q-oscillator from the q-Hermite Polynomial

Yukawa Institute Kyoto DPSU-07-4 YITP-07- 64 arXiv:0710. 2209 [hep-th] Octob er 2007 q -oscillato r from the q -Hermit e P olynomial Satoru Odak e a and Ryu Sasaki b a Departmen t of Ph ysics, Shinsh u Univ ersit y , Matsumoto 390-8621 , Japan b Y uk a wa Institute for Theoretical Ph ysics, Ky oto Univ ersit y , Ky oto 606-85 02, Japan Abstract By factorizat ion of the Hamiltonian describing the qu an tu m mechanics of the con tinuous q -Hermite p olynomial, t he creation and annihilation op erators of the q - oscillato r are o btained. They satisfy a q -oscillator algebra as a consequence of the shap e-inv ariance of the Hamiltonian. A second set of q -oscillato r is deriv ed from the exact Heisenberg op erator solution. No w th e q -oscillator stands on the equal fo oting to the ordinary harmonic oscillator. P A CS : 0 3 .65.-w, 03 .65.Ca, 03.6 5 .Fd, 02.3 0 .Ik, 02.30.Gp, 02.20.Uw 1 In tro duc tion In this Letter, the explicit forms of the generator s of a q -oscillator alg ebra a r e deriv ed from the quan tum mec hanical Hamiltonian [1 , 2] of the q -Hermite p olynomial [3], the q -analogue of the Hermite p olynomial constituting the eigenfunctions of the harmonic oscillator. This is in sharp contrast to the common approac h t o q -oscillators [4], whic h assumes certain forms of the algebras without any dynamical/analytical con ten ts b ehind them. On the other hand, t he ordinar y harmonic oscillator algebra generated by the a nnihilation/creation op erators has ric h analytical structure of differen t ia l op erators related with the classical analysis of the Hermite p olynomial together with the coheren t and squeezed states, etc. 1 Since the annihilation/creation op erators of the ha rmonic oscillator and their algebra are the cornerstone of mo dern quan tum phys ics, their go o d deformation is b ound to pla y an imp ortant role, as evidenced by the represen tation theory of the quan tum gro ups in terms of the q -oscillators. Th us our new results are exp ected to enric h the sub ject b y stim ulat ing the in terplay b etw een (quantum ) alg ebra and ana lysis through new coheren t/squeezed states etc, whic h w ould find applications in quan tum optics and quan t um info rmation theory . Here w e discuss only Rogers’ q -Hermite p olynomial [3], or the so-called con tin uous q -Hermite p olynomial [5, 6] for the pa r a meter range 0 < q < 1. Lik e the Hermite p o lynomial, the q -Hermite p o lynomial has no parameter other t han q . This Letter is organized as follows . The factorized Hamiltonian for the q -Hermite p oly- nomial is presen ted and the q -oscillator comm utation relation is sho wn to b e a simple conse- quence of their structure. After brief exploration o f the eigenfunctions, t he exact Heise n b erg op erator solution [7] is presen ted. A second set of q -oscillator algebra is deriv ed from t he explicit forms of the annihilation/creation op erators whic h are t he p o sitiv e/negative energy parts of the exact Heisen b erg op erator solution. These q -oscillators reduce to the ordinary harmonic o scillator in the q → 1 limit. Relationship to v arious fo rms of q -oscillator alg ebras is explained. The Letter concludes with some historical commen ts and a summary . 