On Symmetry Operations in Quantum Mechanics

Rolf Hagedorn

Jos� F. Cari�ena, Mariano Santander

Geoffrey Brooker

Abstract “Symmetry in quantum mechanics” shows how symmetry properties are helpful in taming problems in quantum mechanics. When two operators commute, wavefunctions may be prefabricated as eigenstates of the simpler operator. There is detailed application to spin–orbit interaction and to helium.

P. Cordero, Sergio A. Hojman

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.

Carl M. Bender, Philip D. Mannheim

In nonrelativistic quantum mechanics and in relativistic quantum field theory, the time coordinate $t$ is a parameter and thus the time-reversal operator $\mathcal{T}$ does not actually reverse the sign of $t$. In contrast, in the five-dimensional approach to relativistic quantum mechanics introduced by Feynman, time $t$ is a quantum-mechanical operator. In this paper it is shown how one can use this five-dimensional approach to extend $\mathcal{T}$ and $\mathcal{P}\mathcal{T}$ symmetry from nonrelativistic to relativistic quantum mechanics and implement time-reversal as an operation that effects $\mathcal{T}\text{ }t\mathcal{T}=\ensuremath{-}t$ just as $\mathcal{P}$ effects $\mathcal{P}x\mathcal{P}=\ensuremath{-}x$, with $\mathcal{P}\mathcal{T}$ thus effecting $\mathcal{P}\mathcal{T}\text{ }{x}_{\ensuremath{\mu}}\mathcal{P}\mathcal{T}=\ensuremath{-}{x}_{\ensuremath{\mu}}$. Some illustrative relativistic quantum-mechanical models are constructed whose associated Hamiltonians are non-Hermitian but $\mathcal{P}\mathcal{T}$ symmetric, and it is shown that for each such Hamiltonian the energy eigenvalues are all real.