Definition 1. The (one–mode) Galilei algebra G is the Lie algebra with generators ξ

THE CURRENT GALILEI ALGEBRA AND ASSOCIATED RANDOM VARIABLES

Luigi Accardi and Andreas Boukas

(Received July, 2011)

Abstract. We compute the Fock kernel for the one-mode (i.e. restricted to a single interval) and current (i.e. extended to simple functions) Galilei algebra.

We also compute the characteristic function of a family of random variables naturally associated with the Galilei algebra.

1. The Galilei and RHPWN Lie Algebras

Definition 1. The (one–mode) Galilei algebra G is the Lie algebra with generators ξ

1

, ξ

2

, ξ

3

, ξ

4

and commutation relations

[ξ

₄

, ξ

₁

] = ξ

₂

; [ξ

₄

, ξ

₂

] = ξ

₃

. All other commutators among generators are equal to zero.

If a

^†

and a are a Boson pair, i.e.

[a, a

^†

] = 1 ; (a)

^∗

= a

^†

then the Lie algebra generated by {1, p, q, q

²

}, where

q = i (a − a

^†

) ; p = a

^†

+ a is a Boson form of G since

[q, p] = i ; [q

²

, p] = 2 i q ; [q

²

, q] = 0 and, in the notation of Definition 1, we may take

ξ

1

= 1

2 q

²

; ξ

2

= −i q ; ξ

3

= −1 ; ξ

4

= p.

Notice that

(q)

^∗

= q ; (q

²

)

^∗

= q

²

; (p)

^∗

= p.

In order to consider the smeared field form of {1, p, q, q

²

}, i.e. the current Galilei algebra, we recall some basic facts about the ∗-Lie algebra of the Renormalized Higher Powers of White Noise (see [3]-[7]).

2010 Mathematics Subject Classification Primary 17B15, 17B56, 60B15, 81S05. Secondary 81R05, 81S25.

Key words and phrases: Galilei algebra, Heisenberg algebra, Central extension, Fock representa- tion, Fock kernel, Renormalized Higher Powers of White Noise.

The quantum white noise functionals a

_t

(creation density) and a

t

(annihilation density) satisfy the Boson commutation relations

[a

_t

, a

^†_s

] = δ(t − s) ; [a

^†_t

, a

^†_s

] = [a

_t

, a

_s

] = 0,

where t, s ∈ R and δ is the Dirac delta function, as well as the duality relation (a

s

)

^∗

= a

^†_s

.

For a test function f and n, k ∈ {0, 1, 2, ...}, the sesquilinear forms

B

_kⁿ

(f ) = Z

R

f (t) a

^†_tⁿ

a

^k_t

dt,

where dt denotes integration with respect to Lebesgue measure µ, with involution (B

_kⁿ

(f ))

^∗

= B

^k_n

( ¯ f )

were defined in [9]. In [3] and [4] we introduced the convolution type renormaliza- tion

δ

^l

(t − s) = δ(s) δ(t − s) ; l = 2, 3, .... (1.1) 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 (RHP W N ) ∗–Lie algebra commutation relations

[B

_kⁿ

(f ), B

_K^N

(g)] = (k N − K n) B

_k+K−1^{n+N −1}

(f g).

2. The RHPWN Form of the Current Galilei Algebra Lemma 1. Let a

^†_t

and a

t

be as in Section 1. Define

q

_t

= i (a

_t

− a

^†_t

) ; p

_t

= a

^†_t

+ a

_t

. Then

[q

t

, p

s

] = i δ(t − s) ; [q

²_t

, p

s

] = 2 i q

t

δ(t − s)

[q

t

, q

s

] = [p

t

, p

s

] = [q

_t²

, q

²_s

] = [q

²_t

, q

s

] = 0 and

(q

s

)

^∗

= q

s

; (q

_s²

)

^∗

= q

²_s

; (p

s

)

^∗

= p

s

.

Proof. The proof follows easily from the commutation and duality relations satis- fied by the quantum white noise functionals a

^†_t

and a

t

.



Proposition 1. For step functions f, g vanishing at zero let

q(f ) = i (B

₁⁰

(f ) − B

₀¹

(f )) p(f ) = B

⁰₁

(f ) + B

₀¹

(f )

q

²

(f ) = 2 B

¹₁

(f ) − B

₂⁰

(f ) − B

₀²

(f ).

