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Reps for JIMWLK: applications of representation theory to a novel approach to the JIMWLK equation
In recent work, R. Moerman and H. Weigert have introduced a truncation scheme for the Balitsky hierarchy, arguing that this is the most general possible method for obtaining finite Nc approximate solutions to the JIMWLK equation, while ensuring that these solutions obey several key properties that a...
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| Format: | Thesis |
| Language: | English |
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Department of Physics
2019
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| _version_ | 1867614521091162112 |
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| access_status_str | Open Access |
| author | Rayner, Jonathan |
| author2 | Weigert, Heribert |
| author_browse | Rayner, Jonathan Weigert, Heribert |
| author_facet | Weigert, Heribert Rayner, Jonathan |
| author_sort | Rayner, Jonathan |
| collection | Thesis |
| description | In recent work, R. Moerman and H. Weigert have introduced a truncation scheme for the Balitsky hierarchy, arguing that this is the most general possible method for obtaining finite Nc approximate solutions to the JIMWLK equation, while ensuring that these solutions obey several key properties that are known to be true of any exact solution to JIMWLK [1]. To carry out this truncation, it becomes necessary to systematically construct an orthogonal basis for the space of color singlets with purely adjoint indices. The primary contribution of this dissertation is to construct a basis that makes significant strides towards this goal, using irreducible representations of the permutation group Sk and recently-developed Hermitian Young projection operators [2–4]. Our method directly produces the basis for these singlets, avoiding the need to construct a basis for all multiplets and project out the singlets, as is common in other approaches. In our basis, orthogonality holds both between elements associated with non-isomorphic and isomorphic representations, with the exception of representations that are identical (and not just isomorphic). In working through the robust mathematical framework that describes this construction, we show that failures of orthogonality are a direct result of these basis elements being associated with identical induced representations arising from derangements with differing cycle structure, which suggests a possible strategy for constructing a fully-orthogonal basis in future research. We also prove that this basis always consists of elements that are real or purely imaginary and show how to determine these properties at the level of representations using characters and Frobenius reciprocity. We then shift gears to prove a small number of analytic properties of the images of commonly-used Wilson line operators. Explicitly, we provide a proof that hasn’t existed in the literature previously that the image of the dipole operator in the complex plane is the hypocycloid with Nc-cusps and we prove that all Wilson line operators that appear in the amplitude matrix used in the JIMWLK evolution of two quark-antiquark pairs are bounded by the unit circle. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/29532 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:53:21.712Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2019 |
| publishDateRange | 2019 |
| publishDateSort | 2019 |
| publisher | Department of Physics |
| publisherStr | Department of Physics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/29532 Reps for JIMWLK: applications of representation theory to a novel approach to the JIMWLK equation Rayner, Jonathan Weigert, Heribert Theoretical Physics In recent work, R. Moerman and H. Weigert have introduced a truncation scheme for the Balitsky hierarchy, arguing that this is the most general possible method for obtaining finite Nc approximate solutions to the JIMWLK equation, while ensuring that these solutions obey several key properties that are known to be true of any exact solution to JIMWLK [1]. To carry out this truncation, it becomes necessary to systematically construct an orthogonal basis for the space of color singlets with purely adjoint indices. The primary contribution of this dissertation is to construct a basis that makes significant strides towards this goal, using irreducible representations of the permutation group Sk and recently-developed Hermitian Young projection operators [2–4]. Our method directly produces the basis for these singlets, avoiding the need to construct a basis for all multiplets and project out the singlets, as is common in other approaches. In our basis, orthogonality holds both between elements associated with non-isomorphic and isomorphic representations, with the exception of representations that are identical (and not just isomorphic). In working through the robust mathematical framework that describes this construction, we show that failures of orthogonality are a direct result of these basis elements being associated with identical induced representations arising from derangements with differing cycle structure, which suggests a possible strategy for constructing a fully-orthogonal basis in future research. We also prove that this basis always consists of elements that are real or purely imaginary and show how to determine these properties at the level of representations using characters and Frobenius reciprocity. We then shift gears to prove a small number of analytic properties of the images of commonly-used Wilson line operators. Explicitly, we provide a proof that hasn’t existed in the literature previously that the image of the dipole operator in the complex plane is the hypocycloid with Nc-cusps and we prove that all Wilson line operators that appear in the amplitude matrix used in the JIMWLK evolution of two quark-antiquark pairs are bounded by the unit circle. 2019-02-14T13:17:58Z 2019-02-14T13:17:58Z 2018 2019-02-14T11:07:10Z Master Thesis Masters MSc http://hdl.handle.net/11427/29532 eng application/pdf Department of Physics Faculty of Science University of Cape Town |
| spellingShingle | Theoretical Physics Rayner, Jonathan Reps for JIMWLK: applications of representation theory to a novel approach to the JIMWLK equation |
| thesis_degree_str | Master's |
| title | Reps for JIMWLK: applications of representation theory to a novel approach to the JIMWLK equation |
| title_full | Reps for JIMWLK: applications of representation theory to a novel approach to the JIMWLK equation |
| title_fullStr | Reps for JIMWLK: applications of representation theory to a novel approach to the JIMWLK equation |
| title_full_unstemmed | Reps for JIMWLK: applications of representation theory to a novel approach to the JIMWLK equation |
| title_short | Reps for JIMWLK: applications of representation theory to a novel approach to the JIMWLK equation |
| title_sort | reps for jimwlk applications of representation theory to a novel approach to the jimwlk equation |
| topic | Theoretical Physics |
| url | http://hdl.handle.net/11427/29532 |
| work_keys_str_mv | AT raynerjonathan repsforjimwlkapplicationsofrepresentationtheorytoanovelapproachtothejimwlkequation |