Quantization of bi-Hamiltonian systems

In 1975, one of the present authors 1 showed how to obtain the quantized levels of the nonlinear Schrodinger equation using the action-angle variables (canonical coordinates) of the AKNS scattering data. The symplectic form used to effect the reduction to canonical coordinates was based on the standard Hamiltonian structure for the nonlinear Schrooinger equation. The method used was a nonlinear generalization of one of the standard methods for the second quantization of the electromagnetic field. As presented in the textbook by Schiff, one takes the classical electromagnetic field and decomposes it into normal modes (Fourier components). The key idea in this approach is that the classical electromagnetic Hamiltonian will decompose into a sum of noninteracting classical Hamiltonians, each of which has just two degrees of freedom and· is easily quantized by itself. This method of quantization bypasses all the inherent difficulties of fully quantizing the system, including the factor-ordering problem, defining the quantum field operators for the fundamental fields, etc. 31t is fundamentally based on the symmetries of the classical system, and reduces the problem to one of quantizing noninteracting particles. In this way, the original difficult second quantization problem is reduced to a simpler set of noninteracting problems. The advantage of this simpler solution is tremendous when one considers the information that one can glean from it. First, one can obtain the spacings of the energy levels. One also discovers which quantum variables will commute, and which modes will have a particle-like behavior. Of course, for a full quantum theory, one still has to deal with a number of remaining difficult problems, including finding a consistent factor-ordering for the quantum operators, evaluating matrix elements, etc. Unfortunately, the solution to this larger quantization problem may well be multivalued. However, in the meantime, one has been able to immediately isolate the above mentioned important features of second quantization, and, very importantly, those quantities which would have the same common solution for every possible consistent second quantization. Thus, any difficulty which would be found at this level would also be present in any quantum field theory. And a study by this method can provide valuable insight into the structure of the more thorny parts of the second-quantization problem. The symplectic form used in Ref. 1 to effect the reduction to canonical coordinates was based on the first Hamiltonian structure for the nonlinear Schrooinger equation. In 1978, Magris showed how many soliton equations, including the nonlinear Schrooinger equation, could be written as biHamiltonian systems, meaning that they have two distinct, but compatible, Hamiltonian structures. Indeed, his fundamental result showed that, subject to some technical hypotheses5.6 any bi-Hamiltonian system is completely integrable in the sense that it has infinitely many conservation laws in involution and corresponding commuting Hamiltonian flows. From the viewpoint of quantum mechanics, the existence of more than one Hamiltonian structure for a given classical mechanical system raises the possibility of there existing more than one quantized version of this system, even at the level of quantization considered in Ref. 1. The resulting ambiguity in the quantization procedure raises serious physical doubts as to the mathematical framework of quantization. However, the main result to be proven here is that, for AKNS soliton equations, both quantized versions are essentially the same. We demonstrate that, in terms of the respective canonical coordinates on the scattering data, the two Hamiltonians have identical expressions, and hence identical quantum versions. Indeed, we conjecture that this phenomenon is true in general: quantization does not depend on the underlying Hamiltonian structure. (The results of Dodonov et aI., 8 in which an ambiguity in the quantization procedure for certain finite-dimensional bi-Hamiltonian systems is supposedly demonstrated, are erroneous, since they fail to incorporate the important topological properties of phase space properly in their picture. Indeed, their ambiguity is just a version of the ambiguity inherent in the quantization of two-dimensional Hamiltonian systems, which we discuss in detail below.) Moreover, we will see that for the other members of the associated hierarchy of soliton equations the only difference in the quantum versions is in the choice of weighting factor for the quantum operators corresponding to the continuous spectrum, the weight being determined by the classical dispersion relation, and the replacement of the bound state Hamiltonians. Thus, the effect of quantizing different members of the soliton hierarchy will only be significant for the bound states/solitons. Our presentation relies heavily on the notation and results in earlier papers by Kaup and NeweU I•9• on the closure of the squared eigenfunctions for the AKNS scattering