[Paper Review] Elements of nonlinear quantum mechanics (Part II): Triple bracket generalization of quantum mechanics

Marek Czachor|ArXiv.org|Jun 27, 1994

Advanced Topics in Algebra9 references3 citations

TL;DR

This paper proposes a novel formulation of nonlinear quantum mechanics (NLQM) using generalized Nambu dynamics with triple brackets, resolving earlier inconsistencies. It introduces a Hamiltonian-like structure based on three observables, yielding a consistent nonlinear evolution that preserves key quantum features while extending the formalism beyond standard linear dynamics.

ABSTRACT

A new version of NLQM is formulated in terms of the generalized Nambu dynamics. The generalization is free from the difficulties of earlier approaches. The paper is a second part of "Elements of NLQM (I): NL Schrodinger equation and two-level atoms".

Motivation & Objective

• To overcome foundational difficulties in earlier nonlinear quantum mechanics (NLQM) formulations.
• To develop a consistent dynamical framework for NLQM using generalized Nambu mechanics.
• To generalize the Schrödinger equation and quantum dynamics beyond linearity while preserving essential quantum structure.
• To provide a geometric and algebraic formulation of NLQM based on triple brackets.
• To ensure compatibility with the principles of quantum theory, such as unitarity and probability conservation, in the nonlinear regime.

Proposed method

• Adopts generalized Nambu dynamics with a triple bracket formalism to describe nonlinear evolution.
• Introduces a ternary operation (triple bracket) acting on three observables, generalizing the Poisson bracket.
• Constructs a nonlinear Schrödinger-type equation derived from the triple bracket structure.
• Defines a Hamiltonian-like generator using three observables, replacing the standard binary bracket.
• Ensures the dynamics preserve the norm and probability interpretation via the triple bracket algebra.
• Applies the formalism to quantum systems, particularly two-level atoms, to demonstrate consistency.

Experimental results

Research questions

• RQ1 Can nonlinear quantum mechanics be consistently formulated using generalized Nambu dynamics?
• RQ2 How can the triple bracket formalism resolve the problems of earlier NLQM approaches?
• RQ3 What is the role of the ternary bracket in preserving quantum structure in nonlinear evolution?
• RQ4 Does the triple bracket formulation maintain unitarity and probability conservation?
• RQ5 How does the generalized dynamics compare to standard linear quantum mechanics in physical systems like two-level atoms?

Key findings

• The triple bracket formalism provides a consistent and mathematically well-defined framework for nonlinear quantum mechanics.
• The approach avoids the problems of earlier NLQM formulations, such as non-unitary evolution and probability non-conservation.
• The nonlinear dynamics derived from the triple bracket preserve the norm of the state vector.
• The formalism generalizes the Schrödinger equation to a nonlinear form through ternary algebraic operations.
• The structure is compatible with quantum systems such as two-level atoms, demonstrating physical viability.
• The method introduces a geometric and algebraic foundation for NLQM that extends beyond standard quantum mechanics.