Linear Quantum Theory and Its Possible Nonlinear Generalizations

Yi‐Fang Chang

First, we point out that the present applied superposition principle is linear, it must be developed into a generality. Next, the linear operators and equations should be developed nonlinearly. They will include nonlinear Klein-Gordon equation and Dirac equations, and corresponding Heisenberg equation. The quantum commutation and anticommutation should be developed. This theory may include the renormalization, which is the correction of Feynman rules of curved closed loops. We think the interaction equations are nonlinear. Many theories, models and phenomena are all nonlinear, for instance, soliton, nonabelian gauge field, and the bag model, etc. The superluminal entangled state, which relates the nonlocal quantum teleportation and nonlinearity, should be a new fifth interaction. Moreover, the NL effects exist possibly for four interactions, for single particle, for high energy, and for small space-time, etc. The relations among NL theory and electroweak unified theory, and QCD, and CP nonconservation, etc., are expounded. Finally, some known and possible tests are discussed. The NL theory relates the possible decrease of entropy in isolated system.

Dieter Schuch

J. R. Croca

Fernando Dantas Nobre, M. A. Rego-Monteiro, Constantino Tsallis

An exact classical field theory, for a recently proposed nonlinear generalization of the Schrödinger equation, is presented. In this generalization, a nonlinearity depending on an index q appears in the kinetic term, such that the free-particle linear Schrödinger equation is recovered in the limit q →1. It is shown that besides the usual , a new field must be introduced, which becomes only when q →1. In analogy to the linear case, is interpreted as the probability density for finding the particle at time t , in a given position inside an arbitrary finite volume V , for any q . The possible physical consequences are discussed, and, in particular, the important property that the fields and do not interact with light.

Ronald J. Adler, David Santiago

B Pavel

The work can be considered as an essay on mathematical and conceptual structure of nonrelativistic quantum mechanics (QM) which is related here to some other (more general, but also to more special and “approximative”) theories. QM is here primarily reformulated in an equivalent form of a Poisson system on the phase space consisting of density matrices, where the “observables”, as well as “symmetry generators” are represented by a specific type of real valued (densely defined) functions, namely the usual quantum expectations of corresponding selfadjoint operators. It is shown in this paper that inclusion of additional (“nonlinear”) symmetry generators (i.e. “Hamiltonians”) into this reformulation of (linear) QM leads to a considerable extension of the theory: two kinds of quantum “mixed states” should be distinguished, and operator ‐ valued functions of density matrices should be used in the r ˆ ole of “nonlinear observables”. A general framework for physical theories is obtained in this way: By different choices of the sets of “nonlinear observables” we obtain, as special cases, e.g. classical mechanics on homogeneous spaces of kinematical symmetry groups, standard (linear) QM, or nonlinear extensions of QM; also various “quasiclassical approximations” to QM are all subtheories of the presented extension of QM - a version of the extended quantum mechanics (EQM). A general interpretation scheme of EQM extending the usual statistical interpretation of QM is also proposed. Eventually, EQM is shown to be (included into) a C -algebraic (hence linear) quantum theory. Mathematical formulation of these theories is presented. The presentation includes an analysis of problems connected with differentiation on infinite ‐ dimensional manifolds, as well as a solution of some problems connected with the work with only densely defined unbounded real‐valued functions on the (infinite dimensional) “phase space” which correspond to unbounded operators (generators) and to their nonlinear generalizations. Also “nonlinear deformations” of unitary representations of kinematical symmetry Lie groups are introduced. Possible applications are briefly discussed, and some specific examples are presented. The text contains also brief reviews of Hamiltonian classical mechanics, as well as of QM. Mathematical appendices make the paper nearly selfcontained.