2 Hamiltonian for the q -Hermite p olyn omial The Hamiltonian of the ‘discrete’ quan tum mec hanics for one degree of freedom has the general structure [2 , 1] H def = p V ( x ) e γ p p V ( x ) ∗ + p V ( x ) ∗ e − γ p p V ( x ) − V ( x ) − V ( x ) ∗ (1) = p V ( x ) q D p V ( x ) ∗ + p V ( x ) ∗ q − D p V ( x ) − V ( x ) − V ( x ) ∗ , (2) in whic h x ∈ R is the co ordinate and p = − i∂ x is the conjugate momen tum. The constan t γ in the presen t case is γ def = log q , 0 < q < 1 and the p oten tia l function f o r the dynamics of the q -Hermite p olynomial is giv en b y V ( x ) def = 1 (1 − z 2 )(1 − q z 2 ) , z def = e ix , (3) 2 with D = p = − i∂ x = z d dz . It is a special case o f the Ask ey-Wilson p olynomial [1, 5]. The Hamiltonian is fa ctorized as H = A † A , (4) A † def = − i  p V ( x ) q D / 2 − p V ( x ) ∗ q − D / 2  , (5) A def = i  q D / 2 p V ( x ) ∗ − q − D / 2 p V ( x )  . (6) With the explicit form of the p otential function V , (3), it is straigh tforw ard to deriv e the q -o scillator comm utation relation AA † − q − 1 A † A = q − 1 − 1 . (7) Sometimes it is written as [ A , A † ] q − 1 = q − 1 − 1 with t he standard notat io n [ A, B ] c def = AB − cB A . W e also hav e [ H , A ] q = ( q − 1 ) A , [ H , A † ] q − 1 = ( q − 1 − 1) A † . (8) The q -oscillator commutation relation (7 ) is also a consequence of the shap e invarian c e with- out shifting parameter [8] among the general Ask ey-Wilson p ot entials [1, 5]. One could also sa y that t he comm utation relation of the harmonic oscillator aa † − a † a = 1 is a manifestation of the shap e-inv aria nce. The g roundstate wa v efunction φ 0 is annihilated by the op erator A : A φ 0 = 0 = ⇒ φ 0 ( x ) def = p ( e 2 ix ; q ) ∞ ( e − 2 ix ; q ) ∞ , (9) in which the standard notation o f q -P o c hhammer symbol ( a ; q ) n is used: ( a ; q ) n def = n Y k =1 (1 − aq k − 1 ) = (1 − a )(1 − aq ) · · · (1 − aq n − 1 ) , (10) including the limiting case n → ∞ . With this ch oice of the groundstate w av efunction, w e can sho w tha t the Hamiltonian (1) is hermitian with resp ect to the inner pro duct ( f , g ) = R π 0 f ( x ) ∗ g ( x ) dx in the Hilb ert space L 2 [0 , π ] [9]. By using the factorization (4) and the q - oscillator relation (7), it is straightforw ard to demonstrate that ( A † ) n φ 0 is an eigenstate of the Hamiltonia n with the geometric sequence sp ectrum: H ( A † ) n φ 0 = E n ( A † ) n φ 0 , E n def = q − n − 1 . (11) 3 3 The q -Hermite p ol ynomial The a nalytical a ppro ac h to the Sc hr¨ odinger equation H φ n = E n φ n , (12) whic h is a differ enc e equation instead of a second o r der differential equation, go es as follows. By similarity transformation in terms of t he groundstate wa v efunction φ 0 , o ne introduces e H def = φ − 1 0 ◦ H ◦ φ 0 = V ( x )( q D − 1) + V ( x ) ∗ ( q − D − 1) , (13) whic h acts on the p olynomial part of the eigenfunction P n ( η ( x )): φ n ( x ) = φ 0 ( x ) P n ( η ( x )) . (14) It is elemen ta r y to sho w e H ( z + 1 /z ) n = ( q − n − 1)( z + 1 /z ) n + lo wer order terms in z + 1 /z , (15) since the r esidues at z = ± 1, z = ± q ± 1 / 2 , and z = ± q ∓ 1 / 2 all v anish. Th us one can find the eigenp olynomial in η ( x ) = cos x = ( z + 1 /z ) / 