On the one-mode quadratic Weyl operators

Accepted Manuscript On the one-mode quadratic Weyl operators

Habib Rebei

PII: DOI: Reference:

S0022-247X(16)00174-8 http://dx.doi.org/10.1016/j.jmaa.2016.02.040 YJMAA 20213

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

24 December 2015

Please cite this article in press as: H. Rebei, On the one-mode quadratic Weyl operators, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.02.040

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On the one-mode quadratic Weyl operators Habib Rebei Department of Mathematics, College of Science Qassim University, KSA [email protected]

Abstract It has been known, in [4], that one-mode quadratic Weyl operators are welldefined as unitary operators acting on the quadratic Fock space. These operators was defined by their action on the finite particles space. However, their action on the domain of the quadratic exponential vectors still unknown. In this paper, we provide a new reformulation (w.r.t [1]) of the quadratic exponential vectors and we compute the action of the one-mode quadratic Weyl operators on the set of these exponential vectors. Then, we prove the independence of the one-mode quadratic Weyl operators parameterized by a such sub-domain of the quadratic Heisenberg group obtained in [4]. This significant contribution to the program of developing the quadratic white noise calculus constitute a step toward the C ∗ –representation of the renormalized square of white noise algebra. Keywords: sl2 (C), renormalized square of white noise, quadratic Fock space, quadratic exponential vectors, quadratic Weyl operators 2000 MSC: primary 60J65, secondary 60J45, 60H40.

1. Introduction The investigations on the stochastic limit of quantum theory in [10] led to the development of quantum white noise calculus as a natural generalization of classical and quantum stochastic calculus. This was initially developed at a pragmatic level, just to the extent needed to solve the concrete physical problems which stimulated the birth of the theory [8],[9], [10], [7]. The first systematic exposition of the theory is contained in the paper [11] and its full development in [13]. This new approach naturally suggested the idea to generalize stochastic calculus by extending it to higher powers of (classical and quantum) white noise. In this sense we speak of nonlinear white noise calculus. Through the natural identification between white noises and quantum fields, the program of developing a nonlinear white noise calculus becomes essentially equivalent to the problem of dealing with the powers of the creation and annihilation densities (local quantum fields).

Preprint submitted to Journal of Mathematical Analysis and Applications February 18, 2016

This attempt led to unexpected connections between mathematical objects and results emerged in different fields of mathematics and at different times, such as white noise, the representation theory of certain famous Lie algebras, the renormalization problem in physics, the theory of independent increment stationary (L´evy) processes and in particular the Meixner classes,... (see [3], [5], [2], [6] for more details) More precisely, when speaking of the renormalization problem, we mean the following: (i) to construct a continuous analogue of the ∗–algebra of differential operators with polynomial coefficients acting on the space C ∞ (Rn , C) of complex valued smooth functions in n ∈ N real variables (here continuous means that the space Rn ≡ {functions {1, ..., n} −→ R} is replaced by the space {functions R −→ R}); (ii) to construct a ∗–representation of this algebra as operators on a Hilbert space H (all spaces considered in this paper will be complex separable and all associative algebras will have an identity, unless otherwise stated); (iii) the ideal goal would be to have a unitary representation, i.e., one in which the skew symmetric elements of this ∗–algebra can be exponentiated, leading to strongly continuous 1–parameter unitary groups. However, in this paper these problems will not arise because we will deal only with the so-called one-mode case which corresponds to one degree of freedom. The results obtained in this paper are framed within this direction and constitute a contribution to this program. More precisely, these results open the way to attempt the goal (iii). This paper is organized as follows: In Section 2, we give a new reformulation of the exponential vectors. These vectors will be defined by a change of parameters with respect to those given in the paper [1]. A short discussion will be given in order to clarify the relation between them. Next, we give the action of the exponential of generators of the Fock representation of sl2 (C) on the set of the exponential vectors. The main result is proved in Section 3, that is the one-mode quadratic Weyl operators, parameterized by a domain DI ⊂ R3 , are linearly independent. This domain will be called the domain of independence of the one-mode quadratic Weyl operators. This very useful property, in the first order case (see [3] for more details), is a simple consequence of the linear independence of the complex exponential functions. In the quadratic case the proof is more subtle. Finally, in Section 4, we define the vacuum state on the C ∗ –algebra generated by the set of the one-mode quadratic Weyl operators, we compute the vacuum distributions of observable in sl2 (C) and we prove that they are of Meixner type.

2

2. The one-mode Fock space 2.1. The quadratic exponential vectors Definition 1. The sl2 (C) algebra is the complex ∗–Lie algebra generated by the set {M, B ± } with the following relations [B − , B + ] = M

[M, B ± ] = ±2B ±

;

− ∗

(B ) = B

+

(1)

∗

;

M =M (2)  The central extension of sl2 (C), denoted by sl2 (C), is the ∗–Lie algebra generated by sl2 (C) and the central self-adjoint element E, i.e., commuting with all other generators of sl2 (C) and satisfying E ∗ = E. It is well-known (see [4]) that for all μ > 0, there exists a unique ∗–  2 (C), called the Fock representation, realized on a Hilbert representation of sl space Γ with an orthonormal basis {Φn , n ∈ N} on which B ± , M and E act as  2 (C) with their images in the representation): follows (we identify elements of sl √ (3) B + Φn = ωn+1 Φn+1 , n ∈ N, √ − ∗ B Φn = ωn Φn−1 , n ∈ N , (4) where Φ−1 = 0 and Φ0 =: Φ is the vacuum vector, M Φn = (2n + μ)Φn

,

n∈N

(5)

EΦn = Φn

(6)

the sequence (ωn ) is uniquely determined to be {ωn = n(μ + n − 1)

,

n = 1, 2, . . .}

,

ω0 := 1.

