Moments of two noncommutative random variables in terms of their joint quantum operators

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1 Journl of Physics: Conference Series OPEN ACCESS Moments of two noncommuttive rndom vribles in terms of their joint quntum opertors To cite this rticle: Gbriel Pop nd Aurel I Stn 013 J. Phys.: Conf. Ser Relted content - Gmm distributed rndom vribles nd their semi-quntum opertors Gbriel Pop nd Aurel I Stn - Moleculr motion in NMR: II T B Smith - Modeling experiments using quntum nd Kolmogorov probbility Krl Hess View the rticle online for updtes nd enhncements. Recent cittions - Gmm distributed rndom vribles nd their semi-quntum opertors Gbriel Pop nd Aurel I Stn This content ws downloded from IP ddress on 04/0/018 t 19:33

2 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 Moments of two noncommuttive rndom vribles in terms of their joint quntum opertors Gbriel Pop 1 nd Aurel I. Stn 1 Deprtment of Physics, College of Arts nd Sciences, 40 Elton Hll, Ohio University, Znesville, OH 43701, USA Deprtment of Mthemtics, Ohio Stte University t Mrion, 1465 Mount Vernon Avenue, Mrion, OH 4330, USA E-mil: popg@ohio.edu, stn.7@osu.edu Abstrct. The computtion of the generting function of the joint moments of two non commuttive gussino poissonin rndom vribles, from the commuttors of their quntum opertors, is presented first. An exmple of such rndom vribles is lso included in the pper. 1. Introduction In this pper we pply the lgorithm for recovering the joint moments from the commuttors of the quntum opertors, nd the first order moments, outlined in 9], to two noncommuttive rndom vribles X nd Y. These re not clssicl rndom vribles, but symmetric opertors densely defined on Hilbert spce. We must lso mention the fct tht, the lgorithm mentioned bove ws extended to q commuttors in 4], nd the uthors of tht pper worked out the joint moments of both commuttive nd noncommuttive rndom vribles. The multi dimensionl cse is much hrder thn the one dimensionl one, nd for this reson we restrict our ttention to the dimension d = in this pper. The pper is structured s follows. In section we present little bit of the bckground concerning Noncommuttive Probbility nd the joint quntum opertors, generted by finite fmily of noncommuttive rndom vribles. In section 3 we pose two dimensionl commuttor problem nd pply the lgorithm, from 9], to find recursive formul for the joint moments of the two rndom vribles involved in tht commuttor problem. In section 4, we use the recursive formul, obtined in the previous section, to compute the joint Lplce trnsform of the two rndom vribles from section 3. Finlly, in section 5, we show the existence of the opertors involved in the commuttor problem from section 3.. Bckground We present in this section bsic bckground of Noncommuttive Probbility nd joint quntum opertors generted by fmily of not necessry commuttive rndom vribles. The definition of noncommuttive probbility spce is tken from 13], nd the fcts bout the joint quntum opertors from 10]. Content from this work my be used under the terms of the Cretive Commons Attribution 3.0 licence. Any further distribution of this work must mintin ttribution to the uthor(s) nd the title of the work, journl cittion nd DOI. Published under licence by IOP Publishing Ltd 1

3 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 Let (H,, ) be Hilbert spce over the field of rel numbers R, nd let X 1, X,..., X d be d symmetric densely defined liner opertors on H. We denote by A the unitl lgebr generted by X 1, X,..., X d. It is cler tht A is in fct, the spce of ll opertors of the form f(x 1, X,..., X d ), where f is polynomil of d noncommuttive vribles x 1, x,..., x d. We mke the ssumption tht there exists nonzero element ϕ of H, such tht ϕ belongs to the domin of g, for ny g A. We cn normlize ϕ nd ssume tht: ϕ = 1, (1) where denotes the norm of the Hilbert spce H. We fix ϕ nd cll it the vcuum vector. Definition.1 We cll the pir (A, ϕ) probbility spce supported by H. We cll every element g of A, rndom vrible, nd define its expecttion s: We lso define the following equivlence reltion (see 10]): Eg] := gϕ, ϕ. () Definition. Let (A, ϕ) nd (A, ϕ ) be two probbility spces supported by two Hilbert spces H nd H, nd let E nd E denote their expecttions. Let X 1, X,..., X d be opertors from A, nd X 1, X,..., X d opertors from A. We sy tht the rndom vectors (X 1, X,..., X d ) nd (X 1, X,..., X d ) re moment equl nd denote this fct by (X 1, X,..., X d ) (X 1, X,..., X d ), if for ny polynomil p(x 1, x,..., x d ) of d noncommuttive vribles, we hve: We define the subspce F, of H, s: E p (X 1, X,..., X d )] = E p ( X 1, X,..., X d)]. (3) F := {gϕ g A}. (4) For ny non negtive integer n, we define the spce F n s the set of ll vectors, of H, of the form f(x 1, X,..., X d )ϕ, where f is polynomil of totl degree less thn or equl to n. We observe tht, since F n is finite dimensionl subspce of H, F n is closed subspce of H, for ll n 0. We hve: nd F 0 F 1 F F H (5) F = n=0f n. (6) We define G 0 := F 0, nd for ll n 1, G n := F n F n1, tht mens, G n is the orthogonl complement of F n1 into F n. For ny n 0, we cll G n the homogenous chos spce of order n generted by X 1, X,..., X d. We lso define the spce: H := n=0g n, nd cll H the chos spce generted by X 1, X,..., X d. It is not hrd to see tht H is the closure of the spce Aϕ := {gϕ g A} in H. We hve the following lemm: Lemm.3 For ny i {1,,..., d} nd ny non negtive integer n: for ll k n 1, n, nd n + 1, where mens orthogonl to. X i G n G k, (7)

