Classical mechanics as nonlinear quantum mechanics

All measurable predictions of classical mechanics can be reproduced from a quantum-like interpretation of a nonlinear Schrodinger equation. The key observation leading to classical physics is the fact that a wave function that satisfies a linear equation is real and positive, rather than complex. This has profound implications on the role of the Bohmian classical-like interpretation of linear quantum mechanics, as well as on the possibilities to find a consistent interpretation of arbitrary nonlinear generalizations of quantum mechanics.Comment: 7 pages, invited talk given at conference Quantum Theory: Reconsideration of Foundations 4, Vaxjo, Sweden, June 11-16, 200

Nambu Quantum Mechanics: A Nonlinear Generalization of Geometric Quantum Mechanics

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.Comment: 15 pages, LaTeX, typos correcte

Time Asymmetric Quantum Mechanics and Shock Waves: Exploring the Irreversibility in Nonlinear Optics

The description of irreversible phenomena is a still debated topic in quantum mechanics. Still nowadays, there is no clear procedure to distinguish the coupling with external baths from the intrinsic irreversibility in isolated systems. In 1928 Gamow introduced states with exponentially decaying observables not belonging to the conventional Hilbert space. These states are named Gamow vectors, and they belong to rigged Hilbert spaces. This review summarizes the contemporary approach using Gamow vectors and rigged Hilbert space formalism as foundations of a generalized “time asymmetric” quantum mechanics. We study the irreversible propagation of specific wave packets and show that the topic is surprisingly related to the problem of irreversibility of shock waves in classical nonlinear evolution. We specifically consider the applications in the field of nonlinear optics. We show that it is possible to emulate irreversible quantum mechanical process by the nonlinear evolution of a laser beam and we provide experimental tests by the generation of dispersive shock waves in highly nonlocal regimes. We demonstrate experimentally the quantization of decay rates predicted by the time-asymmetric quantum mechanics. This work furnishes support to the idea of intrinsically irreversible wave propagation, and to novel tests of the foundations of quantum mechanics

Lie-Nambu and beyond

Linear quantum mechanics can be regarded as a particular example of a nonlinear Nambu-type theory. Some elements of this approach are presented.Comment: revtex; an extended version of the talk given at the workshop "Actual problems in quantum mechanics", Peyresq, July, 199