ S, a ∈ B). We say the
3
elements of P (S) are S–positive.
A pre-Hilbert B–module is a B–B–module E (where 1x = x1 = x) with a sesquilinear
inner product h•, •i : E × E → B, fulfilling the following requirements
hx, xi = 0 ⇒ x = 0
(definiteness),
hx, ybi = hx, yib
(right B–linearity),
hx, byi = hb∗ x, yi
(∗–property),
and the positivity condition that for all choices of b ∈ S and of finitely many xi ∈ E there
exist finitely many bk ∈ S and bki ∈ B, such that
X
hxi , bxj i =
b∗ki bk bkj .
k
If definiteness is missing, then we speak of a semiinner product and a semi-Hilbert module.
A mapping a on a semi-Hilbert B–module E is adjointable, if there exists a mapping
a∗ on E such that hx, ayi = ha∗ x, yi. By Lr (E) and La (E) we denote the spaces of right
B–linear and of adjointable mappings, respectively, on E. Notice that on a pre-Hilbert
module the adjoint is unique, and that La (E) ⊂ Lr (E).
Since 1 ∈ S, the inner product is S–positive (i.e. hx, xi ∈ P (S)), and since S consists
only of self-adjoint elements, the inner product is symmetric (i.e. hx, yi = hy, xi∗ ) and left
anti-linear (i.e. hxb, yi = b∗ hx, yi). It is sufficient to check the positivity on a subset of
E which generates E as a right module; see [AS98]. It is also sufficient, if we show only
existence of bk ∈ P (S).
2.2 Observation. Any semi-Hilbert module over a commutative algebra B has a trivial
left module structure over B where right and left multiplication are just the same. We
denote this trivial left multiplication by br , i.e. br : x 7→ xb, in order to distinguish it from
a possible non-trivial left multiplication.
Let E and F be semi-Hilbert B–modules. Their tensor product over B
E ¯ F = E ⊗ F/{xb ⊗ y − x ⊗ by}
is turned into a semi-Hilbert B–module by setting
hx ¯ y, x0 ¯ y 0 i = hy, hx, x0 iy 0 i.
In general, E ¯ F and F ¯ E may be quite different objects (e.g. one is {0} and the
other is not; see [Ske98, Example 6.7]). Usually, the mapping x ¯ y 7→ y ¯ x is illdefined. Therefore, it is not always possible to construct a symmetric Fock module over an
arbitrary one-particle sector E. The tensor sign ¯ is “transparent” for algebra elements,
4
i.e. x ¯ by = xb ¯ y. Any operator a ∈ La (E) gives rise to a well-defined operator
a ¯ id ∈ La (E ¯ F ). (This embedding is, however, not necessarily injective.) For operators
a on F the situation is not so pleasant. In general, we can define id ¯a, if a is bilinear.
(Also this embedding need not be injective.)
So far we were concerned with semi-Hilbert modules. For several reasons it is desirable
to have a strictly positive inner product. For instance, contrary to a semi-inner product, an
inner product guarantees for uniqueness of adjoints. We provide a quotienting procedure,
which allows to construct a pre-Hilbert module out of a given semi-Hilbert module, if
on B there exists a separating set S ∗ of positive functionals which is compatible with
the positivity structure determined by S. We say a functional ϕ on B is S–positive, if
ϕ(b) ≥ 0 for all b ∈ P (S). Let S ∗ be a set of S–positive functionals. We say S ∗ separates
the points (or is separating), if ϕ(b) = 0 for all ϕ ∈ S ∗ implies b = 0. If S ∗ is a separating
©
ª
set of S–positive functionals on B, then the set N = x ∈ E : hx, xi = 0 is a two-sided
B–submodule of E. Moreover, the quotient module E0 = E/N inherits a pre-Hilbert
B–module structure by setting hx + N, y + Ni = hx, yi.
Before comming to Fock modules, we describe a construction which relates our algebraic definition of positivity with “concrete” positivity of operators on pre-Hilbert
spaces. Let π be a representation of B on a pre-Hilbert space G. In other words, G
is a B–C–module and we can ask, whether (equipping C with convex C–cone generated
by 1 as positive elements, and extending our definition of pre-Hilbert modules to two-sided
modules over different algebras in an obvious fashion) G with its natural inner product
is a pre-Hilbert module. For this it is necessary and sufficient that hg, π(b)gi ≥ 0 for all
g ∈ G and all b ∈ S (see [AS98]). If π has this property, then we say it is S–positive. In
particular, π is S–positive, if it sends elements in S to sums over elements of the form b∗ b
(b ∈ La (G)).