4 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 We present proof of the bove lemm, tht follows exctly the proof of the sme lemm, in the commuttive cse, from 1]. Proof. Let i {1,,..., d} be fixed. If k / {n 1, n, n + 1}, then either k n + or k n. Cse 1: If k n +, then let u G n nd v G k. Since u G n, there exists f polynomil of degree t most n, such tht: u = f(x 1, X,..., X d )ϕ. (8) Since the polynomil x i f(x 1, x,..., x d ) hs degree t most n + 1, we hve: X i u = X i f(x 1, X,..., X d )ϕ F n+1. Becuse n + 1 < k, we hve G k F n+1, nd so: v X i u. Cse : If k n, then let u G n nd v G k. We cn see s before tht X i v F k+1. Using first the fct tht X i is symmetric opertor, nd second tht G n F k+1 since n > k + 1, we obtin: X i u, v = u, X i v = 0. To ese our nottion, we write: X = (X 1, X,..., X d ) nd f(x) = f(x 1, X,..., X d ) for every polynomil f of d vribles. Let n be fixed nonnegtive integer, nd f(x)ϕ G n. Then, ccording to the previous lemm, for ny i {1,,..., d}: X i f(x)ϕ G n1 G n G n+1. (9) Thus, there exist nd re unique f i,n1 (X)ϕ G n1, f i,n (X)ϕ G n, nd f i,n+1 (X)ϕ G n+1, such tht: X i f(x)ϕ = f i,n1 (X)ϕ + f i,n (X)ϕ + f i,n+1 (X)ϕ. (10) We define the following three fmilies of opertors: defined s: D n (i) : G n G n1, (11) D n (i)f(x)ϕ := f i,n1 (X)ϕ. (1) Since f hs degree n, while f i,n1 hs degree n 1, we see tht D n (i) decreses the degree of homogenous polynomil by 1 unit, nd so we cll D n (i) n nnihiltion opertor. D 0 n(i) : G n G n, (13) 3

5 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 defined s: D 0 n(i)f(x)ϕ := f i,n (X)ϕ. (14) Since f hs degree n, nd f i,n hs degree n, too, we see tht D 0 n(i) preserves the degree of homogenous polynomil, nd so we cll D 0 n(i) preservtion opertor. defined s: D + n (i) : G n G n+1, (15) D + n (i)f(x)ϕ := f i,n+1 (X)ϕ. (16) Since f hs degree n, while f i,n+1 hs degree n + 1, we see tht D + n (i) increses the degree of homogenous polynomil by 1 unit, nd so we cll D + n (i) cretion opertor. If we restrict the domins of the opertors X 1, X,..., X d to the spce G n, then we cn see from (10) tht, for ll i {1,,..., d}, we hve: X i Gn = D n (i) + D 0 n(i) + D + n (i). (17) We hve defined, so fr, the nnihiltion, preservtion, nd cretion opertors only seprtely on ech homogenous chos spce G n, for n 0. We cn extend this prtil definition, in liner wy, to the whole spce F = Aϕ. More precisely, if f is polynomil of degree k, where k is nonnegtive integer, then f(x)ϕ F k, nd we know tht: F k = G 0 G 1 G k. (18) Thus, there exist nd re unique f 0 (X)ϕ G 0, f 1 (X)ϕ G 1,..., f k (X)ϕ G k, such tht: We define: f(x)ϕ = f 0 (X)ϕ + f 1 (X)ϕ + + f k (X)ϕ. (19) (i)f(x)ϕ := D 0 (i)f 0(X)ϕ + + D k (i)f k(x)ϕ, (0) nd 0 (i)f(x)ϕ := D 0 0(i)f 0 (X)ϕ + + D 0 k (i)f k(x)ϕ, (1) + (i)f(x)ϕ := D + 0 (i)f 0(X)ϕ + + D + k (i)f k(x)ϕ. () We cll (i) n nnihiltion opertor, 0 (i) preservtion opertor, nd + (i) cretion opertor. We lso cll the opertors { (i)} 1 i d, { 0 (i)} 1 i d, nd { + (i)} 1 i d, the miniml joint quntum opertors generted by the fmily {X i } 1 i d. The domin of the miniml joint quntum opertors is understood to be the spce F. Restricting the domin of ech X i, for i {1,,..., d}, to the spce F, we hve now: X i = (i) + 0 (i) + + (i), (3) for ll i {1,,..., d}. For ech 1 i d, we cll the equlity (3), the miniml quntum decomposition of X i. 4

