Approximate substitutions and the normal ordering problem

Appro ximate substi tutions and the n ormal orde ring problem H Cheballah † , G H E Duc hamp † and K A P enson ♦ † Universit ´ e P aris 13 Lab oratoire d’Informatique Paris Nord, CNRS U MR 7030 99 Av . J-B. Cl´ ement, F 93430 Villetaneuse, F rance ♦ Lab oratoire de Physique Th ´ eorique de la Mati ` ere Co ndens´ ee Universit ´ e Pierre et Mar ie C urie, CNRS UMR 7600 T our 24 - 2e ´ et., 4 p l. Jussieu, F 7525 2 P aris Ce dex 05 , F rance E-mail: hayat.cheb allah@lipn- univ.paris13.fr , ghed@lipn-univ.par is13.fr , penson@lpt l.jussieu.f r Abstract. In this pap er, w e show that the infinite generalis ed S tirling matrices associated with b oson strings wi th one annihilation o p erator are pro jective li mits of appro ximate substitutions, the latter being characterised by a finite set of algebrai c equations. 1. In tro duction The series of pap ers [1, 2, 3] had tw o sequels. First one, algebraic, w as the construction of a Hopf algebra of F eynman-Bender diag rams [10, 11] arising from the pr o duct form ula applied to t wo exp onent ials. Second one w as the construction and description of one parameter group s of infinite matrice s [4, 9 ] and their link with t he co m binatorics of so called Sheffer p olynomials . T h e ob ject of this pap er is to con tinue the inv estigation of those one-p ar ameter gr oups by highlighting the stru ctur e of the group of sub s titutions. It is sho w n here ho w we can see th is group as a pro jectiv e limit of what will b e called appr oximate substitution gr oups . First, we consider Boson creation and annih ilation op erators with the comm u tation relation [ a, a † ] = 1 (1) Recall th at [1] • the annihilation op erator a , in the second qu an tizatio n, repr esents an op erator w hic h c h an ges ea c h state | N i of the F o c k space (conta ining N ≥ 1) particle s to another cont aining N − 1 particles. One h as here a | N i = √ N | N − 1 i (2) • the hermitian conjugate of the annihilation op erator is the creatio n op erator a † whic h c h an ges eac h state | N i of th e F o ck space contai ning N particles to another conta ining N + 1 particles. One h as then a † | N i = √ N + 1 | N + 1 i (3) Starting fr om the v acuum | 0 i , w e can reac h al l the normalized states in the F o ck space. There are indeed giv en by: | N i = ( a † ) N √ N ! | 0 i (4) 2. Normally ordered form The non commutativit y of annih ilation and creation op er ators may cause problems in definin g an op erator fun ction in q u an tum mechanics. T o solv e these problems, we ha v e to fin d some suitable form which allo ws computing, reduction and equ alit y test. One of the simplest and widely used p r o cedure is the fin ding of th e normal ly or der e d form of the b oson op erators in whic h all a † stand to the left of all the factors a [1]. There are t w o w ell known pro cedur es defined on the b oson expr essions: namely N , the normal or dering and :: , the double dot op er ation , [4]. 2.1. Norma l or dering By normal order in g of a general expression F ( a † , a ), we mean N [ F ( a † , a )] which is obtained by mo vin g all the annih ilation op erators a to the right, using the comm u tation relation (1). Example 2.1 L et w ∈ { a, a † } ∗ a wor d given by w = aa † aaa † a . The normal or dering of w i s aa † aaa † a = (1 + a † a ) a (1 + a † a ) a = a 2 + a † a 3 + aa † a 2 + a † a 2 a † a 2 = a 2 + a † a 3 + (1 + a † a ) a 2 + a † a (1 + a † a ) a 2 = a 2 + a † a 3 + a 2 + a † a 3 + a † a 3 + a † aa † a 3 = 2 a 2 + 3 a † a 3 + a † (1 + a † a ) a 3 = 2 a 2 + 3 a † a 3 + ( a † ) 2 a 4 Remark 2.2 Note that al l c o efficie nts of the normal or dering of a wor d (mor e pr e cisely the c o efficients of the de c omp osition of a wor d on the b asis n ( a † ) j a l o ) ar e p ositive inte ge rs. This suggests that these inte gers c ount c ombinatorial obje cts [13]. 