Now let π be S–positive and let E be a pre-Hilbert B–module. Then our tensor
product construction goes through as before (notice that idC constitutes a separating set
of positive functionals on C) and we obtain a pre-Hilbert B–C–module H = E ¯ G. In
other words, H is a pre-Hilbert space with a representation ρ(b) = b ¯ id of B. (Actually,
with the same definition ρ extends to a representation of La (E) on H.) Additionally,
we may interpret elements x ∈ E as mappings Lx : g 7→ x ¯ g in La (G, H) with adjoint
L∗x : y ¯ g 7→ π(hx, yi)g. Of course, Lbxb0 = ρ(b)Lx π(b0 ). Observe that x 7→ Lx is one-toone, if π is faithful. In this case also ρ is faithful.
2.3 Definition. The full Fock module over a two-sided pre-Hilbert B–module E is the
∞
L
two-sided pre-Hilbert B–module F(E) =
E ¯n (where E ¯0 = B with inner product
n=0
hb, b0 i = b∗ b0 and the direct sum is algebraic). We denote the unit of E ¯0 by ω in order to
distinguish it from right or left multiplication by 1 ∈ B.
5
On F(E) we define for each x ∈ E the creation operator `∗ (x) by setting
`∗ (x)xn ¯ . . . ¯ x1 = x ¯ xn ¯ . . . ¯ x1
`∗ (x)ω = x,
and its adjoint, the annihilation operator, by setting
`(x)xn ¯ . . . ¯ x1 = hx, xn ixn−1 ¯ . . . ¯ x1
`(x)ω = 0.
For each bilinear operator a on E we define the operator λ(a) on F(E), by setting
λ(a)xn ¯ . . . ¯ x1 = axn ¯ xn−1 ¯ . . . ¯ x1
+ xn ¯ axn−1 ¯ . . . ¯ x1 + . . . + xn ¯ xn−1 ¯ . . . ¯ ax1
and λ(a)ω = 0.
Fock modules were first considered in [AL92, AL96]. The first formal definitions of
full Fock modules in the framework of Hilbert C ∗ –modules were given in [Pim97, Spe98].
Here we use the extension to the framework of ∗–algebras as given in [AS98].
3
The algebra of number operators of L∞(R)
We introduce the commutative ∗–algebra of number operators on the symmetric Fock
space Γ(L2 (R)) to elements of L∞ (R). Like for the full Fock module, we use an algebraic
definition of symmetric Fock space. For us it is important to consider Γ(L2 (R)) as a
subspace of F(L2 (R)).
3.1 Definition. On the full Fock space F(L2 (R)) (with vacuum denoted by Ω) we define
a projection P by setting
P fn ⊗ . . . ⊗ f1 =
1 X
fσ(n) ⊗ . . . ⊗ fσ(1)
n! σ∈S
n
and P Ω = Ω. The symmetric Fock space Γ(L2 (R)) is P F(L2 (R)).
For a ∈ L∞ (R) we define the number operator Na = P λ(a)P . By
N = alg{Na (a ∈ L∞ (R))}
we denote the unital algebra generated by all Na .
Cearly, Na∗ = Na∗ and Na Na0 = Na0 Na , so that N is a commutative ∗–algebra. By the
multiple polarization formula
1
2n
X
εn . . . ε1 (εn fn + . . . + ε1 f1 )⊗n =
εn ,...,ε1 =±1
X
σ∈Sn
6
fσ(n) ⊗ . . . ⊗ fσ(1)
(3.1)
we see that the vectors Ω and f ⊗n (f ∈ L2 (R), n ∈ N) form a total subset of Γ(L2 (R)).
Observe also that Na = P λ(a) = λ(a)P . This follows easily from P λ(a)P = λ(a)P and
its adjoint.
Let I1 , I2 be two disjoint measurable subsets of R. It is noteworthy that the wellknown factorization Γ(L2 (S1 )) ⊗ Γ(L2 (S2 )) ∼
= Γ(L2 (S1 ∪ S2 )) restricts to our algebraic
domain, i.e. Γ(L2 (S1 )) ⊗ Γ(L2 (S2 )) ∼
= Γ(L2 (S1 ∪ S2 )), cf. the proof of Theorem 5.1. In
this identification we have NχS1 = NχS1 ⊗ id and NχS2 = id ⊗NχS2 . Similar statements
are true for factorization into more than two disjoint subsets.
