The C* Axioms and the Phase Space Fomalism of Quantum Mechanics

Abstract

In 1972, “Algebraic Methods in Statistical Mechanics and Quantum Field Theory” was published by Gerard Emch [8], giving the axioms for a physical system in an algebraic setting using the language of Irving Segal [19], and then continuing to obtain the C*-algebraic formalism for a physical system. Of these axioms, only the fifth contained an assumption that was questionable in its physical content. Bearing in mind that for each observable A and state φ, one obtains a distribution of values for the observed results of measurement, we have:

Axiom 5: For any element A in the set of observables $\mathfrak{A}$ and any non-negative integer n, there is at least one element, denoted Aⁿ, in $\mathfrak{A}$ such that (i) the set of dispersion-free states for Aⁿ is contained in the set of dispersion-free states for A, (ii) < φ; Aⁿ > = < φ; A >ⁿ for all φ in the set of dispersion-free states for A. (Here < φ; B > is the expectation of B in state φ.)

It is proven that the phase space localization operators on the Hilbert spaces of ordinary quantum mechanics provide a set of operators that are physically motivated and form a C* algebra. Then, it is proven that the set of localization operators, when extended, are informationally complete in the original Hilbert spaces.

Dedicated to G. G. Emch.