6 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 One should not forget tht, for ny i {1,,..., d} nd n 0, (i) : G n G n1, 0 (i) : G n G n, nd + (i) : G n G n+1, where G 1 := {0} is the null spce. If for ech n 0, we denote by P n the orthogonl projection of H onto its closed subspce F n, nd restrict the domin of P n to F, then it is not hrd to see tht: (i) = 0 (i) = + (i) = P n1 X i P n, (4) n=1 P n X i P n, (5) n=0 P n+1 X i P n, (6) n=0 for ll i {1,,..., d}. We cn see from these three formuls tht, since X i is symmetric, for ll u nd v in F, nd ll i {1,,..., d}, we hve: + (i)u, v = u, (i)v, (7) 0 (i)u, v = u, 0 (i)v. (8) If U nd V re two opertors, then we define their commuttor U, V ] s: U, V ] := UV V U. (9) It ws proven in 1], ], nd 10], tht X 1, X,..., X d commute mong themselves if nd only if the following conditions hold, for ny i, j {1,,..., d}: (i), (j) ] = 0, (30) (i), 0 (j) ] = (j), 0 (i) ], (31) (i), + (j) ] (j), + (i) ] = 0 (j), 0 (i) ]. (3) Exmple.4 Let X 1, X,..., X d be clssic rndom vribles defined on the sme probbility spce (Ω, F, P ), hving finite moments of ny order. Let H := L (Ω, F, P ) nd ϕ = 1, i.e., the constnt rndom vrible equl to 1. We cn view X 1, X,..., X d s multipliction opertors on the spce A1 H, where A is the lgebr of the rndom vribles of the form f(x 1, X,..., X d ), where f is polynomil of d vribles. It is obvious tht X i X j = X j X i, for ll 1 i < j d. The following definition is tken from 10]. Definition.5 Let H be Hilbert spce, (A, ϕ) probbility spce supported by H, nd {X i } 1 i d elements of A, tht re symmetric opertors. Let { x i } 1 i d, { 0 x i } 1 i d, nd { + x i } 1 i d be three fmilies of liner opertors, defined on subspces of H, such tht, ϕ belongs to the domin of ϵ 1 xi1 ϵ xi ϵ n, for ll n 1, (i xi 1, i n,..., i n ) {1,,..., d} n, nd (ϵ 1, ϵ,..., ϵ n ) {, 0, +} n. We sy tht these fmilies of opertors form joint quntum decomposition of {X i } 1 i d reltive to A, if the following conditions hold: X i = x i + 0 x i + + x i, (33) ( + xi ) Aϕ = xi Aϕ, (34) x i H n H n1, (35) 0 x i H n H n, (36) 5

7 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 for ll 1 i d nd n 0, where H 1 := {0}, H 0 := Rϕ, nd H k is the vector spce spnned by ll vectors of the form + x i1 + x i + x ik ϕ, where i 1, i,..., i k {1,,..., d}, for ll k 1. We cll x i n nnihiltion opertor, 0 x i preservtion opertor, nd + x i cretion opertor, for ll 1 i d. For ll 1 i d, the fct tht ( + x i ) Aϕ = xi Aϕ is understood s: + x i u, v = u, x i v, for ll u nd v in Aϕ, where, denotes the inner product of H. Becuse, for ll i {1,,..., d}, X i is symmetric opertor, it follows from (34), tht: ( 0 xi ) Aϕ = 0 xi Aϕ. (37) The following lemm from 10] refers to the uniqueness of the joint quntum decomposition of finite fmily of symmetric rndom vribles. Lemm.6 Let {X i } 1 i d be fmily of symmetric rndom vribles in noncommuttive probbility spce (A, ϕ), nd { x i } 1 i d, { 0 x i } 1 i d, nd { + x i } 1 i d joint quntum decomposition of {X i } 1 i d reltive to A. Let A be the lgebr generted by {X i } 1 i d, nd { (i)} 1 i d, { 0 (i)} 1 i d, nd { + (i)} 1 i d be the miniml joint quntum decomposition of {X i } 1 i d. Then for ny i {1,,..., d} nd ny ϵ {, 0, +}, we hve: ϵ x i A ϕ = ϵ (i) A ϕ. (38) Moreover, if A denotes the lgebr generted by d i=1 { x i, 0 x i, + x i }, then A ϕ = A ϕ, (39) nd { x i } 1 i d, { 0 x i } 1 i d, nd { + x i } 1 i d is joint quntum decomposition of {X i } 1 i n reltive to A. 3. A commuttor problem Let us consider two symmetric rndom vribles X nd Y, whose joint quntum opertors stisfy the following commuttion reltionships: We ssume tht EX] = EY ] = 0. x, ] y x, + ] x x, + ] y y, + ] y x, 0 x] x, 0 y] y, 0 x] y, 0 y] = 0, (1) = I, () = bi, (3) = I, (4) = p x + p y, (5) = 0, (6) = 0, (7) = q x + q y. (8) 6

8 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 Tking the djoint in both sides of ech of the reltions (1), (3), (5), nd (8), we obtin the following commuttion reltionships: + y, + ] x = 0, (9) y, + ] x 0 x, + ] x 0 y, + ] y From the Jcobi commuttion identity: y, 0 x, + ]] x + 0 x, + x, ]] y + + x, y, 0 x]] = bi, (10) = p + x + p + y, (11) = q + x + q + y. (1) = 0, we conclude tht p = pb. Similrly, we cn see tht q = qb. Thus the commuttion reltionships (5) nd (8) cn be written s: x, 0 x] = p ( x b ) y, (13) y, 0 y] = q ( y b ) x. (14) We ssume tht b = 1, b 0, p 0, nd q 0. XY = Y X, then we would hve: 0 = X, Y ] = x + 0 x + + x, y + 0 y + + ] y = ϵ 1 x, ϵ ] y. (ϵ 1,ϵ ) {,0,+} It follows tht XY Y X, since if In the bove sum of commuttors, we hve ϵ 1 x, ϵ y ] + ϵ x, ϵ 1 y ] = 0, for ll ϵ 1 ϵ. Moreover, x, y ] = + x, + y ] = 0. Thus we obtin: 0 x, 0 y] = X, Y ] = 0. Using now the Jcobi identity: x, 0 x, 0 ]] y + 0 x, 0 y, ]] x + 0 y, x, 0 x]] = 0, nd the fct tht bpq 0, we conclude tht y = b x. Tking the djoint in both sides of this equlity we get + y = b + x. It follows now from () nd (4) tht b = 1, which is not possible since b = 1. Therefore we re deling with non commuttive rndom vribles. We will show t the end of the pper tht, for ech b (1, 1), p 0, nd q 0, there is model in which these commuttion reltionships re relized. The necessry condition b 1 follows from the Schwrz inequlity, s it will be explined below. We exclude the cse b = ±1 from our discussion. Since both X nd Y re symmetric, using Schwrz inequlity, we obtin: EXY ] = XY ϕ, ϕ = Y ϕ, Xϕ ( Y ϕ, Y ϕ ) 1/ ( Xϕ, Xϕ ) 1/ = ( X ϕ, ϕ ) 1/ ( Y, ϕ ) 1/ = ( E X ]) 1/ ( E Y ]) 1/. (15) 7

9 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 Becuse EX] = EY ] = 0, we hve 0 xϕ = 0 yϕ = 0. Since x ϕ = y ϕ = 0, we hve: Similrly, we hve: nd EXY ] = ( x + 0 x + + x )( y + 0 y + + y )ϕ, ϕ = ( x + 0 x + + x ) + y ϕ, ϕ = + y ϕ, ( + x + 0 x + x )ϕ = + y ϕ, + x ϕ = x + y ϕ, ϕ = ( x + y + y x )ϕ, ϕ = x, + y ]ϕ, ϕ = biϕ, ϕ It follows now from (15), (16), (17), nd (18) tht: = b. (16) EX ] = 1 (17) EY ] = 1. (18) b 1. (19) Inside the lgebr A, we cn consider the bi module M(X, Y ) (left module over the ring P(X) of ll polynomils of X, nd right module over the ring P(Y ) of ll polynomils of Y ) generted by the identity opertor I. Tht mens, M(X, Y ) is the vector spce spnned by the monomils of the form {X m Y n } m 0,n 0, in which ll fctors of X re to the left of ll fctors of Y. We show below method for computing the expecttion of ll monomils from M(X, Y ). Lemm 3.1 If X nd Y stisfy the commuttion reltionships (1) (8), then for ny m nd n non negtive integers, with m 1, we hve: where EX m Y n ] = (m 1)EX m Y n ] + nbex m1 Y n1 ] ( ) m 1 +(1 b ) pex m3 Y n ] +p X m3j ( x b y )X j Y n ϕ, ϕ R := 0 j<i m (m 1)bpqR, 0 k n1 X m Y n1k ( y b x )Y k ϕ, ϕ. Proof. Let m nd n be two fixed non negtive integers, such tht m 1. We pply the commuttor method from 9]. According to tht method: EX m Y n ] = X m Y n ϕ, ϕ = ( + x + 0 x + x )X m1 Y n ϕ, ϕ = X m1 Y n ϕ, x ϕ + X m1 Y n ϕ, 0 xϕ + x X m1 Y n ϕ, ϕ = 0 + X m1 Y n ϕ, EX]ϕ + x X m1 Y n ϕ, ϕ = x X m1 Y n ϕ, ϕ, 8