2, whic h is called the contin uous q - Hermite p olynomial in tro duced b y Ro gers [3, 5] e H H n (cos x | q ) = E n H n (cos x | q ) , (16) H n (cos x | q ) def = n X k =0 ( q ; q ) n ( q ; q ) k ( q ; q ) n − k e i ( n − 2 k ) x , H 0 = 1 , H 1 (cos x | q ) = 2 cos x. (17) It has a definite parit y . Reflecting the orthogonality o f the eigenfunctions of the Hamiltonian H , ( φ n , φ m ) ∝ δ nm , it is orthogonal with r esp ect to the w eight function φ 0 ( x ) 2 : Z π 0 φ 0 ( x ) 2 H n (cos x | q ) H m (cos x | q ) dx = δ n m 2 π ( q n +1 ; q ) ∞ , (18) satisfying the thr ee term recurrence relation 2 η H n ( η | q ) = H n +1 ( η | q ) + (1 − q n ) H n − 1 ( η | q ) . (19) 4 The action of the creation A † and annihilation A op erators on the p olynomial H n (cos x | q ) is e A † def = φ − 1 0 ◦ A † ◦ φ 0 , e A def = φ − 1 0 ◦ A ◦ φ 0 , (20) e A † = q − 1 2 − 1 z − z − 1  z − 2 q D / 2 − z 2 q − D / 2  , (21) e A = − 1 z − z − 1  q D / 2 − q − D / 2  , (22) e A † H n (cos x | q ) = q − ( n +1) / 2 H n +1 (cos x | q ) , (23) ( e A † ) n 1 = q − n ( n +1) / 4 H n (cos x | q ) , (24) e A H n (cos x | q ) = ( q − n/ 2 − q n/ 2 ) H n − 1 (cos x | q ) . (25) The similarity transfor med e A (22) is prop ortio nal to the divided difference op erator. 4 Heisen b erg o p e rator solut ion The harmonic o scillator is a typical example for which the Heisen b erg op erator solution is kno wn and the annihilation/creation op erat o rs can also b e extracted as the p ositiv e/negative frequency parts of t he Heisen b erg op erator solution. The situation is par a llel but sligh tly differen t for the q -oscillator. T he exact Heisen b erg operato r solution is deriv ed a nd its p ositiv e/nega t ive frequency parts giv e another set of annihilation/creation op erators a ( ± ) whic h ar e closely related to A a nd A † . (F or the general theory of exact Heisen b erg op erator solutions, see [7, 10] fo r systems of single degree of freedom and [11] for a class of m ulti- particle dynamics.) W e start fr om the closur e r elation [ H , [ H , cos x ] ] = cos x R 0 ( H ) + [ H , cos x ] R 1 ( H ) , (26) R 0 ( H ) def = ( q − 1 2 − q 1 2 ) 2 ( H + 1) 2 , (27) R 1 ( H ) def = ( q − 1 2 − q 1 2 ) 2 ( H + 1) , (28) whic h can b e readily v erified. This relation enables us to express any m ultiple comm utator [ H , [ H , · · · , [ H , cos x ] · · · ]] as a linear com bination of the op erators cos x and [ H , cos x ] with co efficien ts dep ending on the Hamiltonian H o nly . Th us w e arriv e at the exact Heisen b erg op erat or solution for the 5 sinusoidal c o or di n ate η ( x ) def = cos x [7]: e it H cos x e − it H = cos x q e iα + ( H ) t + e iα − ( H ) t 1 + q + [ H , cos x ] e iα + ( H ) t − e iα − ( H ) t ( q − 1 − q )( H + 1) , (29) α ± ( H ) = ( q ∓ 1 − 1)( H + 1) . (30) This simply means that the co ordinate cos x undergo es sinusoidal motions with frequencies α ± ( H ). While factorization of Hamiltonian is kno wn to prov ide the annihilation/creation op- erators only fo r the harmonic oscillator and the q -oscillator, the authen tic definition o f the annihilation/creation op erators is through the p ositiv e/negat ive frequency parts of t he Heisen b erg o p erator solution [7] ( η = cos x ): e it H cos x e − it