Lattice operations between observables in axiomatic quantum mechanics

Frederic W. Shultz

Maciej Mączyński

Stanley Gudder

Diederik Aerts

A. R. Marlow

An axiomatic mathematical structure is presented in terms of which arbitrary mathematical models of a physical system can be rigorously formulated and studied. Physical systems, observables, and states all arise in a natural way from the primitive notion of a physical operation. In particular, the structure is independent of the special mathematical completion defined by the quantum mechanical model, since no lattice theoretical or Hilbert space assumptions are used. The precise relationship between the present structure and the usual quantum mechanical model is investigated in a succeeding paper.

Wawrzyniec Guz

Abstract The paper, summarizing the results of recent papers of the author, is devoted to a detailed analysis of postulates of nonrelativistic quantum theory appearing within two classical axiomatic frameworks the “quantum logic” and the “algebraic approach” respectively, which are presently the two main alternatives for quantum axiomatics. The first part of the paper concerns the structure of two important sets, the set of questions (propositional logic) and the set of pure states. It is shown how the quantum logic axiomatic scheme can be modified and improved in order to overcome the well known troubles connected with the lack of satisfactory physical justification for some postulates of this approach. In particular, an axiom system is developed, related closely to the quantum logic axiomatic scheme, in which the questions of the complete lattice structure, the atomisticity, and the validity of the covering law in the propositional logic do not appear so problematic. The particular attention is directed to the covering law, which is here obtained as a consequence of physically clear properties of the experimental procedures (“filters”) associated with the quantum‐mechanical propositions. In Section 1.5 we formulate an axiom system for quantum mechanics based exclusively on the concepts of pure state, transition probability, and pure filter. In the second part of the paper the axiomatic scheme presented in Part 1 is developed with the aim to analyse the algebraic structure of some subspaces of the set of simple observables. The quantum logic axiomatic framework is here shown closely connected with the Jordan‐Banach algebraic scheme. In the third part of the paper, similarly as it was done in Parts 1 and 2, the correspondence between the propositions and the experimental (measuring) procedures verifying these propositions is put as the basic assumption of the axiomatization of quantum mechanics. As a consequence of this assumption, there is established the structure of a real Jordan‐Banach algebra in the set Ob of the bounded observables associated with a physical system, so that we can apply the GNS representation theorem proved recently for such algebras by Alfsen et al. (1978) to obtain the Hilbert space (or, to be more precise, the C*‐algebraic) representation for Ob.