10 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 since EX] = 0. We swp now x nd X m1 Y n, using the product rule for commuttors: A, B 1 B k ] = k B 1 B i1 A, B i ] B i+1 B k, for ll opertors A, B 1,..., B k, nd the fct tht x ϕ = 0, nd obtin: Let us observe now tht: nd Therefore, we hve: i=1 EX m Y n ] = x X m1 Y n ϕ, ϕ EX m Y n ] = = X m1 Y n x ϕ, ϕ + x, X m1 Y n ]ϕ, ϕ = X mi x, X]X i Y n ϕ, ϕ 0 i m + 0 j n1 X m1 Y n1j x, Y ]Y j ϕ, ϕ. x, X] = x, + x + 0 x + x ] = x, + x ] + x, 0 x] = I + p( x b y ) x, Y ] = x, + y ] + x, 0 y] + x, y ] = bi = bi. 0 i m + 0 j n1 X mi {I + p( x b y )}X i Y n ϕ, ϕ X m1 Y n1j biy j ϕ, ϕ = (m 1)EX m Y n ] + nbex m1 Y n1 ] +p X mi ( x b y )X i Y n ϕ, ϕ. 0 i m Let us swp now the opertors x b y nd X i Y n, using gin the product rule for commuttors, nd the fct tht ( x b y )ϕ = 0. We obtin: EX m Y n ] = (m 1)EX m Y n ] + nbex m1 Y n1 ] +p X mi ( x b y )X i Y n ϕ, ϕ 0 i m = (m 1)EX m Y n ] + nbex m1 Y n1 ] +p X mi x b y, X i Y n ]ϕ, ϕ 0 i m = (m 1)EX m Y n ] + nbex m1 Y n1 ] +p X mi X i1j x b y, X]X j Y n ϕ, ϕ +p 0 j<i m 0 i m 0 k n1 X mi X i Y n1k x b y, Y ]Y k ϕ, ϕ. 9

11 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 Using the simple fcts tht X mi X i1j = X m3j nd X mi X i = X m, we obtin: Let us observe tht: nd Thus, we obtin: EX m Y n ] = (m 1)EX m Y n ] + nbex m1 Y n1 ] +p X m3j x b y, X]X j Y n ϕ, ϕ 0 j<i m +(m 1)p 0 k n1 x b y, X] = x, X] b y, X] X m Y n1k x b y, Y ]Y k ϕ, ϕ. = I + p( x b y ) b I = (1 b )I + p( x b y ) x b y, Y ] = x, Y ] b y, Y ] = bi b{i + q( y b x )} = bq( y b x ). EX m Y n ] = (m 1)EX m Y n ] + nbex m1 Y n1 ] +p X m3j {(1 b )I + p( x b y )}X j Y n ϕ, ϕ 0 j<i m (m 1)bpq 0 k n1 X m Y n1k ( y b x )Y k ϕ, ϕ = (m 1)EX m Y n ] + nbex m1 Y n1 ] ( ) m 1 +(1 b ) pex m3 Y n ] +p X m3j ( x b y )X j Y n ϕ, ϕ 0 j<i m (m 1)bpqR, where R := 0 k n1 X m Y n1k ( y b x )Y k ϕ, ϕ. Lemm 3. Using the nottions from the Lemm 3.1, we hve: R = 1 b q E X m { (Y + qi) n Y n nqy n1}]. 10

12 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 Proof. Swpping y b x nd Y k, we get: R = = = 0 k n1 0 l<k n1 0 l<k n1 X m Y n1k y b x, Y k ]ϕ, ϕ X m Y n1k Y k1l y b x, Y ]Y l ϕ, ϕ X m Y nl {(1 b )I + q( y b x )}Y l ϕ, ϕ ( ) n = (1 b ) EX m Y n ] +q X m Y nl ( y b x )Y l ϕ, ϕ. 0 l<k n1 We repet this procedure, swpping now y b x tht: {( ) n R = (1 b ) EX m Y n ] + + ( n n nd Y l, nd so on, nd obtin in the end ( ) n 3 qex m Y n3 ] + ) } q n EX m Y 0 ] = 1 b q E X m { (Y + qi) n Y n nqy n1}]. Lemm 3.3 If X nd Y stisfy the commuttion reltionships (1) (8), then for ny m nd n non negtive integers, with m 1, we hve: EX m Y n ] (0) = (m 1)EX m Y n ] + nbex m1 Y n1 ] + 1 b E { (X + pi) m1 X m1 (m 1)pX m} Y n] p b(1 b ) E { (X + pi) m1 X m1} { (Y + qi) n Y n nqy n1}]. q Proof. Using Lemms 3.1 nd 3., we get: EX m Y n ] = (m 1)EX m Y n ] + nbex m1 Y n1 ] ( ) m 1 +(1 b ) pex m3 Y n ] +p X m3j ( x b y )X j Y n ϕ, ϕ 0 j<i m b(1 b ) E (m 1)pX m { (Y + qi) n Y n nqy n1}]. q We do now similr procedure s in the proof of Lemm 3.1, by swpping first ( x b y ) nd X j Y n. In this wy we rrive t new sum R 3 which is similr to R. We compute R 3 in the 11

