Additional degrees of freedom associated with position measurements in non-commutative quantum mechanics

TL;DR

This thesis explores how position measurements in non-commutative quantum mechanics require additional degrees of freedom, revealing extended structures and gauge aspects necessary for complete quantum state description.

Contribution

It introduces local formulations with extra degrees of freedom to fully specify states in non-commutative quantum mechanics, linking to physical models of extended objects.

Findings

Local descriptions reveal extended, structured objects in non-commutative space

Additional degrees of freedom are essential for complete state specification

Connections to gauge invariance and physical models like two-charge systems

Abstract

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.