Quantum Painleve Equations: from Continuous to Discrete

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 4 (2008), 051, 9 pages Quan tum P ain lev ´ e Equatio ns: from C on tin uous to Discret e Hajime NA GO Y A † , Ba sil GR AMMA TICOS ‡ and A lfr e d RAMANI § † Gr aduate Scho ol of Mathematic al Scienc es, The University of T okyo , Jap an E-mail: nagoya@ms.u-tokyo.ac.jp ‡ IMNC, U niversit´ e Paris VII & XI, CNRS, UM R 81 65, Bˆ a t. 104, 91406 Orsay, F r anc e E-mail: gr ammati@p aris7.jussieu.fr § Centr e de Physique Th´ eorique, Ec ole Polyte chnique, CNRS, 91128 Palaise au, F r anc e E-mail: r amani@cpht.p olyte chnique.fr Received March 05, 20 08, in f ina l form May 03, 2 0 08; P ublished online J une 09 , 20 08 Original article is av ailable at http: //www .emis .de/journals/SIGMA/2008/051/ Abstract. W e examine quantum extensions of the contin uous P ainlev´ e equations, expr e ssed as systems of f irst-order dif ferential equa tions for non-co mm uting ob jects. W e fo cus on the Painlev ´ e equatio ns II, IV and V. F rom their auto-B¨ acklund transforma tio ns we derive the contiguit y relations which w e int erpret as the quantum a nalogues of the discrete Painlev ´ e equations. Key wor ds: dis c rete s ystems; q uantization; Painlev ´ e equations 2000 Mathematics Subje ct Classific atio n: 34M55; 37 K55; 81S9 9 1 In tro duction The w ord “quan tum” used in the title needs some qualifying. Historically , this term was intro- duced in r elation to th e d iscr eteness of the sp ectrum of op erators lik e the Hamiltonian and the angular momen tum. Ho w ev er, with the blossoming of quantum mec hanics, its use w as generali- sed to the description of non-classical ob jects, typically non-commuting op erators. I t is in this sense that w e are going to u se the term quantum in this pap er: namely , f or the description of equations where the v arious comp onents of the dep enden t v ariable do not commute among themselv es. In th is pap er w e shall f o cus on inte gr able equ ations in v olving n on-comm uting v ariables. Their in terest, in particular, as far as quan tum f ield theories are conce rned, is ob vious. A large lite- rature exists concerning sup ersymmetric or ju st fermionic extensions of integ rable ev olution equations. It is no w clear th at the sp ec ial pr op erties w hic h characte rise inte grabilit y can b e extended to the case where the dep enden t v ariable inv olv es fermionic as w ell as b oso nic comp o- nen ts. The quantisati on of lo w dimensional in tegrable Hamiltonian systems has b een the ob ject of extensiv e inv estigations by Hieta rin ta and collaborators [6, 7]. It w as sho wn, in particular, that th e pr eserv ation of integrabilit y in a quantum setting often necessitates the in tro duction of purely quantum ( i.e. , explicitly ~ d ep enden t) terms in b oth the Hamiltonian and the in v ariants. In some cases, these terms can b e absorb ed by the pr op er ordering and the in tro duction of a non-f lat space metric. Ho w ev er, it is not clear whether th is su f f ices in all cases. The “quan tum” extension of (con tin uous) P ainlev ´ e equ ations has b een int ro du ced by one of us (HN) in [9, 10]. F or these paradigmatic integrable systems, integ rabilit y is not related to the existence of in v ariants, but r ather to the fact that these n on au tonomous equations can be obtained as the compatibilit y condition of a (an o v erdetermined) linear system, the Lax pair. 2 H. Nago y a, B. Grammaticos and A. Ramani (Inciden tally , No viko v [12] refers to this compatibilit y condition as a quantisat ion condition for the sp ectral curve and thus the deautonomisation pro cess can b e considered, in some f ormal sense, as a f irst kind of quan tisation.) Starting fr om “qu an tum” conti n uous P ainlev ´ e equations w e sh all derive their co n tiguit y re- lations whic h, as already shown in the comm uting case, can b e in terpreted as discrete P ainlev ´ e equations [1, 8]. Quan tum and discrete systems p ossess a common charact er as far as the phase- space of their d ynamics is concerned. While in the former case th e surface of an elemen tary cell is f ixed by the relat ion ∆ x ∆ p = ~ in the latter case the el emen tary cel l is rigidly f ixed b y ∆ x = ∆ p = 1 (in the appr op r iate units). The main d if f icult y in quant ising discrete sys tems, in tegrabilit y not withstanding, lies in the fact that one m ust in trod uce a comm utation rule con- sisten t w ith the ev olutio n [13]. T h is is a highly non trivial problem. W e hav e addressed this question in [4] and [16] where we h a v e shown that for the mappings of the Q R T [14] family the follo wing comm utatio n r u le xy = q y x + λx + µy + ν (1) is consisten t with the ev ol ution (with the adequate choice of the parameters). Since discrete P ainlev ´ e equations are n onautonomous extensions of th e QR T mappings we exp ect rule (1) to b e suf f icien t f or th e quan tisation of the cases w e sh all consid er here. In what follo ws, we shall analyse the “quantum” form s of conti n uous Painlev ´ e equations deriv ed b y o ne of us (HN) in [9] and use their auto-B¨ ac klund transformations so as to dedu ce their contig uit y relations. T hus w orking w ith q u an tum forms of P II , P IV and P V w e will deriv e quan tum forms for the discrete P I , P II , P II I and P IV . 2 Non-comm uting v ariables: some basic relations Before pro ceeding to the deriv ation of quan tum discrete Painlev ´ e equations we should p r esen t a sum mary of our f indings in [9]. Our deriv ation of quan tum con tin uous P ainlev ´ e equ ations consists in extending the s y m metrical form of P ainlev ´ e equatio ns prop osed b y Noumi and Y a- mada [11] (see also Willo x et a l. [15]) to non-comm uting ob jects. In the case of the quan tum P II equation w e in tro duce three unknown op erators f 0 , f 1 , f 2 of t and tw o parameters α 0 , α 1 in the complex n um b er f ield C . The comm utatio n ru les are [ f 1 , f 0 ] = 2 ~ f 2 , [ f 0 , f 2 ] = [ f 2 , f 1 ] = ~ . (2) (In a m ore physical parlance w e can say that the ~ app earing in the comm utation relations is the Planc k constant). Generalising these relations to more ob jects so as to describ e h igher quantum P ainlev ´ e equa- tions is straigh tforw ard. F or a p ositiv e n umber l greater than 1, w e in tro d uce l + 1 c-n um b er parameters α i (0 ≤ i ≤ l ) and unknown op erators f i (0 ≤ i ≤ l ) with comm utatio n relations [ f i , f i +1 ] = ~ , [ f i , f j ] = 0 otherwise , (3) where the indices 0 , 1 , . . . , l are u ndersto o d as elemen ts of Z / ( l + 1) Z . In the case of the qu an tum P II equation, the evolutio n equations with resp ect to t for th e unknown op erators f i are ∂ t f 0 = f 0 f 2 + f 2 f 0 + α 0 , ∂ t f 1 = − f 1 f 2 − f 2 f 1 + α 1 , ∂ t f 2 = f 1 − f 0 , (4) and in th e case of the quan tum P IV equation, the quantum P V equation and higher P ainlev ´ e equations, the ev olution equations w ith resp ect to t for the unknown op erators f i are: for l = 2 n Quant um P ainlev ´ e Equations: fr om Conti n uous to Discrete 3 ( n ≥ 1), ∂ t f i = f i   X 1 ≤ r ≤ n f i +2 r − 1   −   X 1 ≤ r ≤ n f i +2 r   f i + α i , (5) and for l = 2 n + 1 ( n ≥ 1), ∂ t f i = f i   X 1 ≤ r ≤ s ≤ n f i +2 r − 1 f i +2 s   −   X 1 ≤ r ≤ s ≤ n f i +2 r f i +2 s +1   f i +   k 2 − X 1 ≤ r ≤ n α i +2 r   f i + α i X 1 ≤ r ≤ n f i +2 r , (6) where k = α 0 + · · · + α l . These (non-comm utativ e) d if ferent ial systems are qu an tizatio n of nonlinear ordin ary dif f eren tial systems p rop osed by Noumi and Y amada in [11], and equal to quan tization of P IV and P V for l = 2 and l = 3, resp ectiv ely . In the in tro duction w e ha v e stressed the necessit y for the comm utation rule to b e consistent with the evo lution. All systems here are consisten t with the corresp ondin g ev olution ∂ t , namely the ev olution ∂ t preserve s the comm utation relations (2) for the case of the quan tum P II equation and (3) for the other quan tum P ainlev ´ e equ ations. The non-comm utativ e dif f eren tial systems (4 ), (5) and (6) admit the af f in e W eyl group actions of typ e A (1) l , resp ectiv ely , as well as the classical case. The actions are as follo ws. In the case of the quantum P II equation, w e hav e s 0 ( f 0 ) = f 0 , s 0 ( f 1 ) = f 1 − f 2 α 0 f 0 − α 0 f 0 f 2 − α 2 0 f 2 0 , s 0 ( f 2 ) = f 2 + α 0 f 0 , s 1 ( f 0 ) = f 0 + f 2 α 1 f 1 + α 1 f 1 f 2 − α 2 1 f 2 1 , s 1 ( f 1 ) = f 1 , s 1 ( f 2 ) = f 2 − α 1 f 1 , s 0 ( α 0 ) = − α 0 , s 0 ( α 1 ) = α 1 + 2 α 0 , s 1 ( α 0 ) = α 0 + 2 α 1 , s 1 ( α 1 ) = − α 1 , π ( f 0 ) = f 1 , π ( f 1 ) = f 0 , π ( f 2 ) = − f 2 , π ( α 0 ) = α 1 , π ( α 1 ) = α 0 . (7) The actions s 0 , s 1 and π preserv e the comm utation relations (2) and giv e a rep r esen tation of the extended af f ine W eyl group of t yp e A (1) 1 , n amely , they satisfy the relations s 2 i = 1 , π 2 = 1 , π s i = s i +1 π . Moreo v er the actions of s 0 , s 1 and π comm ute with the d if ferentiat ion ∂ t , that is, they are B¨ ac klund transformations of the q u an tum P II equation. In the case of the quant um P IV equation ( l = 2), th e quant um P V equation ( l = 3) and higher quan tum Painlev ´ e equations ( l ≥ 4), for i, j = 0 , 1 , . . . , l ( l ≥ 2) we h a v e s i ( f j ) = f j + α i f i u ij , s i ( α j ) = α j − α i a ij , π ( f j ) = f j +1 , π ( α j ) = α j +1 , (8) where u i,i ± 1 = ± 1 , u l 0 = 1 , u 0 l = − 1 , u ij = 0 otherwise , a ii = 2 , a i,i ± 1 = − 1 , a l 0 = a 0 l = − 1 , a ij = 0 otherwise . The actions of s i ( i = 0 , 1 , . . . , l ) and π preserv e the comm utation relations (3) and giv e repr e- sen tations of the extended af f ine W eyl groups of t yp e A (1) l , namely , they satisfy the r elations s 2 i = 1 , ( s i s i +1 ) 3 = 1 , s i s j = s j s i ( j 6 = i ± 1) , π l +1 = 1 , π s i = s i +1 π . 