Since Na Ω = 0 for any a ∈ L∞ (R), the vacuum state ϕΩ (•) = hΩ, •Ωi is a character
for N . Its kernel consists of the span of all monomials with at least one factor Na and its
GNS-pre-Hilbert space is just CΩ.
3.2 Definition. As positivity defining subset of B we choose
©
ª
S = NχI1 . . . NχIn : Ii bounded intervals in R (n ∈ N0 ; i = 1, . . . , n)
3.3 Proposition. The defining representation id of N on Γ(L2 (R)) is S–positive.
P ∗
bi bi where bi are taken (for all
Proof. It is sufficient to show that NχI is of the form
i
¡
¢
I) from a commutative subalgebra of La F(L2 (R)) . Let λni (a) (n ∈ N, 1 ≤ i ≤ n) be
the representation of L∞ (R) which acts on the i–th component of the n–particle sector of
F(L2 (R)). Then
Nχ I =
X
X
λni (χI ) =
1≤i≤n<∞
λni (χI )∗ λni (χI ).
1≤i≤n<∞
In the following section we intend to define a representation of N by assigning to each
Na an operator and extension as algebra homomorphism. The goal of the remainder of
the present section is to show that this is possible, at least, if we restrict to the subalgebra
S(R) of step functions, which is dense in L∞ (R) in a suitable weak topology.
Denote by N the number operator on F(L2 (R)) which sends F ∈ E ⊗n to nF . (As
P N = N P we use the same symbol for the number operator on Γ(L2 (R)).) Then, clearly,
alg{N } is isomorphic to the algebra of polynomials in one self-adjoint indeterminate.
Moreover, for each measurable non-null-set S ⊂ R the algebra alg{NχS } is isomorphic to
alg{N }. Therefore, for any self-adjoint element a in a ∗–algebra A the mapping N 7→ a
extends to a homomorphism alg{N } → A.
Let t = (t0 , . . . , tm ) be a tuple with t0 < . . . < tm . Then by the factorization
Γ(L2 (t0 , tm )) = Γ(L2 (t0 , t1 )) ⊗ . . . ⊗ Γ(L2 (tm−1 , tm )) we find
Nt := alg{Nχ[tk−1 ,tk ] (k = 1, . . . , m)} = alg{Nχ[t0 ,t1 ] } ⊗ . . . ⊗ alg{Nχ[tm−1 ,tm ] }.
7
Therefore, any involutive mapping
Tt : St (R) := span{χ[tk−1 ,tk ] (k = 1, . . . , m)} −→ A
with commutative range defines a unique homorphism ρt : Nt → A fulfilling ρt (Na ) =
Tt (a).
Now we are ready to prove the universal property of the algebra NS :=
S
t
Nt which
shows that NS is nothing but the symmetric tensor algebra over the involutive vector
space S(R).
3.4 Theorem. Let T : S(R) → A be an involutive mapping with commutative range.
Then there exists a unique homorphism ρ : NS → A fulfilling ρ(Na ) = T (a).
Proof. It suffices to remark that NS is the inductive limit of Nt over the set of all tuples
t directed increasingly by “inclusion” of tuples. Denoting by βts the canonical embedding
Ns → Nt (s ≤ t) we easily check that ρt ◦ βts = ρs . In other words, the family ρt extends
as a unique homomorphism ρ to all of NS .
4
Realization of square of white noise
The idea to realize Relations (1.1a) and (1.1b) on a symmetric Fock module is to take
the right-hand side of (1.1a) as the definition of an N –valued inner product on a module
E generated by the elements f ∈ L2 (R) ∩ L∞ (R) and then to define a left multiplication
by elements of N such that the generating elements f fulfill (1.1b). However, the direct
attempt with an inner product determined by (1.1a) fails. Therefore, we start with the
linear ansatz in (4.1) and adjust the constants later suitably.
In view of Theorem 3.4, for the time being, we restrict to elements in NS . By (1.1a)
this makes it necessary also to restrict to elements f ∈ S(R).
On S(R) ⊗ NS with its natural right NS –module structure we define for arbitrary
positive constants β and γ a sesquilinear mapping h•, •i, by setting
hf ⊗ 1, g ⊗ 1i = Mf g
where Ma = β Tr a + γNa
(4.1)
and by right linear and left anti-linear extension.
