We show that the Feinsilver‐Kocik‐Schott (FKS) kernel for the Schrödinger algebra is not positive definite. We show how the FKS Schrödinger kernel can be reduced to a positive definite one through a restriction of the defining parameters of the exponential vectors. We define the Fock space associated with the reduced FKS Schrödinger kernel. We compute the characteristic functions of quantum random variables naturally associated with the FKS Schrödinger kernel and expressed in terms of the renormalized higher powers of white noise (or RHPWN) Lie algebra generators

Accardi, L

Boukas, A

We show that the Feinsilver‐Kocik‐Schott (FKS) kernel for the Schrödinger algebra is not positive definite. We show how the FKS Schrödinger kernel can be reduced to a positive definite one through a restriction of the defining parameters of the exponential vectors. We define the Fock space associated with the reduced FKS Schrödinger kernel. We compute the characteristic functions of quantum random variables naturally associated with the FKS Schrödinger kernel and expressed in terms of the renormalized higher powers of white noise (or RHPWN) Lie algebra generators

ACCARDI, LUIGI

Boukas, A.

English

Random variables and positive definite Kernels associates with the Schrodinger algebra

Archivio della Ricerca - Università di Roma Tor vergata

Random variables and positive definite kernels associated with the Schro¨dinger algebra
Luigi Accardi
Centro Vito Volterra, Universita` di Roma Tor Vergata
via Columbia 2, 00133 Roma, Italy
accardi@volterra.mat.uniroma2.it
http://volterra.mat.uniroma2.it
Andreas Boukas
Department of Mathematics, The American College of Greece
Aghia Paraskevi, Athens 15342, Greece
andreasboukas@acg.edu
1
2Contents
3Abstract We show that the Feinsilver-Kocik-Schott (FKS) kernel for the Schro¨dinger alge-
bra is not positive definite. We show how the FKS Schro¨dinger kernel can be reduced to a
positive definite one through a restriction of the defining parameters of the exponential vectors.
We define the Fock space associated with the reduced FKS Schro¨dinger kernel. We compute
the characteristic functions of quantum random variables naturally associated with the FKS
Schro¨dinger kernel and expressed in terms of the renormalized higher powers of white noise (or
RHPWN) Lie algebra generators.
1. The Schro¨dinger and RHPWN Algebras
The quantum white noise functionals b†t (creation density) and bt (annihilation density) satisfy
the Boson commutation relations
(1.1) [bt, b
†
s] = δ(t− s) ; [b†t , b†s] = [bt, bs] = 0
where t, s ∈ R and δ is the Dirac delta function, as well as the duality relation
(1.2) (bs)
∗ = b†s
In order to consider the smeared fields defined by the higher powers of bt and b
†
t , for a test
function f and n, k ∈ {0, 1, 2, ...} we define the sesquilinear form
(1.3) Bnk (f) =
∫
R
f(t) b†t
n
bkt dt
with involution
(1.4) (Bnk (f))
∗ = Bkn(f¯)
In [?] and [?] we introduced the convolution type renormalization
(1.5) δl(t− s) = δ(s) δ(t− s) ; l = 2, 3, ....
