Physics – Mathematical Physics
Scientific paper
1999-09-17
acta phys. slov. 50 (2000) 1-198
Physics
Mathematical Physics
213 pages, AMSLaTeX, published in: acta phys. slov., vol. 50 (February 2000), pp. 1-198; 4th revised version, corrected and co
Scientific paper
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 -- ``approximative'') theories. QM is here primarily equivalently reformulated in the 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 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: 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 - the extended quantum mechanics (EQM). A general interpretation scheme of EQM extending the usual statistical interpretation of QM is also propo sed. Eventually, EQM is shown to be included into a C*-algebraic (hence linear) quantum theory. Mathematical formulation of these theories is presented. Specifications and applications, as well as mathematical appendices and Indexes are included, making the text easily readable and almost selfcontained.
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