Thesis (MSc (Physics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Due to the minimal length scale induced by non-commuting co-ordinates, it is not clear a priori what is meant by a position measurement on a non-commutative space. It was shown recently in a paper by Scholtz et al. that it is indeed possible to recover the notion of quantum mechanical position measurements consistently on the non-commutative plane. To do this, it is necessary to introduce weak (non-projective) measurements, formulated in terms of Positive Operator-Valued Measures (POVMs). In this thesis we shall demonstrate, however, that a measurement of position alone in non-commutative space cannot yield complete information about the quantum state of a particle. Indeed, the aforementioned formalism entails a description that is non-local in that it requires knowledge of all orders of positional derivatives through the star product that is used ubiquitously to map operator multiplication onto function multiplication in non-commutative systems. It will be shown that there exist several equivalent local descriptions, which are arrived at via the introduction of additional degrees of freedom. Consequently non-commutative quantum mechanical position measurements necessarily confront us with some additional structure which is necessary (in addition to position) to specify quantum states completely. The remainder of the thesis, based in part on a recent publication (\Noncommutative quantum mechanics { a perspective on structure and spatial extent", C.M. Rohwer, K.G. Zloshchastiev, L. Gouba and F.G. Scholtz, J. Phys. A: Math. Theor. 43 (2010) 345302) will involve investigations into the physical interpretation of these additional degrees of freedom. For one particular local formulation, the corresponding classical theory will be used to demonstrate that the concept of extended, structured objects emerges quite naturally and unavoidably there. This description will be shown to be equivalent to one describing a two-charge harmonically interacting composite in a strong magnetic eld found by Susskind. It will be argued through various applications that these notions also extend naturally to the quantum level, and constraints will be shown to arise there. A further local formulation will be introduced, where the natural interpretation is that of objects located at a point with a certain angular momentum about that point. This again enforces the idea of particles that are not point-like. Both local descriptions are convenient, in that they make explicit the additional structure which is encoded more subtly in the non-local description. Lastly we shall argue that the additional degrees of freedom introduced by local descriptions may also be thought of as gauge degrees of freedom in a gauge-invariant formulation of the theory. / AFRIKAANSE OPSOMMING: As gevolg van die minimum lengteskaal wat deur nie-kommuterende ko ordinate ge nduseer word is dit nie a priori duidelik wat met 'n posisiemeting op 'n nie-kommutatiewe ruimte bedoel word nie. Dit is onlangs in 'n artikel deur Scholtz et al. getoon dat dit wel op 'n nie-kommutatiewe vlak moontlik is om die begrip van kwantummeganiese posisiemetings te herwin. Vir hierdie doel benodig ons die konsep van swak (nie-projektiewe) metings wat in terme van 'n positief operator-waardige maat geformuleer word. In hierdie tesis sal ons egter toon dat 'n meting van slegs die posisie nie volledige inligting oor die kwantumtoestand van 'n deeltjie in 'n niekommutatiewe ruimte lewer nie. Ons formalisme behels 'n nie-lokale beskrywing waarbinne kennis oor alle ordes van posisieafgeleides in die sogenaamde sterproduk bevat word. Die sterproduk is 'n welbekende konstruksie waardeur operatorvermenigvuldiging op funksievermenigvuldiging afgebeeld kan word. Ons sal toon dat verskeie ekwivalente lokale beskrywings bestaan wat volg uit die invoer van bykomende vryheidsgrade. Dit beteken dat nie-kommutatiewe posisiemetings op 'n natuurlike wyse die nodigheid van bykomende strukture uitwys wat noodsaaklik is om die kwantumtoestand van 'n sisteem volledig te beskryf. Die res van die tesis, wat gedeeltelik op 'n onlangse publikasie (\Noncommutative quantum mechanics { a perspective on structure and spatial extent", C.M. Rohwer, K.G. Zloshchastiev, L. Gouba and F.G. Scholtz, J. Phys. A: Math. Theor. 