Consistent Histories and Contrary Inferences

Consistent History Approach to quantum theory, introduced in explicit form by Griffiths [1], provides an extension of the interpretation of the formalism of quantum mechanics. While standard quantum theory is based on the concept of event, represented by a projection operator E of the Hilbert spaceH describing the system, CHA is based on the concept of history, which is a finite ordered sequence h = (E1, E2, ..., En) of events. Suitable families of histories can be selected by means of a criterion of consistency. According to CHA, only the histories of a consistent family have physical meaning. The occurrences of histories are the empirical facts the theory is concerned with. One of the aims of CHA is to solve some conceptual difficulties of standard quantum mechanics. Griffiths and Omnès [2] argue that the famous quantum measurement problem [3] is solved by the extended interpretation provided by CHA. However, the new theory raised some criticisms. One of the most debated problems is linked to the so called contrary inferences, illustrated in section 3 below. The debate provoked by this problem did not reach a shared conclusion. We think that a better clarity can be obtained if some natural concepts of CHA are better formalized, so that to allow a more transparent analysis of the debated questions. We do this by extending the formalism of the conceptual basis of CHA with the introduction of the notion of support. Starting from it, new tools can be developed allowing a deeper analysis of some questions. Also the problem of contrary inferences can be submitted to such an analysis. Our result is that contrary inferences do not entail conceptual difficulties. The plan of the paper is the following. In section 2 the basic concepts of CHA are outlined. The problem of contrary inferences is described is section 3. The notion of support of a family of histories is introduced in section 4, where the formalism stemming from this notion is developed. In section 4.2 some general implications of the extended formalism are derived. A deeper undestanding of CHA notion of compatible families is obtained in section 4.3. Finally, the problem of contrary inferences is analyzed in section 5 on the basis of the extended formalism, showing that the related difficulties are ruled out.