2.2. Double dot op e r ation The double d ot op eration : F ( a † , a ) : is a similar procedu re using another comm utation relation i.e. [ a, a † ] = 0 instead of [ a, a † ] = 1, i.e mo ving all ann ihilation op erators a to the right as if they were comm uting with th e cr eation op erators a † . Remark 2.3 The double dot op er ation :: is a line ar op er ator which c an b e dir e ctly define d, for a wor d w ∈ { a, a † } ∗ , by: : w := a † ( | w | a † ) a ( | w | a ) wher e | w | x stands for the numb er of o c cur enc es of a symb ol x in the wor d w . Example 2.4 We take the same wor d as ab ove: w = aa † aaa † a . The double dot op e r ation give s : aa † aaa † a : = a † a † aaaa 3. Com binatorics of t he normal ordering The Bell and Stirling n u m b ers h a ve a pur ely com b inatorial origin [8], but in this comm unication, w e will consider them as co efficient s of the norm al ordering problem [1]. The general w ord w ∈ { a, a † } ∗ with letters in { a, a † } , i.e. Boson s tring, can b e d escrib ed by t wo sequences of non negativ e in tegers r = ( r 1 , r 2 , · · · , r M ) and s = ( s 1 , s 2 , · · · , s M ), so that we define w r , s = ( a † ) r 1 a s 1 ( a † ) r 2 a s 2 · · · ( a † ) r M a s M (5) and d = n X m =1 ( r m − s m ) , n = 1 , · · · , M represen ts the excess ( i. e. the differen ce b et ween the n umber of creations and the num b er of annihilatio ns). Then, the normally ordered form of w n r , s is giv en by N ( w n r , s ) =            ( a † ) nd ∞ X k =0 S r , s ( n, k )( a † ) k a k , if d > 0;  ∞ X k =0 S r , s ( n, k )( a † ) k a k  ( a † ) n | d | , otherwise, where qu an tities S r , s ( n, k ) are generalizatio ns of standard Stirling num b er s [2, 3 ]. The generalized Bell p olynomials B r , s ( n, x ) and the generalized Bell n umbers B r , s ( n ) are defined resp ectiv ely by B r , s ( n, x ) = ∞ X k =0 S r , s ( n, k ) x k = nr X k =0 S r , s ( n, k ) x k (6) B r , s ( n, 1) = B r , s ( n ) = ∞ X k =0 S r , s ( n, k ) = nr X k =0 S r , s ( n, k ) (7) Remark 3.1 Notic e that S r , s ( n, k ) =  1 , for k = nr ; 0 , for k > nr . Example 3.2 Using a c omputer algebr a pr o gr am, we get the fol lowing matric es. • w = a † a , we get the u sual matrix of Stirling numb ers of se c ond kind S ( n, k )[ 8 ]             1 0 0 0 0 0 0 · · · 0 1 0 0 0 0 0 · · · 0 1 1 0 0 0 0 · · · 0 1 3 1 0 0 0 · · · 0 1 7 6 1 0 0 · · · 0 1 15 25 10 1 0 · · · 0 1 31 90 65 15 1 · · · . . . . . . . . . . . . . . . . . . . . .             • w = a † aa † , we have             1 0 0 0 0 0 0 · · · 1 1 0 0 0 0 0 · · · 2 4 1 0 0 0 0 · · · 6 18 9 1 0 0 0 · · · 24 96 72 16 1 0 0 · · · 120 600 600 200 25 1 0 · · · 720 4320 5400 2400 450 36 1 · · · . . . . . . . . . . . . . . . . . . . . .             • w = a † aaa † a † , one gets          1 0 0 0 0 0 0 0 0 · · · 2 4 1 0 0 0 0 0 0 · · · 12 60 54 14 1 0 0 0 0 · · · 144 1296 2232 1296 306 30 1 0 0 · · · 2880 40320 109440 105120 45000 9504 1016 52 1 · · · . . . . . . . . . . . . . . . . . . . . . . . . . . .          Remark 3.3 In e ach c ase, the matrix ( S r , s [ n, k ]) n ≥ 0 ,k ≥ 0 is of a stair c ase f orm and the dimen- sion of the step is the numb er of a ’s in the wor d w . Thus, al l the matric es ar e r ow-finite and ar e unitriangular iff the numb er of its annihilation op er ators is exactly one. Mor e over, the first c olumn is (1 , 0 , · · · , 0 , · · · ) iff w ends with a (this me ans that N ( w n ) has no c onstant term for al l n > 0 ) [4, 9]. A wor d w with only one annihilation op er ator c an b e written in the u ni q ue form w = ( a † ) r − p a ( a † ) p . Then: • i f p = 0 , S w is the matrix of a unip otent substitution. • i f p > 0 , S w is the matrix of a unip otent substitution with pr efunction [ 9]. 