Scattering in soliton models and crossing symmetry
| dc.contributor.advisor | Weigel, Herbert | en_ZA |
| dc.contributor.author | Abdelhady, A. M. H. H. | en_ZA |
| dc.contributor.other | Stellenbosch University. Faculty of Science. Dept. of Physics. | en_ZA |
| dc.date.accessioned | 2012-10-04T08:48:26Z | en_ZA |
| dc.date.accessioned | 2012-12-12T08:08:36Z | |
| dc.date.available | 2012-10-04T08:48:26Z | en_ZA |
| dc.date.available | 2012-12-12T08:08:36Z | |
| dc.date.issued | 2012-12 | en_ZA |
| dc.description | Thesis (MSc)--Stellenbosch University, 2012. | en_ZA |
| dc.description.abstract | ENGLISH ABSTRACT: Crossing symmetry relates scattering and annihilation processes to each other. Its derivation is straightforward in perturbative approaches to quantum field theory: it merely reflects the exchange of in- and outgoing states in Feynman diagram computations. In soliton models, the situation is much more complicated because the scattering and the annihilation processes concern distinct topological sectors that are not related by any continuous transformation. In this thesis a simple soliton model will be employed to address this problem numerically. First, in the unit topological sector we extract asymptotically the phase shift of the scattering process of a wave packet off the kink-solution. To this end we solve the time-dependent equation of motion of the non-integrable '4 field model in (1+1) spacetime dimensions for two distinct initial conditions: the wave packet in a trivial vacuum background and in the background of the kink-solution. Second, in the topologically trivial sector we present numerical solutions of the kink– antikink interaction in the same model. We find that the final state of this interaction varies dramatically with the impact velocity. As result, we analyze our numerical solutions for the kink–antikink collisions system in two regimes. For the initial velocity of the system less than some critical velocity, vc 0:26, the kink and the antikink either annihilate or inelastically scatter. On the other hand, the kink and the antikink always inelastically scatter when the initial velocity of the system is higher than this critical velocity. However, the scattering processes of the kink–antikink with initial velocity below and above the critical velocity are different. Below the critical velocity the kink and the antikink collide and always undergo n-bounces (n 2) before they depart to infinity. When the initial velocity of the system is higher than vc, the kink and the antikink depart to infinity after only one bounce. We present a qualitative description for these bounce effects between the kink and the antikink motivated by earlier studies as well as our numerical simulations. We utilize collective coordinates to study the dynamics of the kink–antikink system in two degrees of freedom. In this regime, we modify the ansätze of the kink–antikink system from earlier studies to account for relativistic effects. We perform a comparison between this approximation and the full system. We end our discussion of this sector by discussing the scattering data for the inelastic scattering and the annihilation processes of the kink–antikink. Third, we compare the extracted scattering data for the scattering process of a wave packet off the kink-solution and the annihilation process of the kink–antikink to each other. Finally, these studies of different sectors allow us to make a conjecture about the validity of crossing symmetry within the non-integrable '4 field model. | en_ZA |
| dc.description.abstract | AFRIKAANSE OPSOMMING: Kruising-simmetrie beskryf ’n verband tussen verstrooiings- en vernietigingsprosesse. Die afleiding daarvan binne die raamwerk van steuringsteorie is eenvoudig: dit behels bloot die omruil van ingaande en uitgaande toestande in die Feynman-diagram. In soliton-modelle is die situasie egter meer ingewikkeld aangesien die verstrooiings- en vernietigingsprosesse in verskillende topologiese sektore plaasvind wat nie deur kontinue transformasies aan mekaar gekoppel is nie. In hierdie tesis word daar van ’n eenvoudige soliton-model gebruik gemaak om hierdie probleem numeries te ondersoek. Eerstens word die faseverskuiwing van die verstrooiingsproses van ’n golfpakkie vanaf ’n kinkoplossing asimptoties in die topologiese eenheidssektor bepaal. Vir hierdie doel word die tydafhanklike bewegingsvergelykings van die klassieke, nie-integreerbare 4-veldeteorie in (1+1) dimensionele ruimte-tyd opgelos. Twee beginkondisies word ondersoek: ’n golfpakkie in die triviale vakuum agtergrond asook in die kinkoplossing agtergrond. Tweedens ondersoek ons ook numeriese oplossings vir die kink-antikink wisselwerking binne die triviale topologiese sektor van dieselfde model. Hier vind ons dat die finale toestand van hierdie wisselwerkingsproses op ’n uiters sensitiewe wyse van die impaksnelheid afhang. Ons ondersoek gevolglik die numeriese oplossings vir die kink-antikink botsings in twee gebiede. Vir beginsnelhede onder die kritieke snelheid vc 0:26 sal die kink en antikink mekaar óf vernietig óf nie-elasties verstrooi. In teenstelling hiermee sal die kink-antikink altyd nie-elastiese verstrooiing ondergaan as die beginsnelheid die kritieke snelheid oorskry. Die aard van die verstrooiingsprosesse vir beginsnelhede bo en onder die kritieke snelheid is egter verskillend. Onder die kritieke snelheid sal die kink en antikink ’n n-bots proses (n 2) ondergaan voor hulle finaal van mekaar weg beweeg. Bo die kritieke snelheid sal die kink-antikink egter net ’n enkele botsing ondergaan en dan uitmekaar beweeg. Ons lewer ’n kwalitatiewe beskrywing vir die bons-effek tussen die kink en antikink wat deur vorige studies asook ons numeriese resultate gemotiveer word. Ons maak gebruik van ’n kollektiewe koördinaatstelsel om die dinamika van die kink-antikink in terme van twee vryheidsgrade te bestudeer. In hierdie gebied pas ons ook die ansatz vir die kink-antikink stelsel aan om relatiwistiese effekte in ag te neem. Ons vergelyk dan hierdie benadering met die oplossing van die volle sisteem. Die bespreking van hierdie sektor word afgesluit met ’n analise van die verstrooiingsdata vir die verstrooiing- en vernietingsprosesse van die kink-antikink. Derdens vergelyk ons die verstrooiingsdata vir die verstrooiing van ’n golfpakkie vanaf ’n kinkoplossing met die van die vernietigingsproses van die kink-antikink. Ons studie van die verskillende sektore laat ons dan toe om ’n vermoede te formuleer oor die geldigheid van kruissing-simmetrie binne die nie-integreerbare 4-model. | af_ZA |
| dc.format.extent | vii, 74 p. : ill. | |
| dc.identifier.uri | http://hdl.handle.net/10019.1/71664 | |
| dc.language.iso | en_ZA | en_ZA |
| dc.language.iso | en_ZA | en_ZA |
| dc.publisher | Stellenbosch : Stellenbosch University | en_ZA |
| dc.rights.holder | Stellenbosch University | en_ZA |
| dc.subject | Symmetry | en_ZA |
| dc.subject | Perturbation theory | en_ZA |
| dc.subject | Soliton models | en_ZA |
| dc.subject | Solitary waves | en_ZA |
| dc.subject | Dissertations -- Physics | en_ZA |
| dc.subject | Theses -- Physics | en_ZA |
| dc.subject.other | Physics | en_ZA |
| dc.title | Scattering in soliton models and crossing symmetry | en_ZA |
| dc.type | Thesis | en_ZA |