Please use this identifier to cite or link to this item:
http://hdl.handle.net/2445/186255
Title: | $p$-adic groups in quantum mechanics |
Author: | Blanco Cabanillas, Anna |
Director/Tutor: | Travesa i Grau, Artur |
Keywords: | Nombres p-àdics Treballs de fi de grau Camps p-àdics Anàlisi p-àdica Teoria quàntica p-adic numbers Bachelor's theses p-adic fields p-adic analysis Quantum theory |
Issue Date: | 22-Jan-2022 |
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. |
Note: | Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Artur Travesa i Grau |
URI: | http://hdl.handle.net/2445/186255 |
Appears in Collections: | Treballs Finals de Grau (TFG) - Matemàtiques |
Files in This Item:
File | Description | Size | Format | |
---|---|---|---|---|
tfg_blanco_cabanillas_anna.pdf | Memòria | 546.14 kB | Adobe PDF | View/Open |
This item is licensed under a
Creative Commons License