The Hamiltonian Form of the KdV Equation: Multiperiodic Solutions and Applications to Quantum Mechanics

In the development of quantum mechanics in the 1920s both matrix mechanics (Born, Heisenberg and Jordon) and Schrödinger wave mechanics (Schrödinger) prevailed. These early cases corresponded to the quantum mechanics of particles. Matrix mechanics was found to lead directly to the Schrödinger equation and likewise, the Schrödinger equation could be used to derive the inverse problem for matrix mechanics. Later emphasis lay with the development of the dynamics of fields, where the classical field equations were quantized (see for example Weinberg). Today, quantum field theory is one of the most successful physical theories ever developed. One of the important properties of quantum mechanics is that it is linear (i.e. the Schrödinger equation is linear), leading to some confusion about how to treat the problem for nonlinear classical field equations. In the present paper we address the case of classical soliton equations which are exactly integrable in terms of the periodic/quasiperiodic inverse scattering transform. This means that all physical spectral solutions of the soliton equations can be computed exactly for these specific boundary conditions. Unfortunately, such solutions are highly nonlinear, leading to difficulties for solving the associated quantum mechanical problems. Here we develop a road to find the exact quantum mechanical solutions for soliton dynamics. To address this difficulty, we address a recently derived result for soliton equations, i.e. that all solutions can be written as quasiperiodic Fourier series. This means that soliton equations, in spite of their nonlinear solutions, are perfectly linearizable with quasiperiodic boundary conditions. We then invoke the result that soliton equations are Hamiltonian and we are able to show that the generalized coordinates and momenta are also quasiperiodic Fourier series, a generalized linear superposition law that is valid in the case of classical dynamics and is here extended to quantum mechanics. This simplification of Hamiltonian dynamics thus leads to matrix mechanics. This completes the main theme of our paper, i.e. that classical, nonlinear soliton field equations, linearizable with quasiperiodic Fourier series, can always be quantized in terms of matrix mechanics. Future work will be formulated in terms of the Schrödinger equation. This work guarantees that all classical, soliton field equations have an exact linearized form for their quantum mechanical properties.