of the higher powers of the Dirac delta function, and by restricting to test functions f(t) such
that f(0) = 0 we obtained the renormalized higher powers of white noise (or RHPWN) ∗–Lie
algebra commutation relations
(1.6) [Bnk (f), B
N
K (g)]RHPWN = (k N −K n) Bn+N−1k+K−1 (f g)
Let I ⊂ R be a set of finite positive measure µ(I). In the notation of [?], the operators
M := B00(χI ) = µ(I) I, K := 2B
2
0(χI ), G := 2B
1
0(χI ), D := B
1
1(χI ), Px :=
1
2
B01(χI ), and
Pt :=
1
8
B02(χI ), satisfy the commutation relations of the Schro¨dinger algebra S given in the
table
4(1.7)
M K G D Px Pt
M 0 0 0 0 0 0
K 0 0 0 −2K −G −D
G 0 0 0 −G −M −Px
D 0 2K G 0 −Px −2Pt
Px 0 G M Px 0 0
Pt 0 D Px 2Pt 0 0
The Feinsilver-Kocik-Schott (FKS) kernel (or Leibniz function), i.e. the inner product of the
exponential vectors
(1.8) ψa,b(χI ) := e
aB20(χI ) ebB
1
0(χI ) Φ = e
a
2
K e
b
2
G Φ
where a, b ∈ C with |a| < 2 and Φ is the Fock vacuum such that B02(χI ) Φ = B01(χI ) Φ = 0, is
given by (see Proposition 6.1 of [?] for c = µ(I) / 2 , m = µ(I) and a, b ∈ R ; a detailed proof
can be found in [?])
〈ψa,b(χI ), ψA,B(χI )〉 =
(
1− a¯ A
4
)−µ(I)
2
e
µ(I)
4
(
a¯ B2+4 b¯ B+b¯2 A
4−a¯ A
)
(1.9)
= e−
µ(I)
2
ln (1− a¯ A4 ) e
µ(I)
4
(
a¯ B2+4 b¯ B+b¯2 A
4−a¯ A
)
For a = A = 0 (??) reduces to the usual inner product of the Heisenberg algebra exponential
vectors, and for b = B = 0 it reduces to the inner product of the sl(2) (Square of White Noise
Lie algebra) exponential vectors.
Using the commutativity of the Schro¨dinger algebra generators on disjoint sets, we may
extend the FKS kernel (??) to exponential vectors of the form
(1.10) Ψ(f, g) :=
∏
i
e
aiB
2
0(χIi
)
e
biB
1
0(χIi
)
Φ = eB
2
0(f) eB
1
0(g) Φ
where f =
∑
i ai χIi and g =
∑
i bi χIi , with Ii ∩ Ij =  for i 6= j, are simple functions with
|f | < 2 and we obtain
(1.11) 〈Ψ(f1, g1),Ψ(f2, g2)〉 = e−
1
2
∫
ln
(
1− f¯1 f2
4
)
dµ
e
1
4
∫ ( f¯1 g22+4 g¯1 g2+g¯21 f2
4−f¯1 f2
)
dµ
Proposition 1. The kernel (??) (and so also (??)) is not positive definite for arbitrary µ(I).
This, in particular, implies the non-positive definiteness of the kernel of Proposition 6.1 of [?]
for arbitrary c.
5Proof. Denoting ‖ . . . ‖2 := 〈. . . , . . .〉, we examine the non–negativity of
Nk := ‖
k∑
i=1
ciψai,bi‖2
where k ∈ {1, 2, ...} and (for simplicity, as in [?]), ai, bi, ci ∈ R with |ai| < 2 for all i ∈
{1, ..., k}.
For k = 1 we clearly have
N1 := |c1|2 ‖ψa1,b1‖2 = |c1|2 e
−µ(I)
2
ln
(
1−a
2
1
4
)
e
µ(I)
4
(
a1 b
2
1+4 b
2
1+b
2
1 a1
4−a21
)
≥ 0
for all intervals I ⊂ R.
For k = 2 we have
N2 =
2∑
i,j=1
ci cj e
−µ(I)
2
ln (1−ai aj4 ) e
µ(I)
4
(
ai b
2
j+4 bi bj+b
2
i aj
4−ai aj
)
= |c1|2 e
−µ(I)
2
ln
(
1−a
2
1
4
)
e
µ(I)
4
(
a1 b
2
1+4 b
2
1+b
2
1 a1
4−a21
)
+ |c2|2 e
−µ(I)
2
ln
(
1−a
2
2
4
)
e
µ(I)
4
(
a2 b
2
2+4 b
2
2+b
2
2 a2
4−a22
)
+ 2 c1 c2 e
−µ(I)
2
ln (1−a1 a24 ) e
µ(I)
4
(
a1 b
2
2+4 b1 b2+b
2
1 a2
4−a1 a2
)
If c1 c2 ≥ 0 then, clearly, N2 ≥ 0. Suppose that c1 c2 < 0. The positive definiteness of N2
can be studied by checking the non-negativity of the determinants d1 and d2 of the principal
minors of the real symmetric matrix
A2 =

