Some Mathematical Foundations of Quantum Mechanics

Anders Wimo

Neumann, J. v., Wheeler, N. A.

Baptiste Schubnel

Alexander A. Pechenkin

Quantum logic as it is presented in Birkhoff-von Neumann’s 1936 paper arose within the framework of the mathematical elaboration of the foundations of quantum mechanics. Birkhoff-von Neumann’s axiomatic was an alternative of the traditional formulation of foundations of quantum mechanics proceeding from the technique of the Hilbert space. It provided more economic formulation of the foundations of the theory. In Mackay’s writings it has been transformed into the axiomatic of quantum theory of probability. However, Mackay came back to the concept of Hilbert space as the fundamental for quantum mechanics. He also adopted the rules of super-selections in his presentation of the foundations of quantum mechanics. The contemporary approach to Birkhoff-von Neumann’s axiomatic is twofold: the criticism of their too rigid construction from the point of view of physics and the development in connection with mathematical intuitionism.

Robert Golub, S. K. Lamoreaux

Abstract Von Neumann’s book, The Mathematical Foundations of Quantum Mechanics, is reviewed in its entirety. The absence of the Dirac delta function in von Neumann’s treatment of quantum mechanics is understood from the historical perspective. Basically, the delta function was not on a firm mathematical foundation at the time. Von Neumann introduced the concept of the density matrix and we discuss its application to his proof of “no hidden parameters (variables)” in quantum mechanics. His proof is often maligned as being incorrect. Von Neumann showed that the existence of hidden parameters would require a reworking of the entire structure of quantum mechanics.

A. Mangoni