Axioms of an experimental system (Q1805696)

The aim of this paper is to develop an axiomatic theory abstracting the key properties of an experimental system, specifically with quantum mechanics in mind. This axiomatisation is based on the notion of a direct product decomposition of Hilbert space. Thus for \(\mathcal H\) a Hilbert space associated with a quantum system, \(P_1, \dots ,P_n\) projection operators corresponding to \(n\) mutually exclusive and exhaustive outcomes of some experiment on this system, and \(v\) a vector in \(\mathcal H\); since the \(P_i\) are mutually orthogonal and span \({\mathcal H}\), any vector \(v\) can be written as a sum \(v = P_1v + \dots + P_nv\) and hence it is argued, each experiment generates a corresponding direct product decomposition \(\mathcal H = \mathcal H_1 \times\dots \times \mathcal H_n\). Much of what follows in this paper is in a sense a re-statement in terms of direct products, of results familiar from discussions of projections and vectors. Indeed the author claims early that ``In a sense the fundamental observation is that an additive structure is not needed to describe a process of superposition''. Thus ``logical'' operations are defined among binary experiments or ``questions'' in such a way as to have the properties of the operations of an orthomodular poset. ``Observable quantities'' are introduced in terms of Boolean subalgebras of this structure, and compatibility derived from this. Lastly probabilities are introduced and finally the analysis is applied to examples from quantum and classical mechanics. The paper is well organised and clearly written, with heuristic as well as technical discussion. It is perhaps a pity though that a wider perspective is missing, with no reference to the tradition of orthomodular posets in discussions of quantum mechanics, a tradition after all which goes back to von Neumann.