e
−µ(I)
2
ln
(
1−a
2
1
4
)
e
µ(I)
4
(
a1 b
2
1+4 b
2
1+b
2
1 a1
4−a21
)
e−
µ(I)
2
ln (1−a1 a24 ) e
µ(I)
4
(
a1 b
2
2+4 b1 b2+b
2
1 a2
4−a1 a2
)
e−
µ(I)
2
ln (1−a1 a24 ) e
µ(I)
4
(
a1 b
2
2+4 b1 b2+b
2
1 a2
4−a1 a2
)
e
−µ(I)
2
ln
(
1−a
2
2
4
)
e
µ(I)
4
(
a2 b
2
2+4 b
2
2+b
2
2 a2
4−a22
)

corresponding to N2. As in the case k = 1, we have
d1 = |c1|2 e
−µ(I)
2
ln
(
1−a
2
1
4
)
e
µ(I)
4
(
a1 b
2
1+4 b
2
1+b
2
1 a1
4−a21
)
≥ 0
Moreover, since the case µ(I) = 0 is trivial, we may divide out µ(I) > 0 and proving d2 ≥ 0
is equivalent to showing that
−1
2
ln
(
1− a
2
1
4
)
+
1
4
(
a1 b
2
1 + 4 b
2
1 + b
2
1 a1
4− a21
)
− 1
2
ln
(
1− a
2
2
4
)
+
1
4
(
a2 b
2
2 + 4 b
2
2 + b
2
2 a2
4− a22
)
6≥ − ln
(
1− a1 a2
4
)
+
1
2
(
a1 b
2
2 + 4 b1 b2 + b
2
1 a2
4− a1 a2
)
which is equivalent to showing that
ln
(
4− a1 a2√
(a21 − 4) (a22 − 4)
)
≥ ((a2 − 2) b1 − (a1 − 2) b2)
2
(a1 − 2) (a2 − 2) (a1 a2 − 2)
which is true since
((a2 − 2) b1 − (a1 − 2) b2)2
(a1 − 2) (a2 − 2) (a1 a2 − 2) ≤ 0
and
4− a1 a2√
(a21 − 4) (a22 − 4)
≥ 1⇔ 4 (a1 − a2)2 ≥ 0
Thus N1 ≥ 0 and N2 ≥ 0. However, for k = 3 the situation is different. We have
N3 =
3∑
i,j=1
ci cj e
−µ(I)
2
ln (1−ai aj4 ) e
µ(I)
4
(
ai b
2
j+4 bi bj+b
2
i aj
4−ai aj
)
(1.12)
with associated real symmetric matrix
A3 =

〈ψa1,b1(χI ), ψa1,b1(χI )〉 〈ψa1,b1(χI ), ψa2,b2(χI )〉 〈ψa1,b1(χI ), ψa3,b3(χI )〉
〈ψa2,b2(χI ), ψa1,b1(χI )〉 〈ψa2,b2(χI ), ψa2,b2(χI )〉 〈ψa2,b2(χI ), ψa3,b3(χI )〉
〈ψa3,b3(χI ), ψa1,b1(χI )〉 〈ψa3,b3(χI ), ψa2,b2(χI )〉 〈ψa3,b3(χI ), ψa3,b3(χI )〉

As before, d1 ≥ 0 and d2 ≥ 0. However, for µ(I) ≈ 0.21, (a1, a2, a3) = (1,−1,−1) and
(b1, b2, b3) = (0, 10, 0) we find that d3 ≈ −3.4 < 0 and, for (c1, c2, c3) = (3,−0.1,−2.82), we
have N3 ≈ −0.29 < 0. In fact, for this choice of (a1, a2, a3), (b1, b2, b3) and (c1, c2, c3) we find
that d3, N3 < 0 for 0 < µ(I) < m0, and d3, N3 ≥ 0 for µ(I) ≥ m0, where m0 ≈ 0.275. This is in
analogy with the no–go theorems of [?]–[?] and [?] for the impossibility of a Fock representation
of RHPWN , where the existence of negative–norm (or ”ghost”) vectors was proved for µ(I)
below a certain threshold.

