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.

Jan 1, 200018 citations

Applications to quantum physics

Stanisław Goldstein, Louis E. Labuschagne

Abstract The closing chapter details the application of noncommutative Orlicz spaces to Quantum Physics. We specifically show that the (Köthe dual) pair of Orlicz spaces Llog⁡(L+1) and Lcosh−1 may be better suited as home for regular states and observables respectively than the more standard pair of spaces L∞ and L∞. Specifically the geometry of the space L1∩Llog⁡(L+1) is shown to be well suited to identifying those states which admit a well-defined notion of entropy. In fact the geometry of this space may be used to posit a clear definition of entropy valid for the most general von Neumann algebras. Parallel to this the space Lcosh−1 proves to be home for possibly unbounded observables which nevertheless are very regular in the sense of for example having all their moments finite. It is further pointed out that in the case of B(H) this approach yields exactly the Dirac style elementary Quantum Mechanical formalism. The technology thus developed is also used to gain a clearer understanding of relative entropy of states. The fact that the space Lcosh−1 admits unbounded observables allows for an application to the necessarily unbounded field operators of Quantum Field Theory. We specifically show that under very natural criteria, a large number of quantum fields which satisfy a so-called generalised H-bound condition automatically also satisfy an Lcosh−1 regularity condition namely that one may canonically embed the field operators into Lcosh−1 - the natural superspace for the regular observables.

Jun 19, 20250 citations