We define a left action of Ma , by setting
Ma (f ⊗ 1) = f ⊗ Ma + αaf ⊗ 1
and right linear extension to all elements of S(R) ⊗ NS . Here α is an arbitrary real
constant. Observe that the scalar term in Ma does not change this commutation relation.
8
Therefore, Na fulfills the same commutation relations with α replaced by
α
.
γ
In view of
(1.1b) this fraction should be equal to 2.
By definition, mutliplication by Ma from the left is a right linear mapping on S(R) ⊗
NS . One easily checks that Ma∗ = Ma∗ is an adjoint with respect to the sesquilinear
mapping (4.1). By Theorem 3.4 this left action extends to a left action of all elements of
NS .
4.1 Proposition. (4.1) is a semi-inner product so that S(R) ⊗ NS is a semi-Hilbert
NS –module.
Proof. We have to check only the positivity condition, because the remaining properties
are obvious. We remarked already that it is sufficient to check positivity for elements
of the form χIi ⊗ 1, because these elements generate S(R) ⊗ NS as a right module.
Additionally, we may assume that Ii ∩ Ij = ∅ for i 6= j. Then hχIi ⊗ 1, b(χIj ⊗ 1)i = 0
for i 6= j, whatever b ∈ NS might be. Now let b be in S. We may assume (possibly
Q
after modifying the Ii suitably) that b has the form NχnIi where ni ∈ N0 . Observe that
i
i
NχIi (χIj ⊗ 1) = (χIj ⊗ 1)NχIi for i 6= j, and that (proof by induction)
NχnI (χI ⊗ 1) =
n
X
¡n¢¡ α ¢(n−k)
k
γ
(χI ⊗ NχkI ).
k=0
It follows that
hχIi ⊗ 1, b(χIj ⊗ 1)i = δij MχIi
ni
X
¡ni ¢¡ α ¢(ni −k)
k
γ
k=0
NχkI
i
Y
NχnI` .
`6=i
`
Let us define bk = hχIk ⊗1, b(χIk ⊗1)i and bki = δki 1. Then hχIi ⊗1, b(χIj ⊗1)i =
(and, of course, bk ∈ P (S)).
P
k
b∗ki bk bkj
We may divide out the length-zero elements so that
E := S(R) ⊗ NS /NS(R)⊗NS
is a two-sided pre-Hilbert NS –module. We use the notation f ⊗ b + NS(R)⊗NS = f b.
Clearly, we have
Ma f = f Ma + αaf.
(4.2)
On the generating subset fn ¯ . . . ¯ f1 of E ¯n we find by repeated application of (4.2)
Ma fn ¯ . . . ¯ f1 = fn ¯ . . . ¯ f1 Ma + αλ(a)fn ¯ . . . ¯ f1 .
Therefore, on the full Fock module F(E) we have the relation
Ma = Mar + αλ(a)
9
(4.3)
where Mar denotes multiplication by Ma from the right in the sense of Observation 2.2.
Now we try to define the symmetric Fock module over E analogously to Definition
3.1. The basis for the symmetrization is a flip which exchanges in a tensor f ⊗ g the order
of factors. As we already remarked, we may not hope to define a flip on E ¯ E by just
sending x ¯ y to y ¯ x for all x, y ∈ E. We may, however, hope to succeed, if we, as done
in [Ske98] for centered modules, define such a flip only on such x, y which come from a
suitable generating subset of E. In order to build general permutations on E ¯n we also
must be sure that the flip is bilinear.
4.2 Proposition. The mapping
τ : f ¯ g 7−→ g ¯ f
for all f, g ∈ S(R) ⊂ E
extends to a unique bilinear unitary (i.e. inner product preserving and surjective) isomorphism E ¯ E → E ¯ E.
Proof. We find
hf ¯ g, f 0 ¯ g 0 i = hg, hf, f 0 ig 0 i = hg, Mf f 0 g 0 i = hg, g 0 Mf f 0 + αf f 0 g 0 i
= Mf f 0 Mgg0 + αMgf f 0 g0 = Mgg0 Mf f 0 + αMf gg0 f 0 = hg ¯ f, g 0 ¯ f 0 i.
The elements f ¯ g form a (right) generating subset of E ¯ E. Therefore, τ extends as
a well-defined isometric (i.e. inner product preserving) mapping to E ¯ E. Clearly, this
extension is surjective so that τ is unitary.
