Thesis (MSc (Physics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Non-commutative quantum mechanics is a generalisation of quantum mechanics which incorporates
the notion of a fundamental shortest length scale by introducing non-commuting
position coordinates. Various theories of quantum gravity indicate the existence of such
a shortest length scale in nature. It has furthermore been realised that certain condensed
matter systems allow effective descriptions in terms of non-commuting coordinates. As a
result, non-commutative quantum mechanics has received increasing attention recently.
A consistent formulation and interpretation of non-commutative quantum mechanics,
which unambiguously defines position measurement within the existing framework of quantum
mechanics, was recently presented by Scholtz et al. This thesis builds on the latter
formalism, extends it to many-particle systems and links it up with non-commutative
quantum field theory via second quantisation. It is shown that interactions of particles,
among themselves and with external potentials, are altered as a result of the fuzziness
induced by non-commutativity. For potential scattering, generic increases are found for
the differential and total scattering cross sections. Furthermore, the recovery of a scattering
potential from scattering data is shown to involve a suppression of high energy
contributions, disallowing divergent interaction forces. Likewise, the effective statistical
interaction among fermions and bosons is modified, leading to an apparent violation of
Pauli’s exclusion principle and foretelling implications for thermodynamics at high densities. / AFRIKAANSE OPSOMMING: Nie-kommutatiewe kwantummeganika is ’n veralgemening van kwantummeganika wat die
idee van ’n fundamentele kortste lengteskaal invoer d.m.v. nie-kommuterende ko¨ordinate.
Verskeie teorie¨e van kwantum-grawitasie dui op die bestaan van so ’n kortste lengteskaal
in die natuur. Dit is verder uitgewys dat sekere gekondenseerde materie sisteme effektiewe
beskrywings in terme van nie-kommuterende koordinate toelaat. Gevolglik het die veld
van nie-kommutatiewe kwantummeganika onlangs toenemende aandag geniet.
’n Konsistente formulering en interpretasie van nie-kommutatiewe kwantummeganika,
wat posisiemetings eenduidig binne bestaande kwantummeganika raamwerke defineer, is
onlangs voorgestel deur Scholtz et al. Hierdie tesis brei uit op hierdie formalisme, veralgemeen
dit tot veeldeeltjiesisteme en koppel dit aan nie-kommutatiewe kwantumveldeteorie
d.m.v. tweede kwantisering. Daar word gewys dat interaksies tussen deeltjies en met
eksterne potensiale verander word as gevolg van nie-kommutatiwiteit. Vir potensiale verstrooi
¨ıng verskyn generiese toenames vir die differensi¨ele and totale verstroi¨ıngskanvlak.
Verder word gewys dat die herkonstruksie van ’n verstrooi¨ıngspotensiaal vanaf verstrooi¨ıngsdata
’n onderdrukking van ho¨e-energiebydrae behels, wat divergente interaksiekragte verbied.
Soortgelyk word die effektiewe statistiese interaksie tussen fermione en bosone verander,
wat ly tot ’n skynbare verbreking van Pauli se uitsluitingsbeginsel en dui op verdere gevolge
vir termodinamika by ho¨e digthede.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/5255 |
| Date | 12 1900 |
| Creators | Hafver, Andreas |
| Contributors | Scholtz, Frederik G., University of Stellenbosch. Faculty of Science. Dept. of Physics. |
| Publisher | Stellenbosch : University of Stellenbosch |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 85 p. |
| Rights | University of Stellenbosch |
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