Then

[q(f ), p(g)] = 2 i Z

R

f g dt ; [q

²

(f ), p(g)] = 4 i q(f g) [q(f ), q(g)] = [p(f ), p(g)] = [q

²

(f ), q

²

(g)] = [q

²

(f ), q(g)] = 0 and

(q(f ))

^∗

= q( ¯ f ) ; (q

²

(f ))

^∗

= q

²

( ¯ f ) ; (p(f ))

^∗

= p( ¯ f ).

Proof. For t, s ∈ R

q

_t

q

_s

δ(t − s) = −(a

_t

− a

^†_t

)(a

_s

− a

^†_s

) δ(t − s)

= −a

^†_t

a

^†_s

δ(t − s) + a

^†_t

a

s

δ(t − s) + a

t

a

^†_s

δ(t − s) − a

t

a

s

δ(t − s)

= −a

^†_t

a

^†_s

δ(t − s) + a

^†_t

a

s

δ(t − s) + a

^†_s

a

t

δ(t − s) + δ

²

(t − s) − a

t

a

s

δ(t − s) Taking R

R

R

R

f (s) ... ds dt of both sides and using the renormalization (1.1) we have that

Z

R

Z

R

f (s) δ

²

(t − s) ds dt = Z

R

Z

R

f (s) δ(s) δ(t − s) ds dt = f (0) = 0 and so

q

²

(f ) = Z

R

f (t) q

²_t

dt = 2 B

¹₁

(f ) − B

₂⁰

(f ) − B

₀²

(f ).

For p(f ) and q(f ) we directly find that q(f ) =

Z

R

f (t) q

_t

dt = Z

R

f (t) i (a

_t

− a

^†_t

) dt = i (B

₁⁰

(f ) − B

₀¹

(f )) and

p(f ) = Z

R

f (t) p

t

dt = Z

R

f (t) (a

t

+ a

^†_t

) dt = B

₁⁰

(f ) + B

¹₀

(f ).

Moreover

[q(f ), p(g)] = i [B

₁⁰

(f ) − B

₀¹

(f ), B

₁⁰

(g) + B

₀¹

(g)]

= i [B

₁⁰

(f ), B

₀¹

(g)] − i [B

¹₀

(f ), B

₁⁰

(g)]

= 2 i [B

₁⁰

(f ), B

₀¹

(g)]

= 2 i B

₀⁰

(f g)

= 2 i Z

R

f g dt

and

[q

²

(f ), p(g)] = [2 B

₁¹

(f ) − B

⁰₂

(f ) − B

₀²

(f ), B

₁⁰

(g) + B

₀¹

(g)]

= 2 [B

₁¹

(f ), B

₁⁰

(g)] + 2 [B

¹₁

(f ), B

₀¹

(g)] − [B

₂⁰

(f ), B

₀¹

(g) − [B

²₀

(f ), B

⁰₁

(g)

= −2 B

₁⁰

(f g) + 2 B

₀¹

(f g) − 2 B

₁⁰

(f g) + 2 B

¹₀

(f g)

= 4 B

₀¹

(f g) − B

₁⁰

(f g)

= 4 i q(f g).

Finally

q

²

(f )

∗

= 2 B

₁¹

(f ) − B

₂⁰

(f ) − B

²₀

(f )

∗

= 2 B

¹₁

( ¯ f ) − B

₀²

( ¯ f ) − B

⁰₂

( ¯ f )

= q

²

( ¯ f ) and, similarly,

(q(f ))

^∗

= q( ¯ f ) ; (p(f ))

^∗

= p( ¯ f ).

 Definition 2. The current Galilei algebra is the ∗–Lie algebra generated by

{q(f ), p(f ), q

²

(f ), 1 ; f ∈ S

₀

} where S

₀

is the set of step functions vanishing at zero.