It remains to show that τ is bilinear. Again it is sufficient to show this on a generating
subset and, of course, to show it only for the generators Ma of NS . We find
τ (Ma f ¯ g) = τ (f ¯ gMa + α(af ¯ g + f ¯ ag))
= g ¯ f Ma + α(g ¯ af + ag ¯ f ) = Ma g ¯ f = Ma τ (f ¯ g).
Now we are in a position to define the symmetric Fock module Γ(E) precisely as in
Definition 3.1. The preceding proposition also shows that P Ma = Ma P , i.e. P is a bilinear
projection. Again, we have P λ(a) = λ(a)P = P λ(a)P . Consequently, (4.3) remains true
also on the symmetric Fock module. In the sequel, we do not distinguish between λ(a)
and its restriction to Γ(E). In both cases we denote the number operator by N := λ(1).
As λ(a) is bilinear, so is N and, of course, N P = P N . Also here we find by (3.1) that
the symmetric tensors form a generating subset.
For x ∈ E we define the creation operator on Γ(E) as a∗ (x) =
√
N P `∗ (x). Clearly,
x 7→ a∗ (x) is a bilinear mapping, because x 7→ `∗ (x) is. We find the commutation relation
Ma a∗ (f ) = a∗ (Ma f ) = a∗ (f Ma + αaf ) = a∗ (f )Ma + αa∗ (af ).
10
√
Of course, a∗ (x) has an adjoint, namely, a(x) = `(x)P N .
Now we restrict our attention to creators a∗ (f ) and annihilators a(f ) to elements f in
S(R) ⊂ E. Their actions on symmetric tensors g ¯n (g ∈ S(R)) are
n
∗
a (f )g
¯n
X
1
=√
g ¯i ¯ f ¯ g ¯(n−i)
n + 1 i=0
a(f )g ¯n =
√
nMf g g ¯(n−1) .
Clearly, a∗ (f )a∗ (g) = a∗ (g)a∗ (f ). However, nothing like this is true for a∗ (x) and a∗ (y)
for more general elements in x, y ∈ E.
For the CCR we have to compute a(f )a∗ (f 0 ) and a∗ (f 0 )a(f ). We find
n
∗
0
a(f )a (f )g
¯n
X
1
=√
a(f )
g ¯i ¯ f ¯ g ¯(n−i)
n+1
i=0
=Mf f 0 g
¯n
+ Mf g
n−1
X
g ¯i ¯ f 0 ¯ g ¯(n−1−i)
i=0
and
¢
√
√ ¡
a∗ (f 0 )a(f )g ¯n = na∗ (f 0 )Mf g g ¯(n−1) = n Mf g a∗ (f 0 ) − αa∗ (f gf 0 ) g ¯(n−1)
=Mf g
n−1
X
g ¯i ¯ f 0 ¯ g ¯(n−1−i) − αλ(f f 0 )g ¯n .
i=0
Taking the difference, the sums over i dissappear. Taking into account that g ¯n is arbitrary and also (4.3), we find
[a(f ), a∗ (f 0 )] = Mf f 0 + αλ(f f 0 ) = 2Mf f 0 − Mfrf 0 = β Tr(f f 0 ) + 2γNf f 0 − γNfrf 0 .
In other words, putting β = 2c and γ = 2 we have realized (1.1a) by operators a∗ (f ),
however, only modulo some right multiplication by certain elements of NS . (Notice that
this is independent of the choice of α. Putting α = 4 we realize also (1.1b).)
So we have to do two things. Firstly, we must get rid of contributions of Nar in the
above relation. Secondly, in order to compare with the construction in [ALV99] we must
interpret our construction in terms of pre-Hilbert spaces. Both goals can be achieved at
once by the following construction. We consider the tensor product H = Γ(E) ¯ CΩ
of Γ(E) with the pre-Hilbert NS –C–module CΩ which is the pre-Hilbert space carrying
the GNS-representation of the vacuum state ϕΩ on NS . This tensor product is possible,
because by Proposition 3.3 the defining representation of N on Γ(L2 (R)) and, therefore,
any subrepresentation on an invariant subspace is S–positive. Thus, H is a pre-Hilbert
space and carries a representation of La (Γ(E)). In this representation all operators Nar
are represented by 0. Indeed, Nar commutes with everything, so that we put it on the
right, and
Nar g ¯n ¯ Ω = g ¯n ¯ Nar Ω = 0.
11
By Bf+ we denote the image of a∗ (f ) in La (H). The image of Na coincides with the image
of 4λ(a). We denote both by the same symbol Na . By Φ = ω ¯ Ω we denote the vacuum
in H.
