Quantum Measurements in the Presence of Symmetry

This thesis concerns how symmetries impinge on quantum mechanical measurements, and preclude certain self adjoint operators from representing observable quantities. After developing the requisite mathematical machinery and aspects of quantum measurement theory necessary for our analysis, we proceed to critically review the literature surrounding the remarkable theorem of Wigner, Araki and Yanase (WAY) that prohibits accurate and repeatable measurements of any observable not commuting with an additive conserved quantity, as well as discussing the conditions under which approximate measurements with approximate degrees of repeatability can be achieved. We strengthen the original statement of the WAY theorem and generalise it to the case of position measurements obeying momentum conservation, leading to a solution of a long-standing problem of Stein and Shimony. A superselection rule appearing as the existence of an observable which commutes with all others gives rise to a stronger restriction than the WAY theorem, yielding self adjoint operators which do not represent observable quantities. We analyse various perspectives on superselection rules, aiming to clarify different viewpoints appearing in the literature since the inception of the topic in 1952. We exploit an explicit description of relative phase observables which have been lacking in other contributions, delineating conditions under which relative and (prohibited) absolute phases become statistically close. By providing simple models we are able to mimic a number of attempts to overcome superselection rules, in order to highlight the generic features of such attempts. We show that the statistical proximity of absolute and relative quantities arises only when there is a highly localised phase reference, and that the superselection rule compatible relative phase factors between certain superpositions takes on the appearence of a forbidden relative phase factor in this limit. However, we argue that these relative phase factors can be determined fully within the confines of a superselection rule.