3. The Fock Kernel for the One-Mode Galilei Algebra

Definition 3. Let I ⊂ R with µ(I) > 0 and a, b, c ∈ C. The exponential vector ψ

a,b,c

(I) is defined by

ψ

a,b,c

(I) = e

^{a q2(χI)}

e

^{b q(χI)}

e

^{c p(χI)}

Φ

= e

^a

(

^{2 B1}1(χ_I)−B₂⁰(χ_I)−B²₀(χ_I)

) e

^{i b}

(

^B1⁰(χ_I)−B¹₀(χ_I)

) e

^c

(

^B1⁰(χ_I)+B¹₀(χ_I)

) Φ where the Fock vacuum vector Φ is such that

B

₁⁰

(χ

I

) Φ = B

₂⁰

(χ

I

) Φ = 0 ; B

₁¹

(χ

I

) Φ = µ(I)

2 Φ

and χ

I

denotes the characteristic function of the set I.

Notice that

q

²

(χ

I

) ψ

a,b,c

(I) = ∂

∂ a ψ

a,b,c

= ∂

∂  |

=0

ψ

a+,b,c

(I) q(χ

_I

) ψ

_a,b,c

(I) = ∂

∂ b ψ

_a,b,c

= ∂

∂  |

=0

ψ

_a,b+,c

(I) p(χ

_I

) ψ

_a,b,c

(I) = ∂

∂ c ψ

_a,b,c

= ∂

∂  |

_=0

ψ

_a,b,c+

(I)

Lemma 2. For |a| < 1

(1 − a)

^−B11(χI)

Φ = (1 − a)

^−µ(I)2

Φ.

Proof.

(1 − a)

^−B11(χI)

Φ = e

− ln(1−a) B₁¹(χ_I)

Φ

=

∞

X

n=0

(− ln(1 − a))

ⁿ

n! B

₁¹

(χ

I

)

ⁿ

Φ

=

∞

X

n=0

(− ln(1 − a))

ⁿ

n!

 µ(I) 2



ⁿ

Φ

= e

^{− ln(1−a)µ(I)2}

Φ

= (1 − a)

^−µ(I)2

Φ.

 Lemma 3. (i) If x, d, N and h satisfy the oscillator algebra commutation relations [d, x] = h ; [d, h] = [x, h] = 0 ; [N, x] = x ; [d, N ] = d (3.1) then for all a, b, t, s ∈ C

e

^{t d}

e

^{a x}

= e

^{a x}

e

^{t d}

e

^{a t h}

(3.2)

e

s (x+a d+b h)

= e

^{s x}

e

^{s a d}

e

^{(s b+s2a/2)h}

(3.3) and

e

^{a N}

e

^{b x}

= e

^{eab x}

e

^{a N}

. (3.4) (ii) If ∆, R and ρ satisfy the sl(2) algebra commutation relations

[∆, R] = ρ ; [ρ, R] = 2 R ; [∆, ρ] = 2 ∆ (3.5) then

e

^{t ∆}

e

^{a R}

= e

^{1−a ta R}

(1 − a t)

^−ρ

e

^{1−a tt ∆}

(3.6) and

e

^{a (ρ−R−∆)}

= e

^{a−1a R}

(1 − a)

^−ρ

e

^{a−1a ∆}

. (3.7) Proof. (3.2), (3.3), (3.4) and (3.6) are, respectively, Propositions 2.2.1, 4.1.1 2.4.2 (for N = x D) and 3.3.2 of [10]. For (3.7) we notice that by Proposition 4.3.1 of [10], for all A, B, s ∈ C

e

s (R+A ρ+B ∆)

= e

^{V (s) R}

e

^{H(s) ρ}

e

^{U (s) ∆}

where

V

⁰

(s) = 1 + 2 A V (s) + B V (s)

²

; V (0) = 0 (Riccati ODE) H

⁰

(s) = A + B V (s) ; H(0) = 0

U

⁰

(s) = B e

^{2 H(s)}

; U (0) = 0.

For A = −1 and B = 1 we obtain

V

⁰

(s) = 1 − 2 V (s) + V (s)

²

; V (0) = 0 H

⁰

(s) = −1 + V (s) ; H(0) = 0

U

⁰

(s) = e

^{2 H(s)}

; U (0) = 0 which imply that

V (s) = U (s) = s

s + 1 ; H(s) = − ln(s + 1) i.e.

e

^{s (R− ρ+∆)}

= e

^{s+1s R}

e

− ln(s+1) ρ

e

^{s+1s ∆}

from which (3.7) follows by letting s = −a.