4.3 Theorem. The operators Bf+ , Na ∈ La (H) (f, a ∈ S(R)) fulfill Relations (1.1a),
n
(1.1b), and [Bf+ , Bg+ ] = [Na , Na0 ] = 0. Moreover, the vectors Bf+ Φ (f ∈ S(R), n ∈ N0 )
span H.
4.4 Remark. It follows that H is precisely the pre-Hilbert space as constructed in
[ALV99]. However, in [ALV99] the inner product on the total set of vectors was defined a priori and it was quite tedious to show that it is positive. Here positivity and also
well-definedness of the representation are automatic.
n
4.5 Remark. Putting Hn = span{Bf+ Φ (f ∈ S(R))}, we see that H =
∞
L
Hn is an in-
n=0
teracting Fock space with creation operators Bf+ as introduced in [ALV97] in the notations
of [AS98].
4.6 Theorem. The realization of Relations (1.1a) and (1.1b) extends to elements f ∈
∞
L
Hn .
L2 (R) ∩ L∞ (R) and a ∈ L∞ (R) as a representation by operators on
n=0
Proof. We extend the definition of the operators Bf+ and Na formally to f ∈ L2 (R) ∩
n
L∞ (R) and a ∈ L∞ (R), considering them as operators on vectors of the form Bf+ Φ
(f ∈ L2 (R) ∩ L∞ (R), n ∈ N0 ). The inner product of such vectors we define by continuous
extension of the inner product of those vectors where f ∈ S in the σ–weak topology of
L∞ (R) (which, clearly, is possible and unique). Positivity of this inner product follows by
approximation with inner products, and well-definedness of our operators follows, because
all operators have formal adjoints.
5
The associated product system
Let I ⊂ R be a finite union of intervals. Denote by HI the subspace of H spanned
n
by vectors of the form Bf+ Φ (f ∈ S(I), n ∈ N0 ). In particular, for 0 ≤ t ≤ ∞ set
Ht := H[0,t) . (This means that H0 = H∅ = CΦ.) Notice that HI does not depend on
whether the intervals in I are open, half-open, or closed.
Denote by I + t the time shifted set I. Denote by ft the time shifted function ft (s) =
n
n
f (s − t). Obviously, by sending Bf+ Φ to Bf+t Φ we define an isomorphism HI → HI+t .
Observe that by Relation (1.1a) the operators Bf and Bg+ to functions f ∈ S(I) and
g ∈ S(R\I) commute. Define NI := alg{Na (a ∈ S(I))}. Then by Relation (1.1b) also
the elements of NI commute with all Bg to functions g ∈ S(R\I).
12
5.1 Theorem. Let I, J ⊂ R be finite unions of intervals such that I ∩ J is a null-set.
Then
n
n
m
m
UIJ : Bf+ Bg+ Φ 7−→ Bf+ Φ ⊗ Bg+ Φ
for
f ∈ HI , g ∈ HJ
extends as an isomorphism HI∪J → HI ⊗ HJ . Of course, the composition of these isomorphisms is associative in the sense that (UIJ ⊗ id) ◦ U(I∪J)K = (id ⊗UJK ) ◦ UI(J∪K) .
n
n
m
m
Proof. Of course, the vectors Bf+ Bg+ Φ are total in HI∪J and the vectors Bf+ Φ⊗Bg+ Φ
are total in HI ⊗ HJ . Thus, it suffices to show isometry. We have
­ m
­ +n +m
n0
m0 ®
n0
m0 ®
Bf Bg Φ, Bf+0 Bg+0 Φ = Bg+ Φ, Bfn Bf+0 Bg+0 Φ .
Without loss of generality suppose that n ≥ n0 . We have
0
n
Bfn Bf+0
0
=
0
0
0
n
Bfn−n Bfn Bf+0
=
0
Bfn−n
n
X
k
bk Bf+0 Bfk
k=0
where bk ∈ NI . As Bf commutes with Bg+ and Bf Φ = 0, the only non-zero contribution
0
comes from Bfn−n b0 . On the other hand, also b0 commutes with Bg+ and b0 Φ = ΦϕΩ (b0 ) =
0
ΦhΦ, b0 Φi. Therefore, also Bfn−n (commuting with Bg+ ) comes to act directly on Φ and
gives 0, unless n = n0 . Henceforth, the only non-zero contributions appear for n = n0 and
m = m0 . We obtain
­ +n +m
­ m
­ m
n
m ®
m ®
m ® ­
n
n ®
Bf Bg Φ, Bf+0 Bg+0 Φ = Bg+ Φ, Bg+0 Φ hΦ, b0 Φi = Bg+ Φ, Bg+0 Φ Bf+ Φ, Bf+0 Φ
as desired.
