Symmetry projection, geometry and choice of the basis

R. Lemus and A.O. Hernández-Castillo

Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, Circuito Exterior, C.U., 04510 México, D.F., México.

Received 04 June 2015; ]]> Accepted 10 August 2015

Abstract

A geometrical point of view of symmetry adapted projection to irreducible subspaces is presented. The projection is applied in two stages. The first step consists in projecting over subspaces spanning irreducible representations (irreps) of the symmetry group, while the second projection is carried out over the irreps of a subgroup defined through a suitable group chain. It is shown that choosing different chains is equivalent to propose alternative bases (passive point of view), while changing the projected function corresponds to the active point of view where the vector to be projected is rotated. Because of the importance of choosing the appropriate basis, an approach based on the concept of invariant operators to obtain the basis for discrete groups is presented. We show that this approach is analogue to the case of continuum groups and it is closely related to the definition of quantum numbers. The importance of these concepts is illustrated through the effect of symmetry breaking. Because of the deep insight into the group theory concepts, we believe the presented viewpoint helps to understand the main ingredients involved in group representation theory using the latest advances in the subject for discrete groups.

Keywords: Symmetry projection; quantum numbers; discrete groups; eigenfunction approach; symmetry breaking.

Resumen

Se presenta un punto de vista geométrico de la proyección a subespacios que portan representaciones irreducibles. La proyección se lleva a cabo en dos pasos. Primero se efectúa la proyección sobre subespacios que portan representaciones irreducibles del grupo de simetría, para posteriormente efectuar la proyección con respecto a un subgrupo definido a través de una cadena apropiada de subgrupos. Se muestra que la selección de diferentes cadenas es equivalente a proponer bases alternativas (punto de vista pasivo), mientras que el cambio de la función a proyectar equivale al punto de vista activo, donde el vector a proyectar es rotado. Debido a la importancia de seleccionar una base apropiada, se presenta un método de proyección basado en el concepto de operadores invariantes en el caso de grupos discretos. Se muestra que este método es análogo al caso de grupos continuos e íntimamente relacionado con el mismo concepto de número cuántico. La importancia de estos conceptos es ilustrada mediante el concepto de rompimiento de simetría. Creemos que dada la profundidad del marco teórico presentado, este material será de gran ayuda en la comprensión de los conceptos de teoría de representaciones de grupos, en donde se ha incluido la esencia de los últimos métodos de proyección desarrollados para grupos discretos.

Palabras clave: Proyección de simetría; números cuánticos; grupos discretos; método de funciones propias; rompimiento de simetría.

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Acknowledgments

This work is partially supported by DGAPA-UNAM under project No. IN109113.

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