An Axiomatic Basis for Quantum Mechanics

Remarks on Unsharp Quantum Observables, Objectification, and Modal Interpretations

“…Chapter VI of [1]). It has been argued on general consistency grounds that macroscopic observables displaying permanent definite values are to be described by POV measures that are not PV measures [3]. More precisely, the notion of quantities having definite values at all times, as it is described in classical theories typically applying to macroscopic objects, can only be represented approximately within a quantum mechanical many-body theory, and the quantum representatives of such quantities are non-PV POV measures.…”

Spectrum of quantum dynamical systems: Subsystems and entropy

“…As the set of noncompatible events does not ful l the axioms of Boolean algebra, Boolean algebras are replaced by orthomodular lattices or posets ( [6,22]). If a quantum mechanical system F is represented in the usual way by a Hilbert space H, then self-adjoint operators on H satisfying 0 ≤ ≤ correspond to e ects for F ( [18,19]). E ects are of signi cance in representing unsharp measurements or observations on the system F ( [5]), and e ect valued measures play an important role in stochastic quantum mechanics ( [1,23]).…”

Derivation of quantum theory with superselection rules

“…However, as he pointed out "I do not believe in Hilbert space anymore" [2], it is thus noteworthy that this formulation is far from a clear understanding of the physical structure of quantum theory. For a proper understanding of the reason why the complex Hilbert space (or C * -algebraic) formalism is relevant in describing the microscopic world, Birkoff and von Neumann [3], Zierler [4], Mackey [5], Jauch and Piron [6], Ludwig [7], and many other researchers have investigated the reconstruction of the mathematical structure of quantum theory (simply referred to as the derivation of quantum theory) from physically meaningful principles.…”