13 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 sme wy tht we clculted R in the proof of Lemm 3., nd so on, obtining in the end tht: EX m Y n ] = (m 1)EX m Y n ] + nbex m1 Y n1 ] {( ) ( ) m 1 m 1 +(1 b ) pex m3 Y n ] + p EX m4 Y n ] + 3 ( ) } m 1 + p m EX 0 Y n ] m 1 b(1 ( b ) m 1 E )px m { (Y + qi) n Y n nqy n1}] q 1 b(1 ( b ) m 1 E )p X m3 { (Y + qi) n Y n nqy n1}] q b(1 b ) E q ( m 1 )p m1 X 0 { (Y + qi) n Y n nqy n1}] m 1 = (m 1)EX m Y n ] + nbex m1 Y n1 ] + 1 b E { (X + pi) m1 X m1 (m 1)pX m} Y n] p b(1 b ) E { (X + pi) m1 X m1} { (Y + qi) n Y n nqy n1}]. q 4. Joint left X right Y Lplce trnsform of X nd Y In this section, we compute the joint Lplce trnsform of the rndom vribles X nd Y from the previous section. We strt first by presenting some importnt bounds for the joint moments of X nd Y. Lemm 4.1 Let X = x + 0 x + + x nd Y = 0 y + 0 y + + y be joint quntum decompositions of two rndom vribles, whose terms stisfy the commuttion reltionships (1) (8), where b, p, p, q, nd q re rel numbers, such tht b < 1, p = pb, nd q = qb. We ssume tht EX] = EY ] = 0. Then there exists positive constnt C, such tht, for ny m nd n non negtive integers, we hve: EX m Y n ] Cm!n!. (1) Proof. Indeed, if we define, for ll non negtive integers m nd n, C m,n := mx{ EX i Y j ] /(i!j!) 0 i m, 0 j n}, () then the sequence {C m,n } m 0,n 0 is left nd right incresing, in the sense tht if we fix n, then it is incresing with respect to m, nd vice vers. We hve: E { (X + pi) m1 X m1 (m 1)pX m} Y n] m1 (m 1)!n! i= m1 (m 1)!n!C m3,n EX m1i Y n ] (m 1 i)!n! i= p i i! (e p 1 p) C m3,n (m 1)!n!. pi i! 1

14 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 Similrly, we hve: E { (X + pi) m1 X m1} { (Y + qi) n Y n nqy n1}] (e p 1) (e q 1 q) C m,n (m 1)!n!. It follows now from our recursive formul (0) tht: EX m Y n ] (m 1)C m,n (m )!n! + nbc m1,n1 (m 1)!(n 1)! + 1 b p + b (1 b ) q (e p 1 p) C m3,n (m 1)!n! (e p 1) (e q 1 q) C m,n (m 1)!n!. Since the sequence {C m,n } m 0,n 0 is left nd right incresing, we conclude now tht there is constnt K depending on b, p, nd q, such tht: EX m Y n ] KC m1,n (m 1)!n!, for ll m 1, n 0. Dividing both sides of this inequlity by m!n!, we obtin: EX m Y n ] m!n! for ll m 1 nd n 0. Thus, we hve: K m C m1,n, C m,n = mx { C m1,n, EX m Y 0 ] /(m!0!),..., EX m Y n ] /(m!n!) } { mx C m1,n, K m C m1,0,..., K } m C m1,n { = mx C m1,n, K } m C m1,n = C m1,n, for ll m K + 1 nd n 0. On the other hnd we hve C m,n C m1,n, for ll m 1 nd ll n 0. Thus, the sequences {C m,0 } m 1, {C m,1 } m 1,... re uniformly sttionry, in the sense tht there is nturl number m 0, tht is the sme for ll n 0, such tht C m0,n = C m0 +1,n = C m0 +,n =. On the other hnd, since X nd Y re symmetric opertors, we hve: EY n X m ] = Y n X m ϕ, ϕ = ϕ, X m Y n ϕ = EX m Y n ]. Thus similr rgument will imply now tht there exists n n 0 1, such tht for ll m 0, C m,n0 = C m,n0 +1 = C m,n0 + =. Since the sequence {C m,n } m 0,n 0 is left nd right incresing, we conclude now tht, for ll m 0 nd ll n 0, we hve C m,n C m0,n 0. Thus, for ll m 0 nd n 0, EX m Y n ] C m0,n 0 m!n!. We re now redy to compute the joint Lplce trnsform of X nd Y. 13

