  In this thesis we shall demonstrate that a measurement of position alone in
non-commutative space cannot yield complete information about the quantum state
of a particle. Indeed, the formalism used entails a description that is
non-local in that it requires all orders of positional derivatives through the
star product that is used ubiquitously to map operator multiplication onto
function multiplication in non-commutative systems. It will be shown that there
exist several equivalent local descriptions, which are arrived at via the
introduction of additional degrees of freedom. Consequently non-commutative
quantum mechanical position measurements necessarily confront us with some
additional structure which is necessary to specify quantum states completely.
The remainder of the thesis, will involve investigations into the physical
interpretation of these additional degrees of freedom. For one particular local
formulation, the corresponding classical theory will be used to demonstrate
that the concept of extended, structured objects emerges quite naturally and
unavoidably there. This description will be shown to be equivalent to one
describing a two-charge harmonically interacting composite in a strong magnetic
field found by Susskind. It will be argued that these notions also extend
naturally to the quantum level, and constraints will be shown to arise there. A
further local formulation will be introduced, with an interpretation in terms
of objects located at a point with a certain angular momentum about that point.
This again enforces the idea of particles that are not point-like. Both local
descriptions make explicit the additional structure which is encoded more
subtly in the non-local description. Additional degrees of freedom introduced
by local descriptions may also be thought of as gauge degrees of freedom in a
gauge-invariant formulation of the theory.
