Roberto Beneduci

Since its birth quantum mechanics has inspired new logical ideas. The most important “logical revolution” of the theory concerns a divergence between the concepts of maximal information and logically complete information. This is a natural consequence of Heisenberg’s uncertainty principle: any quantum pure state represents a maximal information that cannot be consistently extended to a richer knowledge about the physical system under consideration. At the same time, such information is logically incomplete, since it cannot decide all relevant properties of the system in question. Birkhoff and von Neumann’s quantum logic represents a natural logical abstraction from the algebraic structure of all events (that may occur to a quantum system $$\textbf{S}$$ ), mathematically represented as projection operators of the Hilbert space $$\varvec{\mathcal {H}}^{\textbf{S}}$$ associated to $$\textbf{S}$$. Projections of $$\varvec{\mathcal {H}}^{\textbf{S}}$$ represent sharp quantum events that satisfy the logical non-contradiction principle. Following Ludwig, the sharp concept of quantum event can be generalized to the unsharp concept of quantum effect, that gives rise to possible violations of the non-contradiction principle. The concepts of observable and of compatibility between observables have different behaviors in the framework of sharp and of unsharp quantum theory. Canonical examples of observables (say, position and momentum), that are incompatible in sharp quantum theory, turn out to be jointly measurable in the case of unsharp quantum theory. Interestingly enough, a famous thought experiment (proposed by Heisenberg) about the possibility of an approximate simultaneous measurement of position and momentum can be described and justified in the formal language of unsharp quantum theory.

A recent letter1 by van Kampen stimulated an interesting debate2–4 on the interpretation of quantum mechanics. The central theme of the debate was a criticism to the (also recent) presence in the literature of “voluminous discussions” about the interpretation of quantum mechanics. That there still exists such discussion is what van Kampen calls the “scandal of quantum mechanics.” We claim that a weak point in the debate was a lack of a definition of the term interpretation. In the present note, we would like to make this meaning precise and to show how such a clarification is necessary in order to avoid misunderstandings. The concept of interpretation plays a key role in the history of physics, so we hope the present analysis might be helpful to students. Max Jammer,5 in his famous book, The philosophy of quantum mechanics, distinguishes four different meanings of the word interpretation. In what follows, we restrict our attention to only three different meanings, which we now try to clarify.