72. Random variables associated with the Schro¨dinger algebra
For a fixed interval I, we will compute the vacuum characteristic function 〈Φ, ei sX Φ〉 of the
self-adjoint operator
(2.1) X := xB20(χI ) + x¯ B
0
2(χI ) + y B
1
0(χI ) + y¯ B
0
1(χI )
where x, y ∈ C with x 6= 0.
For reasons explained in [?] we define the action of the RHPWN generators Bnk (f) on the
Fock vacuum vector Φ by
(2.2) Bnk (f) Φ :=

0 if n < k or n · k < 0
Bn−k0 (f) Φ if n > k ≥ 0
1
n+1
∫
R f(t) dtΦ if n = k
Thus, in particular, B02(χI ) Φ = B
0
1(χI ) Φ = 0 and B
1
1(χI ) Φ = µ(I) Φ.
Lemma 1. (i) If x, d, N and h satisfy the oscillator algebra commutation relations
(2.3) [d, x] = h ; [d, h] = [x, h] = 0 ; [N, x] = x ; [d,N ] = d
then for all analytic functions f and for all a ∈ C
(2.4) [N, f(x)] = x f ′(x) ; aN f(x) = f(a x) aN
Moreover, for all n,m ∈ N
(2.5) dn xm =
n∧m∑
j=0
(
n,m
j
)
xm−j dn−j hj
where
(2.6)
(
n,m
j
)
=
(
n
j
)(
m
j
)
j!
(ii) If ∆, R and ρ satisfy the sl(2) algebra commutation relations
(2.7) [∆, R] = ρ ; [ρ,R] = 2R ; [∆, ρ] = 2 ∆
then
(2.8) et∆ eaR = e
aR
1−a t (1− a t)−ρ e t∆1−a t
8Proof. The proofs of (i) and (ii) can be found in [?].

