dc.contributor.advisor | Scholtz, F. G. | en_ZA |
dc.contributor.author | Williams, Paul Henry | en_ZA |
dc.contributor.other | Stellenbosch University. Faculty of Science. Dept. of Physics. | en_ZA |
dc.date.accessioned | 2015-12-14T07:43:41Z | |
dc.date.available | 2015-12-14T07:43:41Z | |
dc.date.issued | 2015-12 | |
dc.identifier.uri | http://hdl.handle.net/10019.1/97990 | |
dc.description | Thesis (MSc)--Stellenbosch University, 2015. | en_ZA |
dc.description.abstract | ENGLISH ABSTRACT: A brief review of non-relativistic and relativistic quantum mechanics is carried
out, with particular attention to concepts that are modified in the noncommutative
setting. The Hilbert-Schmidt operator formulation of non-relativistic,
non-commutative quantum mechanics is generalized to the relativistic
setting of 4-dimensional non-commutative space-time. The generators of
Lorentz transformations are derived in this formalism and compared with the
commutative case. It is shown that the non-interacting, non-commutative
Dirac equation is Lorentz invariant. Lorentz invariance can be maintained
when electromagnetic interactions are included provided that the gauge and
Dirac fields are composed through an appropriate star product. The appropriate
gauge transformations for the non-commutative Dirac equation are found.
The non-commutative C, P, T symmetries of the interacting Dirac equation
are investigated. The free Dirac equation and the interacting Dirac equation
for the case of a constant background magnetic field are studied on the operator
level. Systems confined to a specific space-time volume, and the associated
boundary conditions, are studied using appropriate projection operators. As
a specific example the Dirac equation in an infinitely long cylinder is considered.
An operator valued action, which yields the interacting Dirac equation
as the equation of motion, is derived and evaluated in a coherent state basis.
The constant background magnetic field is revisited in the coherent state
basis. This establishes the link to the standard star product formulation of
non-commutative quantum field theories. Useful properties of the star product
are derived in an appendix. | en_ZA |
dc.description.abstract | AFRIKAANSE OPSOMMING: 'n Kort hersiening van nie-relatiwistiese kwantummeganika op nie-kommutatiewe
ruimtes word gedoen, met die fokus op konsepte wat verander in die niekommutatiewe
raamwerk. Die Hilbert-Schmidt operator formulering van nierelatiwistiese,
nie-kommutatiewe kwantummeganika word veralgemeen na die
relatiwistiese raamwerk van 4-dimensionele nie-kommutatiewe ruimte-tyd. Die
generatore van Lorentz transformasies word herlei in hierdie fomulering en
word vergelyk met die kommutatiewe geval. Dit word aangetoon dat die vrye
nie-kommutatiewe Dirac vergelyking Lorentz invariant is. Lorentz invariansie
kan behou word, in die teenwoordigheid van elektromagnetiese wisselwerkings,
as die yk en Dirac velde met die gepaste ster produck saamgestel word. Die
regte yk transformasies van die nie-kommutatiewe Dirac vergelyking word gevind.
Die nie-kommutatiewe C, P, T simmetrieë van die Dirac vergelyking
met wisselwerkings word ondersoek. Die vrye Dirac vergelyking en die Dirac
vergelyking 'n konstante agtergrond magneetveld word opgelos op die operator
vlak. Stelsels beperk tot 'n spesifieke ruimte-tyd volume, en die geassosieerde
randvoorwaardes, word bestudeer met gepaste projeksie operatore. Die Dirac
vergelyking in 'n oneindig lang silinder word as 'n eksplisiete toepassing
aangebied. 'n Operator aksie wat die Dirac vergelyking met wisselwerkings as
bewegingvergelyking lewer, is gevind en in 'n koherentetoestand basis geëvalueer.
Die geval van 'n konstante agtergrond magneetveld is in die koherentetoestand
basis herondersoek. Dit vestig 'n verband met die standaard ster produk
formulering van nie-kommutatiewe kwantumveldeteorieë. Nuttige eienskappe
van die ster produk word in 'n bylaag herlei. | af_ZA |
dc.format.extent | vi, 65 pages | en_ZA |
dc.language.iso | en_ZA | en_ZA |
dc.publisher | Stellenbosch : Stellenbosch University | en_ZA |
dc.subject | Relativistic quantum mechanics | en_ZA |
dc.subject | Dirac equation | en_ZA |
dc.subject | Hall effect | en_ZA |
dc.subject | Electromagnetic interactions | en_ZA |
dc.subject | UCTD | en_ZA |
dc.title | Relativistic quantum mechanics on non-commutative spaces | en_ZA |
dc.type | Thesis | en_ZA |
dc.rights.holder | Stellenbosch University | en_ZA |