4 H. Nago y a, B. Grammaticos and A. Ramani Moreo v er the actions of s i ( i = 0 , 1 , . . . , l ) and π comm ute with the dif ferent iation ∂ t , that is, they are B¨ ac klund trans f ormations of the corresp onding non-comm utativ e dif ferential systems. W e stress that the auto-B¨ ac klund transformations p reserv e the comm utation r elations in eac h case so that a discrete ev olution wh ic h is constructed from these auto-B¨ ac klu nd transformations also p reserv es the commutati on relations. W e remark that eac h d if ferential sy s tem (4), (5) or (6) h as r elations ∂ t ( f 0 + f 1 + f 2 2 ) = k for the qu an tum P II case, ∂ t l X r =0 f r ! = k for the l = 2 n ( n ≥ 1) case and ∂ t l X r =0 f 2 r ! = k 2 l X r =0 f 2 r , ∂ t l X r =0 f 2 r +1 ! = k 2 l X r =0 f 2 r +1 for the l = 2 n + 1 ( n ≥ 1) case, where k = α 0 + α 1 + · · · + α l . F or simplicit y , w e n ormalize k = 1 in the f ollo w ing. 3 The con tin uous quan tum Pa inlev ´ e I I and the related discrete equation As explained in Section 2 the Pa inlev ´ e I I case can b e obtained w ith three non-comm uting ob jects f 0 , f 1 , f 2 as describ ed in (2) and parameters α 0 , α 1 . F r om the dy n amical equ ations f ′ 0 = f 0 f 2 + f 2 f 0 + α 0 , f ′ 1 = − f 1 f 2 − f 2 f 1 + α 1 , f ′ 2 = f 1 − f 0 , (where the prime ′ denotes dif ferentiat ion with resp ec t to t ), eliminating f 0 , f 1 w e obtain the equation for the quantum P II f ′′ 2 = 2 f 3 2 − tf 2 + α 1 − α 0 , whic h h as the same expression as the comm utativ e P II . On the other hand , if we eliminate f 0 , f 2 w e f ind the quan tum version of P 34 . W e obtain f ′′ 1 = 1 2 f ′ 1 f − 1 1 f ′ 1 − 4 f 2 1 + 2 tf 1 − 1 2 ( α 2 1 − ~ 2 ) f − 1 1 . The dif ference of this “quantum” v ersion w ith th e comm utativ e P 34 is clearly seen in the term quadratic in the f irst deriv ative (whic h wo uld hav e b een f ′ 2 1 /f 1 in the commuta tiv e case) b ut also in the last term. As a matter of fact, the coef f icient of the term prop ortional to 1 /f 1 is a p erf ect square in the comm utati v e case. T he app earance of the − ~ 2 in the α 2 1 − ~ 2 co ef f icien t is a consequence of the non-comm utativ e c haract er of the f i ’s. In ord er to derive the discrete equation o btained as a contig uit y relation of the solutions of the quantum P II w e shall use the relations (7) presen ted in the previous section. W e def ine an Quant um P ainlev ´ e Equations: fr om Conti n uous to Discrete 5 ev olution in the parameter space of P II b y ¯ f ≡ s 1 π f and as a consequen ce the reverse evol ution is f ¯ ≡ π s 1 f . Using the relations (7) we f ind ¯ f 2 + f 2 = α 1 f − 1 1 (9) and f 1 ¯ + f 1 = t − f 2 2 . (10) In the same w a y we compu te the ef fect of the transformations on the parameters α . W e f ind ¯ α 1 = α 1 + 1 (and similarly α 1 ¯ = α 1 − 1). Thus applying n times the transformation s 1 π on α 1 , w e f ind that α 1 b ecomes α 1 + n . W e eliminate f 1 b et w een the tw o equations (9) and (1 0) and obtain an equation go v erning the ev olution in th e p arameter space. W e f ind α 1 ( ¯ f 2 + f 2 ) − 1 + α 1 ¯ ( f 2 ¯ + f 2 ) − 1 = t − f 2 2 . This is the quan tum v ersion of the equatio n kno wn, in the