In the context of the RHPWN generators, in a manner consistent with the RHPWN repre-
sentation of the Schro¨dinger algebra generators M,K,G,D, Px, Pt given in the previous section,
(2.9) R = K = 2B20(χI ) ; ∆ = Pt =
1
8
B02(χI ) ; ρ = D = B
1
1(χI )
and
(2.10) x = G = 2B10(χI ) ; d = Px =
1
2
B01(χI ) ; N = D = B
1
1(χI ) ; h = M = B
0
0(χI )
Lemma 2. For all a, b ∈ C
(i)
B02(χI ) e
aB20(χI ) = 4 a2B20(χI ) e
aB20(χI ) +(2.11)
4 a eaB
2
0(χI ) B11(χI ) + e
aB20(χI ) B02(χI )
(ii)
(2.12) B11(χI ) e
bB10(χI ) Φ =
(
bB10(χI ) +
µ(I)
2
)
ebB
1
0(χI ) Φ
(iii) For all n ≥ 0
(2.13) B02(χI ) (B
1
0(χI ))
n Φ = n (n− 1)µ(I) (B10(χI ))n−2 Φ
(iv)
(2.14) B02(χI ) e
bB10(χI ) Φ = b2 µ(I) ebB
1
0(χI ) Φ
(v) For all n ≥ 0
(2.15) B01(χI ) (B
2
0(χI ))
n = 2nB10(χI ) (B
2
0(χI ))
n−1 + (B20(χI ))
nB01(χI )
(vi)
(2.16) B01(χI ) e
aB20(χI ) = 2 aB10(χI ) e
aB20(χI ) + eaB
2
0(χI ) B01(χI )
9Proof. (i) By (??) and lemma ??(ii)
B02(χI ) e
aB20(χI ) =
∂
∂ t
|t=0
(
etB
0
2(χI ) eaB
2
0(χI )
)
=
∂
∂ t
|t=0
(
e
a
1−4 t a B
2
0(χI ) (1− 4 t a)−B11(χI ) e t1−4 t a B02(χI )
)
= 4 a eaB
2
0(χI ) B11(χI ) + e
aB20(χI ) B02(χI )
(ii) By (??), (??) and lemma ??(i)
B11(χI ) e
bB10(χI ) Φ =
(
[B11(χI ), e
bB10(χI )] + ebB
1
0(χI ) B11(χI )
)
Φ
=
(
bB10(χI ) +
µ(I)
2
)
ebB
1
0(χI ) Φ
(iii) For n = 0, both sides of (??) are zero. For n = 1, since
B02(χI )B
1
0(χI ) = 2B
0
1(χI ) +B
1
0(χI )B
0
2(χI )
again both sides of (??) are zero. For n ≥ 2 we have
an := B
0
2(χI ) (B
1
0(χI ))
n Φ
= B02(χI )B
1
0(χI ) (B
1
0(χI ))
n−1 Φ
= (2B01(χI ) +B
1
0(χI )B
0
2(χI )) (B
1
0(χI ))
n−1 Φ
= 2B01(χI ) (B
1
0(χI ))
n−1 Φ +B10(χI ) an−1
By (??)
B01(χI ) (B
1
0(χI ))
n−1 = (n− 1) (B10(χI ))n−2B00(χI ) + (B10(χI ))n−1B01(χI )
Thus
10
an = 2µ(I) (n− 1) (B10(χI ))n−2 Φ +B10(χI ) an−1
= 2µ(I) ((n− 1) + (n− 2)) (B10(χI ))n−2 Φ + (B10(χI ))2 an−2
= 2µ(I) ((n− 1) + (n− 2) + (n− 3)) (B10(χI ))n−2 Φ + (B10(χI ))3 an−3
= ...
= 2µ(I) ((n− 1) + (n− 2) + (n− 3) + ...+ (n− (n− 1))) (B10(χI ))n−2 Φ + (B10(χI ))n−1 a1
= 2µ(I) ((n− 1) + (n− 2) + (n− 3) + ...+ (n− (n− 1))) (B10(χI ))n−2 Φ
= 2µ(I) (n (n− 1)− (1 + 2 + ...+ (n− 1)) (B10(χI ))n−2 Φ
= 2µ(I)
(
n (n− 1)− n (n− 1)
2
)
(B10(χI ))
n−2 Φ
= µ(I)n (n− 1) (B10(χI ))n−2 Φ
(iv) Using (iii)
B02(χI ) e
bB10(χI ) Φ = B02(χI )
∞∑
n=0
bn
n!
(B10(χI ))
n
Φ
=
∞∑
n=0
bn
n!
B02(χI ) (B
1
0(χI ))
n
Φ
=
∞∑
n=0
bn
n!
n (n− 1)µ(I) (B10(χI ))n−2 Φ
=
∞∑
n=2
bn
n!
n (n− 1)µ(I) (B10(χI ))n−2 Φ
= b2 µ(I)
∞∑
n=2
bn−2
(n− 2)! (B
1
0(χI ))
n−2 Φ
= b2 µ(I) ebB
1
0(χI ) Φ
(v) As in the proof of (iii)
11
bn := B
0
1(χI ) (B
2
0(χI ))
n
= B01(χI )B
2
0(χI ) (B
2
0(χI ))
n−1
= (2B10(χI ) +B
2
0(χI )B
0
1(χI )) (B
2
0(χI ))
n−1
= 2B10(χI ) (B
2
0(χI ))
n−1 +B20(χI ) bn−1
= 4B10(χI ) (B
2
0(χI ))
n−1 + (B20(χI ))
2 bn−2
= ...
= 2nB10(χI ) (B
2
0(χI ))
n−1 + (B20(χI ))
n b0
= 2nB10(χI ) (B
2
0(χI ))
n−1 + (B20(χI ))
nB01(χI )
(vi)
B01(χI ) e
aB20(χI ) = B01(χI )
∞∑
n=0
an
n!
(B20(χI ))
n
=
∞∑
n=0
an
n!
B01(χI ) (B
2
0(χI ))
n
=
∞∑
n=0
an
n!
(2nB10(χI ) (B
2
0(χI ))
n−1 + (B20(χI ))
nB01(χI ))
= 2 aB10(χI )
∞∑
n=0
an−1
(n− 1)! (B
2
0(χI ))
n−1
+
∞∑
n=0
an
n!
(B20(χI ))
n
B01(χI )
= 2 aB10(χI ) e
aB20(χI ) + eaB
2
0(χI )B01(χI )