4. Appro ximate substitutions W e define here the sp ace of transformation matrices and its top ology , and then w e concen trate on the Rior dan sub gr oup [7] ( i.e transformations which are sub s titutions with prefactor func- tions) [4, 9]. Let C N b e the vecto r s pace of all complex sequences, endo w ed with the F rechet pro duct top ology [6, 12]. The algebra L ( C N ) of all con tinuous op erators C N − → C N is the space of ro w -finite matrices with complex co efficient s (a subs pace of C N × N ). Let M b e a matrix of this space. F or a sequence A = ( a n ) n ≥ 0 , the transform ed sequence B = M A is give n b y B = ( b n ) n ≥ 0 with: b n = X k ≥ 0 M ( n, k ) a k . (8) W e will also asso ciate to A (see paragraph (4.2)) its exp onential generating function X n ≥ 0 a n z n n ! . (9) 4.1. Substitutions with pr efunctions W e now examine an im p ortant class of transformations: the su bstitutions with pr efunctions. W e consider, for a generating fu nction f , the transf orm ation T g ,φ [ f ]( x ) = g ( x ) f  φ ( x )  (10) where g ( x ) = 1 + P ∞ n =1 g n x n and φ ( x ) = x + P ∞ n =2 φ n x n are arbitrary formal p o w er series. The mapping T g ,φ is a linear application [9], the matrix of this transformation M g ,φ is given by the transforms of the monomials x k k ! hence X n ≥ 0 M g ,φ ( n, k ) x n n ! = T g ,φ h x k k ! i = g ( x ) φ ( x ) k k ! (11) Remark 4.1 If f ( x ) = e y x , Eq. (11) c omes down to the Sheffer c ondition on the matrix of T . It amounts to the statement that [1]: X n,k ≥ 0 T ( n, k ) x n n ! y k = g ( x ) e y φ ( x ) (12) wher e T ∈ L ( C N ) is a matrix with non-zer o two first c olumns. 4.2. Appr oximate sub stitutions In this section, we defin e appro ximate substitutions matrices and giv e a w a y to determine whether an unip oten t (lo we r triangular with all diagonal elemen ts equal to 1) matrix is a matrix of an appro ximate subs titution. Definition 4.2 L et M ∈ C [0 ··· n ] × [0 ··· n ] b e a unip otent matrix, M = M [ i, k ] 0 ≤ i,k ≤ n = ( a ik ) 0 ≤ i,k ≤ n ; is c al le d matrix of appr oximate substitution if it satisfies the fol lowing c ondition: c k = h c 0  c 1 c 0  k k ! i n , for al l 0 ≤ k ≤ n (13) wher e c k = n X i =0 M [ i, k ] x i i ! M =            1 0 0 · · · · · · · · · · · · a 1 , 0 1 0 · · · · · · · · · · · · a 2 , 0 a 2 , 1 1 0 · · · · · · · · · a 3 , 0 a 3 , 1 · · · 1 0 · · · · · · a 4 , 0 a 4 , 1 · · · a 4 ,k · · · · · · · · · a 5 , 0 a 5 , 1 · · · a 5 ,k · · · 1 · · · . . . . . . . . . . . . . . . . . . . . .            c 0 c 1 c k Thus c k r epr ese nts the exp onential gener ating series (her e a p olynomia l) of the k th c olumn (henc e c 0 , c 1 ar e r esp e ctively the exp onential gener ating series of the 1 st and the 2 nd c olumn) and h i n is the trunc ation, at or der n , of a series. W e consider n o w the set of matrice s with complex co efficien ts noted b y C N × N [5] and let C [0 ··· n ] × [0 ··· n ] b e the set of all matrices of size ( n + 1) × ( n + 1). Let also r n b e the truncation of the matrices taking the upp er left principal submatrix of dimension ( n + 1), hence r n ( M ) =  M [ i, k ]  0 ≤ i,k ≤ n . Thus, w e get a linear mapping r n : C N × N − → C [0 ··· n ] × [0 ··· n ] It is clear th at r n is not a morphism for the (partially d efined) multiplica tion ( i.e . r n ( AB ) 6 = r n ( A ) r n ( B ) in general). W e consider now LT ( N , C ) the algebra of low er triangular matrices