comm utat iv e case, a s the alternate discrete P ainlev ´ e I. 4 The con tin uous quan tum Pa inlev ´ e IV and the related discrete equation W e turn no w to th e case of the quantum P IV whic h is obtained from the equations p resen ted in Section 2 for l = 2. Again we ha v e three dep end en t v ariables f 0 , f 1 , f 2 . Th e quan tum con tin uous P IV equation is f ′ 0 = f 0 f 1 − f 2 f 0 + α 0 , f ′ 1 = f 1 f 2 − f 0 f 1 + α 1 , f ′ 2 = f 2 f 0 − f 1 f 2 + α 2 , whic h is ve ry similar to the one ob tained in the commutati v e case. It is int eresting to eliminate f 0 and f 2 and giv e th e equation f or f 1 . After a somewh at length y calculation we f in d f ′′ 1 = 1 2 f ′ 1 f − 1 1 f ′ 1 + 3 2 f 3 1 − 2 tf 2 1 +  t 2 2 + α 2 − α 0  f 1 − 1 2 ( α 2 1 − ~ 2 ) f − 1 1 , where w e h a v e used the identit y P i f i = t . Again w e r emark that this equ ation, w ith resp ect to the commutativ e P IV , con tains an explicit quantum correction. Before p ro ceeding to th e constru ction of the discrete sys tem related to this equation w e deriv e some auxiliary resu lts. Starting from the action of s i on f j , whic h f rom (8) is just s i ( f j ) = f j + α i f − 1 i u ij w e f in d s i ( P j f j ) = P j f j b ecause P j u ij = 0. Similarly π ( P j f j ) = P j f j . Th us P j f j is conserved u nder an y combination of the transformations π a nd s i of (8). W e can n o w d ef ine the evolutio n in the parameter sp ace just as we did in the case of the quan tum P II . W e put ¯ f ≡ s 1 s 0 π f and for the reve rse ev olution is f ¯ ≡ π − 1 s 0 s 1 f . Using the relations (8) we f ind ¯ f 2 + f 1 + f 2 = t − α 1 f 1 (11) and similarly f 1 + f 2 + f 1 ¯ = t + α 2 f 2 . (12) 6 H. Nago y a, B. Grammaticos and A. Ramani W e now study the ef fect of the u p -shift op erator on α 1 and − α 2 whic h pla y the role of th e indep en d en t v ariable. (The minus sign in front of α 2 guaran tees that the t w o equations ha v e the same form). Usin g the rela tions in (8) a nd the fact that the sum of the α ’s is constan t we f ind ¯ α 1 = α 1 + 1 and ( − ¯ α 2 ) = ( − α 2 ) + 1. Th us in cremen ting the indep endent v ariable under rep eated applications of the up -shift op erato r leads to a lin ear dep endence on the num b er of iterations. Still, since the starting p oin t is dif f eren t we ha v e tw o free parameters. Th us the system (11), (12) is exa ctly the quan tum analogue of the equation kno wn (in the comm utativ e case) as the “asymmetric discrete P ainlev ´ e I”, which is, in fact, a d iscrete f orm of P II [5]. 5 The con tin uous quan tum Pa inlev ´ e V and the related discrete systems Finally w e examine th e case of the quan tum P V , corresp onding to the case l = 3 in Secti on 2. The quan tum contin uous P V equation is f ′ 0 = f 0 f 1 f 2 − f 2 f 3 f 0 +  1 2 − α 2  f 0 + α 0 f 2 , f ′ 1 = f 1 f 2 f 3 − f 3 f 0 f 1 +  1 2 − α 3  f 1 + α 1 f 3 , f ′ 2 = f 2 f 3 f 0 − f 0 f 1 f 2 +  1 2 − α 0  f 2 + α 2 f 0 , f ′ 3 = f 3 f 0 f 1 − f 1 f 2 f 3 +  1 2 − α 1  f 3 + α 3 f 1 . W e remark that f ′ 0 + f ′ 2 = ( f 0 + f 2 ) / 2 and s imilarly f ′ 1 + f ′ 3 = ( f 1 + f 3 ) / 2. Thus t w o of the v ariables can b e easily eliminated by in trod ucing explicitly the time v ariable thr ough the exp onential e t/ 2 . Just as