author={Marek Czachor},
  journal={Physics Letters A},
  • M. Czachor
  • Published 16 January 1996
  • Mathematics, Physics
  • Physics Letters A
Abstract A nonlinear generalization of the Dirac equation extending the standard paradigm of nonlinear Hamiltonians is discussed. “Faster-than-light telegraphs” are absent for all theories formulated within the new framework. A new metric for infinite-dimensional Lie algebras associated with Lie-Poisson dynamics is introduced. 
A generalization of the Lax equation
Abstract We propose a generalization of the standard Lax equation defined by means of an arbitrary action of a Lie algebra on a matrix differential manifold. We analyze properties of obtained
Darboux integration of
Abstract A Darboux-type method of solving a class of nonlinear von Neumann equations i ρ =[H,f(ρ)] is developed. An explicit construction demonstrates that self-scattering solutions are a generic
Lie-Nambu and Beyond
Linear quantum mechanics can be regarded as aparticular example of a nonlinear Nambu-type theory.Some elements of this approach are presented.
Investigation of the time-dependent Schroedinger-Newton equation
The Schrödinger-Newton equation is a highly non-linear integro-partial differential equation that arises from both the theory of semiclassical gravity of Møller and Rosenfeld and as a possible
Vacuum Cherenkov effect in logarithmic nonlinear quantum theory
Abstract We describe the radiation phenomena which can take place in the physical vacuum such as Cherenkov-type shock waves. Their macroscopical characteristics – cone angle, flash duration,
Quantum morphogenesis: a variation on Thom's catastrophe theory.
Noncommutative propositions are characteristic of both quantum and nonquantum (sociological, biological, and psychological) situations. In a Hilbert space model, states, understood as correlations
Correlation experiments in nonlinear quantum mechanics
We show how one can compute multiple-time multi-particle correlation functions in nonlinear quantum mechanics in a way which guarantees locality of the formalism.
Informal Resource Letter - Nonlinear quantum mechanics on arXiv up to August 2004
I compiled a list of articles on arXiv that deal with possible fundamental quantum nonlinearities or examine the origins of its linearity. The list extends til August 2004.
Abstract DNA-type systems
It is explained why non-Kolmogorovian probability models occurring in soliton kinetics are naturally associated with chemical reactions and why fluctuations based on Darboux transformations will not destroy the dynamics but only switch between a finite number of helical structures.
Nambu-Lie 3-algebras on fuzzy 3-manifolds
We consider Nambu-Poisson 3-algebras on three dimensional manifolds 3, such as the Euclidean 3-space R3, the 3-sphere S3 as well as the 3-torus T3. We demonstrate that in the Clebsch-Monge gauge, the


Generalized Hamiltonian dynamics
Taking the Liouville theorem as a guiding principle, we propose a possible generalization of classical Hamiltonian dynamics to a three-dimensional phase space. The equation of motion involves two
Relation between Nambu and Hamiltonian mechanics
The connection between Nambu's generalization of classical dynamics and conventional Hamiltonian ideas is explored. In particular, the possibility of embedding the dynamics of a Nambu triplet in a
Quantum mechanics as a generalization of Nambu dynamics to the Weyl-Wigner formalism
Abstract It is shown that Nambu dynamics can be generalized to any number of dimensions by replacing the O(3) algebra, a prominent feature of Nambu's formulation, by an arbitrary Lie algebra. For the
Relativistic models of nonlinear quantum mechanics
I present and discuss a class of nonlinear quantum-theory models, based on simple relativistic field theories, in which the parameters depend on the state of the system via expectation values of
On a general nonlinear Schrödinger equation admitting diffusion currents
Abstract Some fundamental considerations of quantum theory suggest a general, complex nonlinear Schrodinger equation outside the classes most often studied. The equation follows from admitting
Geometrization of quantum mechanics
Quantum mechanics is cast into a classical Hamiltonian form in terms of a symplectic structure, not on the Hilbert space of state-vectors but on the more physically relevant infinite-dimensional
Probing Weinberg's nonlinear quantum mechanics
Abstract A search for nonlinear representations of propositions is made by assuming that the multiplication rule Weinberg uses to express the Lie bracket as a commutator can also be used to calculate
Classical Trajectories and Quantum Spectra
A classical model of the Schrodinger's wave packet is considered. The problem of finding the energy levels corresponds to a classical manipulation game. It leads to an approximate but
Dynamical symmetries and Nambu mechanics
It is shown that several Hamiltonian systems possessing dynamical or hidden symmetries can be realized within the framework of Nambu's generalized mechanics. Among such systems are the
Realization of Nambu mechanics: A particle interacting with an SU(2) monopole
We study the system of a particle bearing isospin degrees of freedom interacting with an SU(2) 't Hooft-Polyakov monopole. We show that its equation of motion can be cast into the form of Nambu's