43 (2010) 345302) gebaseer is, behels 'n ondersoek na die siese interpretasie van hierdie bykomende strukture. Ons sal toon dat vir 'n spesi eke lokale formulering die beeld van objekte met struktuur op 'n natuurlike wyse in die ooreenstemmende klassieke teorie na vore kom. Hierdie beskrywing is inderdaad ekwivalent aan die van Susskind wat twee gelaaide deeltjies, gekoppel deur 'n harmoniese interaksie, in 'n sterk magneetveld behels. Met behulp van verskeie toepassings sal ons toon dat hierdie interpretasie op 'n natuurlike wyse na die kwantummeganiese konteks vertaal waar sekere dwangvoorwaardes na vore kom. 'n Tweede lokale beskrywing in terme van objekte wat by 'n sekere punt met 'n vaste hoekmomentum gelokaliseer is sal ook ondersoek word. Binne hierdie konteks sal ons weer deur die begrip van addisionele struktuur gekonfronteer word. Beide lokale beskrywings is gerie ik omdat hulle hierdie bykomende strukture eksplisiet maak, terwyl dit in die nie-lokale beskrywing deur die sterproduk versteek word. Laastens sal ons toon dat die bykomende vryheidsgrade in lokale beskrywings ook as ykvryheidsgrade van 'n ykinvariante formulering van die teorie beskou kan word.

Thesis (PhD (Physics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: The aim of this thesis is threefold. Firstly, I give an overview of non-commutative quan- tum mechanics and build up a description of non-commutative piecewise constant poten- tial wells in this context. Secondly, I look at some of the stationary properties of a finite non-commutative well using the mathematical tools laid out in the first part. Lastly, I in- vestigate how non-commutativity affects the tunneling rate through a barrier. Throughout this work I give the normal commutative descriptions and results for comparsion. / AFRIKAANSE OPSOMMING: Die doel van hierdie tesis is drievoudig. Eerstens gee ek ’n oorsig van niekommutatiewe kwantummeganika en bou daarmee ’n beskrywing van niekommutatiewe deelswyskon- stante potensiaal putte op. Tweedens kyk ek na ’n paar van die stasionˆere eienskappe van ’n eindige niekommutatiewe potensiaal put deur die wiskunde te gebruik wat in die eerste deel uiteengesit is. Laastens ondersoek ek hoe niekommutatiwiteit die spoed van tonneling deur ’n potensiaal wal be¨ınvloed. Dwarsdeur die hierdie hele tesis gee ek die normale kommutatiewe beskrywings en resultate vir maklike vergelyking.

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.

Orientador: Prof. Dr. Vladislav Kupriyanov / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2015. / No presente trabalho estudamos as consequências físicas da não-comutatividade dependente da posição e rotacionalmente invariante em duas dimensões [x, y] = iq f (x2 + y2), usando a teoria de perturbações em mecânica quântica e considerando os modelos exatamente solúveis como o oscilador harmônico isotrópico e o problema de Landau. Nós demonstramos a consistência da abordagem proposta, em particular, derivamos a versão não-comutativa da equação de continuidade e mostramos que a probabilidade é conservada na nossa abordagem. Pesquisamos três formas gerais diferentes para a f (r): constante, monomial de r2 e exponencial Gaussiana. Obtendo resultados diversos de acordo com as características específicas de cada f (e. g. a potência do monomio, largura da Gaussiana). Para a maior parte das escolhas da f , temos encontrado quebra da degenerescência. / In the present work we study the physical consequences of the position dependent rotationally invariant noncommutativity in two dimensions [x, y] = iq f (x2 + y2), using the perturbation theory in quantum mechanics and considering the exactly solvable models in standard quantum mechanics: isotropic harmonic oscillator and Landau problem. We demonstrate the consistency of the proposed approach, in particular, we derive the noncommutative continuity equation and show that the probability is conserved in our approach. We investigate three different general forms of f (r): constant, monomial of r2 and Gaussian exponential. Obtaining diverse results according to specific characteristics of each f (e. g. monomial power and Gaussian width). Degeneracy breaking is found in most of the cases.