in the case of P II I and P IV w e can eliminate one further v ariable and obtain the qu an tum form of P V expressed in terms of a single v ariable. In the present case it is more co n v enien t to in tro duce an auxiliary v ariable w = 1 − e t/ 2 /f 0 . W e obtain thus for w the equation w ′′ = w ′  1 w − 1 + 1 2 w  w ′ + ( w − 1) 2  α 2 0 − ~ 2 2 w + ~ 2 − α 2 2 2 w  + ( α 3 − α 1 ) e t w + e 2 t w ( w + 1) 2(1 − w ) . (13) As in the previous cases, the equation (13) con ta ins an explicit quan tum correction, w ith resp ect to the commutati v e P V , as w ell as a symmetrisation of the term quadr atic in the f irst deriv at iv e. Just as in the case of comm utativ e P V w e can d ef ine s everal dif feren t ev olutions in the parameter space giving rise to dif ferent discrete equations. F or th e f irst equation w e introduce the f irst up-sh ift op erator R = π s 3 s 2 s 1 from (8) for l = 3. If we def ine x = f 0 + f 2 and y = f 1 + f 3 a careful application of the ru les (8) sho ws th at Rx = y and Ry = x . Next we in tro duce an auxiliary v ariable g d ef ined by g = f 3 − α 0 f − 1 0 = y − f 1 − α 0 f − 1 0 and w e seek an equatio n in terms of th e v ariables f 1 , f 0 , g . Calling ¯ f ≡ Rf , we f in d ¯ f 1 + f 0 = x − α 0 + α 3 g (14) complemen ted b y the equation coming from the d ef inition of g f 1 + g = y − α 0 f 0 (15) Quant um P ainlev ´ e Equations: fr om Conti n uous to Discrete 7 and f inally g ¯ + f 0 = x + α 1 f 1 . (16) In order to introd u ce the pr op er indep endent v a riable w e co nsider the action of R on the α ’s: ¯ α 0 = α 0 + 1, ¯ α 1 = α 1 − 1, ¯ α 2 = α 2 and ¯ α 3 = α 3 . Thus we can choose z = α 0 (whic h gro ws linearly with the successiv e applications of R ). The sy s tem (14), (15) and (16) is the qu an tum analogue of the asymmetric, ternary , d iscrete Painlev ´ e I, which, as w e ha v e sho wn in [3] is a d iscrete form of the Pai nlev ´ e IV equation. (A t this p oin t we should p oin t out that th e alternating constan ts x , y do not int ro du ce an extra degree of freedom. As a matter of fact by c ho osing an approp r iate gauge of the dep enden t v ariables and rescaling of the indep en d en t ones we can bring these constan ts to an y non-zero v alue). F or the second equation w e introduce a new up-sh if t op erato r T = s 1 π s 3 s 2 , the action of whic h is again obtained with the help of (8). T he v ariables x and y are def ined in the same wa y as in the previous p aragraph and again we ha v e T x = y and T y = x . An auxiliary v ariable is necessary in this case also and thus we introd uce h = f 3 + α 2 f − 1 2 . W e denote the action of T b y a tilde: ˜ f ≡ T f . W e seek an equation f or f 1 , f 2 , h . W e f in d f ˜ 1 + f 2 = x + α 2 + α 3 h . (17) The def inition of h imp lies h + f 1 = y + α 2 f 2 (18) and f inally w e ha v e f 2 + ˜ h = x − α 1 f 1 . (19) The actio n of T on the α ’s is ˜ α 0 = α 0 , ˜ α 1 = α 1 + 1, ˜ α 2 = α 2 − 1 and ˜ α 3 = α 3 . Com bining the t w o systems ab o v e w e can obtain a n icer, and more familiar, form. First, comparing (16 ) and (19), w e f ind that g ¯ = f 2 + α 1 f − 1 1 = x − ˜ h = T ( y − h ), or equiv alen tly g = RT ( y − h ). W e are th us led to in tro duce the oper ator S = RT , and w e denote its action b y a “hat” acce nt ˆ f = S f . S ubtracting (15) from (18) and el iminating g thr ough the relation g = y − ˆ h , w e obtain h + ˆ h = y + α 0 f 0 − α 2 f 0 − x . (20) F or the second equation w e start from (17) and apply the op erator RT to it. W e f ind ¯ f 1 + ˆ f 2 = x − α 0 + α 1 ˆ h (21) where we ha v e used the fact that RT ( α 2 + α 3 ) = − ( α 0 + α 1 ). Next w e su btract (21) from (14), use the d ef inition of x in order to eliminate f 2 and f ind f in ally f 0 + ˆ f 0 = x + α 0 + α 3 ˆ h − y + α 0 + α 1 ˆ h . (22) A careful application of the rules (8 ) sho ws th at the a ction of the op erator RT on eac h of th e n umerators of the fractions in the r ight hand side of (20) and (22) results in an increase b y exactly 1 and thus an indep enden t v ariable linear in the num b er of iterations of RT can b e in tro duced. As in th e case of th e f irs t system (14), (15) and (16) presented in this s ection 8 H. Nago y a, B. Grammaticos and A. Ramani the alternating constan ts x , y do n ot introdu ce an extra degree of freedom. A b etter choic e of the dep endent v a riables w ould b e h → h − y / 2 an d f 0 → f 0 − x/ 2. Moreo v er, by c h o osing an approp r iate gauge of the dep enden t v ariables and rescaling of the indep en d en t ones we can bring these constan ts to an y non-zero v alue, for instance x = y = 2. The system no w b ecomes h + ˆ h = α 0 f 0 + 1 − α 2 f 0 − 1 , (23) f 0 + ˆ f 0 = α 0 + α 3 ˆ h − 1 + α 0 + α 1 ˆ h + 1 . (24) Under this form one recognizes immediately the structure of the “a symmetric discrete P ainle- v ´ e II” e quation, in trodu ced in [2] and which is a discrete analogue of P II I . Th us (23) and (24) constitute the qu an tum extension of the latter. 6 Conclusions In this p ap er, w e ha ve analysed the “quan tum” f orms of Pai nlev ´ e equations deriv ed b y one of us (HN) in [9]. Th e deriv ation consists in extending the symmetrical form of P ainlev ´ e equations prop osed b y Noumi and Y amada to non-commuting v ariables. W e ha v e fo cused here on P II , P IV and P V and derived their more “familiar” forms expressed in terms of a single v ariable. The non- comm utativ e c haracter manifests itself in the fact that the dep enden t f unction do es not comm ute with its f irst deriv ative . As a consequen ce (and despite the fact that a symmetrised form of the term quadratic in the f irst der iv ativ e is used) exp licit qu an tum correctiv e terms ap p ear in the equation, prop ortional to the square of the Planck constan t. Using the auto-B¨ ac klund transformations of the contin u ous P ainlev ´ e equ ations we deriv e their con tiguit y relations wh ic h are just the quantum forms for the discrete P I , P II , P II I and P IV . References [1] F ok as A., Grammaticos B. , Ramani A., F rom continuous to discrete Painlev ´ e equations, J. Math. Anal. Appl. 180 (1993), 342–360. [2] Grammaticos B., Nijhof f F.W., Papageorg iou V., Ramani A ., Satsuma J., Linearizatio n and solutions of the discrete Pai nlev´ e II I equ ation, Phys. Le tt. A 185 (1994), 446–45 2, solv-int/93 10003. [3] Grammaticos B., Ramani A., Pa pageorgiou V., Discrete dressing transformations and P ainlev´ e equations, Phys. L ett. A 235 (1997), 475–479. 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