 Corollary 1. For I ⊂ R and a, b, A, t, T, s ∈ C

e

^{t B01(χI)}

e

^{a B10(χI)}

= e

^{a B01(χI)}

e

^{t B10(χI)}

e

^{a t µ(I)}

e

^{s (B10(χI)+a B01(χI)+b µ(I))}

= e

^{s B01(χI)}

e

^{s a B01(χI)}

e

^{(s b+s2a/2) µ(I)}

e

^{a B11(χI)}

e

^{b B10(χI)}

= e

^{eab B01(χI)}

e

^{a B11(χI)}

(3.8) e

^{T B20(χI)}

e

^{A B20(χI)}

= e

^{1−4 A TA B02(χI)}

(1 − 4 A T )

^−B11(χI)

e

^{1−4 A TT B20(χI)}

e

^{A (2 B11(χI)−B02(χI)−B02(χI))}

= e

^{2 A−1A B02(χI)}

(1 − 2 A)

^−B11(χI)

e

^{2 A−1A B02(χI)}

. In particular (3.8) implies that, for |a| < 1,

(1 − a)

^−B11(χI)

e

^{b B01(χI)}

= e

^{1−ab B10(χI)}

(1 − a)

^−B11(χI)

Proof. The proof follows from Lemma 3 by noticing that for a fixed I ⊂ R, the operators x, D, N, h defined by B

¹₀

(χ

I

) = x, B

₁⁰

(χ

I

) = D and B

₁¹

(χ

I

) = N and µ(I) 1 = h, satisfy the oscillator algebra commutation relations (3.1) while the operators R, ∆, ρ defined by B

₀²

(χ

I

) = 2 R, B

⁰₂

(χ

I

) = 2 ∆ and B

₁¹

(χ

I

) = ρ, satisfy the sl(2) algebra commutation relations (3.5) and by letting

^a₂

= A and

₂^t

= T in (3.6) and (3.7).

 Lemma 4. For I ⊂ R and a, b ∈ C

e

^{a B20(χI)}

e

^{b B10(χI)}

Φ = e

^{a b2µ(I)}

e

^{b B10(χI)}

Φ.

Proof. In [1] we showed that

B

₂⁰

(χ

_I

) e

^{b B01(χI)}

Φ = b

²

µ(I) e

^{b B10(χI)}

Φ.

Iterating we find that, for n ≥ 0, B

⁰₂

(χ

_I

)

ⁿ

e

^{b B01(χI)}

Φ = b

²

µ(I)

ⁿ

e

^{b B10(χI)}

Φ.

Therefore

e

^{a B20(χI)}

e

^{b B01(χI)}

Φ =

∞

X

n=0

a

ⁿ

n! B

₂⁰

(χ

_I

)

n

e

^{b B10(χI)}

Φ

=

∞

X

n=0

a

ⁿ

n! b

²

µ(I)

ⁿ

e

^{b B01(χI)}

Φ

= e

^{a b2µ(I)}

e

^{b B10(χI)}

Φ.

 Lemma 5. For all a, b, c ∈ C and I ⊂ R with µ(I) > 0 and |a| <

¹2

ψ

_a,b,c

(I) = (1 − 2 a)

^−µ(I)2

exp  b

²

+ c

²

− 4 a c

²

+ 2 i b c

2 − 4 a µ(I)



exp

 a

2 a − 1 B

₀²

(χ

_I

)



exp  c − i b

1 − 2 a B

¹₀

(χ

_I

)

 Φ

Proof. For brevity, in what follows, we drop the use of χ

_I

. Using Lemmas 2 and 4, Corollary 1, and the fact that, for all λ ∈ C, e

^{λ B10}

Φ = e

^{λ B02}

Φ = Φ we have

ψ

a,b,c

(I) = e

^a

(

^{2 B}1¹(χ_I)−B⁰₂(χ_I)−B²₀(χ_I)

) e

^{i b}

(

^B1⁰(χ_I)−B¹₀(χ_I)

) e

^c

(

^B01(χ_I)+B₀¹(χ_I)

