Axioms for Non-Relativistic Quantum Mechanics

Abstract

In the approach to the axiomatization of quantum mechanics of George W. Mackey [7], a series of plausible axioms is completed by a final axiom that is more or less ad hoc. This axiom states that a certain partially ordered set — the set P of all two-valued observables — is isomorphic to the lattice of all closed subspaces of Hilbert space. The question arises as to whether this axiom can be deduced from others of a more a priori nature, or, more generally, whether the lattice of closed subspaces of Hilbert space can be characterized in a physically meaningful way. Our central result is a characterization of this lattice which may serve as a step in the indicated direction, although there is not now a precise sense in which our axioms are more plausible than his. Its principal features may be described as follows.

The work reported in this paper was performed while the author was a member of the staff of Lincoln Laboratory, Massachusetts Institute of Technology. He is now with the Arcon Corporation, Lexington, Massachusetts.