Symmetries in Quantum Mechanics

Bipin R. Desai

Jeffrey H. Boyd

Although quantum mathematics is the most successful science ever, that does not mean we live in the universe described by quantum mechanics. This article is entirely based on symmetry. Two symmetrical universes could have exactly the same mathematics, but differ in other respects. The motivation for seeking symmetry inside quantum mathematics is that the QM picture of nature is bizarre. Richard Feynman says no one can understand it. We propose that the quantum world is not bizarre. QM portrays the wrong universe: the symmetrical one, not the one we inhabit. If quantum waves travel in the opposite direction as what is expected, then we would have the same math but a different universe, one that is recognizable and familiar. Wave equations are symmetrical with respect to time reversal. This means they are symmetrical with respect to wave direction reversal (with time going forwards). This wave equation symmetry is the basis of the symmetry of two models of the universe, only one of which is congruent with the universe we inhabit.

Walter Greiner, Berndt Müller

Walter Greiner, Berndt Müller

Asao Arai

A fundamental symmetry principle in quantum mechanics is formulated in the framework of the standard axiomatic quantum mechanics and a new philosophical interpretation for quantum mechanics, which dissolves "difficulties" in the conventional interpretations for quantum mechanics, is presented. Moreover, philosophical phases of the fundamental symmetry principle are discussed in connection with Plato's philosophy and Oriental philosophies, in particular, Zen Buddhism.

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.