) Φ

= e

^{b2 +c22 µ(I)}

e

^{2 a−1a B02}

(1 − 2 a)

^−B11

e

^{2 a−1a B20}

e

^{−i b B10}

e

^{i b B01}

e

^{c B10}

e

^{c B01}

Φ

= e

_{b2 +c2}

2 +i b c

µ(I)

e

^{2 a−1a B20}

(1 − 2 a)

^−B11

e

^{2 a−1a B02}

e

^{−i b B01}

e

^{c B01}

e

^{i b B10}

Φ

= e

_{b2 +c2}

2 +i b c

µ(I)

e

^{2 a−1a B20}

(1 − 2 a)

^−B11

e

^{2 a−1a B02}

e

^{c−i b B10}

Φ

= e



b2 +c2

2 +i b c+^{a (c−i b)2}_{2 a−1}



µ(I)

e

^{2 a−1a B20}

(1 − 2 a)

^−B11

e

^{c−i b B10}

Φ

= e

b2 +c2 −4 a c2 +2 i b c

2−4 a µ(I)

e

^{2 a−1a B02}

e

^{1−2 ac−i b B10}

(1 − 2 a)

^−B11

Φ

= (1 − 2 a)

^−µ(I)2

e

b2 +c2 −4 a c2 +2 i b c

2−4 a µ(I)

e

^{2 a−1a B02(χI)}

e

^{1−2 ac−i bB01(χI)}

Φ.



Proposition 2. (i) Let I ⊂ R with µ(I) > 0 and a, b, c, A, B, C ∈ C with |a| <

¹₂

and |A| <

¹₂

. Then, the Galilei Fock space inner product of the exponential vectors

ψ

_A,B,C

(I) and ψ

_a,b,c

(I) is given by

hψ

A,B,C

(I), ψ

_a,b,c

(I)i (3.9)

= (1 − 2 a)

^−µ(I)2

(1 − 2 ¯ A)

^−µ(I)2



1 −

_{4 (2 ¯}a−1) (2 A−1)^{¯a A}



−^µ(I)₂

exp 

b²+c²−4 a c²+2 i b c 2−4 a

µ(I) 

exp 

_¯

B²+ ¯C²−4 ¯A ¯C²−2 i ¯B ¯C 2−4 ¯A

µ(I)  exp

^µ(I)₄

¯ a (C−i B)2

(2 ¯a−1) (2 A−1)2+4^(¯_{(2 ¯}^{c+i ¯}a−1) (2 A−1)^{b) (C−i B)}+ ^{A (¯c+i ¯b)2}

(2 ¯a−1)2 (2 A−1)

4−_{(2 ¯}a−1) (2 A−1)^{a A¯}

! .

(ii) If I ∩ J =  then

hψ

A,B,C

(I), ψ

a,b,c

(J )i = 0

Proof. To prove (i), as in Lemma 5, we drop the use of χ

_I

. By Lemma 5

hψ

_A,B,C

, ψ

_a,b,c

i

= (1 − 2 a)

^−µ(I)2

(1 − 2 ¯ A)

^−µ(I)2

e

b2 +c2 −4 a c2 +2 i b c

2−4 a µ(I)

e

B2 + ¯¯ C2 −4 ¯A ¯C2 −2 i ¯B ¯C

2−4 ¯A µ(I)

he

^{2 a−1a B20}

e

^{1−2 ac−i b B10}

Φ, e

^{2 A−1A B02}

e

^{C−i B1−2 AB10}

Φi.

Using the Feinsilver-Kocik-Schott Fock kernel for the Schr¨ odinger algebra (see [12]), in [2] we showed that for all a, b, A, B ∈ C, with B

¹1

(χ

I

) Φ =

^µ(I)₂

Φ,

he

^{a B20}

e

^{b B01}

Φ, e

^{A B20}

e

^{B B10}

Φi =

 1 − a A ¯

4



−^µ(I)₂

e

^{µ(I)4 ¯a B2 +4 ¯4−¯b B+¯a A b2 A}

. Therefore

he

^{2 a−1a B20}

e

^{1−2 ac−i b B10}

Φ, e

^{2 A−1A B02}

e

^{C−i B1−2 AB01}

Φi

= 

1 −

_{4 (2 ¯}a−1) (2 A−1)^{¯a A}



−^µ(I)₂

exp

^µ(I)₄

¯ a (C−i B)2

(2 ¯a−1) (2 A−1)2+4^(¯_{(2 ¯}^{c+i ¯}a−1) (2 A−1)^{b) (C−i B)}+ ^{A (¯c+i ¯b)2}

(2 ¯a−1)2 (2 A−1)