(7)

The number operator is defined by N Φn = 2nΦn .

(8)

or equivalently N = M − μE. Definition 2. The space Γ is called the one-mode quadratic Fock space. From these prescriptions it follows that the finite particle space, defined by Γ0 := algebraic linear span of the set {Φn , n ∈ N} is invariant under the action of the operators B ± and M . Moreover, it is a dense subspace of the one-mode quadratic Fock space. It was known from [4], that the finite particle vectors are analytic vectors for B + , B − and M so that the operators euB+ , evM , ewB are well-defined on the domain Γ0 for complex numbers u, v, w such that |u| < 1. In particular, we have evM Φn = e2n+μ Φn

∀v ∈ C, ∀n ∈ N.

(9)

In the following we will introduce a dense domain in Γ invariant under the action of the exponential of B ± and M . Now, we introduce some notations and we recall some known results which will be used in the remain of this paper. 3

1. For all α ∈ C and for all sequence (ωn )n ⊂ C, let us denote (α)n =

n 

(α + k − 1)

;

ωn ! =

k=1

n 

ωk , n ≥ 1.

(10)

k=1

From the definition of (ωn )n in (7), we get ωn ! =

n 

ωk =

k=1

n 

k(μ + k − 1) = n!(μ)n ·

k=1

2. It is well-known that for all z ∈ C with |z| < 1, one has (1 − z)−μ := e−μ log(1−z) =

+∞  (μ)n n z , n! n=0

(11)

where the above series converges in norm for all |z| < 1 and where log is the principal determination of the Logarithm. Lemma 3. The function ξ given by ξ(z) =

+∞ 



n=0

(μ)n n z n!

(12)

is well-defined for all z ∈ C such that |z| < 1.  n Proof. Let Un = (μ) n! . Then  Un+1 μ+n = 1. = lim lim n→+∞ Un n→+∞ n+1 This gives that the radius of convergence of the above series is equal to 1. Lemma 4. For all complex number z such that |z| < 1, the exponential vector +

Φ(z) := ezB Φ =

+∞ n  z (B + )n Φ n! n=0

is well-defined, where the series converges in Γ. Proof. From Eq.(3), one has +∞  n=0



zn + n (B ) Φ

n!

=

=

=

+∞  n=0 +∞ 



zn  ωn !Φn

n!

zn  ωn !| n! n=0  +∞  (μ)n n |z| = ξ(|z|) < +∞. n! n=0

This ends the proof. 4

|

(13)

2.2. Connection with the renormalized square of white noise algebra Recall that the renormalized square of white noise algebra (RSWN) over the Hilbert algebra K = L2 (R) ∩ L∞ (R) is the unital ∗–Lie algebra with generators 1 (central element), Bf+ , Bf− , Nf ; f ∈ K, which are linearly independent (in the sense that f0 1 + Bf+1 , Bf−2 + Nf4 = 0 with f0 ∈ C and fj ∈ K (j = 1, 2, 3), if and only if, f0 = 0 and f1 = f2 = f3 = 0) and relations [Bf− , Bg+ ] = 2c f, g1 + 4Nf¯g ,

(14)

[Nf , Bg+ ] = 2Bf+g ,

(15)

[Nf , Bg− ] = −2Bf− ¯g ,

(16)

[Bf+ , Bg+ ]

[Nf , Ng ] =

=

[Bf− , Bg− ]

= 0,

(17)

(Bf− )∗ = Bf+ .

Nf∗ = Nf¯ ;

(18)

Note that the constant c > 0 appeared in the equation (14) come from the following renormalization δ 2 (t − s) = cδ(t − s). The quadratic exponential vectors have been defined in [1] by +∞  (Bf+ )n

+

Ψf := eBf Φ =

n=0

n!

Φ , f ∈ K, f <

1 . 2

The one-mode renormalized square of white noise algebra denoted RSWN(1), is obtained when we take the test function space to be the complex algebra of the multiple of the characteristic function of the bounded interval I ⊂ R, i.e., KI = CχI . Then the commutation relations become [Bχ−I , Bχ+I ] = 2c|I|1 + 4NχI

;

[NχI , Bχ±I ] = ±2Bχ±I ,

(the other commutation relations vanish) and the involution will be given by (Bχ−I )∗ = Bχ+I

;

(NχI )∗ = NχI

1∗ = 1.

;

 2 (C) given by relations from (3) to (6) Lemma 5. The representation of sl induces a Fock representation ρ of the algebra RSWN(1) in the following way: ρ : Bχ±I −→ 2B ± where μ =

;

NχI −→ M − μE

;

1 −→ E,

c|I| 2 .

