Completeness Criteria

Abstract

The aim of this chapter is to study the role of completeness of underlying inner product space in quantum measure theory. The research in this field has brought a rather surprising result: ”Natural” subspace structures of an inner product space, S, admit a nonzero completely additive measure if, and only if, S is complete. As a consequence, the probability theory on incomplete spaces is virtually empty and so the Hilbert space is the only inner product space on which a reasonable measure and probability theory can be built. This once again advocates the role of the (topological) completeness assumption in mathematical foundations of quantum theory and illustrates the unique position of the Hilbert space lattice in the realm of various ordered structures of subspaces that describe quantum systems. At the beginning of the effort to build mathematical formalism of quantum physics, as witnessed by the work by J. von Neumann and others, the completeness of inner product space was adopted for purely mathematical and technical reasons. In contrast to this, the results in this chapter will clearly show that completeness can be justified entirely on physical grounds. In fact, upon adopting the probability structure of quantum theory we automatically accept completeness as a consequence of the intrinsic character of quantum world.