Lemma 3. For all s ∈ R
ei sX Φ = ew1(s)B
2
0(χI ) ew2(s)B
1
0(χI ) ew3(s) Φ
where X is as in (??) and
w1(s) =
i
2
√
x
x¯
tanh (2 s |x|)(2.17)
w2(s) =
i
2
y¯
x¯
(1− sech (2 s |x|)) + y
2 |x| tanh(2 s |x|)(2.18)
and
12
(2.19)
w3(s) =
( |y|2 + i s (y2 x¯+ x y¯2)
4 |x|2 −
i (y2 x¯+ x y¯2)
8 |x|3 tanh (2 s |x|)−
|y|2
4 |x|2 sech (2 s |x|)
)
µ(I)
Proof. We will show that w1(s), w2(s) and w3(s) satisfy the differential equations
w′1(s) = 4 i x¯ w1(s)
2 + i x ; w1(0) = 0 (Riccati ODE)(2.20)
w′2(s) = 4 i x¯ w1(s)w2(s) + 2 y¯ w1(s) + y ; w2(0) = 0 (Linear ODE)(2.21)
w′3(s) = (i x¯ w2(s)
2 + y¯ w2(s) + 2 i x¯ w1(s))µ(I) ; w3(0) = 0(2.22)
whose solutions are given by (??), (??) and (??) respectively.
In so doing, let
(2.23) E Φ := es(xB
2
0(χI )+x¯ B
0
2(χI )+y B
1
0(χI )+y¯ B
0
1(χI )) Φ
and also
(2.24) E Φ := ew1(s)B
2
0(χI ) ew2(s)B
1
0(χI ) ew3(s) Φ
where wi(0) = 0 for i ∈ {1, 2, 3}. Then, (??) implies that
(2.25)
∂
∂ s
E Φ =
(
w′1(s)B
2
0(χI ) + w
′
2(s)B
1
0(χI ) + w
′
3(s)
)
E Φ
and, since by lemma ??,
B02(χI )E Φ = (4w1(s)
2B20(χI ) + 4w1(s)w2(s)B
1
0(χI )(2.26)
+(2w1(s) + w2(s)
2)µ(I))E Φ
and
B01(χI )E Φ =
(
2w1(s)B
1
0(χI ) + µ(I)w2(s)
)
E Φ(2.27)
from (??) we also obtain
∂
∂ s
E Φ = ((4 i x¯ w1(s)
2 + i x)B20(χI ) + (4 i x¯ w1(s)w2(s) + 2 y¯ w1(s) + y)B
1
0(χI )(2.28)
+(i x¯ w2(s)
2 + y¯ w2(s) + 2 i x¯ w1(s))µ(I))E Φ
From (??) and (??), by equating coefficients of B20(χI ), B
1
0(χI ) and 1, we obtain (??), (??)
and (??) thus completing the proof.

13
Proposition 2. (Characteristic Function) For all s ∈ R
〈Φ, ei sX Φ〉 =(2.29)
e
(
|y|2+i s(y2 x¯+x y¯2)
4 |x|2 −
i(y2 x¯+x y¯2)
8 |x|3 tanh (2 s |x|)−
|y|2
4 |x|2 sech (2 s |x|)
)
µ(I)
Proof. Using lemma ?? and the fact that for any z ∈ C
(2.30) ez B
0
2(χI ) Φ = ez B
0
1(χI ) Φ = Φ
we have
〈Φ, ei sX Φ〉 = 〈Φ, ew1(s)B20(χI ) ew2(s)B10(χI ) ew3(s) Φ〉(2.31)
= 〈ew¯2(s)B01(χI ) ew¯1(s)B02(χI ) Φ, ew3(s) Φ〉
= 〈Φ, ew3(s) Φ〉
= ew3(s) 〈Φ,Φ〉
= ew3(s)
= e
(
|y|2+i s(y2 x¯+x y¯2)
4 |x|2 −
i(y2 x¯+x y¯2)
8 |x|3 tanh (2 s |x|)−
|y|2
4 |x|2 sech (2 s |x|)
)
µ(I)

