INTRODUCTION: Symmetry, Quantum Mechanics, Group Theory, and Special Functions in a Nutshell

Paul F. McMillan, Anthony C. Hess

Douglas C. Youvan

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.

P. Mc Millan, A. C. Hess

Florian A. Scheck

Bojan Bistrović

This article shows that one can consistently incorporate nonunitary representations of at least one group into the ``ordinary'' nonrelativistic quantum mechanics. This group turns out to be Lorentz group thus giving us an alternative approach to QFT for combining the quantum mechanics and special theory of relativity which keeps the concept of wave function (belonging to some representation of Lorentz group) through the whole theory. Scalar product has been redefined to take into the account the nonunitarity of representations of Lorentz group. Understanding parity symmetry turns out to be the key ingredient throughout the process. Instead of trying to guess an equation or a set of equations for some wave functions or fields (or equivalently trying to guess a Lagrangian for the same), one derives them based only on the superposition principle and properties of wave functions under Lorentz transformations and parity. The resulting model has striking similarities with the standard quantum field theory and yet has no negative energy states, no zitterbewegung effects, symmetric energy momentum tensor and angular momentum density tensor for \emph{all} representations of Lorentz group (unifying the theoretical description of all particles), as well as clear physical interpretation. It also offers a possible interpretation why particles and antiparticles have opposite quantum numbers.