Boolean powers and quantum measurements (Q1317472)

This paper concerns the mathematical representation of coupled orthomodular posets, representing coupled quantum mechanical systems. In particular it concerns the case where one quantum system is coupled to one classical measuring device, and it is argued here that in this case an orthomodular poset \(L\) is coupled with a Boolean algebra \(B\), and the most appropriate representative structure of the composite system is a ``Boolean power of the orthomodular logic \(L\)'' \(L[B]\). Such a structure is defined here and the principal result established in this paper is that in this special case the Boolean power \(L[B]\) is an orthomodular poset with complete embeddings from \(L\) and from \(B\), into \(L[B]\) which meet certain structure-preserving conditions. In the latter part of the paper the earlier use of a Boolean power and the embeddings is applied to the specific issues of classical measuring devices, using the formalism developed by \textit{P. Busch}, \textit{P. J. Lahti} and \textit{P. Mittelstaedt} in: The quantum theory of measurement (Lect. Notes Phys., New Ser. 2) (1991), whereby a measurement is represented by a 5-tuple, defined in terms of the orthomodular `logic' \(L\) and the Boolean measuring system \(B\), states of the individual and composite systems, and a `pointer function' relating the observable associated with the measuring device to the observable actually measured. The paper ends by applying this analysis specifically to the cases of observables with discrete and to observables with continuous spectra.