H = a (+) e iα + ( H ) t + a ( − ) e iα − ( H ) t , (31) a ( ± ) = ± 1 q − 1 − q  [ H , η ] q ± 1 + (1 − q ± 1 ) η  ( H + 1) − 1 , a ( − ) † = a (+) . (32) Their action on the full eigenfunction is ( φ n ( x ) def = φ 0 ( x ) H n (cos x | q )): a ( − ) φ n = 1 2 (1 − q n ) φ n − 1 , a (+) φ n = 1 2 φ n +1 , (33) to b e compared with A φ n = q − n 2 (1 − q n ) φ n − 1 , A † φ n = q − n +1 2 φ n +1 . (34) F rom these and (18), it is easy to c hec k the hermiticit y ( φ n − 1 , a ( − ) φ n ) = ( a (+) φ n − 1 , φ n ) , (35) ( φ n − 1 , A φ n ) = ( A † φ n − 1 , φ n ) . (36) They satisfy commutation relations [ a ( − ) , a (+) ] = 1 4 (1 − q )( H + 1) − 1 , (37) [ H , a ( ± ) ] = ( q ∓ 1 − 1) a ( ± ) ( H + 1) . (38) By remo ving the Hamiltonian from the r.h.s. they can b e cast in to another q -oscillator fo rm a ( − ) a (+) − q a (+) a ( − ) = 1 4 (1 − q ) , (39) H a ( ± ) − q ∓ 1 a ( ± ) H = ( q ∓ 1 − 1) a ( ± ) . (40) 6 It should b e noted that t he q -oscillator relations (39) -(40) also hold for the con tin uous big q -Hermite p o lynomial [5, 7]. W e will rep or t on this topic elsewhere. The tw o t yp es of creation- annihilation op erator s are closely relat ed with each other [7] a (+) = A † X , a ( − ) = X † A , (41) with X = − i 2 q  z p V ( x ) q D / 2 − z − 1 p V ( x ) ∗ q − D / 2  ( H + 1) − 1 , (42) X † = i 2 q ( H + 1 ) − 1  q D / 2 z − 1 p V ( x ) ∗ − q − D / 2 z p V ( x )  , (43) and t he op erators X and X † map the eigenfunction φ n to itself: X φ n = 1 2 q ( n +1) / 2 φ n , X † φ n = 1 2 q ( n +1) / 2 φ n . (44) The structure of these op erat o rs is b etter understo o d b y the similarit y transformation in terms of the groundstate w a vefunc tion φ 0 e X def = φ − 1 0 ◦ X ◦ φ 0 , e X † def = φ − 1 0 ◦ X † ◦ φ 0 . (45) In fa ct, their actions on p olynomials { H n (cos x | q ) } are essen tially iden tical: e X = 1 2 q 1 2 − 1 z − z − 1 ( z − 1 q D / 2 − z q − D / 2 )( e H + 1) − 1 , (46) e X † = 1 2 q 1 2 ( e H + 1) − 1 − 1 z − z − 1 ( z − 1 q D / 2 − z q − D / 2 ) . (47) The ma in part of e X a nd e X † , defined by D q = − 1 z − z − 1 ( z − 1 q D / 2 − z q − D / 2 ) , (48) w as a lso in tro duced by Atakishiy ev-Klim yk [12] eq(9). It satisfies the relation D q H n (cos x | q ) = q − n/ 2 H n (cos x | q ) , (49) and it factorizes e H a nd e H + 1: ( D q − 1)( D q + 1) = e H , ( D q ) 2 = e H + 1 . (50) The coheren t state of the harmonic oscillator is defined a s the eigen v ector of the anni- hilation op erator; aψ = αψ , whic h is the generating function o f t he Hermite p o lynomials. 7 W e encoun ter a parallel situation here. The eigenv ector of the op erator a ( − ) , a ( − ) ψ ( x ; α ) = αψ ( x ; α ), is give n b y ψ ( x ; α ) = φ 0 ( x ) ∞ X n =0 (2 α ) n ( q ; q ) n H n (cos x | q ) (51) = φ 0 ( x ) 1 (2 α e ix ; q ) ∞ (2 α e − ix ; q ) ∞ . (52) The second factor is the generating function of the q -Hermite p olynomials [5, 6]. The coheren t state define d b y the other annihilation op erator A , A ψ ′ ( x ; α ) = αψ ′ ( x ; α ), has a similar structure: ψ ′ ( x ; α ) = φ 0 ( x ) ∞ X n =0 α n q 1 4 n ( n +1) ( q ; q ) n H n (cos x | q ) . (53) 5 Limit to th e ordinary harmonic o scillator The q -oscillators reduce to the ordinary harmonic oscillator in the q → 1 limit. T o