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Lorentz symmetry in non-commutative field theories : commutative/non-commutative dualities and manifest covariance
Thesis (PhD)--Stellenbosch University, 2020.
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University.
2020
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| _version_ | 1867614015738347520 |
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| access_status_str | Open Access |
| author | Williams, Paul Henry |
| author2 | Scholtz, Frederik G. |
| author_browse | Scholtz, Frederik G. Williams, Paul Henry |
| author_facet | Scholtz, Frederik G. Williams, Paul Henry |
| author_sort | Williams, Paul Henry |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University. |
| description | Thesis (PhD)--Stellenbosch University, 2020. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/107984 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:45:19.124Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2020 |
| publishDateRange | 2020 |
| publishDateSort | 2020 |
| publisher | Stellenbosch : Stellenbosch University. |
| publisherStr | Stellenbosch : Stellenbosch University. |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/107984 Lorentz symmetry in non-commutative field theories : commutative/non-commutative dualities and manifest covariance Williams, Paul Henry Scholtz, Frederik G. Kriel, Johannes N. Stellenbosch University. Faculty of Science. Dept. of Physics. Lorentz equations Field theory (Physics) Commutative and non-commutative field theories Commutative relations (Quantum mechanics) Symmetry (Physics) Scalar field theory Space and time Analysis of covariance UCTD Thesis (PhD)--Stellenbosch University, 2020. ENGLISH ABSTRACT: We cover two approaches to building Lorentz invariant non-commutative field theories. First we construct dualities between commutative and non-commutative field theories. This construction exploits a generalization of the exact renormalization group equation (ERG). We review ERG dualities for the two dimensional quantum mechanical Landau problem. We also review the idea of non-canonical field theories. From this we build an ERG duality for the free non-canonical complex scalar field theory. This approach allows us to track the Lorentz symmetry, we show this explicitly for the free theory. Finally we construct dualities for the φ4 interacting theory. Second we build a manifestly Lorentz invariant 2 + 1 dimensional field theory living in SU(1, 1) fuzzy space-time. Here the commutation relations themselves respect the Lorentz symmetry. We start by reviewing the nonrelativistic construction of SU(2) fuzzy space and quantum mechanics. We briefly discuss a potential field theory extension of this. We then begin our SU(1, 1) construction by reviewing the SU(1, 1) group and make the connection to space-time. We then build a quantum theory living on this space-time and introduce dynamics with the Klein-Gordon equation. From this we can move on to a scalar field theory in terms of functions on the group manifold. Finally we make the connection to commutative theories by introducing the symbols of operators. From this we are able to compute different correlators and compare with commutative theories. AFRIKAANSE OPSOMMING:Ons beskryf twee benaderings tot die konstruksie van Lorentz invariante nie-kommutatiewe veldeteorieë. Eerstens beskryf ons dualiteite tussen kommutatiewe en nie-kommutatiewe veldeteorieë. Hierdie konstruksie berus op ’n veralgemening van die Eksakte Renormerings Groep (ERG). Ons hersien dualiteite vir die twee-dimensionele kwantum meganiese Landau probleem. Dit word opgevolg deur ’n hersiening van nie-kanoniese veldeteorieë. Met dit as basis, konstrueer ons ’n ERG dualiteit vir die vrye nie-kanoniese komplekse skalare veld. Hierdie benadering stel ons in staat om die Lorentz simmetrie na te volg, en dit word eksplisiet vir die vrye teorie gedemonstreer. Ten slotte, konstrueer ons dualiteite vir die φ4 wisselwerkende teorie. Tweedens, konstrueer ons ’n eksplisiete Lorentz invariante 2 + 1 dimensionele veldeteorie in SU(1,1) wasige ruimte-tyd. In hierdie geval respekteer die kommutasieverbande self die Lorentz simmetrie. Ons begin met ’n hersiening van die nie-relativistiese konstruksie van wasige ruimte en die formulering van kwantum meganika op die ruimtes. Ons bespreek dan kortliks ’n veldteoretiese veralgemening hiervan. Ons begin dan met die SU(1,1) konstruksie deur die hersiening van die SU(1,1) groep en die konneksie met ruimte-tyd. Ons konstrueer dan ’n kwantum veldeteorie op die ruimte-tyd en voer dinamika in deur die Klein-Gordon vergelyking. Met dit as basis, kan ons dan ’n vrye skalare teorie opbou in terme van funksies op die groep manifold. Ten slotte maak ons die verbintenis met kommutatiewe toerië deur die invoering van die simbole van operatore. Die gebruik hiervan stel ons in staat om verskillende korrelators te bereken en met kommutatiewe teorieë te vergelyk. Doctoral 2020-02-19T06:26:01Z 2020-04-28T12:12:52Z 2020-02-19T06:26:01Z 2020-04-28T12:12:52Z 2020-04 Thesis http://hdl.handle.net/10019.1/107984 en_ZA Stellenbosch University. viii, 100 pages application/pdf Stellenbosch : Stellenbosch University. |
| spellingShingle | Lorentz equations Field theory (Physics) Commutative and non-commutative field theories Commutative relations (Quantum mechanics) Symmetry (Physics) Scalar field theory Space and time Analysis of covariance UCTD Williams, Paul Henry Lorentz symmetry in non-commutative field theories : commutative/non-commutative dualities and manifest covariance |
| title | Lorentz symmetry in non-commutative field theories : commutative/non-commutative dualities and manifest covariance |
| title_full | Lorentz symmetry in non-commutative field theories : commutative/non-commutative dualities and manifest covariance |
| title_fullStr | Lorentz symmetry in non-commutative field theories : commutative/non-commutative dualities and manifest covariance |
| title_full_unstemmed | Lorentz symmetry in non-commutative field theories : commutative/non-commutative dualities and manifest covariance |
| title_short | Lorentz symmetry in non-commutative field theories : commutative/non-commutative dualities and manifest covariance |
| title_sort | lorentz symmetry in non commutative field theories commutative non commutative dualities and manifest covariance |
| topic | Lorentz equations Field theory (Physics) Commutative and non-commutative field theories Commutative relations (Quantum mechanics) Symmetry (Physics) Scalar field theory Space and time Analysis of covariance UCTD |
| url | http://hdl.handle.net/10019.1/107984 |
| work_keys_str_mv | AT williamspaulhenry lorentzsymmetryinnoncommutativefieldtheoriescommutativenoncommutativedualitiesandmanifestcovariance |