Este trabalho está dedicado a estudar a consistência global da dinâmica quântica de sistemas não-comutativos. Nosso ponto de partida é a teoria de sistemas vinculados, dado que esta provê uma descrição uni cada da dinâmica clássica e quântica para os modelos a serem investigados. Analisamos o problema relacionado com a existência da série de Born e unitariedade e focamos, na seqüência, na formulação funcional da dinâmica quântica dos sistemas não-comutativos. A compatibilidade entre as abordagens funcional e operatorial é substanciada de forma geral. Subseqüentemente, a transformada de Weyl generalizada de índice α é usada para implementar a de nição "via time-slicing" da integral de caminho no espaço de fase, o que nos permite calcular o correspondente propagador de Feynman. Como esperado, esta representação para o propagador de Feynman não é única, mas rotulada pelo parâmetro real α. Provamos que as contribuições dependentes de α desaparecem no limite quando o "slice" de tempo tende a zero, tal qual é requerido pela consistência da formulação. Esta prova é intrincada pois o Hamiltoniano envolve, necessariamente, produtos de operadores não comutantes. A anti-simetria da matriz que parametriza a não-comutatividade joga um papel fundamental no mecanismo de cancelamento dos termos dependentes de α. Por m, estudamos a implementação do processo formulado por Batalin, Fradkin e Tyutin (BFT), o qual permite transformar esses sistemas em uma teoria de calibre Abeliana exibindo apenas vínculos de primeira classe. A adequação da imersão BFT, como aplicada neste trabalho, é veri cada demonstrando que existe um mapeamento isomór co que conecta o modelo de segunda classe com o setor invariante de calibre da teoria de calibre. Como é sabido, a quantização funcional de uma teoria de calibre exige a eliminação da liberdade de calibre. Então, temos a nossa disposição um conjunto in nito de descrições alternativas para a mecânica quântica não-comutativa, uma para cada calibre. Estudamos as características relevantes deste in nito conjunto de correspondências. A quantização funcional da teoria de calibre é explicitamente realizada para dois calibres diferentes e os resultados comparados com o correspondente ao sistema de segunda classe. Dentro do quadro operatorial, a teoria de calibre é quantizada utilizando-se o método de Dirac. / This work is concerned with the global consistency of the quantum dynamics of noncommutative systems. Our point of departure is the theory of constrained systems, since it provides a uni ed description of the classical and quantum dynamics for the models under investigation. We then analise the problem concerned with the su cient conditions for the existence of the Born series and unitarity and turn, afterwards, into studying the functional quantization of non-commutative systems. The compatibility between the operator and the functional approaches is established in full generality. Subsequently, the generalized Weyl transform of index α is used to implement the time-slice de nition of the phase space path integral yielding the Feynman kernel in the case of noncommutative quantum mechanics. As expected, this representation for the Feynman kernel is not unique but labeled by the real parameter α. We succeed in proving that the α-dependent contributions disappear at the limit where the time slice goes to zero. This proof of consistency turns out to be intricate because the Hamiltonian necessarily involves products of noncommuting operators. The antisymmetry of the matrix parameterizing the noncommutativity plays a key role in the cancelation mechanism of the α-dependent terms. Finally, we study the embedding procedure formulated by Batalin, Fradkin and Tyutin (BFT) which enables one to transform these noncommutative systems into an Abelian gauge theory exhibiting only rst class constraints. The appropriateness of the BFT embedding, as implemented in this work, is veri ed by showing that there exists a one to one mapping linking the second class model with the gauge invariant sector of the gauge theory. As is known, the functional quantization of a gauge theory calls for the elimination of its gauge freedom. Then, we have at our disposal an in nite set of alternative descriptions for noncommutative quantum mechanics, one for each gauge. We study the relevant features of this in nite set of correspondences. The functional quantization of the gauge theory is explicitly performed for two di erent gauges and the results compared with that corresponding to the second class system. Within the operator framework the gauge theory is quantized by using Dirac's method.