5.2 Corollary. We have Hs ⊗ Ht ∼
= H[0,s)+t ⊗ Ht ∼
= Hs+t . Also here the isomorphisms
Ust : Hs ⊗ Ht → Hs+t compose associatively. In other words, the Ht form a tensor product
system of pre-Hilbert spaces in the sense of [BS99].
The definition in [BS99] is purely algebraic. Of course, it is interesting to ask, whether
the Ht form a tensor product system in the stronger sense of Arveson [Arv89], who
introduced the notion. To that goal all Ht must be separable (infinite-dimensional) and
some measurability conditions must be checked. We proceed more indirectly.
If Ht is an Arveson system, then naturally the question arises what type it has. Type
I product systems are precisely those which come from a symmetric Fock space and the
characteristic property of the symmetric Fock space (among other Arveson systems) is
that it is spanned by exponential vectors. In other words, we must find all families ψt ∈ Ht
(so-called units) which compose under tensor product like exponential vectors to indicator
functions, i.e. ψs ⊗ ψt = ψs+t .
Good canditates for exponential vectors are
∞
∞
+ n
X
Φ X
Bρχ
ρn + n
t
ψρ (t) =
=
B Φ
n!
n! χt
n=0
n=0
13
where ρ ∈ C and χt := χ[0,t] . (If we replace B + by creators on the usual symmetric Fock
space we obtain precisely the usual exponential vector ψ(ρχt ).) Whenever ψρ0 (t) exists,
then it is an analytic vector-valued function of ρ with |ρ| < |ρ0 |. It is not difficult to
check that whenever ψρ (s) and ψρ (t) exist, then also ψρ (s + t) exists and equals (in the
factorization as in Corollary 5.2) ψρ (s) ⊗ ψρ (t). Moreover, as ψρ (t) is analytic in ρ, we
¯
n
dn ¯
may differentiate. It follows that Bχ+t Φ = dρ
n ρ=0 ψρ (t) is in the closed linear span of
ψρ (t) (|ρ| < |ρ0 |). Therefore, if for each t > 0 there exists ρ0 > 0 such that ψρ0 (t) exists,
then the vectors ψρ (t) and their time shifts form a total subset of H∞ .
5.3 Lemma. ψρ (t) exists, whenever |ρ| < 12 . Moreover, we have
ct
hψρ (t), ψσ (t)i = e− 2
ln(1−4ρσ)
(5.1)
where the function
c
κ : (ρ, σ) 7→ − ln(1 − 4ρσ)
2
is a positive definite kernel on U 1 (0) × U 1 (0).
2
2
Proof. First, we show that the left-hand side of (5.1) exists in the simpler case σ = ρ
showing, thus, existence of ψρ (t).
Set f = ρχt . Then (f f )f = |ρ|2 f . This yields the commutation relation Nf f Bf+ =
Bf+ Nf f + 2 |ρ|2 Bf+ . Moreover, 2c Tr(f f ) = 2c |ρ|2 t. We find
Bf Bf+ =Bf+ Bf Bf+
n
n−1
+ (2c |ρ|2 t + 4Nf f )Bf+
=Bf+ Bf Bf+
n−1
+ Bf+
n−1
n−1
(2c |ρ|2 t + 8 |ρ|2 (n − 1)) + Bf+
n
n−1
=Bf+ Bf + nBf+ 4Nf f
¡
n−1
+ Bf+ 2 |ρ|2 (ct + 4(n −
n
n−1
=Bf+ Bf + nBf+ 4Nf f +
n−1
4Nf f
1)) + (ct + 4(n − 2)) + . . . + (ct + 0)
Bf+
n−1
¢
2n |ρ|2 (ct + 2(n − 1)).
If we apply this to the vacuum Φ, then the first two summands dissappear. We find the
recursion formula
+
³ ct
hBf+ Φ, Bf+ Φi
Φ, Bf+ Φi
n − 1 ´ hBf
2
= 4 |ρ|
+
.
(n!)2
2n
n
((n − 1)!)2
n
It is clear that
∞
P
n=0
n
n
n
hBf+ Φ,Bf+ Φi
(n!)2
n−1
n−1
converges, if and only if 4 |ρ|2 < 1 or |ρ| < 12 .
