The wavefront set and its applications in quantum field theory
This work explores the relevance of the wavefront set in Quantum Field Theory (QFT) with a focus on addressing the problem of divergences. It begins with an introduction to QFT, detailing the interaction picture, Dyson series, and Wick’s theorems, which are fundamental tools for understanding partic...
- Autores:
-
Granados Rodríguez, Juana
- Tipo de recurso:
- Trabajo de grado de pregrado
- Fecha de publicación:
- 2024
- Institución:
- Universidad de los Andes
- Repositorio:
- Séneca: repositorio Uniandes
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.uniandes.edu.co:1992/74846
- Acceso en línea:
- https://hdl.handle.net/1992/74846
- Palabra clave:
- Wavefront Set
Quantum Field Theory (QFT)
Divergences
Distribution Theory
Hörmander’s Condition
Wightman Propagator
Feynman Propagator
Epstein-Glaser Renormalization
Renormalization Methods
Mathematical Physics
Física
- Rights
- openAccess
- License
- Attribution 4.0 International
| id |
UNIANDES2_699c5d187598b6f2e3a52b955db4e4f4 |
|---|---|
| oai_identifier_str |
oai:repositorio.uniandes.edu.co:1992/74846 |
| network_acronym_str |
UNIANDES2 |
| network_name_str |
Séneca: repositorio Uniandes |
| repository_id_str |
|
| dc.title.eng.fl_str_mv |
The wavefront set and its applications in quantum field theory |
| dc.title.alternative.spa.fl_str_mv |
El conjunto frente de onda y sus aplicaciones en teoría cuántica de campos |
| title |
The wavefront set and its applications in quantum field theory |
| spellingShingle |
The wavefront set and its applications in quantum field theory Wavefront Set Quantum Field Theory (QFT) Divergences Distribution Theory Hörmander’s Condition Wightman Propagator Feynman Propagator Epstein-Glaser Renormalization Renormalization Methods Mathematical Physics Física |
| title_short |
The wavefront set and its applications in quantum field theory |
| title_full |
The wavefront set and its applications in quantum field theory |
| title_fullStr |
The wavefront set and its applications in quantum field theory |
| title_full_unstemmed |
The wavefront set and its applications in quantum field theory |
| title_sort |
The wavefront set and its applications in quantum field theory |
| dc.creator.fl_str_mv |
Granados Rodríguez, Juana |
| dc.contributor.advisor.none.fl_str_mv |
Reyes Lega, Andrés Fernando |
| dc.contributor.author.none.fl_str_mv |
Granados Rodríguez, Juana |
| dc.contributor.jury.none.fl_str_mv |
Téllez Acosta, Gabriel |
| dc.subject.keyword.eng.fl_str_mv |
Wavefront Set Quantum Field Theory (QFT) Divergences Distribution Theory Hörmander’s Condition Wightman Propagator Feynman Propagator Epstein-Glaser Renormalization Renormalization Methods Mathematical Physics |
| topic |
Wavefront Set Quantum Field Theory (QFT) Divergences Distribution Theory Hörmander’s Condition Wightman Propagator Feynman Propagator Epstein-Glaser Renormalization Renormalization Methods Mathematical Physics Física |
| dc.subject.themes.spa.fl_str_mv |
Física |
| description |