It follows that if we omit the symbol ρ from notations and we identify elements of RSWN(1) with their images under ρ, we can write Bχ±I = 2B ±

;

NχI = M − μE 5

;

1 = E.

Consequently, the quadratic exponential vector will take the form +

ΨzχI = ezBχI Φ =

+∞ n  z (Bχ+I )n Φ. n! n=0

Then it can be reexpressed as follows: Ψ(z) := ΨzχI =

+∞ n +∞   z (2z)n + n (2B + )n Φ = (B ) Φ = Φ(2z) n! n! n=0 n=0

(19)

which is defined for |z| < 12 · Proof. (of Lemma 5) Clearly that for all X ∈ RSWN(1), ρ(X) acts on the Fock space Γ . It sufficient to prove the ∗–Lie algebra homomorphism property on the generators of RSWN(1). We have [ρ(NχI ), ρ(Bχ±I )]

[ρ(Bχ−I ), ρ(Bχ+I )]

[ρ(1), ρ(Bχ±I )]

=

[M − μE, 2B ± ] = 2[M, B ± ] − 2μ[E, B ± ]

=

±2(2B ± ) = ±2ρ(Bχ±I ) = ρ(±2Bχ±I )

=

ρ([NχI , Bχ±I ]),

(20)

[2B − , 2B + ] = 4[B − , B + ] = 4M = 4(M − μE) + 4μE 4ρ(NχI ) + 4μρ(1) = ρ(4NχI + 2c|I|1) ρ([Bχ−I , Bχ+I ]), (21)

= = = =

[ρ(1), ρ(NχI )]

[E, 2B ± ] = 2[E, B ± ] = 0 = ρ(0) = ρ([1, Bχ±I ]),

(22)

[E, M − μE] = 0 = ρ(0) = ρ([1, NχI ]).

(23)

=

For the ∗-property, we have (ρ(Bχ±I ))∗ = (2B ± )∗ = 2B ∓ = ρ(Bχ∓I ) = ρ((Bχ±I )∗ )

(24)

(ρ(1))∗ = E ∗ = E = ρ(1) = ρ(1∗ )

(25)

(ρ(NχI ))∗ = (M − μE)∗ = M ∗ − μE ∗ = M − μE = ρ(NχI ) = ρ(Nχ∗I )

(26)

The equations from (20) to (26) prove that ρ is a ∗–Lie algebra homomorphism which ends the proof.

6

2.3. Exponential of generators Proposition 6. Let z ∈ C. Then, for all u, v, w ∈ C satisfying conditions given in (27), (28) and (29) respectively, the exponential vectors Φ(z) are in the + − domains of the operators euB , ewB and evM . Moreover, we have +

euB Φ(z) = Φ(z + u) −

ewB Φ(z) = (1 − wz)−μ Φ( evM Φ(z) = eμv Φ(e2v z)

;

|z| < 1 − |u| ,

z ) 1 − wz ;

;

|z| <

(27)

1 , 1 + |w|

|z| < min(1, e−2(v) ) ·

(28) (29)

Remark 1. If (v) ≤ 0, then the condition in (29) is verified for all |z| < 1, so that all exponential vectors are in the domain of evM . Corollary 7. For all z, w ∈ D(0, 1) ⊂ C, the inner product of two exponential vectors is given by

Φ(z), Φ(w) = e−μ log(1−zw) · (30) Remark 2. Recall that the inner product of two exponential vectors was given in [1] by  +∞ c

Ψ(f ), Ψ(g) = e− 2 −∞ log(1−4f (s)g(s))ds . In the one mode case, if we take f = zχI and g = wχI , then their corresponding exponential vectors will be given by Ψ(z) = ΨzχI and Ψ(w) = ΨwχI respectively. This gives that c

Ψ(z), Ψ(w) = e− 2 |I| log(1−4¯z w) · In the other hand, when comparing this inner product with that obtained in (30), we get c

Ψ(z), Ψ(w) = Ψ(2z), Ψ(2w) = e−μ log(1−2z2w) = e− 2 |I| log(1−4¯z w) , where z, w ∈ D(0, 12 ). Therefore, in this case, the definition of the exponential vectors is a slight modification of the parameters w.r.t that obtained in the paper [1]. Our choice is justified by the fact that this parametrization is simpler. Proof. 

Φ(z), Φ(w)



= = =

 + − ezB Φ, Φ(w) = Φ, ezB Φ(w)   w w ) = (1 − zw)−μ ) Φ, Φ( ) Φ, (1 − zw)−μ )Φ( 1 − zw 1 − zw = (1 − zw)−μ = e−μ log(1−zw) ·



7

Corollary 8. For all u, v, w ∈ C, |u| < 1, the quadratic exponential vector Φ(z) is in the domain of the quadratic exponential operator +

−

Γ(u, v, w) := euB evM ewB , if and only if, |z| < r(u, v, w) :=

1 max(1,e2(v) ) 1−|u|

+ |w|

·

(31)

In this case, we have Γ(u, v, w)Φ(z) = eμv (1 − wz)−μ Φ(u +

e2v z ) 1 − wz

|z| < r(u, v, w).