and LT ([0 · · · n ] , C ) the matrices of size ([0 · · · n ] × [0 · · · n ]) obtained b y the tru ncation τ n . Then τ n : LT ( N , C ) − → LT ([0 · · · n ] , C ) and, this time, τ n preserve s m ultiplication ( τ n is a morph ism). One has the d iagram C N × N r n / / C [0 ··· n ] × [0 ·· · n ] LT ( N , C ) τ n / / J 1 O O LT ([0 · · · n ] , C ) J 2 O O where J 1 and J 2 are tw o canonical injections. Remark 4.3 We c an write LT ( N , C ) = lim ← − ( LT ([0 · · · n ] , C )) (14) which me ans that LT ([0 · · · n ] , C ) is the pr oje ctive limit of LT ( N , C ) 4.3. R andom gener ation Our motiv ation, in this section, consists in appr oximati ng the matrices of in fi nite su bstitutions b y fin ite matrices of (appr o ximate) substitutions. W e are th en inte rested in the prob ab ilistic study of these matrices. T o this end, we randomly generate unip oten t (u nitriangular) matrices and we observ e the n um b er of o ccurr ences of matrices of substitutions. The construction of unip oten t matrices is done as follo ws: (i) Create an identit y matrix. (ii) Fill randomly the lo wer part of the iden tity matrix (strictly un der the d iagonal) using n umbers w hic h follo w a un iform la w in [0 , 1]. (iii) Multiply those num b ers by the range pr eviously c hosen. (iv) T est if the built unip oten t matrix satisfies Eq.(13). W e start b y giving some examples of our exp eriment whic h are su mmarized in the table b elo w: Size Num b ers of dra wing Range of v ariables P robabilit y [3 × 3] 300 [1 · · · 10] 1 [1 · · · 100 ] 1 [1 · · · 100 00] 1 [4 × 4] 275 [1 · · · 10] 0 . 0473 [1 · · · 100 ] 0 . 0001 [1 · · · 100 00] 0 + [10 × 10] 1500 [1 · · · 10] 0 . 0327 [1 · · · 100 ] 0 + [1 · · · 100 00] 0 + According to the results obtained, we obs er ve that the su bstitutions matrices are not very frequent . How ev er, in meeting certain conditions su c h as size, the n umber of d ra w ings and the range of the v ariables, w e can obtain p ositiv e probabilities that these matrices app ear. Let us note that the smaller the size of the m atrix th e more p r obable on e obtains a m atrix of substitution in a reasonable num b er of d ra wings. W e also notice that, if we v ary the r ange of v ariables, and this in an increasing wa y and by k eepin g un c h anged th e num b er of dr awings and size, the probabilit y tends to zero. W e also notice that the unip oten t matrices of size 3 are all matrices of approximat e sub stitutions. Th is is easy to see b ecause the exp onen tial generating series of the 3 r d column will alwa ys ha v e the form c k = x 2 2! . Th us, we can sa y that th e test actually starts from th e matrices of size higher or equal to 4. Result 4.4 L et r r epr esent the c ar dinality of the r ange of variables and n × n b e the size of the matrix. A c c or ding to the r esults obtaine d; we c an say that the pr ob ability p n of app e ar anc e of the matric es of substitutions dep ends on r and n and we have the fol lowing u pp e r b ound: p n ≤ r 2 n − 3 r n ( n − 1) 2 (15) which shows that p n − → 0 as n − → ∞ (16) Conjecture 4.5 One c an c onje ctur e that the effe ct of the r ange sele ction vanishes when n tends to infinity. Mor e pr e cisely: p n ∼ r 2 n − 3 r n ( n − 1) 2 (17) References [1] P. Blasiak , Combinatorics of normal ordering and some applications, PhD Dissertation, U niversit y of P aris 6 (200 5). [2] P. Blasiak, K. A . Penson, A. I. Solomon . The Gene r al Boson Normal Or dering Pr oblem Phys. L ett. A 309 198 (2003). [3] P. Blasiak, G. H. E. Duchamp, A. Horzela, K. A.Penson, A. I. Solomon . Combinatorial Physics, Normal Or der and Mo del F eynman Gr aphs . 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