This work explores the relevance of the wavefront set in Quantum Field Theory (QFT) with a focus on addressing the problem of divergences. It begins with an introduction to QFT, detailing the interaction picture, Dyson series, and Wick’s theorems, which are fundamental tools for understanding particle interactions and the emergence of divergences. The work transitions into an examination of distribution theory, providing the necessary mathematical framework to handle the infinities associated with these divergences. The wavefront set is defined and examined through the lens of Hörmander's condition, with detailed analyses of wavefront sets for common distributions. The wavefront sets of both the Wightman and the Feynman propagator are also explored, demonstrating the relevance of these mathematical tools in the field. The final section discusses the integration of the wavefront set into a more modern version of the Epstein-Glaser renormalization method, emphasizing the practical benefits and challenges of using the wavefront set in this context. The study concludes with reflections on the potential of this approach to offer deeper insights and more robust solutions to the persistent issues of renormalization in QFT. |
| publishDate |
2024 |
| dc.date.accessioned.none.fl_str_mv |
2024-07-31T21:52:40Z |
| dc.date.available.none.fl_str_mv |
2024-07-31T21:52:40Z |
| dc.date.issued.none.fl_str_mv |
2024-07-30 |
| dc.type.none.fl_str_mv |
Trabajo de grado - Pregrado |
| dc.type.driver.none.fl_str_mv |
info:eu-repo/semantics/bachelorThesis |
| dc.type.version.none.fl_str_mv |
info:eu-repo/semantics/acceptedVersion |
| dc.type.coar.none.fl_str_mv |
http://purl.org/coar/resource_type/c_7a1f |
| dc.type.content.none.fl_str_mv |
Text |
| dc.type.redcol.none.fl_str_mv |
http://purl.org/redcol/resource_type/TP |
| format |
http://purl.org/coar/resource_type/c_7a1f |
| status_str |
acceptedVersion |
| dc.identifier.uri.none.fl_str_mv |
https://hdl.handle.net/1992/74846 |
| dc.identifier.instname.none.fl_str_mv |
instname:Universidad de los Andes |
| dc.identifier.reponame.none.fl_str_mv |
reponame:Repositorio Institucional Séneca |
| dc.identifier.repourl.none.fl_str_mv |
repourl:https://repositorio.uniandes.edu.co/ |
| url |
https://hdl.handle.net/1992/74846 |
| identifier_str_mv |
instname:Universidad de los Andes reponame:Repositorio Institucional Séneca repourl:https://repositorio.uniandes.edu.co/ |
| dc.language.iso.none.fl_str_mv |
eng |
| language |
eng |
| dc.relation.references.none.fl_str_mv |
J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics. Boston: Addison-Wesley, 2nd ed., 2011. F. Scheck, Quantum Physics: With 102 Exercises, Hints and Solutions. Berlin Heidelberg: Springer, 2007. N. N. Bogoliubov and D. V. Shirkov, Quantum Fields. Massachusetts, U.S.A.: Benjamin/Cummings Publishing Company, 1st english ed., 1982. C. Brouder, N. V. Dang, and F. Hélein, “A smooth introduction to the wavefront set,” Journal of Physics A: Mathematical and Theoretical, vol. 47, p. 443001, Nov. 2014. L. Hörmander, The Analysis of Linear Partial Differential Operators I. Springer Berlin, Heidelberg, 2003. G. Tellez, Métodos Matemáticos. Bogotá, Colombia: Ediciones Uniandes, 2nd ed., 2022. A. Giniatoulline, Introducción a las Distribuciones (Funciones Generalizadas). 1999. J. Rauch, Partial Differential Equations. Springer Science & Business Media, Dec 2012. M. Reed and B. Simon, Methods of Modern Mathematical Physics. Volume II: Fourier Analysis, Self-Adjointness. No. v. 2 in II: Fourier Analysis, Self-Adjointness, Elsevier Science, 1975. J. M. Gracia-Bondía, “The epstein-glaser approach to qft,” AIP Conference Proceedings, vol. 809, pp. 24–43, Jan 2006. G. Popineau and R. Stora, “A pedagogical remark on the main theorem of perturbative renormalization theory,” Nuclear Physics B, vol. 912, pp. 70–78, 2016. Mathematical Foundations of Quantum Field Theory: A volume dedicated to the Memory of Raymond Stora. S. A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time. Cambridge University Press, 1989. |