;

(32)

Proof. Since 1 − |u| ≤ 1, then r(u, v, w)

= ≤ ≤

1 max(1,e2(v) ) 1−|u|

+ |w|

1 max(1, e2(v) ) + |w| 1 · 1 + |w|

This gives that for all |z| < r(u, v, w), we have |z| <

1 1+|w| ·

Then we get (28).

z instead of z. Then it is Now, we want to apply the identity (29) to 1−wz z sufficient to prove that the condition | 1−wz | < min(1, e−2(v) ) is satisfied for all |z| < r(u, v, w).

From the definition of r(u, v, w) in (31), we get

|

z | 1 − wz

≤ < = = =

|z| 1 − |w||z| r(u, v, w) 1 − |w|r(u, v, w) 1 1 r(u,v,w) − |w| 1 |w| +

max(1,e2(v) ) 1−|u|

− |w|

1 − |u| max(1, e2(v) )

= (1 − |u|) min(1, e−2(v) ) ≤ min(1, e−2(v) ). 8

(33)

z this gives | 1−wz | < min(1, e−2(v) ).

Combining (28) and (29), we get

− evM ewB Φ(z) = evM (1 − wz)−μ Φ(

 z e2v z ) = eμv (1 − wz)−μ Φ( ). 1 − wz 1 − wz (34) e2v z It remains to combine (27) with (34). But, we need to check if 1−wz verifies the condition in (27). From (33), we deduce that |

e2v z z 1 − |u| | = e2(v) | | < e2(v) ≤ 1 − |u|. 1 − wz 1 − wz max(1, e2(v) )

By combination of (27) with (34), we obtain Γ(u, v, w)Φ(z) = euB

+



eμv (1−wz)−μ Φ(

e2v z  e2v z ) = eμv (1−wz)−μ Φ(u+ )· 1 − wz 1 − wz

Proof. (of Proposition 6). +

1. To prove that Φ(z) ∈ Dom(euB ), it is sufficient to prove that the series S + :=

+∞ +∞  



k=0 n=0

uk z n + k+n (B ) Φ

k! n!

converges. Using Eq.(3), we get

(B + )k+n Φ =

  ωn+k !Φn+k = ωn+k ! .

Then S+

=

=

=

+∞ +∞  



k=0 n=0 +∞ +∞  

uk z n + k+n (B ) Φ

k! n!

|u|k |z|n

(B + )k+n Φ

k! n! n=0

k=0 +∞ +∞   k=0

|u|k |z|n  ωn+k ! . k! n! n=0

Note that if (Uk,n )n,k is a double sequence in a Banach space, then we have +∞ +∞  +∞  n   Uk,n = Uk,n+k (35) n=0 k=0

k=0 n=0

9

in the cases of convergence or divergence. k |z|n−k √ Taking Uk,n = |u| k! (n−k)! ωn !, , we get S+

=

=

= =

+∞  n  |u|k |z|n−k  ωn ! k! (n − k)! n=0 k=0 +∞ √ n   ωn !  n! |u|k |z|n−k n! (n − k)!k! n=0 k=0  +∞  (μ)n (|u| + |z|)n n! n=0

ξ(|u| + |z|) < +∞. +

This proves that Φ(z) ∈ Dom(euB ). Moreover, one has +

+

+

+

euB Φ(z) = euB ezB Φ = e(u+z)B Φ = Φ(z + u)· −

2. To prove that Φ(z) ∈ Dom(ewB ), it is sufficient to prove that the series S − :=

+∞ +∞  



k=0 n=0

wk z n − k + n (B ) (B ) Φ

k! n!

converges. Using Eq.(4), we get

(B − )k (B + )n Φ = 

ωn ! ωn ! Φn−k =  ωn−k ! ωn−k !

;

and vanishes for k > n. Then S−

=

=

=

+∞ +∞   k=0 n=0 +∞ +∞  



wk z n − k + n (B ) (B ) Φ

k!n!

|w|k |z|n

(B − )k (B + )n Φ

k!n! n=0

k=0 +∞  n 

|w|k |z|n ωn !  . k!n! ωn−k ! n=0 k=0

Using Eq.(35), we get S−

=

+∞  +∞  |w|k |z|n+k ωn+k ! √ k!(n + k)! ωn ! n=0 k=0

=

+∞  +∞  |w|k |z|n+k (μ)n+k  k! n!(μ)n n=0 k=0

10

k≤n

+∞  +∞  |w|k |z|n+k (μ)n (μ + n)k  k! n!(μ)n n=0 k=0  +∞ +∞   (μ)n n  (μ + n)k |z| |wz|k n! k! n=0 k=0  +∞  (μ)n |z|n (1 − |wz|)−(μ+n) n! n=0  +∞  (μ)n |z| n (1 − |wz|)−μ n! 1 − |wz| n=0

|z|  < +∞. (1 − |wz|)−μ ξ 1 − |wz|

=

=

=

= =

−

This proves that Φ(z) ∈ Dom(ewB ). Moreover, from the above computations and with help of Eq.(35), one deduces

−

ewB Φ(z)

