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.