show this, let us in tro duce tw o para meters ( L and c ) and a new co ordinate x ′ : x = π 2 − π L x ′  ⇒ − L 2 < x ′ < L 2  , q = e − 2 π cL . (5 4) The momen tum op erator conjugate to x ′ is p ′ = − i d dx ′ = − π L p . Then the desired limit is obtained by setting L = π c a nd taking c → ∞ limit: c 2 H → x ′ 2 + p ′ 2 − 1 , c 2 E n → 2 n, (55) c A † c A  → x ′ ∓ ip ′ , ca ( ± ) → 1 2 ( x ′ ∓ ip ′ ) , X † X  → 1 2 , (56) c cos x → x ′ ( −∞ < x ′ < ∞ ) , c 4 R 0 ( H ) → 4 , c 2 R 1 ( H ) → 0 , (57) ( q ; q ) ∞ φ 0 ( x ) 2 2 √ π c → e − x ′ 2 , (58) c n H n (cos x | q ) = c n H n  sin x ′ c   e − 2 c 2  → H n ( x ′ ) . (59) Here w e hav e used the Ja cobi’s triple pro duct iden tit y [6] and it s mo dula r t r a nsformation prop erty (the S -tr a nsformation) fo r deriving (58), and the three term recurrence relations for (59 ). 6 Other forms o f q -osci llators Here w e will discuss the relationship b et w een our intrins ic q -oscillator a lgebra (7)- ( 8) and those in tro duced purely algebraically for quan tum gro up represe n tat io ns aro und 1 989-90 [4]. 8 First let us in tro duce the num b er op erator N through the energy sp ectrum formula (11), ( H + 1 ) ∓ 1 = q ±N , N φ n = nφ n , n ∈ Z + , ( 6 0) whic h counts t he lev el from the groundstate. Sev eral differen t forms of q - oscillator algebras are intro duced, among whic h w e list tw o t ypical ones: bb † − q − 1 b † b = q N , (61) bb † − q b † b = q −N . (62) If we define b and b † b y b = A q N / 4 ( q − 1 2 − q 1 2 ) 1 2 , b † = q N / 4 A † ( q − 1 2 − q 1 2 ) 1 2 , (63) it is straightforw ard to v erify bb † − q − 1 2 b † b = q N / 2 , (64) whic h b ecomes (61) by iden tification q → q 2 . Lik ewise, the q -oscillator algebra of a ( ± ) (39) is related to (62) by similar transformatio ns. 7 Commen ts and summary Some historical commen ts are in order. There were attempts to relate q - oscillator algebras to the difference equation of the q -Hermite p olynomial. None o f t hem is ba sed on a Hamilto nian, th us hermiticit y is not manifest and the logic for factorizatio n is unclear. Here w e list a few suc h att empts. Atakish iy ev and Suslo v in 1 9 90 [13] wrot e down an algebra bb + − q − 1 b + b = 1 , H = b + b, (65) whic h is related to our q -o scillator algebra (7) b y a similarity transformatio n p q − 1 − 1  b b +  = 1 √ sin x ◦  A A †  ◦ √ sin x. (66) Floreanini, LeT ourneux and Vinet presen ted in 1994 [14] a q -oscillator alg ebra ( e N def = φ − 1 0 ◦ N ◦ φ 0 ) A − A + − q − 1 A + A − = 1 , K A ± = q ∓ 1 2 A ± K, K = q − e N / 2 , (67) 9 whic h is in our no t ation A + = − e A † , A − = − 1 q − 1 − 1 e A , K = D q . (68) In 20 03 Borzov and Da ma skinsky [15 ] wrote do wn a − q a + q − q a + q a − q = 1 , (69) starting from the three term recurrence relation of the q -Hermite p olynomial and defining the annihilatio n/creation op erators in their own w a y . In summary: w e hav e deriv ed tw o q -oscillator algebras (7) and (39) from the Hamiltonian of the q -Hermite p olynomial (4 ) – (6) [1, 7, 10], whic h is a sp ecial case of the Ask ey-Wilson p olynomial [5, 6]. The generators are gen uine a nnihilat io n/creation op erators and the her- miticit y is manifest. 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