15 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 Theorem 4. Let X = x + 0 x + + x nd Y = 0 y + 0 y + + y be joint quntum decompositions of two rndom vribles, whose terms stisfy the commuttion reltionships (1) (8), where b, p, p, q, nd q re rel numbers, such tht b < 1, p = pb, nd q = qb. We ssume tht EX] = EY ] = 0. Then, for ll (s, t) (1, 1), the series: E e sx e ty ] := m 0 n 0 s m t n EX m Y n ] m!n! (3) is bsolutely convergent, nd the joint left X right Y Lplce trnsform is: E e sx e ty ] = e s +bst+t e 1b p (e ps 1ps 1 p s 1b ) e q (e qt 1qt 1 q t ) ) e b(1b (e pq ps 1ps)(e qt1qt). (4) If p = 0 or q = 0, then formul (4) mkes sense by tking the limit s p 0 or q 0 in its right hnd side. Moreover, there exist two commuting rndom vribles U nd V, such tht (X, Y ) (U, V ), if nd only if bpq = 0. If bpq = 0, then there exist n invertible mtrix T, vector (u, v) in R, nd two independent clssic rndom vribles X nd Y, tht re either Gussin or Poisson, such tht (X, Y ) (X, Y )T + (u, v). Proof. It follows from Lemm 4.1 tht the series m 0,n 0 sm t n EX m Y n ]/m!n! is bsolutely convergent for ll s nd t complex numbers, such tht s < 1 nd t < 1. We denote this series by Ee sx e ty ], to be in greement with the clssic commuttive cse. Let φ(s, t) := Ee sx e ty ]. It follows from Weierstrss M test tht the double series, from the definition of φ(s, t), cn be differentited term by term, with respect to both s nd t, for s < 1 nd t < 1. If we multiply both sides of the recursive reltion (0) by s m1 t!/(m1)!n!] nd sum up the resulting reltions from m = 1 to, nd from n = 0 to, then we obtin: φ (s, t) = sφ(s, t) + btφ(s, t) (5) s + 1 b (e ps 1 ps)φ(s, t) p b(1 b ) (e ps 1)(e qt 1 qt)φ(s, t). q In prticulr, for t = 0, we conclude tht the function g(s) := Ee sx ] stisfies the differentil eqution: Since g(0) = 1, we must hve: dg 1 b (s) = sg(s) + (e ps 1 ps) g(s). ds p ( ) s g(s) = e + 1b p e ps 1ps p s. Similrly, we cn see tht: ( ) Ee ty t ] = e + 1b q e qt 1qt q t. 14

16 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 Solving now the differentil eqution, in the vrible s, (5), we obtin: ( ) φ(s, t) = K(t)e s e bst 1b p e e ps 1ps p s e b(1b ) (e pq ps 1ps)(e qt 1qt). Setting s = 0 in this eqution nd using the fct tht φ(0, t) = Ee ty ], we conclude tht: Thus, we hve: ( ) t K(t) = e + 1b q e qt 1qt q t. Ee sx e ty ] = e t + 1b q (eqt 1qt q t ) e s e bst e 1b p (eps 1ps p s ) e b(1b pq ) (e ps 1ps)(e qt 1qt) = e s +bst+t e 1b p (eps 1ps p s ) e 1b q (eqt 1qt q t ) e b(1b ) (e pq ps 1ps)(e qt1qt). 5. The existence of the opertors X nd Y We close the pper by showing tht, for ny b < 1, nd ny rel numbers p nd q, we cn construct two rndom vribles stisfying the commuttion reltionships from the previous theorem. Let b, p, nd q be fixed, such tht b < 1. Let U nd V be two jointly Gussin clssic (commuttive) rndom vribles, with men 0, vrince 1, nd covrince b, defined on the sme probbility spce (Ω, F, P ). We hve U = u + + u nd V = v + + v, u, + u ] = v, + v ] = I nd u, + v ] = v, + u ] = EUV ]I = bi. See 5], 6], nd 8]. Let A be the unitl lgebr generted by u, + u, v, nd + v. This is n lgebr of opertors tht re densely defined on the Hilbert spce H := L (Ω, σ(u, V ), P ), where σ(u, V ) represents the smllest sub sigm lgebr of F, with respect to which both U nd V re mesurble functions. Inside the lgebr A, let us consider the following opertors: nd X := u + α 1 + u u + β 1 ( + u v + + v u ) + γ 1 + v ] v + + u (6) Y := v + α + v v + β ( + v u + + u v ) + γ + u u ] + + v, (7) where α i, β i, nd γ i, 1 i, re rel constnts tht will be determined such tht X nd Y stisfy the commuttion reltionships (1) (8). The coefficients of + x y nd + y x re the sme in the definitions of X nd Y, β 1 nd β, respectively, to ensure tht X nd Y re both symmetric opertors. Let ϕ := 1 H, be the unit vector with respect to which the 15