For fixed ρ ∈ U 1 (0) the function hψρ (t), ψρ (t)i is the uniform limit of entire functions on
2
t and, therefore, itself an entire function on t. In particular, since ψρ (s+t) = ψρ (s)⊗ψρ (t),
there must exist a number κ ∈ R (actually, in R+ , because hψρ (t), ψρ (t)i ≥ 1) such that
hψρ (t), ψρ (t)i = eκt . We find κ by differentiating at t = 0. The only contribution in
¯
³ ct
³
´
d ¯¯
n − 1´
2
2 ct
4 |ρ|
+
· . . . · 4 |ρ|
+0
dt ¯t=0
2n
n
2
14
comes by the Leibniz rule, if we differentiate the last factor and put t = 0 in the remaining
ones. We find
¯
∞
X
d ¯¯
c
1c
hψρ (t), ψρ (t)i =
(4 |ρ|2 )n
= − ln(1 − 4 |ρ|2 ).
¯
dt t=0
n2
2
n=1
The remaining statements follow essentially by the same computations, replacing |ρ|2 with
ρσ. Cleary, ρσ is a positive definite kernel. Then by Schur’s lemma also the function
κ(ρ, σ) as a limit of positive linear combinations of powers of ρσ is positive definite.
5.4 Remark. The function κ is nothing but the covariance function of the product
system in the sense of Arveson [Arv89], which is defined on the set of all units, restricted
to the set of special units ψρ (t). In the set of all units we must take into account also
multiples ect of our units, and the covariance function on this two parameter set is only a
conditionally positive kernel.
Let
r ³
´
c
(2ρ)2
(2ρ)n
vρ =
2ρ , √ , . . . , √ , . . . ∈ `2 .
2
n
2
Then hvρ , vσ i = − 2c ln(1 − 4ρσ) and the vectors vρ are total in `2 . In other words, the
Kolmogorov decomposition for the covariance function is the pair (`2 , ρ 7→ vρ ). The
following theorem is a simple corollary of Lemma 5.3.
5.5 Theorem. There is a unique time shift invariant isomorphism H∞ → Γ(L2 (R+ , `2 )),
fulfilling
ψρ (t) 7−→ ψ(vρ χt ).
Consequently, Ht is a type I Arveson product system.
5.6 Remark. Defining EI as the submodule of E generated by S(I), we find (for disjoint
I and J) Γ(EI∪J ) = Γ(EI ) ¯ Γ(EJ ) precisely as in the proof of Theorem 5.1. (Here the
isomorphism sends a∗ (f )n a∗ (g)m ω to a∗ (f )n ω ¯ a∗ (g)m ω and the argument is the same
with the exception that, even more simply, bω = ωb.) Clearly, setting Et = E[0,t) , we find
¡
¢
a tensor product system Γ(Et ) of pre-Hilbert NS –NS –modules in the sense of [BS99].
6
Connections with the finite difference algebra
After the rescaling c → 2 and ρ →
ρ
,
2
the right-hand side of (5.1), extended in the
obvious way from indicator functions to step functions, is the kernel used by Boukas
[Bou88, Bou91a] to define a representation space for Feinsilver’s finite difference algebra
[Fei87]. Therefore, Boukas’ space and ours coincide.
15
Once established that the representation spaces coincide, it is natural to ask, whether
the algebra of square of white noise contains elements fulfiling the relations of the finite
difference algebra. Indeed, setting c = 2 and defining
1
Qf = (Bf+ + Nf )
2
1
Pf = (Bf + Nf )
2
Tf = 1 Tr f + Pf + Qf
(6.1)
for f ∈ L2 (R) ∩ L∞ (R), we find
[Pf , Qg ] = [Tf , Qg ] = [Pf , Tg ] = Tf g .
(6.2)
Specializing to f = f ∈ S these are precisely the relations of the finite difference algebra.
In fact, the operators Qf , Pf , Tf are precisely those found by Boukas. However, it is
not clear whether the relation Tf = 1 Tr f + Pf + Qf follows already from (6.2), or is
independent. (In the second case, Boukas’ representation has no chance to be faithful.)
In all cases, the operators Qf , Pf , Tf are not sufficient to recover Bf , Bf+ , Nf . (We
can only recover the operators Bf+ − Bf and Bf+ + Bf + 2Nf .) Whereas the algebra of
square of white noise is generated by creation, annihilation, and number operators, the
representation of the finite difference algebra is generated by certain linear combinations
of such.
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17