Is the C*-Algebraic Approach to Quantum Mechanics an Alternative Formulation to the Dominant One?

Abstract

Since 1947 a foundation of Quantum Mechanics (QM) on functional analysis was suggested by Segal. By defining the C*-algebra of the observables, then the Gelfand-Naimark-Segal theorem faithfully represents this algebra into Hilbert space. In the 70’s Emch has reiterated this formulation and improved it. Recently Strocchi improved it even more. First, he suggested an axiomatization of the paradigmatic Dirac-von Neumann’s formulation of QM to which he addresses two basic criticisms, i.e. a weak linkage with the experimental basis of theoretical physics and the obscurity about the separation mark between classical mechanics and QM. Afterwards, through an analysis of the experimental basis of a physical theory he suggests an explanation of Segal’s restriction of the operators to be bounded. Eventually, he represents this algebra into Hilbert space and at last, by means of Weyl algebra he obtains the symmetries of the dynamics of a particle theory. In fact, several characteristic features of this formulation correspond to those determined by the two choices which are the alternative ones to the choices of the dominant formulation. It is a problem-based theory, since it starts rather from than axioms a problem (i.e. the indeterminacy); then, it argues through both doubly negated propositions and an ad absurdum proof. Moreover, its theoretical development is similar to that of an alternative classical theory since it put, before the geometry, the algebra; the bounded operators are represented by a polynomial algebra; which pertains to constructive mathematics. Eventually, he obtains the symmetries of the theory. The problems to be overcome in order to accurately re-construct his formulation according to the two alternative choices which are listed. It is concluded that rather an alternative role, it plays a complementary role to the paradigmatic formulation.