| dc.rights.en.fl_str_mv |
Attribution 4.0 International |
| dc.rights.uri.none.fl_str_mv |
http://creativecommons.org/licenses/by/4.0/ |
| dc.rights.accessrights.none.fl_str_mv |
info:eu-repo/semantics/openAccess |
| dc.rights.coar.none.fl_str_mv |
http://purl.org/coar/access_right/c_abf2 |
| rights_invalid_str_mv |
Attribution 4.0 International http://creativecommons.org/licenses/by/4.0/ http://purl.org/coar/access_right/c_abf2 |
| eu_rights_str_mv |
openAccess |
| dc.format.extent.none.fl_str_mv |
58 páginas |
| dc.format.mimetype.none.fl_str_mv |
application/pdf |
| dc.publisher.none.fl_str_mv |
Universidad de los Andes |
| dc.publisher.program.none.fl_str_mv |
Física |
| dc.publisher.faculty.none.fl_str_mv |
Facultad de Ciencias |
| dc.publisher.department.none.fl_str_mv |
Departamento de Física |
| publisher.none.fl_str_mv |
Universidad de los Andes |
| institution |
Universidad de los Andes |
| bitstream.url.fl_str_mv |
https://repositorio.uniandes.edu.co/bitstreams/e249704e-c233-4923-bfc9-767cf5c112b9/download https://repositorio.uniandes.edu.co/bitstreams/b8e0e231-e120-4d07-b384-c64cfb403437/download https://repositorio.uniandes.edu.co/bitstreams/60940037-30fe-4f8d-9d7c-02c9649d993b/download https://repositorio.uniandes.edu.co/bitstreams/0e1938c5-1cf1-4065-8cae-ed834dfa7948/download https://repositorio.uniandes.edu.co/bitstreams/e6127523-b79c-4c2b-8677-28ca84e657ac/download https://repositorio.uniandes.edu.co/bitstreams/b0c711dd-aaca-4cf0-8df6-56aff3c19f9b/download https://repositorio.uniandes.edu.co/bitstreams/03c52dbc-fc6b-40fe-b993-20aa3a7ebb0b/download https://repositorio.uniandes.edu.co/bitstreams/aea588de-8a33-476b-a628-9e7561bade72/download |
| bitstream.checksum.fl_str_mv |
0175ea4a2d4caec4bbcc37e300941108 15962d02bf1780f4a832a20a213fa045 21cf1e19d4de3b23efc10ac7ab40d46e ae9e573a68e7f92501b6913cc846c39f 11e3314f39c9b4faf6060a70934e0ed7 a2758b084c919b4fe729c54fa846cd19 8e34e24544eb0560f0835cc2709114b2 049980e6d7301a16f5a0a9168971e448 |
| bitstream.checksumAlgorithm.fl_str_mv |
MD5 MD5 MD5 MD5 MD5 MD5 MD5 MD5 |
| repository.name.fl_str_mv |
Repositorio institucional Séneca |
| repository.mail.fl_str_mv |
adminrepositorio@uniandes.edu.co |
| _version_ |
1812133915686600704 |
| spelling |
Reyes Lega, Andrés Fernandovirtual::19402-1Granados Rodríguez, JuanaTéllez Acosta, Gabrielvirtual::19403-12024-07-31T21:52:40Z2024-07-31T21:52:40Z2024-07-30https://hdl.handle.net/1992/74846instname:Universidad de los Andesreponame:Repositorio Institucional Sénecarepourl:https://repositorio.uniandes.edu.co/This work explores the relevance of the wavefront set in Quantum Field Theory (QFT) with a focus on addressing the problem of divergences. It begins with an introduction to QFT, detailing the interaction picture, Dyson series, and Wick’s theorems, which are fundamental tools for understanding particle interactions and the emergence of divergences. The work transitions into an examination of distribution theory, providing the necessary mathematical framework to handle the infinities associated with these divergences. The wavefront set is defined and examined through the lens of Hörmander's condition, with detailed analyses of wavefront sets for common