Sylvia Pulmannova — Page 2

arXiv2008

Representation of State Property Systems

A 'state property system' is the mathematical structure which models an arbitrary physical system by means of its set of states, its set of properties, and a relation of 'actuality of a certain property for a certain state'. We work out a new axiomatization for standard quantum mechanics, starting with the basic notion of state property system, and making use of a generalization of the standard quantum mechanical notion of 'superposition' for state property systems.

math-ph↗

arXiv2007

Abstract Hermitian Algebras I. Spectral Resolution

We refer to the real Jordan Banach algebra of bounded Hermitian operators on a Hilbert space as a Hermitian algebra. We define an abstract Hermitian algebra (AH-algebra) to be the directed group of an e-ring that contains a semitransparent element, has the quadratic annihilation property, and satisfies a Vigier condition on pairwise commuting ascending sequences. All of this terminology is explicated in this article, where we launch a study of AH-algebras. Here we establish the fundamental properties of AH-algebras, including the existence of polar decompositions and spectral resolutions, and we show that two elements of an AH-algebra commute if and only if their spectral projections commute. We employ spectral resolutions to assess the structure of maximal pairwise commuting subsets of an AH-algebra.

math.FA↗