Este trabalho está dedicado a estudar a consistência global da dinâmica quântica de sistemas não-comutativos. Nosso ponto de partida é a teoria de sistemas vinculados, dado que esta provê uma descrição uni cada da dinâmica clássica e quântica para os modelos a serem investigados. Analisamos o problema relacionado com a existência da série de Born e unitariedade e focamos, na seqüência, na formulação funcional da dinâmica quântica dos sistemas não-comutativos. A compatibilidade entre as abordagens funcional e operatorial é substanciada de forma geral. Subseqüentemente, a transformada de Weyl generalizada de índice α é usada para implementar a de nição "via time-slicing" da integral de caminho no espaço de fase, o que nos permite calcular o correspondente propagador de Feynman. Como esperado, esta representação para o propagador de Feynman não é única, mas rotulada pelo parâmetro real α. Provamos que as contribuições dependentes de α desaparecem no limite quando o "slice" de tempo tende a zero, tal qual é requerido pela consistência da formulação. Esta prova é intrincada pois o Hamiltoniano envolve, necessariamente, produtos de operadores não comutantes. A anti-simetria da matriz que parametriza a não-comutatividade joga um papel fundamental no mecanismo de cancelamento dos termos dependentes de α. Por m, estudamos a implementação do processo formulado por Batalin, Fradkin e Tyutin (BFT), o qual permite transformar esses sistemas em uma teoria de calibre Abeliana exibindo apenas vínculos de primeira classe. A adequação da imersão BFT, como aplicada neste trabalho, é veri cada demonstrando que existe um mapeamento isomór co que conecta o modelo de segunda classe com o setor invariante de calibre da teoria de calibre. Como é sabido, a quantização funcional de uma teoria de calibre exige a eliminação da liberdade de calibre. Então, temos a nossa disposição um conjunto in nito de descrições alternativas para a mecânica quântica não-comutativa, uma para cada calibre. Estudamos as características relevantes deste in nito conjunto de correspondências. A quantização funcional da teoria de calibre é explicitamente realizada para dois calibres diferentes e os resultados comparados com o correspondente ao sistema de segunda classe. Dentro do quadro operatorial, a teoria de calibre é quantizada utilizando-se o método de Dirac. / This work is concerned with the global consistency of the quantum dynamics of noncommutative systems. Our point of departure is the theory of constrained systems, since it provides a uni ed description of the classical and quantum dynamics for the models under investigation. We then analise the problem concerned with the su cient conditions for the existence of the Born series and unitarity and turn, afterwards, into studying the functional quantization of non-commutative systems. The compatibility between the operator and the functional approaches is established in full generality. Subsequently, the generalized Weyl transform of index α is used to implement the time-slice de nition of the phase space path integral yielding the Feynman kernel in the case of noncommutative quantum mechanics. As expected, this representation for the Feynman kernel is not unique but labeled by the real parameter α. We succeed in proving that the α-dependent contributions disappear at the limit where the time slice goes to zero. This proof of consistency turns out to be intricate because the Hamiltonian necessarily involves products of noncommuting operators. The antisymmetry of the matrix parameterizing the noncommutativity plays a key role in the cancelation mechanism of the α-dependent terms. Finally, we study the embedding procedure formulated by Batalin, Fradkin and Tyutin (BFT) which enables one to transform these noncommutative systems into an Abelian gauge theory exhibiting only rst class constraints. The appropriateness of the BFT embedding, as implemented in this work, is veri ed by showing that there exists a one to one mapping linking the second class model with the gauge invariant sector of the gauge theory. As is known, the functional quantization of a gauge theory calls for the elimination of its gauge freedom. Then, we have at our disposal an in nite set of alternative descriptions for noncommutative quantum mechanics, one for each gauge. We study the relevant features of this in nite set of correspondences. The functional quantization of the gauge theory is explicitly performed for two di erent gauges and the results compared with that corresponding to the second class system. Within the operator framework the gauge theory is quantized by using Dirac's method.