4−_{(2 ¯}a−1) (2 A−1)^{a A¯}

!

and so

hψ

A,B,C

, ψ

a,b,c

i = (1 − 2 a)

^−µ(I)2

(1 − 2 ¯ A)

^−µ(I)2

exp 

b²+c²−4 a c²+2 i b c 2−4 a

µ(I) 

exp 

_¯

B²+ ¯C²−4 ¯A ¯C²−2 i ¯B ¯C 2−4 ¯A

µ(I) 



1 −

_{4 (2 ¯}a−1) (2 A−1)^{¯a A}



−^µ(I)₂

exp

^µ(I)₄

¯ a (C−i B)2

(2 ¯a−1) (2 A−1)2+4^(¯_{(2 ¯}^{c+i ¯}a−1) (2 A−1)^{b) (C−i B)}+ ^{A (¯c+i ¯b)2}

(2 ¯a−1)2 (2 A−1)

4−_{(2 ¯}a−1) (2 A−1)^{¯a A}

! .

The proof of (ii) follows from the fact that B

₂⁰

and B

₁⁰

commute with B

₀²

and B

₀¹

on disjoint intervals and also B

₂⁰

Φ = B

₁⁰

Φ = 0.



4. The Fock Kernel for the Current Galilei Algebra

Using the commutativity of the Galilei algebra generators on disjoint sets, we may extend the kernel (3.9) to exponential vectors of the form

Ψ(f, g, h) = e

^{q2(f )}

e

^q(g)

e

^p(h)

Φ

= Y

i

e

^aiq

2(χ

Ii)

e

^biq(χIi)

e

^cip(χIi)

Φ

= Y

i

ψ

_ai,bi,ci

(I

_i

) where f = P

i

a

_i

χ

_Ii

, g = P

i

b

_i

χ

_Ii

and h = P

i

c

_i

χ

_Ii

with I

_i

∩ I

_j

=  for i 6= j, are simple functions with |f | <

¹₂

and, using Proposition 2 (ii) we obtain

hΨ(f

1

, g

1

, h

1

), Ψ(f

2

, g

2

, h

2

)i = Q

i

hψ

a1i,b1i,c1i

(I

i

), ψ

a2i,b2i,c2i

(I

i

)i

= exp



−

¹₂

R ln



(1 − 2 f

1

) 1 − 2 ¯ f

2



1 −

^f¯1f2

4

(

^{1−2 ¯f}1

)

^{(1−2 f}2)

  dµ



exp  R 

_g2

1+h²₁−4 f1h²₁+2 i g₁h₁

2−4 f₁

+

^{¯g22+¯h22−4 ¯}_{2−4 ¯}^f2¯h_f^{22−2 i ¯g2¯h2}

2

 dµ  exp

¹₄

R

f1 (h2−i g2)¯ 2

(2 ¯f1−1) (2 f2−1)2+4^{(¯h1+i ¯}g1) (h2−i g2)

(2 ¯f1−1) (2 f2−1) + ^{f2 (¯h1+i ¯g1)2}

(2 ¯f1−1)2 (2 f2 −1)

4− ^{f1 f2¯}

(2 ¯f1−1) (2 f2−1)

dµ

! .

