Uniformization of Nonlinear Hamiltonian Systems of Vlasov and Hartree Type

TL;DR

This paper introduces a generalized uniformization method for nonlinear Hamiltonian systems like Vlasov and Hartree equations, extending second quantization concepts to algebraic structures and developing a functional calculus for observables.

Contribution

It generalizes the second quantization method to arbitrary Hamiltonian algebras and develops a unified functional calculus for uniformized observables.

Findings

Derived nonlinear kinetic equations for classical and quantum systems.

Developed a functional calculus unifying classical, Bosonian, and Fermionian variables.

Explored approximate solutions via linear uniformized equations.

Abstract

Nonlinear Hamiltonian systems describing the abstract Vlasov and Hartree equations are considered in the framework of algebraic Poissonian theory. The concept of uniformization is introduced; it generalizes the method of second quantization of classical systems to arbitrary Hamiltonian (Lie-Jordan, or Poissonian) algebras, in particular to the algebras of operators in indefinite spaces. A functional calculus is developed for the uniformized observables, which unifies the calculi of generating functionals for classical, Bosonian and Fermionian multiparticle variables. The nonlinear kinetic equation of Vlasov and Hartree type are derived for both classical and quantum multiparticle Hamiltonian systems in the mean field approximation. The possibilities for finding approximate solutions of these equations from the solutions of uniformized linear equations are investigated.