Journal of Nonlinear Mathematical Physics 1998, V.5, N 1, 1–7. Letter On the Moyal Quantized BKP Type Hierarchies

Quantization of BKP type equations are done through the Moyal bracket and the formalism of pseudo-differential operators. It is shown that a variant of the dressing operator can also be constructed for such quantized systems. Quantization of integrable system is one of the most important aspect of present day research on nonlinear systems. In two dimensions, a well established methodology was suggested by Faddeev and his collaborators [1], which goes by the name of Quntum In-verse Spectral Transform (QISM) [1]. On the other hand, no such technique is known (which utililes the inverse scattering framework) for integrable systems in three dimen-sions. Recently the formalism of pseudo-differential operators was used extensively to study such systems in 3 (three) dimensions [2] and an effective way was found to study the Bi-Hamiltonian structures of such three dimensional systems. An ingeneous way to derive the quantum version for the KP system was suggested by Kupershmidt [3] using the tools of p pseudo-differential algebra in conjuction with the idea of the Moyal bracket [4]. The basic idea is that the quantization is a deformation of the classical situation. Here, in this communication, we study a quantization of the BKP like hierarchy [5] using the Moyal bracket approach. It is demonstrated that such systems can possess an infinite number of conservation laws; essential for the complete integrability. Furthermore, we 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 (pi, qi), the Moyal quantization dictates that the product be defined according to the following rule: f ∗ g = exp