Coordinate and Momentum Observables in Axiomatic Quantum Mechanics

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Abstract
An axiomatic model for quantum mechanics is formulated using physically significant axioms. The model contains a slight strengthening of Mackey's first six axioms, together with two axioms which ensure the existence of coordinate and momentum observables. The symmetries or rigid motions are an essential part of the structure, and a link is constructed between these and the quantum proposition system. Coordinate and momentum observables are defined in terms of abstract coordinate systems and one‐parameter groups of motions. It is then shown that as far as the statistical properties of these observables in certain canonical states are concerned, the abstract model may be represented by the usual Hilbert space formulation. Spectral properties of σ‐homomorphisms are also investigated. Also included are two appendices of a technical nature: the first considers one‐parameter groups of derivables, and the second absolutely continuous σ‐homomorphisms.
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