Este trabalho está dedicado a estudar a consistência global da dinâmica quântica de sistemas não-comutativos. Nosso ponto de partida é a teoria de sistemas vinculados, dado que esta provê uma descrição uni cada da dinâmica clássica e quântica para os modelos a serem investigados. Analisamos o problema relacionado com a existência da série de Born e unitariedade e focamos, na seqüência, na formulação funcional da dinâmica quântica dos sistemas não-comutativos. A compatibilidade entre as abordagens funcional e operatorial é substanciada de forma geral. Subseqüentemente, a transformada de Weyl generalizada de índice α é usada para implementar a de nição "via time-slicing" da integral de caminho no espaço de fase, o que nos permite calcular o correspondente propagador de Feynman. Como esperado, esta representação para o propagador de Feynman não é única, mas rotulada pelo parâmetro real α. Provamos que as contribuições dependentes de α desaparecem no limite quando o "slice" de tempo tende a zero, tal qual é requerido pela consistência da formulação. Esta prova é intrincada pois o Hamiltoniano envolve, necessariamente, produtos de operadores não comutantes. A anti-simetria da matriz que parametriza a não-comutatividade joga um papel fundamental no mecanismo de cancelamento dos termos dependentes de α. Por m, estudamos a implementação do processo formulado por Batalin, Fradkin e Tyutin (BFT), o qual permite transformar esses sistemas em uma teoria de calibre Abeliana exibindo apenas vínculos de primeira classe. A adequação da imersão BFT, como aplicada neste trabalho, é veri cada demonstrando que existe um mapeamento isomór co que conecta o modelo de segunda classe com o setor invariante de calibre da teoria de calibre. Como é sabido, a quantização funcional de uma teoria de calibre exige a eliminação da liberdade de calibre. Então, temos a nossa disposição um conjunto in nito de descrições alternativas para a mecânica quântica não-comutativa, uma para cada calibre. Estudamos as características relevantes deste in nito conjunto de correspondências. A quantização funcional da teoria de calibre é explicitamente realizada para dois calibres diferentes e os resultados comparados com o correspondente ao sistema de segunda classe. Dentro do quadro operatorial, a teoria de calibre é quantizada utilizando-se o método de Dirac. / This work is concerned with the global consistency of the quantum dynamics of noncommutative systems. Our point of departure is the theory of constrained systems, since it provides a uni ed description of the classical and quantum dynamics for the models under investigation. We then analise the problem concerned with the su cient conditions for the existence of the Born series and unitarity and turn, afterwards, into studying the functional quantization of non-commutative systems. The compatibility between the operator and the functional approaches is established in full generality. Subsequently, the generalized Weyl transform of index α is used to implement the time-slice de nition of the phase space path integral yielding the Feynman kernel in the case of noncommutative quantum mechanics. As expected, this representation for the Feynman kernel is not unique but labeled by the real parameter α. We succeed in proving that the α-dependent contributions disappear at the limit where the time slice goes to zero. This proof of consistency turns out to be intricate because the Hamiltonian necessarily involves products of noncommuting operators. The antisymmetry of the matrix parameterizing the noncommutativity plays a key role in the cancelation mechanism of the α-dependent terms. Finally, we study the embedding procedure formulated by Batalin, Fradkin and Tyutin (BFT) which enables one to transform these noncommutative systems into an Abelian gauge theory exhibiting only rst class constraints. The appropriateness of the BFT embedding, as implemented in this work, is veri ed by showing that there exists a one to one mapping linking the second class model with the gauge invariant sector of the gauge theory. As is known, the functional quantization of a gauge theory calls for the elimination of its gauge freedom. Then, we have at our disposal an in nite set of alternative descriptions for noncommutative quantum mechanics, one for each gauge. We study the relevant features of this in nite set of correspondences. The functional quantization of the gauge theory is explicitly performed for two di erent gauges and the results compared with that corresponding to the second class system. Within the operator framework the gauge theory is quantized by using Dirac's method.