5. Random Variables Associated with the Galilei Algebra

Definition 4. Let a

^†

and a be a Boson pair as in Section 1. The Boson-Schr¨ odinger algebra is the Lie algebra generated by {1, a, a

^†

, a

²

, a

^†2

, a

^†

a} with commutation re- lations given in the table (see also [12])

a a

^†

a

²

a

^†2

a

^†

a 1

a 0 1 0 2 a

^†

a 0

a

^†

−1 0 −2 a 0 −a

^†

0

a

²

0 2 a 0 2 + 4 a

^†

a 2 a

²

0

a

^†2

−2 a

^†

0 −2 − 4 a

^†

a 0 −2 a

^†2

0

a

^†

a −a a

^†

−2 a

²

2 a

^†2

0 0

1 0 0 0 0 0 0

Lemma 6. Let a

^†

and a be a Boson pair. Let also L ∈ R and M, N ∈ C. Then for all s ∈ R

e

^{i s}

(

^{L a2+L a† 2−2 L a†}a−L+M a+N a^†

) Φ = e

^{w1(s) a† 2}

e

^{w2(s) a†}

e

^w3(s)

Φ where

w

₁

(s) = L s 2 L s − i

w

2

(s) = i L (M + N ) s

²

+ N s

2 L s − i

and

w

3

(s) = (M + N )

²

(L

²

s

⁴

− 2 i L s

³

) − 3 M N s

²

6 (2 i L s + 1) − ln (2 i L s + 1)

2 .

Proof. The proof can be found in [2].

 Corollary 2. Let I ⊂ R with µ(I) > 0. Let also α ∈ R and u, v ∈ C. Then for all s ∈ R

e

^{i s}

(

^{α B0}2(χI)+α B²₀(χI)−2 α B₁¹(χI)+u B⁰₁(χI)+v B₀¹(χI)

) Φ

= e

^{w1(s) B02(χI)}

e

^{w2(s) B10(χI)}

e

^w3(s)

Φ where

w

1

(s) = α s 2 α s − i

w

₂

(s) = i α (u + v) s

²

+ v s 2 α s − i

p µ(I) and

w

₃

(s) = (u + v)

²

(α

²

s

⁴

− 2 i α s

³

) − 3 u v s

²

6 (2 i α s + 1) µ(I) − ln (2 i α s + 1)

2 .

Proof. Using the correspondence,

B

¹₀

(χ

I

) = pµ(I) a

^†

; B

⁰₁

(χ

I

) = pµ(I) a ; B

⁰₀

(χ

I

) = µ(I) B

₀²

(χ

I

) = a

^†2

; B

₂⁰

(χ

I

) = a

²

; B

¹₁

(χ

I

) = a

^†

a +

¹₂

we see that

α B

₂⁰

(χ

_I

) + α B

₀²

(χ

_I

) − 2 α B

₁¹

(χ

_I

) + u B

₁⁰

(χ

_I

) + v B

₀¹

(χ

_I

)

= L a

²

+ L a

^†2

− 2 L a

^†

a − L + M a + N a

^†

where L = α, M = u pµ(I), N = v pµ(I) and the proof follows from Lemma

6. 

Proposition 3. (Characteristic Function) Let λ

i

∈ R ; i = 1, 2, 3. In the notation of Corollary 2, and in view of Proposition 1, consider the random variable (i.e.

self-adjoint operator on the Galilei Fock space)

X = λ

1

q

²

(χ

I

) + λ

2

q(χ

I

) + λ

3

p(χ

I

)

= α B

₂⁰

(χ

_I

) + α B

₀²

(χ

_I

) − 2 α B

₁¹

(χ

_I

) + u B

₁⁰

(χ

_I

) + v B

₀¹

(χ

_I

) where α = −λ

₁

, u = λ

₃

+ i λ

₂

, v = λ

₃

− i λ

2

.

Then, for all s ∈ R, the (vacuum) characteristic function of the random variable

X is given by

hΦ, e

^{i s X}

Φi = (1 − 2 i λ

₁

s)

^−1/2

exp  4 λ

²₃

(λ

²₁

s

⁴

+ 2 i λ

₁

s

³

) − 3 (λ

²₂

+ λ

²₃

) s

²

6 (1 − 2 i λ

1

s) µ(I)

 . Proof. Using Corollary 2 and the fact that for all z ∈ C

e

^{z B20(χI)}

Φ = e

^{z B01(χI)}

Φ = Φ we have

hΦ, e

^{i s X}

Φi = hΦ, e

^{w1(s) B02(χI)}

e

^{w2(s) B10(χI)}

e

^w3(s)

Φi

= he

^{w¯2(s) B01(χI)}

e

^{w¯1(s) B20(χI)}

Φ, e

^w3(s)

Φi

= hΦ, e

^w3(s)

Φi

= e

^w3(s)

hΦ, Φi

= e

^w3(s)

.