=

=

=

=

=

= =

+∞ +∞   wk z n − k + n (B ) (B ) Φ k!n! n=0

k=0 +∞  n 

w k z n ωn !  Φn−k k!n! ωn−k ! n=0 k=0 +∞  +∞  wk z n+k ωn+k ! √ Φn k!(n + k)! ωn ! n=0 k=0  +∞ +∞   (μ)n n  (μ + n)k z (wz)k Φn n! k! n=0 k=0  +∞  (μ)n n z (1 − wz)−(μ+n) Φn n! n=0  +∞  (μ)n z n −μ (1 − wz) Φn n! 1 − wz n=0

z  (1 − wz)−μ Φ · 1 − wz

vM is well-defined on Γ0 . Moreover, in view of (9), one 3. From [4], we have e +∞ (μ)n n vM has Φ(z) := n=0 , if and only if, n! z Φn is in the domain of e

S :=

+∞  n=0

 |

(μ)n n v(2n+μ) 2 z e | < +∞· n!

11

(36)

But, we have S

=

+∞ 

 |

n=0

(μ)n n (2n+μ)(v) 2 z e | n!

=

e4μ

+∞  (μ)n 2n 4n(v) |z| e n! n=0

=

e4μ

+∞  (μ)n 2 4(v) n |z| e n! n=0

(37)

From Eq.(11), the sum (37) is finite, if and only if, |z|2 e4(v) < 1 which equivalent to |z| < e−2(v) . But, z shall verify |z| < 1. This is equivalent to condition |z| < min(1, e−2(z) ) which proves that Φ(z) ∈ Dom(evM ). From Eqs.(12) and (9), one has e

vM

Φ(z)

=

e

vM

+∞





n=0

=

=

= =

+∞ 



n=0  +∞ 

(μ)n n  z Φn n!

(μ)n n vM z e Φn n!

(μ)n n v(2n+μ) z e Φn n! n=0  +∞  (μ)n vμ (ze2v )n Φn e n! n=0 evμ φ(ze2v ).

This ends the proof.

2.4. Independence and totality of set of the quadratic exponential vectors Let us denote by E, the set of the exponential vectors that is given by   E := Φ(z) ; z ∈ D(0, 1) . Proposition 9. The set of the exponential vectors is linearly independent,i.e., for all z1 , · · · , zN ∈ D(0, 1) satisfying zj = zk ∀j = k, the set   Φ(z1 ), Φ(z2 ), · · · , Φ(zN ) is linearly independent. Moreover, it is a total set in Γ.

12

Proof. Let α1 , · · · , αN ∈ C, such that N 

αk Φ(zk ) = 0.

(38)

k=1

Let us consider the function f defined on ] − 1, 1[ by N   αk Φ(zk ) . f (t) := Φ(t), k=1

From (30), we get f (t) =

N 

N   αk Φ(t), Φ(zk ) = αk (1 − tzk )−μ .

k=1

k=1

Taking in account the identity (38), we get N 

αk (1 − tzk )−μ = f (t) = 0

∀t ∈] − 1, 1[.

(39)

k=1

Taking the pth –derivative of (39) at (t = 0), we get 0 = f (p) (0) =

N 

αk (μ)p (zk )p

∀p ∈ N.

k=1

Since (μ)p = 0 ∀p ∈ N, then N 

αk (zk )p = 0 ∀p ∈ N.

k=1

In particular, for all p = 0, 1, · · · , N − 1, we have N 

αk (zk )p = 0

k=1

which is equivalent to ⎡ ⎢ ⎢ AX := ⎢ ⎣

1 z1 .. .

(z1 )N −1

1 z2 .. .

(z2 )N −1

··· ··· .. . ···

⎤⎡

1 zN .. .

⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

(zN )N −1

α1 α2 .. .

⎤ ⎥ ⎥ ⎥ = 0. ⎦

αN

But, we have det(A) = V andermonde(z1 , · · · , zN ) =

N  j
13

(zk − zj ) = 0.

Then the matrix A is invertible which gives X = 0 or equivalently α1 = α2 = · · · = αN = 0. To prove the totality of E, let us consider ϕ ∈ Γ satisfying:

ϕ, Φ(z) = 0 Then

∀z ∈ D(0, 1).

+∞ n  z

ϕ, (B + )n Φ = 0 ∀z ∈ D(0, 1) n! n=0

which gives

+∞ n   z

ϕ, wn !Φn  = 0 n! n=0

∀z ∈ D(0, 1).

Taking the nth –derivative of the above equation at (z = 0), we get  wn ! ϕ, Φn  = 0 ∀n ∈ N. ⊥

⊥

This is equivalent to ϕ ∈ Γ0 . Since Γ0 is a dense subspace of Γ, then Γ0 = {0}. This implies that ϕ = 0 which ends the proof. 3. Independence of the one-mode quadratic Weyl operators Notations 1. For all w ∈ R ∪ iR \ i( π2 + πZ), thc(w) :=

th(w) , w

(40)

where th(w) denotes the hyperbolic tangent of w and thc(w) is defined by continuity at w = 0 to be equal to 1. Extending a terminology used in the physical literature, the left hand side of (40) is called the cardinal tangent of w. 2. Notice that thc(w) ∈ R thc(w) := thc(−w)

