$p$-adic groups in quantum mechanics
Blanco Cabanillas, Anna
Travesa i Grau, Artur
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.
2022-06-02T10:06:29Z
2022-06-02T10:06:29Z
2022-01-22
info:eu-repo/semantics/bachelorThesis
http://hdl.handle.net/2445/186255
eng
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
info:eu-repo/semantics/openAccess
cc-by-nc-nd (c) Anna Blanco Cabanillas, 2022
Treballs Finals de Grau (TFG) - Matemàtiques