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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
Descripción:: 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.
Fuente:: Treballs Finals de Grau (TFG) - Matemàtiques
Idioma:: English
Autor/Productor:: Blanco Cabanillas, Anna
Otros colaboradores/productores:: Travesa i Grau, Artur
Derechos:: cc-by-nc-nd (c) Anna Blanco Cabanillas, 2022; http://creativecommons.org/licenses/by-nc-nd/3.0/es/; info:eu-repo/semantics/openAccess
Fecha:: 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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10. < mods:accessCondition type =" useAndReproduction " > cc-by-nc-nd (c) Anna Blanco Cabanillas, 2022 </ mods:accessCondition >
11. < mods:titleInfo >
  1. < mods:title > $p$-adic groups in quantum mechanics </ mods:title >
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1. < dc:title > $p$-adic groups in quantum mechanics </ dc:title >
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 >
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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 >
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  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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