∀w ∈ R ∪ iR

; ;

∀w ∈ R ∪ iR

π w∈ / i (2Z + 1) 2

, ,

π w∈ / i (2Z + 1) 2

thc(w) = 0, w ∈ iπZ∗ w = iw ∈ iR \ i(

tan(w ) π + πZ) ⇒ thc(w) = . 2 w

14

(41)

3. Since, for all Z = (z, λ) ∈ C × R, the square roots of |z|2 − λ2 are opposite and belonging to R ∪ iR, we can usually chose the complex square root which belongs to R+ ∪ iR+ . We denote this square root by w(Z) :=  |z|2 − λ2 . 4. In the following, we will often identify an element z = x + iy ∈ C to the element (x, y) ∈ R2 and we will freely use the identification C × R ≡ R3 · 5. Let us define the following subsets of R3 : D := {(x, y, z) ∈ R3 \ x2 + y 2 − z 2 > − ∂+ D := {(x, y, z) ∈ R3 \ x2 + y 2 − z 2 = −

π2 } 4

π2 , z > 0} 4

D+ := D ∪ ∂+ D Notice that D+ can be identified to the set {Z = (z, λ) ∈ C × R : w(Z) ∈ R+ ∪ i(0,

π π ] with λ > 0 if w = i }· 2 2

Definition 10. It was proved in [4], that for all Z := (z, λ) ∈ R × C, the following operator Wr (Z) := ei(zB

+

+¯ z B − +λM )

∈ Un(Γ)

is well-defined. It is called the re-scaled Weyl operator. The operator + − W (Z) := ei(zB +¯z B +λN ) = e−iμλ Wr (Z)

(42)

(43)

is called the one-mode quadratic Weyl operator. Now, we recall the splitting formula given in [4],(Theorem 3.1). Proposition 11. For all Z = (z, λ) ∈ C × R, we denote Z¯ = (¯ z , λ) and  2 w(Z) := |z| − λ2 the positive part of the complex square root of |z|2 − λ2 . Then the following identity holds: +

¯

−

Wr (Z) = eF (Z)B eG(Z)M eF (Z)B ,

(44)

where the functions F, G are uniquely defined as follows: F (Z) := G(Z) := Moreover, one has |F (Z)| < 1

izthc(w(Z)) , 1 − iλthc(w(Z))

1 − th2 (w(Z))

 1 · log 2 (1 − iλthc(w(Z)))2 ∀Z ∈ C × R. 15

(45) (46)

Remark 3.

1. From Corollary 8 it follows

 ¯ . Wr (Z) = Γ F (Z), G(Z), F (Z)

Then, by combining (32) and (43) we get:

W (Z)Φ(t) = eμ

G(Z)−iλ



1 − tF (Z)

−μ

te2G(Z)  Φ F (Z) + 1 − tF (Z)

(47)

¯ for all t ∈ C satisfying |t| < r(Z) := r(F (Z), G(Z), F (Z)). Note that such r(Z) exists and satisfies 0 < r(Z) < 1. 2. It was proven in [4] (Proposition 4.1), that for all Z = (z, λ) ∈ C × R, there exists a unique Z0 = (z0 , λ0 ) ∈ D+ , such that Wr (Z) = Wr (Z0 ). For this, the manifold D+ is called the principal domain of the one-mode quadratic Weyl operators. Theorem 12. Let DI be the subset of D+ given by 

 DI = Z ∈ D+ , G(Z) ∈ R + i[0, π[ . Then, for all Z1 , · · · , ZN ∈ DI satisfying ∀ j, k = 1, 2, ..., N , j = k,

Zj = Zk ,

(48)

the set of the one-mode quadratic Weyl operators   W (Z1 ), W (Z2 ), · · · , W (ZN ) is linearly independent. Definition 13. The domain DI will be called the domain of the independence of the one-mode quadratic Weyl operators. Lemma 14. For Z ∈ C × R, we consider the function fZ (t) = F (Z) +

e2G(Z) t 1 − tF (Z)

which is well-defined on ] − 1, 1[. Then, for all Z, Z  ∈ DI such that Z = Z  , the equation fZ (t) = fZ  (t) has a finite number of roots in ] − 1, 1[. Proof. For t ∈] − 1, 1[, we have

16

fZ (t) = fZ  (t) 

te2G(Z ) te2G(Z) ⇔ F (Z) + = F (Z  ) + 1 − tF (Z) 1 − tF (Z  )

 

 e2G(Z ) − F (Z  )F (Z  ) t + F (Z  ) e2G(Z) − F (Z)F (Z) t + F (Z) = ⇔ 1 − tF (Z) 1 − tF (Z  )   at+b at + b = ⇔ 1 − ct 1 − c t ⇔ (a c − ac )t2 + (a − a + cb − bc )t + b − b = 0, (49) where we have denoted ⎧ a = e2G(Z) − F (Z)F (Z), ⎪ ⎪  ⎪ ⎪ a = e2G(Z ) − F (Z  )F (Z  ), ⎪ ⎪ ⎨ b = F (Z), b = F (Z  ), ⎪ ⎪ ⎪ ⎪ ⎪ c = F (Z), ⎪ ⎩  c = F (Z  ). Then, if the equation fZ (t) = fZ  (t) has an infinite number of roots in ] − 1, 1[, the polynomial (a c − ac )t2 + (a − a + cb − bc )t + b − b has an infinite number of zeros which implies that is null. This is equivalent to ⎧  ⎨ ac − a c = 0, a − a + cb − bc = 0, ⎩ b − b = 0. This gives that

⎧  ⎨ ac = a c, a + bc = a + b c , ⎩ b = b .