distributions. The wavefront sets of both the Wightman and the Feynman propagator are also explored, demonstrating the relevance of these mathematical tools in the field. The final section discusses the integration of the wavefront set into a more modern version of the Epstein-Glaser renormalization method, emphasizing the practical benefits and challenges of using the wavefront set in this context. The study concludes with reflections on the potential of this approach to offer deeper insights and more robust solutions to the persistent issues of renormalization in QFT.Este trabajo explora la relevancia del conjunto de ondas en la Teoría Cuántica de Campos (QFT) con un enfoque en abordar el problema de las divergencias. Comienza con una introducción a la QFT, detallando el cuadro de interacción, la serie de Dyson y los teoremas de Wick, que son fundamentales para entender las interacciones de partículas y la emergencia de divergencias. El trabajo transita hacia un examen de la teoría de distribuciones, proporcionando el marco matemático necesario para manejar los infinitos asociadas con estas divergencias. Se define y examina el conjunto frente de onda a través del criterio de Hörmander, con análisis detallados de los conjuntos frente de onda para distribuciones comunes. También se exploran los conjuntos frente de onda del propagador de Wightman y el de Feynman, demostrando la relevancia de estas herramientas matemáticas en el campo. La sección final discute la integración del conjunto de ondas en una versión más moderna del método de renormalización de Epstein-Glaser, enfatizando los beneficios prácticos y los desafíos de usar el conjunto de ondas en este contexto. El estudio concluye con reflexiones sobre el potencial de este enfoque para ofrecer percepciones más profundas y soluciones más robustas a los problemas persistentes de renormalización en la QFT.Pregrado58 páginasapplication/pdfengUniversidad de los AndesFísicaFacultad de CienciasDepartamento de FísicaAttribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2The wavefront set and its applications in quantum field theoryEl conjunto frente de onda y sus aplicaciones en teoría cuántica de camposTrabajo de grado - Pregradoinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_7a1fTexthttp://purl.org/redcol/resource_type/TPWavefront SetQuantum Field Theory (QFT)DivergencesDistribution TheoryHörmander’s ConditionWightman PropagatorFeynman PropagatorEpstein-Glaser RenormalizationRenormalization MethodsMathematical PhysicsFísicaJ. J. Sakurai and J. Napolitano, Modern Quantum Mechanics. Boston: Addison-Wesley, 2nd ed., 2011.F. Scheck, Quantum Physics: With 102 Exercises, Hints and Solutions. Berlin Heidelberg: Springer, 2007.N. N. Bogoliubov and D. V. Shirkov, Quantum Fields. Massachusetts, U.S.A.: Benjamin/Cummings Publishing Company, 1st english ed., 1982.C. Brouder, N. V. Dang, and F. Hélein, “A smooth introduction to the wavefront set,” Journal of Physics A: Mathematical and Theoretical, vol. 47, p. 443001, Nov. 2014.L. Hörmander, The Analysis of Linear Partial Differential Operators I. Springer Berlin, Heidelberg, 2003.G. Tellez, Métodos Matemáticos. Bogotá, Colombia: Ediciones Uniandes, 2nd ed., 2022.A. Giniatoulline, Introducción a las Distribuciones (Funciones Generalizadas). 