Comments on Mielnik's generalized (non linear) quantum mechanics

William M. Cornette, Stanley Gudder

Convex prestructures, a mathematical framework that extends the usual concept of convex subset of a real linear space, are employed to generalize methods used in the study of axiomatic quantum mechanics. A brief summary of the mathematical framework of convex prestructures is given. Convex prestructures are classified, and those which are isomorphic to a convex subset of a real linear space are characterized. The operational quantum mechanics of Davies and Lewis is generalized within the framework of convex prestructures and the existence of a physically motivated orthomodular poset is given. Mielnik's beams and filters are also discussed within the framework of convex prestructures. An error in Mielnik's formulation of beam mixtures is pointed out and it is shown that his beam mixtures and one classification of convex prestructures are equivalent. Also his concept of a filter is generalized in the framework of convex prestructures and geometric requirements needed on the set of normalized states so that they may correspond to a physical system are investigated. Finally, Mackey's axioms are discussed and reformulated in the language of P-convex structures.

Diederik Aerts, Sven Aerts, Bart D’Hooghe +2 more

Jerzy Kijowski, Centrum Fizyki, Teoretycznej Pan

The criticism formulated by Mielnik and Torres-Vega in their paper, does not apply to my derivation of the time of arrival in Quantum Mechanics but to our present understanding of the quantum measurement theory.

Werner Stulpe

P. Nattermann

Motivated by the problems of interpretation of a nonlinear evolution equation in quantum mechanics we discuss in this contribution the concept of nonlinear gauge transformations, that has recently been introduced in joint work with Doebner and Goldin, in the framework of Mielnik's Generalized Quantum Mechanics. Using these gauge transformations we construct linear quantum systems in a ``nonlinear disguise'', and a gauge generalization of these (in analogy to the minimal coupling of orthodox quantum mechanics) leads to a unification of Bialynicki-Birula--Mycielski and Doebner--Goldin evolution equations for the quantum system. The notion of nonlinear observables introduced by Lücke is finally discussed in the same framework.