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Journal of Nonlinear MathematicalPhysics 1998, V.5, N 1, 1–7. Letter On the Moyal Quantized BKP Type Hierarchies Dolan Chapa SEN and A. Roy CHOWDHURY High Energy Physics Division, Department of Physics Jadavpur University, Calcutta 700 032, India 8 9 Received June 30, 1997; Accepted July 17, 1997 9 1 Abstract n a Quantization of BKP type equations are done through the Moyal bracket and the J formalism of pseudo-differential operators. It is shown that a variant of the dressing 1 operator can also be constructed for such quantized systems. 1 v 5 Quantization of integrable system is one of the most important aspect of present day 0 research on nonlinear systems. In two dimensions, a well established methodology was 2 1 suggested by Faddeev and his collaborators [1], which goes by the name of Quntum In- 0 verse Spectral Transform (QISM) [1]. On the other hand, no such technique is known 8 (which utililes the inverse scattering framework) for integrable systems in three dimen- 9 / sions. Recently the formalism of pseudo-differential operators was used extensively to h study such systems in 3 (three) dimensions [2] and an effective way was found to study p - the Bi-Hamiltonian structures of such three dimensional systems. An ingeneous way to h t derive the quantum version for the KP system was suggested by Kupershmidt [3] using a the tools of p pseudo-differential algebra in conjuction with the idea of the Moyal bracket m [4]. The basic idea is that the quantization is a deformation of the classical situation. : v Here, in this communication, we study a quantization of the BKP like hierarchy [5] using i X the Moyal bracket approach. It is demonstrated that such systems can possess an infinite r number of conservation laws; essential for the complete integrability. Furthermore, we a also show that an extension of the usual dressing operator [6] is possible even for these quantized or deformed systems. A quantization is a rule which assigns to every polynomial in variables p and q, a polynomial in operators p and q. The rule must be a linear map satisfying a few natural properties. For any two operators f and g depending upon the canonical set of variables (p ,q ), i i the Moyal quantization dictates that the product be defined according to the following rule: ∂ ∂ ∂ ∂ f ∗g = exp ε − (fg), (1) " ∂pf ∂qg ∂qf ∂pg!# Copyright (cid:13)c1998 by D.C. Sen and A.R. Chowdhury 2 D.C. Sen and A.R. Chowdhury where the operator: ∂ ∂ ∂ ∂ v = − (2) ∂p ∂q ∂q ∂p f g f g acts according to the rule ∂f ∂g ∂f ∂ v(fg) = − . (3) ∂p ∂q ∂q ∂p For our situtation, f and g are some pseudo-differential operators. Before proceeding to theactual problemof thequantized BKPequation, we give below somealgebraic formulae which will be useful later. Using equation (1) we get i pi∗a= εsa(s)pi−s, s ! s=0 X i a∗pi = (−ε)sa(s)pi−s, s ! s=0 X i bpi∗a= εsa(s)bpi−s, s ! sX=0 (4) i pi∗apm = εsa(s)pi+m−s, s ! s=0 X i apm∗pi = (−ε)sa(s)pi+m−s, s ! s=0 X εs s i m bpi∗apm = (s−k) k(−1)kb(k)a(s−k)pi+m−s. s k ! s−k ! k ! s,k=0 X Here a(k) = ∂ka. etc., a, b are any two arbitrary functions of q only, and ∂k represents k-th order derivative with respect to q, that is ∂k/∂qk. The Moyal commutator is defined as: [f,g] = f ∗g−g∗f. (5) For example, it can be easily seen that equation (4) leads to i [pi,a] = 2 a2k+1a(2k+1)pi−2k−1, 2k+1 ! kX=0 (6) i [bpi,a] = 2 ε2k+1a(2k+1)bpi−2k−1, 2k+1 ! k=0 X and similarly for others. c In the above expressions denotes a Binomial coefficient with the following usefull d ! properties: i+2k , when k = 2k −i−1  2k ! 0 = (7) k0 ! (−1) i+2k+1 , when k = 2k+1. 2k+1 0 !  