{"id":{"repo_id":"toronto-retro","oai_identifier":"oai:utoronto.scholaris.ca:1807/77389"},"canonical_url":"https://search.dev.ndltd.org/etd/toronto-retro/oai:utoronto.scholaris.ca:1807/77389","repository":{"repo_id":"toronto-retro","name":"University of Toronto","base_url":"https://utoronto.scholaris.ca/server/oai/request"},"display":{"title":"Quantomorphisms and Quantized Energy Levels for Metaplectic-c Quantization","abstract":"Metaplectic-c quantization was developed by Robinson and Rawnsley as an alternative to the classical Kostant-Souriau quantization procedure with half-form correction. This thesis extends certain properties of Kostant-Souriau quantization to the metaplectic-c context. We show that the Kostant-Souriau results are replicated or improved upon with metaplectic-c quantization. We consider two topics: quantomorphisms and quantized energy levels. If a symplectic manifold admits a Kostant-Souriau prequantization circle bundle, then its Poisson algebra is realized as the space of infinitesimal quantomorphisms of that circle bundle. We present a definition for a metaplectic-c quantomorphism, and prove that the space of infinitesimal metaplectic-c quantomorphisms exhibits all of the same properties that are seen in the Kostant-Souriau case. Next, given a metaplectic-c prequantized symplectic manifold ( M, ω) and a function H ∈ C ∞(M), we propose a condition under which E, a regular value of H, is a quantized energy level for the system (M, ω, H). We prove that our definition is dynamically invariant: if two functions on M share a regular level set, then the quantization condition over that level set is identical for both functions. We calculate the quantized energy levels for the n-dimensional harmonic oscillator and the hydrogen atom, and obtain the quantum mechanical predictions in both cases. Lastly, we generalize the quantization condition to a level set of a family of Poisson-commuting functions, and show that in the special case of a completely integrable system, it reduces to a Bohr-Sommerfeld condition.","abstract_html":"Metaplectic-c quantization was developed by Robinson and Rawnsley as an alternative to the classical Kostant-Souriau quantization procedure with half-form correction. This thesis extends certain properties of Kostant-Souriau quantization to the metaplectic-c context. We show that the Kostant-Souriau results are replicated or improved upon with metaplectic-c quantization. We consider two topics: quantomorphisms and quantized energy levels. If a symplectic manifold admits a Kostant-Souriau prequantization circle bundle, then its Poisson algebra is realized as the space of infinitesimal quantomorphisms of that circle bundle. We present a definition for a metaplectic-c quantomorphism, and prove that the space of infinitesimal metaplectic-c quantomorphisms exhibits all of the same properties that are seen in the Kostant-Souriau case. Next, given a metaplectic-c prequantized symplectic manifold ( M, ω) and a function H ∈ C ∞(M), we propose a condition under which E, a regular value of H, is a quantized energy level for the system (M, ω, H). We prove that our definition is dynamically invariant: if two functions on M share a regular level set, then the quantization condition over that level set is identical for both functions. We calculate the quantized energy levels for the n-dimensional harmonic oscillator and the hydrogen atom, and obtain the quantum mechanical predictions in both cases. Lastly, we generalize the quantization condition to a level set of a family of Poisson-commuting functions, and show that in the special case of a completely integrable system, it reduces to a Bohr-Sommerfeld condition.","abstract_has_math":false,"creators":["Vaughan, Jennifer Jaye"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Mathematics","school":null,"contributors":[],"advisors":["Karshon, Yael"],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-11","date_published":"2016-11","updated_at":"2026-07-27T21:28:16Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1807/77389","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Karshon, Yael"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Vaughan, Jennifer Jaye"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2016-11"]},{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2017-06-04T17:00:27Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2017-06-04T17:00:27Z"]},{"key":"dc:date.issued","label":"Date","values":["2016-11"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1807/77389"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Metaplectic-c quantization was developed by Robinson and Rawnsley as an alternative to the classical Kostant-Souriau quantization procedure with half-form correction. This thesis extends certain properties of Kostant-Souriau quantization to the metaplectic-c context. We show that the Kostant-Souriau results are replicated or improved upon with metaplectic-c quantization. We consider two topics: quantomorphisms and quantized energy levels. If a symplectic manifold admits a Kostant-Souriau prequantization circle bundle, then its Poisson algebra is realized as the space of infinitesimal quantomorphisms of that circle bundle. We present a definition for a metaplectic-c quantomorphism, and prove that the space of infinitesimal metaplectic-c quantomorphisms exhibits all of the same properties that are seen in the Kostant-Souriau case. Next, given a metaplectic-c prequantized symplectic manifold ( M, ω) and a function H ∈ C ∞(M), we propose a condition under which E, a regular value of H, is a quantized energy level for the system (M, ω, H). We prove that our definition is dynamically invariant: if two functions on M share a regular level set, then the quantization condition over that level set is identical for both functions. We calculate the quantized energy levels for the n-dimensional harmonic oscillator and the hydrogen atom, and obtain the quantum mechanical predictions in both cases. Lastly, we generalize the quantization condition to a level set of a family of Poisson-commuting functions, and show that in the special case of a completely integrable system, it reduces to a Bohr-Sommerfeld condition."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Quantomorphisms and Quantized Energy Levels for Metaplectic-c Quantization"]}]}],"canonical_facts":{"dc:contributor.advisor":["Karshon, Yael"],"dc:contributor.department":["Mathematics"],"dc:creator":["Vaughan, Jennifer Jaye"],"dc:date":["2016-11"],"dc:date.accessioned":["2017-06-04T17:00:27Z"],"dc:date.available":["2017-06-04T17:00:27Z"],"dc:date.issued":["2016-11"],"dc:description.abstract":["Metaplectic-c quantization was developed by Robinson and Rawnsley as an alternative to the classical Kostant-Souriau quantization procedure with half-form correction. This thesis extends certain properties of Kostant-Souriau quantization to the metaplectic-c context. We show that the Kostant-Souriau results are replicated or improved upon with metaplectic-c quantization. We consider two topics: quantomorphisms and quantized energy levels. If a symplectic manifold admits a Kostant-Souriau prequantization circle bundle, then its Poisson algebra is realized as the space of infinitesimal quantomorphisms of that circle bundle. We present a definition for a metaplectic-c quantomorphism, and prove that the space of infinitesimal metaplectic-c quantomorphisms exhibits all of the same properties that are seen in the Kostant-Souriau case. Next, given a metaplectic-c prequantized symplectic manifold ( M, ω) and a function H ∈ C ∞(M), we propose a condition under which E, a regular value of H, is a quantized energy level for the system (M, ω, H). We prove that our definition is dynamically invariant: if two functions on M share a regular level set, then the quantization condition over that level set is identical for both functions. We calculate the quantized energy levels for the n-dimensional harmonic oscillator and the hydrogen atom, and obtain the quantum mechanical predictions in both cases. Lastly, we generalize the quantization condition to a level set of a family of Poisson-commuting functions, and show that in the special case of a completely integrable system, it reduces to a Bohr-Sommerfeld condition."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["http://hdl.handle.net/1807/77389"],"dc:title":["Quantomorphisms and Quantized Energy Levels for Metaplectic-c Quantization"],"dc:type":["Thesis"]},"updated_at":"2026-07-27T21:28:16Z"}