17 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 noncommuttive expecttion will be computed. The terms of joint quntum decomposition of X nd Y (reltive to A), re: x = u, 0 x = α 1 + u u + β 1 ( + u v + + v u ) + γ 1 + v v, + x = + u, y = v, 0 y = α + v v + β ( + v u + + u v ) + γ + u u, + y = + v. It is now cler tht (1), (), (3), (4) hold. Also, becuse in the formuls of 0 x nd 0 y, the nnihiltion opertors u nd v re lwys to the right of the cretion opertors + u nd + v, we conclude tht the conditions EX] = 0 nd EY ] = 0 hold. To compute the commuttors x, 0 x], x, 0 y], y, 0 x], nd y, 0 y], we pply gin the product rule for commuttors. For exmple, one commuttor involved in the computtion of x, 0 x] is u, + v u ], which cn be computed s follows: u, + v u ] = u, + v ] u + + v u, u ] = bi u + 0 = b x. Doing ll these types of commuttors, we obtin: x, 0 x] x, 0 y] y, 0 x] y, 0 y] = (α 1 + bβ 1 ) x + (β 1 + bγ 1 ) y, = (γ + bβ ) x + (β + bα ) y, = (β 1 + bα 1 ) x + (γ 1 + bβ 1 ) y, = (β + bγ ) x + (α + bβ ) y. If we wnt the commuttion reltionships (5), (6), (7), nd (8) to hold, then identifying the coefficients of x nd y, we should hve: α 1 + bβ 1 = p, β 1 + bγ 1 = pb, γ + bβ = 0, β + bα = 0, β 1 + bα 1 = 0, γ 1 + bβ 1 = 0, β + bγ = qb, α + bβ = q. This is system of eight equtions nd six unknowns, tht cn be split up into two sub systems of four equtions nd three unknowns ech. Ech of these two sub systems is consistent nd hs unique solution: (α 1, β 1, γ 1 ) = p 1 b (1, b, b ) 16

18 XXIst Interntionl Conference on Integrble Systems nd Quntum Symmetries (ISQS1) IOP Publishing Journl of Physics: Conference Series 474 (013) 0107 nd (α, β, γ ) = q 1 b (1, b, b ). Thus we hve proven the existence of two noncommuttive rndom vribles X nd Y, for which the commuttion reltionships (1) (8) nd the condition EX] = EY ] = 0 hold. The left X right Y Lplce trnsform Ee sx e ty ] is generting function only for the joint moments of the form EX m Y n ], m 0, n 0. Becuse X nd Y do not commute, we lso need to compute other Lplce trnsforms like Ee s1x e t1y e sx e ty ], Ee s1x e t1y e sx e ty e s3x e t3y ] nd so on, to generte the other joint moments like EX m 1 Y n 1 X m Y n ], EX m 1 Y n 1 X m Y n X m 3 Y n 3 ],.... Our commuttor method cn be pplied to compute these other Lplce trnsforms, but the clcultions re more complicted. References 1] Accrdi L, Kuo H-H nd Stn A I 004 Chrcteriztion of probbility mesures through the cnoniclly ssocited intercting Fock spces Infin. Dimens. Anl. Quntum Probb. Relt. Top. 7 No ] Accrdi L, Kuo H-H nd Stn A I 007 Moments nd commuttors of probbility mesures Infin. Dimens. Anl. Quntum Probb. Relt. Top. 10 No ] Chihr T S 1978 An Introduction to Orthogonl Polynomils (New York: Gordon & Brech) 4] Drożdżewicz K nd Mtysik W 011 Moments nd q-commuttors of noncommuttive rndom vectors Infin. Dimens. Anl. Quntum Probb. Relt. Top. 14 No ] Jnson S 1997 Gussin Hilbert Spces Cmbridge University Press 6] Kuo H-H 1996 White Noise Distribution Theory (Boc Rton, FL: CRC Press) 7] Meixner J Orthogonle 1934 Polynomsysteme mit einer besonderen Gestlt der erzeugenden Funktion J. London Mth. Soc ] Obt N 1994 White Noise Clculus nd Fock Spce (Berlin Heidelberg: Springer-Verlg) 9] Stn A I nd Whitker J J 009 A study of probbility mesures through commuttors J. Theoret. Probb. No ] Stn A I 011 Two-dimensionl Meixner rndom vectors of clss M L J. Theoret. Probb. 4 No ] Sunder V S 1986 An invittion to von Neumnn Algebrs Springer Verlg (New York: Springer-Verlg) 1] Szegö M 1975 Orthogonl Polynomils Coll. Publ. 3 Amer. Mth. Soc. 13] Voiculescu D V, Dykem KJ nd Nic A 198 Free Rndom Vribles 1 (Providence, RI: CRM Series, Amer. Mth. Soc.) 17