’The Logic of Quantum Mechanics” appeared as the title of a scientific work in 1936, with the paper of Garrett Birkhoff and John von Neumann (9). The use of the same title for this volume outlines the fact that a great part of the subject matter we shall deal with pertains to a research field that originated in that historic paper. This title, however, is not to be interpreted as focusing on the propositional calculus that mirrors the structure of quantum mechanics, the so-called “quantum logic” with the word “logic” used in technical sense; rather, the title should be interpreted as focusing on the mathematical foundations of quantum mechanics. The complex edifice of this theory contains simpler substructures that have direct physical bases; each of them can explain some aspects of the behavior of quantum systems. This was also the idea of the classical books by G. W. Mackey (3), J. M. Jauch (3), and V. S. Varadarajan (3), which appeared in the sixties. The present volume includes results of more than a decade of active research which followed these classical works.

The volume is divided into three parts. The first contains an exposition of the basic formalism of quantum mechanics using the theory of Hilbert spaces and of linear operators in these spaces. We shall not follow the old tradition of striving to use concepts of classical mechanics to explain quantum facts (as in the so-called principle of correspondence or the wave-particle dualism)—a tradition that reflects the unusually long delay suffered by quantum mechanics before it acquired autonomy and internal coherence, breaking with the language of the theory to be superseded. The second part follows the program of decomposing quantum theory into its conceptual constituents, singling out the basic mathematical structures, isolating what may be founded on direct empirical evidence, and controlling how single assumptions contribute to shape the theory. In the third part we face the problem of recovering the Hilbert-space formulation of quantum mechanics, starting from the simpler and more general theoretical schemes examined in the second part.