Elements of Nonlinear Quantum Mechanics (II): Triple bracket generalization of quantum mechanics

An extension of quantum mechanics to a generalized Nambu dynamics leads to a new version of nonlinear quantum mechanics. The time evolution of states is given in here by a triple bracket generalization of the Liouville-von Neumann equation, where one of the generators is an average energy, and the other is a measure of entropy. A nonlinear evolution can occur only for mixed states, and for systems that are described by Rényi α-entropies with α 6= 2. The case α = 2 corresponds to ordinary, linear quantum mechanics. Since α = 2 entropy is the only entropy characterizing systems which cannot gain information, the nonlinear dynamics corresponds to “observers”, that is, systems that can gain information. The new formulation of nonlinear quantum mechanics is free from difficulties found in earlier attempts. The connection of linearity with possibilities of gaining information is in a striking agreement with the ideas of Wigner formulated in his paradox of a friend. 03.65Bz, Ca Typeset using REVTEX 1