Positive Operator Measures, Generalised Imprimitivity Theorem and Their Applications

Introduction In the common textbook presentation of quantum mechanics the observables of a quantum system are represented by selfadjoint operators, or, equivalently , by spectral measures. The origin of this point of view dates back to the very beginning of quantum theory. Its rigorous mathematical formulation is mainly due to von Neumann [53], and, for a more recent and complete review, we refer to the book of Varadarajan [52]. But it is quite well known that, when particular quantum systems are considered, this approach can not give a satisfactory description of some of their physical properties. A famous example (dating back to Dirac [28]) is provided by the phase of the electromagnetic field, which is a well defined quantity in classical physics, but can not be described by any selfadjoint operator in quantum mechanics [38], [40]. This drawback in the conventional formulation of quantum theory becomes even more evident when one attempts to define a position observable for the photon. In fact, a theorem of Wightman ([56], [52]) asserts that there does not exists any selfadjoint operator describing the localisation property of the photon. Positive operator measures were introduced just to overcome difficulties of this kind arising from the von Neumann formulation of quantum theory. Quite soon after their introduction, it became also clear [26], [38], [40], [47] that the most general description of the observables of quantum mechanics is provided by positive operator measures rather than by spectral measures. In this extended setting, the phase observable and the localisation observable for the photon are associated to positive operator measures that are not spectral maps, and thus can not be represented by any selfadjoint operator. It also turned out that joint measurements of quantities which are incompatible in the von Neumann framework can be described in terms of positive operator measures (an example is provided in section 5.2). The aim of this thesis is to give a complete characterisation of an important class of positive operator measures, namely the positive operator measures that are covariant with respect to unitary representations of a group. Indeed, in quantum physics the observables that describe a particular physical quantity are defined by means of their property of covariance with respect to a specific symmetry group. Thus, covariant positive operator measures naturally acquire a privileged role. 3 Since the characterisation of covariant positive operator measures requires a few deep results from abstract harmonic analysis and …