3. Positive definite restrictions of the Schro¨dinger kernel
To overcome the problem of the non-positive definiteness of the FKS kernel (??) we look for
a restriction of the parameters a, b ∈ C, |a| < 2, in the definition (??) of the exponential vectors
ψa,b(χI ). We notice that if a, b, A,B ∈ C are such that
(3.1) a¯ B2 + b¯2A = 0
i.e. if (a, b), (A,B) ∈ Sλ where, for given λ ∈ R,
(3.2) Sλ := {(a, b) ∈ C2 / a = i λ b2, |λ| |b|2 < 2}
then the FKS kernel (??) reduces to
〈ψa,b(χI ), ψA,B(χI )〉 =
(
1− a¯ A
4
)−µ(I)
2
e
µ(I)
4
b¯ B
1− a¯ A4(3.3)
=
(
1− a¯ A
4
)−µ(I)
2
e
µ(I)
4 (b¯ B)
∑∞
n=0 ( a¯ A4 )
n
14
The kernels Ki : C2 × C2 → C defined for i ∈ {1, 2, 3} by
K1((a, b), (A,B)) :=
(
1− a¯ A
4
)−µ(I)
2
(3.4)
K2((a, b), (A,B)) :=
µ(I)
4
b¯ B(3.5)
K3((a, b), (A,B)) := lim
ρ→∞
K3,ρ((a, b), (A,B))(3.6)
where, for ρ ≥ 1,
(3.7) K3,ρ((a, b), (A,B)) :=
ρ∑
n=0
(
a¯ A
4
)n
are positive definite, since K1 is the well-known sl(2) kernel, K2 and K3,ρ are Heisenberg-type
kernels and K3 is a limit of positive definite kernels . Therefore K1 e
K2 K3 , i.e. (??), is also a
positive definite kernel.
For given λ ∈ R and I ⊂ R we introduce the notation
ψ(b) := ψi λ b2,b(χI ) = e
i λ b2B20(χI ) ebB
1
0(χI ) Φ(3.8)
〈ψ(b), ψ(B)〉λ := 〈ψi λ b2,b(χI ), ψi λB2,B(χI )〉 =
(
1− λ
2
4
(b¯ B)2
)−µ(I)
2
e
b¯ B
4−λ2 (b¯ B)2 µ(I)(3.9)
and its extension to simple functions f =
∑
j bj χIj
Ψ(f) := Ψi λ f2,f =
∏
j
e
i λ b2j B
2
0(χIj
)
e
bj B
1
0(χIj
)
Φ = ei λB
2
0(f
2) eB
1
0(f) Φ(3.10)
〈Ψ(f),Ψ(g)〉λ := 〈Ψ(i λ f 2, f),Ψ(i λ g2, g)〉 = e
− 1
2
∫
ln
(
1−λ2 (f¯ g)2
4
)
dµ
e
∫ f¯ g
4−λ2 (f¯ g)2 dµ(3.11)
If f = b χI then Ψ(f) = ψ(b).
For λ ∈ R we define the (restricted) Schro¨dinger Fock space FS(λ) as the closure of the
linear span of the exponential vectors Ψ(f) defined in (??) with respect to the inner product
〈Ψ(f),Ψ(g)〉λ defined in (??).
The study of FS(λ) and the random variables that can be defined on it, is in progress.
15
References
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[2] : The emergence of the Virasoro and w∞ Lie algebras through the renormalized higher powers of
quantum white noise , International Journal of Mathematics and Computer Science, 1, No.3, (2006) 315–342.
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Random variables and positive definite Kernels associates with the Schrodinger algebra

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