Nambu Quantum Mechanics: A Nonlinear Generalization of Geometric Quantum Mechanics

TL;DR

This paper introduces a nonlinear extension of geometric quantum mechanics inspired by Nambu dynamics, featuring elliptic function-based evolution and connecting to string theory, offering new insights into quantum structure and quantization.

Contribution

It presents a novel nonlinear quantum mechanics framework based on Nambu dynamics, extending geometric quantum mechanics with elliptic function evolution and links to string theory.

Findings

Recover standard quantum mechanics when elliptic moduli are zero.

Identify geometric features of the nonlinear quantum extension.

Suggest relevance to string theory and quantization processes.

Abstract

We propose a generalization of the standard geometric formulation of quantum mechanics, based on the classical Nambu dynamics of free Euler tops. This extended quantum mechanics has in lieu of the standard exponential time evolution, a nonlinear temporal evolution given by Jacobi elliptic functions. In the limit where latter's moduli parameters are set to zero, the usual geometric formulation of quantum mechanics, based on the Kahler structure of a complex projective Hilbert space, is recovered. We point out various novel features of this extended quantum mechanics, including its geometric aspects. Our approach sheds a new light on the problem of quantization of Nambu dynamics. Finally, we argue that the structure of this nonlinear quantum mechanics is natural from the point of view of string theory.