Moyal Quantized BKP Type Hierarchies 3 For positive values c, d these are c! . d!(c−d)! The standart pseudo-differential operator (ΨDO) is written as: L = ∂ + a ∂−m, (8) m m=1 X where a are functions of (x, y and t). In the following we shall rewrite L as m L = p+ a p−m. (9) m m=1 X The deformed or quantized hierarchy of equations are to be obtained from the Lax equation: L = [P ,L]= P ∗L−L∗P . (10) t + + + Here P = [Lm] , where m is any positive integer and (+) denotes the only terms with + + positive powers to be retained. One must also note that Lm = L∗L∗L∗L∗···∗L (m times). (11) For the construction of integrable systems, the time flows can be easily build up with the help of equation (11). For example, L∗p = p+ a p−m ∗ p+ a p−l m l ! ! mX=1 Xl=1 (12) = p2+2 a p−m+1+ εs(−1)k −m −i a(k)a(s−k)p−m−i−s m s−k k m i m s,k ! ! X X or L∗2 = p2+2a +2a p−1+··· 1 2 L∗3 = p3+3a p+3a +(3a +3a2+ε2a(2))p−1+··· 1 2 3 1 1 L∗4 = p4+4a p2+4a p+(4a +6a2+ε24a(2)) 1 2 3 1 1 +(4a +12a a +4ε2a(2))p−1+··· 4 1 2 2 (13) L∗5 = p5+5a p3+5a p2+(5a +10a2 +10ε2a(2))p 1 2 3 1 2 +(5a +20a a +10ε2a(2)) 4 1 2 2 + 5a +20a a +10a2 +10a3+ε(8a(1) +24a a(1) +24a a(1)) 5 1 3 2 1 4 1 2 2 1 +εh2(10a(2) +20a a(2)+10a(1)a(1))+ε38a(3) +ε4a(4) p−1+··· . 3 1 1 1 1 2 1 i The quantized BKP system is then defined through the Lax equations ∂L ∂L = [L3,L] , = [L5,L] . (14) ∂y + ∗ ∂t + ∗ 4 D.C. Sen and A.R. Chowdhury Using the above expressions we immediately get: a = ε(6a(1) +12a a(1))+2ε3a(3), 1y 3 1 1 1 a = ε(6a(1) +12a(1)a +12a a(1))+2ε3a(3), 2y 4 1 2 1 2 2 (1) (3) (1) (1) a = ε(6a +6a a +18a a +12a a ) 3y 5 1 3 1 3 2 2 (15) +ε3(2a(3) +6a(3)a +6a(2)a(1)), 3 1 1 1 1 (1) (1) (1) (1) a = (6a +6a a +24a a +18a a ) 4y 6 1 4 1 4 2 3 +ε3(2a(3) +18a(2)a(1)+24a(3)a +6a a(3)). 4 1 2 1 2 1 2 Similar equations for time evolution. Actually, (14) leads to an infinite set of coupled nonlinear equations in 3 dimensions. To restrict it to a BKP hierarchy we impose the constraint that (L∗)2n+1 will not contain any constant level term. Whence we get a = + 2 a = 0, along with 4 a = ε(6a(1) +6a a(1)+18a(1)a )+ε3(2a(3) +6a(3)a +6a(2)a(1)), (16) 3y 5 1 3 1 3 3 1 1 1 1 a = 2ε5a(5)+ε3(20a(3) +40a(3)a +80a(2)a(1) 1t 1 3 1 1 1 1 (17) +ε(10a(1) +40a(1)a +40a a(1)+60a2a(1)). 5 3 1 3 1 1 1 By eliminating a and a we obtain 3 5 32 40 100 20 a = − ε5a(5)+ε3 − a(3)a − a(1)a(2) +ε2 a(2) 1t 9 1 3 1 1 3 1 1 9 1y (cid:20) (cid:21) (18) 5 5 −10εa2a(1) + (a a +a(1)∂−1a )+ ∂−1a , 1 1 3 1 1y 1 1y 18ε 1y whichistherequirednonlinearequation in(2+1)-dimensions fora (xyt). Onecan observe 1 that this equation is also fifth order in derivative with respect to x as in the case of the usual BKP system. It is possible to generate other types of nonlinear system from different choices of y and t flows in equation (14). For example, consider ∂L = [L∗2,L] , ∗ ∂y (19) ∂L = [L∗5,L] , ∗ ∂t where equations (15) are modified to: (1) a = 4εa , 1y 2 (1) (1) a = 4εa +4εa a , 2y 3 1 1 (20) (1) (1) a = 4εa +8εa a , 3y 4 1 2 a = 4εa(1) +12εa(1)a +4ε3a a(3) 4y 5 1 3 1 1 along with a = 2ε5a(5)+ε3(20a(3) +40a(3)a +80a(2)a(1)) 1t 1 3 1 1 1 1 (21) +ε(10a(1) +40a(1)a +30a a(1)+40a a(1)+60a2a(1)). 5 3 1 3 1 2 2 1 1 Moyal Quantized BKP Type Hierarchies 5 It is interesting to note that one can eliminate all the variables other than a , without 1 imposing any extra condition on L2n+1, since equation (20) yields: + 1 a = ∂−1a , 2 1y 4ε 1 1 a = ∂−2a − a2, (22) 3 16ε2 1yy 2 1 1 a = ∂−1a −2∂−1(a(1)a ). 4 4ε 3y 1 2 Thus, from (21) we at once obtain 5 a = 2ε5a(5)+ε3 10a a(3)+20a(1)a(2) +ε 15a2a(1) + a(1) 1t 1 1 1 1 1 1 1 4 1yy h i (cid:20) (cid:21) 5 + 3∂−1(a a +a a )+2a ∂−1a −3a(1)∂−2a +4a ∂−1a (23) 8ε 1y 1y 1y 1yy 1y 1y 1 1yy 1 1yy h i 5 − ∂−3a . 