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

[摘要] ENGLISH ABSTRACT: Due to the minimal length scale induced by non-commuting co-ordinates, it is not clear a prioriwhat is meant by a position measurement on a non-commutative space. It was shown recently ina paper by Scholtz et al. that it is indeed possible to recover the notion of quantum mechanicalposition measurements consistently on the non-commutative plane. To do this, it is necessary tointroduce weak (non-projective) measurements, formulated in terms of Positive Operator-ValuedMeasures (POVMs). In this thesis we shall demonstrate, however, that a measurement of positionalone in non-commutative space cannot yield complete information about the quantum stateof a particle. Indeed, the aforementioned formalism entails a description that is non-local in thatit requires knowledge of all orders of positional derivatives through the star product that is usedubiquitously to map operator multiplication onto function multiplication in non-commutativesystems. It will be shown that there exist several equivalent local descriptions, which are arrivedat via the introduction of additional degrees of freedom. Consequently non-commutative quantummechanical position measurements necessarily confront us with some additional structurewhich is necessary (in addition to position) to specify quantum states completely. The remainderof the thesis, based in part on a recent publication (\Noncommutative quantum mechanics{ a perspective on structure and spatial extent, C.M. Rohwer, K.G. Zloshchastiev,L. Gouba and F.G. Scholtz, J. Phys. A: Math. Theor. 43 (2010) 345302) will involveinvestigations into the physical interpretation of these additional degrees of freedom. Forone particular local formulation, the corresponding classical theory will be used to demonstratethat 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 harmonicallyinteracting composite in a strong magneticeld found by Susskind. It will be argued throughvarious applications that these notions also extend naturally to the quantum level, and constraintswill be shown to arise there. A further local formulation will be introduced, where thenatural interpretation is that of objects located at a point with a certain angular momentumabout that point. This again enforces the idea of particles that are not point-like. Both localdescriptions are convenient, in that they make explicit the additional structure which is encodedmore subtly in the non-local description. Lastly we shall argue that the additional degrees offreedom introduced by local descriptions may also be thought of as gauge degrees of freedom ina gauge-invariant formulation of the theory.

[发布机构] Stellenbosch University