arXiv:hep-th/9406174

Elements of nonlinear quantum mechanics (Part II): Triple bracket generalization of quantum mechanics

Marek Czachor

Published Jun 27, 1994• 2 citations

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".

DOI: 10.48550/arxiv.hep-th/9406174

Related papers

Triple bracket generalization of quantum mechanics

Marek Czachor

An extension of the Liouville-von Neumann dynamics to a Nambu-type dynamics is proposed. The resulting theory is the first version of nonlinear QM which is free from internal inconsistencies.

arXiv:quant-ph/9501008Jan 9, 19951 citation

Nambu mechanics and its quantization

Debendranath Sahoo, M. C. Valsakumar

The algebra of observables inherent in the Nambu formalism [Phys. Rev. D 7, 2405 (1973)] for a generalization of classical Hamiltonian dynamics is investigated. A consistency requirement of time evolution of the Nambu bracket leads to a five-point identity. Two types of algebras are possible at the classical level. Their composition properties under a tensor product are considered and the physical implications are analyzed. A quantum generalization of these algebras is shown to be impossible.

Physical Review AOct 1, 199244 citations

Nambu Quantum Mechanics: A Nonlinear Generalization of Geometric Quantum Mechanics

Minic, D, Hsiung Chia Tze

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.

Feb 25, 200231 citations

Lecture 3: Axioms, States, Observables, Measurement, Difficulties

Gianfausto Dell’Antonio

In “ Lecture 1: Elements of the History of Quantum Mechanics I” and “ Lecture 2: Elements of the History of Quantum Mechanics II” we have presented experimental results, theoretical analysis, models, analogies and insights that were accumulated for the description of physics at the scale of atoms. We described also the theoretical formulations that were developed in very few years and are the basis of a new theory for the dynamics of atomic systems: Quantum Mechanics.

Atlantis studies in mathematical physicsJan 1, 20150 citations

Quantum Mechanics as a Generalization of Nambu Dynamics to the Weyl-Wigner Formalism

Iwo Białynicki‐Birula, P. J. Morrison

Abstract It is shown that Nambu dynamics can be generalized to any number of dimensions by replacing the 0(3) algebra, a prominent feature of Nambu's formulation, by an arbitrary Lie algebra. For the infinite dimensional algebra of rotations in phase space one obtains quantum mechanics in the Weyl-Wigner representation from the generalized Nambu dynamics. Also, this formulation can be cast into a canonical Hamiltonian form by a natural choice of canonically conjugate variables.

Zeitschrift für Naturforschung AFeb 1, 19970 citations

Quantum deformations of the Heisenberg equations of motion

Alan Chodos, Daniel G. Caldi

The authors consider various formulations of what may be called q-quantum mechanics. One involves replacing the commutator that appears in the Heisenberg equations of motion by the 'quommutator': (A,B) q =qAB-(1/q)BA. They formally integrate the equations of motion, study some simple examples and discuss various apparent difficulties such as lack of conservation of energy, lack of unitarity and the unboundedness of the norm. They also consider another formulation in which the quantum bracket of the time derivative of an operator is used instead of just the time derivative, but the commutator is unchanged, in the Heisenberg equations of motion. This nonlinear treatment while preserving conservation of energy, has its own set of difficulties due to the nonlinearities. They also suggest that the quantum parameter may function as a kind of regulator for the time evolution of the system.

Journal of Physics A Mathematical and GeneralDec 7, 19917 citations