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$p$-adic groups in quantum mechanics

Identificadores del recurso

http://hdl.handle.net/2445/186255

Procedencia

(Dipòsit Digital de la Universitat de Barcelona)

Ficha

Título:: $p$-adic groups in quantum mechanics
Tema:: Nombres p-àdics; Camps p-àdics; Anàlisi p-àdica; Teoria quàntica; Treballs de fi de grau; p-adic numbers; p-adic fields; p-adic analysis; Quantum theory; Bachelor's theses
Descrición:: Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Artur Travesa i Grau; [en] Number theory is being used in physics as a mathematical tool more and more. At the end of the 20th century, $p$-adic numbers made its appearance in quantum gravitational theories like string theory. This was motivated by the non-archimedian nature of space time at Planck scale. In this work we aim to formalize the basis of $p$-adic physics by exploring how to translate complex Quantum Mechanics to $p$-adic Quantum mechanics. This will be done using Weyl's formalism, which defines bounded operators and allows to relate different time-evolution pictures in quantum mechanics. This is done by the means of representation theory. We will be exploring the representation theory of $p$-adic reductive groups, specially induced, supercuspidal and projective representations. With that knowledge we will define the $p$-adic Heisenberg group that encodes the information on the $p$-adic phase space and study the Schrödinger representation. We will explain the importance of the Stone-von Neumann theorem that states uniqueness up to equivalence and we will study the Maslov indices of the group.
Fonte:: Treballs Finals de Grau (TFG) - Matemàtiques
Idioma:: English
Autor/Productor:: Blanco Cabanillas, Anna
Otros colaboradores/productores:: Travesa i Grau, Artur
Dereitos:: cc-by-nc-nd (c) Anna Blanco Cabanillas, 2022; http://creativecommons.org/licenses/by-nc-nd/3.0/es/; info:eu-repo/semantics/openAccess
Data:: 2022-06-02T10:06:29Z; 2022-01-22
Tipo de recurso:: info:eu-repo/semantics/bachelorThesis
Formato:: 49 p.; application/pdf

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2. <dc:contributor.advisor>Travesa i Grau, Artur</dc:contributor.advisor>
3. <dc:creator>Blanco Cabanillas, Anna</dc:creator>
4. <dc:subject.classification>Nombres p-àdics</dc:subject.classification>
5. <dc:subject.classification>Camps p-àdics</dc:subject.classification>
6. <dc:subject.classification>Anàlisi p-àdica</dc:subject.classification>
7. <dc:subject.classification>Teoria quàntica</dc:subject.classification>
8. <dc:subject.classification>Treballs de fi de grau</dc:subject.classification>
9. <dc:subject.other>p-adic numbers</dc:subject.other>
10. <dc:subject.other>p-adic fields</dc:subject.other>
11. <dc:subject.other>p-adic analysis</dc:subject.other>
12. <dc:subject.other>Quantum theory</dc:subject.other>
13. <dc:subject.other>Bachelor's theses</dc:subject.other>
14. <dcterms:abstract>[en] Number theory is being used in physics as a mathematical tool more and more. At the end of the 20th century, $p$-adic numbers made its appearance in quantum gravitational theories like string theory. This was motivated by the non-archimedian nature of space time at Planck scale. In this work we aim to formalize the basis of $p$-adic physics by exploring how to translate complex Quantum Mechanics to $p$-adic Quantum mechanics. This will be done using Weyl's formalism, which defines bounded operators and allows to relate different time-evolution pictures in quantum mechanics. This is done by the means of representation theory. We will be exploring the representation theory of $p$-adic reductive groups, specially induced, supercuspidal and projective representations. With that knowledge we will define the $p$-adic Heisenberg group that encodes the information on the $p$-adic phase space and study the Schrödinger representation. We will explain the importance of the Stone-von Neumann theorem that states uniqueness up to equivalence and we will study the Maslov indices of the group.</dcterms:abstract>
15. <dcterms:dateAccepted>2022-06-02T10:06:29Z</dcterms:dateAccepted>
16. <dcterms:available>2022-06-02T10:06:29Z</dcterms:available>
17. <dcterms:created>2022-06-02T10:06:29Z</dcterms:created>
18. <dcterms:issued>2022-01-22</dcterms:issued>
19. <dc:type>info:eu-repo/semantics/bachelorThesis</dc:type>
20. <dc:identifier>http://hdl.handle.net/2445/186255</dc:identifier>
21. <dc:language>eng</dc:language>
22. <dc:rights>http://creativecommons.org/licenses/by-nc-nd/3.0/es/</dc:rights>
23. <dc:rights>info:eu-repo/semantics/openAccess</dc:rights>
24. <dc:rights>cc-by-nc-nd (c) Anna Blanco Cabanillas, 2022</dc:rights>
25. <dc:source>Treballs Finals de Grau (TFG) - Matemàtiques</dc:source>
</qdc:qualifieddc>

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  1. <dc:title>$p$-adic groups in quantum mechanics</dc:title>
  2. <dc:creator>Blanco Cabanillas, Anna</dc:creator>
  3. <dc:contributor>Travesa i Grau, Artur</dc:contributor>
  4. <dc:description>Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Artur Travesa i Grau</dc:description>
  5. <dc:description>[en] Number theory is being used in physics as a mathematical tool more and more. At the end of the 20th century, $p$-adic numbers made its appearance in quantum gravitational theories like string theory. This was motivated by the non-archimedian nature of space time at Planck scale. In this work we aim to formalize the basis of $p$-adic physics by exploring how to translate complex Quantum Mechanics to $p$-adic Quantum mechanics. This will be done using Weyl's formalism, which defines bounded operators and allows to relate different time-evolution pictures in quantum mechanics. This is done by the means of representation theory. We will be exploring the representation theory of $p$-adic reductive groups, specially induced, supercuspidal and projective representations. With that knowledge we will define the $p$-adic Heisenberg group that encodes the information on the $p$-adic phase space and study the Schrödinger representation. We will explain the importance of the Stone-von Neumann theorem that states uniqueness up to equivalence and we will study the Maslov indices of the group.</dc:description>
  6. <dc:date>2022-06-02T10:06:29Z</dc:date>
  7. <dc:date>2022-06-02T10:06:29Z</dc:date>
  8. <dc:date>2022-01-22</dc:date>
  9. <dc:type>info:eu-repo/semantics/bachelorThesis</dc:type>
  10. <dc:identifier>http://hdl.handle.net/2445/186255</dc:identifier>
  11. <dc:language>eng</dc:language>
  12. <dc:rights>http://creativecommons.org/licenses/by-nc-nd/3.0/es/</dc:rights>
  13. <dc:rights>info:eu-repo/semantics/openAccess</dc:rights>
  14. <dc:rights>cc-by-nc-nd (c) Anna Blanco Cabanillas, 2022</dc:rights>
  15. <dc:source>Treballs Finals de Grau (TFG) - Matemàtiques</dc:source>
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