Axiomatic Approach to Quantum Mechanics

Luca Salasnich

Gianni Cassinelli, PEKKA J. LAHTI

P. Enders

Antonino Drago

Authoritative appraisals qualified this book as an axiomatic theory. However, being its essential content no more than an analogy, its theoretical organization cannot be an axiomatic one. In fact, in the first edition Dirac declares to avoid an axiomatic presentation. Moreover, I show that the text is aimed at solving a basic problem: How quantum mechanics is similar to classical mechanics. A previous paper analyzed all past theories of physics, chemistry and mathematics, presented by the respective authors in a non-axiomatic way. Four characteristic features of a new model of organizing a theory have been recognized. An accurate inspection of Dirac text proves that it actually applied this kind of organization of a theory. This fact gives a formal reason of what Kronz and Lupher suggested through intuitive categories, pragmatism and rigor, i.e. Dirac formulation of Quantum mechanics represents a distinct theoretical attitude from von Neumann axiomatic attitude. However, since the second edition Dirac changed his attitude; although relying again on an analogy, his theory was referred to the axiomatic method. Some considerations on the odd paths which led to the present formulation of QM are added. They suggest to look for a new foundation of this theory.

Honors Thesis

Yvon Gauthier

We discuss the Hilbert program for the axiomatization of physics in the context of what Hilbert and von Neumann came to call the analytical apparatus and its conditions of reality.We suggest that the idea of a physical logic is the basis for a physical mathematics and we use quantum mechanics as a paradigm case for axiomatics in the sense of Hilbert.Finite probability theory requires finite derivations in the measurement theory of QM and we give a polynomial formulation of local complementation for the metric induced on the topology of the Hilbert space.The conclusion hints at a constructivist physics.