128ε3 14y It each case, as in the case of usual ΨDO approach, the conserved quantities are obtained as Res.(L2n+1), where Res denote the coefficient of p−1. We have checked that such residues turn out to be combinations of total derivatives with respect to (x,y,t). We now report on intriguing fact about such Moyal quantized systems. This is related to the so-called modified system. In an interesting paper it was observed by Kupershmidt that modified equations can be generated if one uses dL = [(Lm) ,L] (24) ≥1 dt instead of (Lm) . (In some cases one can also consider (Lm) .) Here we observe that, ≥0 ≥2 if we consider ∂L ∂L = [L3 ,L] , = [L5 ,L] , (25) ∂y ≥1 ∗ ∂t ≥1 ∗ and using (L3) = p3+3a p, ≥1 1 (26) (L∗5) = p5+5a p3+5a p2+(5a +10a2 +ε210a(2))p ≥1 1 2 3 1 1 in these equations, we get (1) a = 0, 2 a = 6εa(1) +12εa a(1) +2ε3a(3), 1y 3 1 1 1 (27) (1) (2) a = 6εa +12εa a , 2y 4 2 1 a = 6εa(1) +6εa a(1)+18εa(1)a +ε3(2a(3) +6a a(3)+6a(1)a(2)) 3y 5 1 3 1 3 3 1 1 1 1 alongwith a = 2ε5a(5)+ε3(20a(3) +40a a(3) +80a(1)a(2)) 1t 1 3 1 1 1 1 +ε(40a a(1) +40a(1)a )+60a2a(1)+10a(1). 1 3 1 3 4 1 5 6 D.C. Sen and A.R. Chowdhury Again eliminating variables other than a , we get 1 a = 2ε5a(5)+ε3(20a(3) +40a(3)a +80a(2)a(1)) 1t 1 3 1 1 1 1 (28) +ε(40a a(1) +40a(1)a +60a2a(1) +10a(1)). 1 3 1 3 1 1 5 So we get back equation (21), but this time without any restriction on the constant level term of (L∗)2n+1. From the structure of the nonlinear equations discuseed so far, it appears that one can define an analogue of a dressing operator even in the Moyal quantized systems. Let us set ∞ s = 1+ ω p−m (29) m m=1 X for the deressing operator. Now ∞ s−1 = 1+ u p−m, (30) m m=1 X with s+s−1 = 1, leads to (u +ω )p−m+εs(−1)k −m −j ω(k)u(s−k)p−m−js = 0, (31) m m j −k k m j ! ! from which one can express the u’s in terms of the ω’s. For example, u = −ω , 1 1 u = −ω +ω2, (32) 2 2 1 u = −ω −ω2+2ω ω , ..., etc. 2 3 1 1 2 Now we demand that s∗p∗s−1 =L = p+ a p−m , (33) m m=1 X which immediately leads to a = −2εω(1), 1 1 a = ε(−2ω(1) +2ω ω(1)), 2 2 1 1 (34) a = u +ω +ω u +ω u +ω u 3 4 4 1 3 2 2 3 1 +ε(u(1) −ω(1)+ω(1)u −ω u(1))+ε2ω(1)u(1), ..., etc. 3 3 1 2 2 1 1 1 On the other hand, if we compute s∗p3∗s−1 and s∗p5∗s−1 via the rules (4), we get s∗p3∗s−1 = 1+ ω p−m ∗p3∗ 1+ u p−l m l m ! l ! (35) X X = p3+(ω +u )p2+(u +ω +ω u +3εu(1) −3εω(1))p+··· . 1 1 2 2 1 1 1 1 Whence, using equations (32) and (34), we get (s∗p3∗s−1) = p3+3a p+(3εu(1) −3εω(1)) + 1 2 2 = p3+3a p+3a (36) 1 2 = L3 (as given in equation (13)). Moyal Quantized BKP Type Hierarchies 7 Using the same methodology, but with a more laborious computation, we can prove that (s∗p5∗s−1) = p5+5a p3+5a p2+(5a +10a2+10a2a(2))p + 1 2 3 1 1 1 (37) +(5a +20a a +10ε2a(2)). 4 1 2 2 Intheaboveanalysiswehaveshownthatnonlinearsystemsin(2+1)-dimensions, which are in the category of the BKP equation, can be quantized using the pseudo-differential operators and Moyal bracket formalism. They are completely integrable since there are an infinite number of conserved quantities with them. Oneoftheauthors(D.C.Sen)isgratefulltoCSIR(Govt. ofIndia)forsuppportthough a SRF grant. References [1] Faddeev L.D. and Takhtajn L.A., Russ. Math. Survey, 1979, V.34, p.11. [2] Kupershmidt B.A., Rev. Asterrisque, 1985, V.123, Paris. [3] Kupershmidt B.A., Lett. Math. Phys., 1990, V.20, p.19. [4] Moyal J.E., Proc. Camb. Phil. Soc., 1949, V.45, p.99. [5] Grosh C. and Chowdhury A.R., J. Phys. Soc. Japan, 1993, V.62, p.851. [6] SatoM.andSatoY.,SolitonEquationsandDynamicalSystemsonInfiniteDimensionalGrossmanian Manifold – Nonlinear PDE in App.Math, North Holland, 1982.