Using the formula for w

3

(s) provided in Corollary 2 we find that

hΦ, e

^{i s X}

Φi

= (2 i α s + 1)

^−1/2

exp 

_(u+v)2(α²s⁴−2 i α s³)−3 u v s² 6 (2 i α s+1)

µ(I) 

= (1 − 2 i λ

₁

s)

^−1/2

exp 

_{4 λ}2

3(λ²₁s⁴+2 i λ1s³)−3 (λ²₂+λ²₃) s² 6 (1−2 i λ1s)

µ(I) 

.

 References

[1] L. Accardi, and A. Boukas, Random variables and positive definite kernels as- sociated with the Schroedinger algebra, Proceedings of the VIII International Workshop Lie Theory and its Applications in Physics, Varna, Bulgaria, June 16-21, 2009, pages 126-137, American Institute of Physics, AIP Conference Pro- ceedings 1243.

[2] L. Accardi, and A. Boukas, The centrally extended Heisenberg algebra and its connection with the Schr¨ odinger, Galilei and renormalized higher powers of quantum white noise Lie algebras, Proceedings of the VIII International Work- shop Lie Theory and its Applications in Physics, Varna, Bulgaria, June 16-21, 2009, pages 115-125, American Institute of Physics, AIP Conference Proceed- ings 1243.

[3] L. Accardi, and A. Boukas, Renormalized higher powers of white noise (RH- PWN) and conformal field theory, Infinite Dimensional Analysis, Quantum Probability, and Related Topics, 9 (3) (2006) 353-360.

[4] L. Accardi, and A. Boukas, 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 (3) (2006) 315–342.

[5] L. Accardi, and A. Boukas, Fock representation of the renormalized higher pow- ers of white noise and the Virasoro–Zamolodchikov–w

∞

∗–Lie algebra, J. Phys.

A: Math. Theor., 41 (2008).

[6] L. Accardi, and A. Boukas, Cohomology of the Virasoro–Zamolodchikov and renormalized higher powers of white noise ∗–Lie algebras, Infinite Dimensional Anal. Quantum Probab. Related Topics, 12 (2) (2009) 193–212.

[7] L. Accardi, and A. Boukas, Quantum probability, renormalization and infinite dimensional ∗–Lie algebras, SIGMA (Symmetry, Integrability and Geometry:

Methods and Applications), 5 (2009), 056, 31 pages.

[8] L. Accardi, and A. Boukas, On the Fock representation of the central extensions of the Heisenberg algebra, Australian Journal of Mathematical Analysis and Applications, 6 (2) (2009) 1–10.

[9] L. Accardi, Y. G. Lu and I. V. Volovich, White noise approach to classical and quantum stochastic calculi, Lecture Notes of the Volterra International School of the same title, Trento, Italy, (1999), Volterra Center preprint 375, Universit` a di Roma Torvergata.

[10] P. J. Feinsilver and R. Schott, Algebraic Structures and Operator Calculus, Volumes I and III, Kluwer, 1993.

[11] P. J. Feinsilver and R. Schott, Differential relations and recurrence formulas for representations of Lie groups, Stud. Appl. Math., 96 (4) (1996) 387–406.

[12] P. J. Feinsilver, J. Kocik, R. Schott, Representations of the Schroedinger alge- bra and Appell systems, Fortschr. Phys. 52 (4) (2004) 343–359.

[13] W. I. Fushchych, R. Z Zhdanov, On new nepresentations of Galilei groups, Nonlinear Mathematical Physics, 4 (34) (1997) 426–435.

[14] Ovando, G.: Four dimensional symplectic Lie algebras, Beitrage Algebra Geom. 47 (2) (2006), 419–434.

Luigi Accardi Centro Vito Volterra,

Universit`a di Roma Tor Vergata via Columbia 2, 00133 Roma, Italy accardi@volterra.mat.uniroma2.it

Andreas Boukas Centro Vito Volterra,

Universita di Roma Tor Vergata via Columbia 2, 00133 Roma, Italy andreasboukas@yahoo.com