Non-commutative quantum mechanics in 3-dimensional fuzzy space and its applications

Groenewald, Hendrikus Wilhelm (2018-03)

Thesis (PhD)--Stellenbosch University, 2018.

Thesis

ENGLISH SUMMARY: As the structure of space-time at very short length scales comparable to the Planck length remains a contentious issue, this study aims to determine whether non-commutative space-time would prove to be an appropriate candidate for space-time at short length scales. Also, it aims to test whether it would be possible to observe non-commutative e ects on macroscopic scales. By providing a formulation for quantum mechanics where fuzzy commutation relations for spatial coordinates are assumed, this study was able to make notable progress toward this goal. Exact solutions to the non-commutative free particle and spherical well problems were achieved and e ects exclusive to non-commutative systems were observed, in particular an upper bound on the kinetic energy of a particle and a nite number of bound states in the spherical well problem. Signi cant deviations from the normal behaviour of commutative systems were also observed in the study of scattering states of the fuzzy well where states with high incident energies experience the spherical well as a repulsive potential. Finally, thermodynamic studies of non-interacting fermions con ned in fuzzy space provided drastically di erent results for high energy, high particle density and low temperature systems in comparison to standard results. These results include an incompressibility limit for the fermion gas, as well as an apparent duality between high density and low density systems.

AFRIKAANSE OPSOMMING: Aangesien die struktuur van ruimtetyd op baie kort lengteskale, vergelykbaar met die Plancklengte, steeds 'n omstrede saak bly, beoog hierdie studie om vas te stel of nie-kommutatiewe ruimtetyd 'n gepaste kandidaat vir ruimte-tyd op kort lengte skale sal wees. Dit beoog ook om te toets of dit moontlik is om nie-kommutatiewe e ekte op makroskopiese skale waar te neem. Deur 'n formulering vir kwantummeganika te verskaf waar \fuzzy" kommutasieverbande vir ruimtelike ko ordinate aanvaar word, kon hierdie studie merkbare vordering maak in die rigting van hierdie doelwit. Eksakte oplossings vir die nie-kommutatiewe vrye deeltjies en sferiese put probleme is verkry en e ekte eksklusief tot nie-kommutatiewe stelsels is waargeneem, veral 'n bogrens op die kinetiese energie van 'n deeltjie en 'n eindige aantal gebonde toestande in die sferiese put probleem. Beduidende afwykings van die normale gedrag van kommutatiewe stelsels is ook waargeneem in die studie oor verstrooiingstoestande van die \fuzzy"-put waar toestande met ho e invalenergie e die sferiese put as 'n afstotende potensiaal ervaar. Ten slotte het termodinamiese studies van nie-wisselwerkende fermione inbeperk in \fuzzy"-ruimte drasties verskillende resultate vir ho e energie, ho e deeltjie digtheid en lae temperatuur stelsels gelewer in vergelyking met standaard resultate. Hierdie resultate sluit in 'n onsaampersbaarheidslimiet vir die fermiongas, sowel as 'n duidelike dualiteit tussen ho e- en laedigtheidstelsels.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/103520
This item appears in the following collections: