Extended Quantum Mechanics

Abstract

We are dealing in this work with such formal and conceptual extensions of nonrelativistic quantum mechanics (QM) which contain QM with its standard formalism and interpretation as a subtheory. QM is here primarily equivalently reformulated in the form of a Poisson system on the infinite-dimensional phase space consisting of all density matrices. It is shown 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 role of "nonlinear observables". A general framework for physical theories is obtained in this way, containing various nonlinear versions of QM, as well as some of "approximations" to standard QM as subtheories. Questions of interpretation are considered and formally consistently solved. Specifications and illustrations, as well as mathematical Appendices and two functioning Indexes are included, making the text almost selfcontained.