1999.J. Rauch, Partial Differential Equations. Springer Science & Business Media, Dec 2012.M. Reed and B. Simon, Methods of Modern Mathematical Physics. Volume II: Fourier Analysis, Self-Adjointness. No. v. 2 in II: Fourier Analysis, Self-Adjointness, Elsevier Science, 1975.J. M. Gracia-Bondía, “The epstein-glaser approach to qft,” AIP Conference Proceedings, vol. 809, pp. 24–43, Jan 2006.G. Popineau and R. Stora, “A pedagogical remark on the main theorem of perturbative renormalization theory,” Nuclear Physics B, vol. 912, pp. 70–78, 2016. Mathematical Foundations of Quantum Field Theory: A volume dedicated to the Memory of Raymond Stora.S. A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time. Cambridge University Press, 1989.202015134Publicationhttps://scholar.google.es/citations?user=04V0g64AAAAJvirtual::19402-1https://scholar.google.es/citations?user=1JHuoIAAAAAJvirtual::19403-10000-0002-6357-260Xvirtual::19403-1https://scienti.minciencias.gov.co/cvlac/visualizador/generarCurriculoCv.do?cod_rh=0000055174virtual::19402-1https://scienti.minciencias.gov.co/cvlac/visualizador/generarCurriculoCv.do?cod_rh=0000077224virtual::19403-19cfe3fb3-ca67-4abc-bf3f-6ceb7f9f4adfvirtual::19402-19cfe3fb3-ca67-4abc-bf3f-6ceb7f9f4adfvirtual::19402-198ad9270-ae12-4bd0-9e5e-6ff873936418virtual::19403-198ad9270-ae12-4bd0-9e5e-6ff873936418virtual::19403-1CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8908https://repositorio.uniandes.edu.co/bitstreams/e249704e-c233-4923-bfc9-767cf5c112b9/download0175ea4a2d4caec4bbcc37e300941108MD51ORIGINALWavefront Set and its Applications in QuantumField Theory.pdfWavefront Set and its Applications in QuantumField Theory.pdfapplication/pdf702596https://repositorio.uniandes.edu.co/bitstreams/b8e0e231-e120-4d07-b384-c64cfb403437/download15962d02bf1780f4a832a20a213fa045MD52formato_autorizacion-firmado.pdfformato_autorizacion-firmado.pdfHIDEapplication/pdf328543https://repositorio.uniandes.edu.co/bitstreams/60940037-30fe-4f8d-9d7c-02c9649d993b/download21cf1e19d4de3b23efc10ac7ab40d46eMD54LICENSElicense.txtlicense.txttext/plain; charset=utf-82535https://repositorio.uniandes.edu.co/bitstreams/0e1938c5-1cf1-4065-8cae-ed834dfa7948/downloadae9e573a68e7f92501b6913cc846c39fMD53TEXTWavefront Set and its Applications in QuantumField Theory.pdf.txtWavefront Set and its Applications in QuantumField Theory.pdf.txtExtracted texttext/plain104028https://repositorio.uniandes.edu.co/bitstreams/e6127523-b79c-4c2b-8677-28ca84e657ac/download11e3314f39c9b4faf6060a70934e0ed7MD55formato_autorizacion-firmado.pdf.txtformato_autorizacion-firmado.pdf.txtExtracted texttext/plain1980https://repositorio.uniandes.edu.co/bitstreams/b0c711dd-aaca-4cf0-8df6-56aff3c19f9b/downloada2758b084c919b4fe729c54fa846cd19MD57THUMBNAILWavefront Set and its Applications in QuantumField Theory.pdf.jpgWavefront Set and its Applications in QuantumField Theory.pdf.jpgGenerated Thumbnailimage/jpeg7025https://repositorio.uniandes.edu.co/bitstreams/03c52dbc-fc6b-40fe-b993-20aa3a7ebb0b/download8e34e24544eb0560f0835cc2709114b2MD56formato_autorizacion-firmado.pdf.jpgformato_autorizacion-firmado.pdf.jpgGenerated Thumbnailimage/jpeg11044https://repositorio.uniandes.edu.co/bitstreams/aea588de-8a33-476b-a628-9e7561bade72/download049980e6d7301a16f5a0a9168971e448MD581992/74846oai:repositorio.uniandes.edu.co:1992/748462024-09-12 15:51:53.125http://creativecommons.org/licenses/by/4.0/Attribution 4.0 Internationalopen.accesshttps://repositorio.uniandes.edu.coRepositorio institucional Sénecaadminrepositorio@uniandes.edu.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 |