From the above notations, we deduce that ⎧      ⎪ ⎪ e2G(Z) − F (Z)F (Z) F (Z  ) = e2G(Z ) − F (Z  )F (Z  ) F (Z), ⎪ ⎨

    e2G(Z) − F (Z)F (Z) + F (Z)F (Z) = e2G(Z ) − F (Z  )F (Z  ) + F (Z  )F (Z  ),



⎪ ⎪ ⎪ ⎩

F (Z) = F (Z  ),

or equivalently,

⎧ 2G(Z)  F (Z  ) = e2G(Z ) F (Z), ⎨ e 2G(Z) 2G(Z  ) e =e , ⎩ F (Z) = F (Z  ) 17

which implies that F (Z) = F (Z  )

F (Z  ) = F (Z)

;

;



e2G(Z) = e2G(Z ) .

(50)



Since e2G(Z) = e2G(Z ) , then there exists k ∈ Z, such that 2G(Z  ) = 2G(Z) + 2ikπ. This gives G(Z  ) = G(Z) + ikπ. But, we have Z, Z  ∈ DI , then G(Z  ) = G(Z). From (50), we deduce that Wr (Z) = Wr (Z  ). Then, in view of Remark 3, we deduce that Z = Z  . Proof. (of Theorem 12) Let Z1 = (z1 , λ1 ), · · · , ZN = (zN , λN ) ∈ DI satisfying the condition (48) and let α1 , · · · , αN ∈ C, such that N 

αk W (Zk ) = 0.

(51)

k=1

Let r0 = inf{r(Zk ), k = 1, ..., N } ∈]0, 1[. Applying the identity (51) to Φ(t) for t ∈] − r0 , r0 [, we get N 

αk W (Zk )Φ(t) = 0.

k=1

Then, by using Eq.(47), we get N 

αk eμ

G(Zk )−iλk



1−tF (Zk )

k=1

−μ

te2G(Zk )  Φ F (Zk )+ =0 1 − tF (Zk )

∀t ∈]−r0 , r0 [. (52)

Let us consider the functions ] − r0 , r0 [ t −→ fk (t) := F (Zk ) +

te2G(Zk ) . 1 − tF (Zk )

Equation (52) implies N 

αk eμ

G(Zk )−iλk



1 − tF (Zk )

−μ

Φ(fk (t)) = 0

∀t ∈] − r0 , r0 [·

(53)

k=1

Assuming by contradiction that: ∀t ∈] − r0 , r0 [ Then ] − r0 , r0 [⊂

∃ j, k = 1, 2, ..., N, j = k : fj (t) = fk (t).

(54)

  Ij,k t ∈] − r0 , r0 [, fj (t) = fk (t) =

(55)

 j=k

j=k

18

  where Ij,k = t ∈]−r0 , r0 [, fj (t) = fk (t) is the set of the zeros of the equation fj (t) = fk (t) in the interval ] − r0 , r0 [. Since for all j, k = 1, 2, ..., N, j = k, the set Ij,k is finite ( this follows from Lemma 14), then their union is finite too. This is in contradiction with (55). Therefore, the assumption (54) is false. Consequently, there exists t0 ∈] − r0 , r0 [ separating fk ’s which means that fj (t0 ) = fk (t0 )

∀j = k ∈ {1, 2, · · · , N }.

For this t0 , denoting



−μ βk := αk eμ G(Zk )−iλk 1 − t0 F (Zk )

and

wk = fk (t0 ).

From (53), we get N 

βk Φ(wk ) = 0

k=1

with wk = wj ∀j, k = 1, 2, ..., N ; j = k. By the fact that the exponential vectors are linearly independent , we conclude that βk = 0 ∀k = 1, · · · , N which implies that αk = 0 for all k = 1, · · · , N . 4. The vacuum distribution of observable in sl2 (C) In this section, we give the vacuum distribution of the observable X = zB + + z¯B − + λM. Proposition 15. The vacuum distribution of the observable X = zB + + z¯B − + λM is given by the identity E(eisX ) = Φ, eisX Φ = eψ(s) , where

 1 − th2 (ws) μ log . 2 2 (1 − iλs thc(ws)) Moreover, it recovers the three non standard Meixner classes. ψ(s) =

Proof. E(eisX )

:=

Φ, eisX Φ

=

Φ, ei(szB +s¯zB +sλM ) Φ

Φ, Wr (sZ)Φ

Φ, eμG(sZ) Φ(F (sZ))

= = = =

+

eμG(sZ)

e

μ 2

log

19

−

1−th2 (ws) (1−iλs thc(ws))2

 ·

(56)

