Relativistic quantum mechanics on non-commutative spaces

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.

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.