According to the parameter w, we distinguish three cases: Case 1. (|z|2 − λ2 = w2 > 0) It is well-known from [14] or [12], that the Meixner (a, b, d, m) distribution has a characteristic function of the form

cos( b ) 2d 2 E(eisX ) = eism (57) ch( as−ib 2 ) with mean M = m + da tan( 2b ), where a, d > 0, −π < b < π, m ∈ R. Then we can write

as b as  E(eisX ) = exp ims − 2d log ch( ) − i tan( )sh( ) . 2 2 2 Then its characteristic exponent has the form

as b as  ψa,b,d,m (s) = ims − 2d log ch( ) − i tan( )sh( ) . 2 2 2 But in our case, Eq. (56) can be written as follow:

 1 λ λ ψ(s) = −i s − log ch(ws) − i sh(ws) . 2 2 w

(58)

(59)

If we look to Eqs. (58) and (59), we deduce that a λ b 1 λ = w; = tan( ); d = ; m = − . 2 w 2 4 2 Case 2. (|z|2 − λ2 = w2 = 0) In this case, the characteristic exponent is of the form 1 λ ψ(s) = −i s − log(1 − iλs) 2 2

(60)

which corresponds to the re-scaled Gamma distribution. In fact it is well known (see [14]) that the characteristic exponent of the Gamma distribution Γ(a, b), a, b > 0 is given by ψa,b (s) = −a log(1 − ibs). Then we can choose a = and b = ±λ, depending on the sign of λ. Case 3. (|z|2 − λ2 = −w < 0) After some not difficult calculations, the characteristic exponent is expressed by μ 2 2

ψ(s) = i

1 w−λ s + log 2 2 1−

2w  w+λ · λ−w 2iws λ+w e

(61)

But also we know from [14], that the characteristic exponent of the negative binomial distribution N B(p, r), r > 0, p ∈]0, 1[, has the form ψp,r (s) = r log 20

1−p  · 1 − peis

Then we can choose r = μ2 and p = λ−w λ+w which is in ]0, 1[, if λ > w. In the case of λ < −w, the characteristic exponent is given by ψ(s) = −i

1 w+λ s + log 2 2 1−

2w  w−λ · λ+w −2iws λ−w e

(62)

λ+w ∈]0, 1[. In the two cases the characteristic It is convenient to take p = λ−w exponents (61) and (62) corresponds to the re-scaled negative binomial distribution.

[1] L. Accardi, A. Dhahri: Quadratic exponential vectors, J. Math. Phys, Vol. 50, 122103 (2009). [2] L. Accardi, U. Franz, M. Skeide: Renormalized squares of white noise and other non–Gaussian noises as L´evy processes on real Lie algebras, Comm. Math. Phys. 228 (2002) 123–150. Preprint Volterra, N. 423 (2000). [3] L. Accardi, H. Ouerdiane, H. Rebei: L´evy processes through time schift on oscillator Weyl algebra, Communications on stochastic analysis, Vol.6 (1), 125-155 (2012) [4] L. Accardi, H. Ouerdiane, H. Rebei: On the quadratic Heisenberg group, Infin. Dimens. Anal.Quantum Probab. Relat. Top., Vol.13 (4), 551-587 (2010) [5] L. Accardi, H. Ouerdiane, H. Rebei: Renormalized square of white noise quantum time shift, Communications on Stochastic Analysis, Vol.6 (2), 177-191 (2012) [6] L. Accardi, H. Rebei, A. Riahi, The Quantum Decomposition of Random Variables Without Moments , Infin. Dimens. Anal.Quantum Probab. Relat. Top.Vol: 16 no.2 (2013). [7] L. Accardi, I. Volovich: Quantum white noise with singular non-linear interaction, in Developments of Infinite-Dimensional Noncommutative Analysis (Kyoto, 1998), Editor N. Obata, no. 1099, Research Institute for Mathematical Sciences, Kyoto, 1999, 61–69. [8] L. Accardi, Y.G. Lu, N. Obata: Towards a non-linear extension of stochastic calculus, Quantum Stochastic Analysis and Related Fields (Kyoto, 1995), Research Institute for Mathematical Sciences, Kyoto, 1996, 1– 15. [9] L. Accardi, Y.G. Lu, I. Volovich: Non-linear extensions of classical and quantum stochastic calculus and essentially infinite-dimensional analysis Probability Towards (2000) (October 2–6, 1995, Columbia University, New York), Editors L. Accardi and C. Heyde, Lecture Notes in Statist., Vol. 128, Springer, New York, (1998), 1–33.

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[10] L. Accardi, Y.G. Lu, I. Volovich: Quantum theory and its stochastic limit, Springer-Verlag, Berlin, (2002). [11] L. Accardi, Y.G. Lu, I. Volovich: White noise approach to classical and quantum stochastic calculus, Lecture Notes of the Volterra International School, Trento, Italy, (1999). [12] D. Appelebaum, L´evy processes and stochastic calculus, Cambridge University Press (2009) [13] W. Ayed: White noise approach to quantum stochastic calculus, Thesis Doctor of Mathematics, University of Tunis El Manar, Tunis, (2006). [14] K. Sato, L´evy Processes and Infinitely Divisible Distributions, Cambridge Univ. Press (1999).

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