Representation theory of SU(2) (original) (raw)
Die Darstellungstheorie der Lie-Algebra ist von grundlegender Bedeutung in Mathematik und Physik. In der Mathematik ist sie der einfachste Fall in der Klassifikation der Darstellungen halbeinfacher Lie-Algebren, in der Physik spielt sie eine zentrale Rolle in der Quantenmechanik, weil sie die Darstellungen der Drehimpulsalgebra klassifiziert.
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| dbo:abstract | Die Darstellungstheorie der Lie-Algebra ist von grundlegender Bedeutung in Mathematik und Physik. In der Mathematik ist sie der einfachste Fall in der Klassifikation der Darstellungen halbeinfacher Lie-Algebren, in der Physik spielt sie eine zentrale Rolle in der Quantenmechanik, weil sie die Darstellungen der Drehimpulsalgebra klassifiziert. (de) Dentro del estudio de la teoría de la representación del grupo de Lie, el estudio de las representaciones del grupo unitario especial es fundamental para el estudio de las representaciones de . Es el primer caso de un grupo de Lie que es tanto un como . La primera condición implica que la teoría de la representación es discreta: las representaciones son de una colección de representaciones irreducibles básicas (gobernadas por el teorema de Peter-Weyl). El segundo significa que habrá representaciones irreducibles en dimensiones superiores a 1. SU(2) es el grupo de recubrimiento del grupo de rotación SO(3), por lo que su teoría de representación incluye la de este último, debido al homomorfismo suprayectivo existente entre ambos. Esto subraya la importancia de SU(2) para la descripción no relativista del espín en la física teórica. Como se muestra a continuación, las representaciones irreducibles de dimensión finita de SU(2) se indexan mediante un entero no negativo y tienen una dimensión . En la literatura sobre física, las representaciones están etiquetadas por la cantidad , donde es entonces un número entero o semientero, y la dimensión es . (es) In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abelian group. The first condition implies the representation theory is discrete: representations are direct sums of a collection of basic irreducible representations (governed by the Peter–Weyl theorem). The second means that there will be irreducible representations in dimensions greater than 1. SU(2) is the universal covering group of SO(3), and so its representation theory includes that of the latter, by dint of a surjective homomorphism to it. This underlies the significance of SU(2) for the description of non-relativistic spin in theoretical physics; see for other physical and historical context. As shown below, the finite-dimensional irreducible representations of SU(2) are indexed by a non-negative integer and have dimension . In the physics literature, the representations are labeled by the quantity , where is then either an integer or a half-integer, and the dimension is . (en) |
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| rdfs:comment | Die Darstellungstheorie der Lie-Algebra ist von grundlegender Bedeutung in Mathematik und Physik. In der Mathematik ist sie der einfachste Fall in der Klassifikation der Darstellungen halbeinfacher Lie-Algebren, in der Physik spielt sie eine zentrale Rolle in der Quantenmechanik, weil sie die Darstellungen der Drehimpulsalgebra klassifiziert. (de) Dentro del estudio de la teoría de la representación del grupo de Lie, el estudio de las representaciones del grupo unitario especial es fundamental para el estudio de las representaciones de . Es el primer caso de un grupo de Lie que es tanto un como . La primera condición implica que la teoría de la representación es discreta: las representaciones son de una colección de representaciones irreducibles básicas (gobernadas por el teorema de Peter-Weyl). El segundo significa que habrá representaciones irreducibles en dimensiones superiores a 1. (es) In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abelian group. The first condition implies the representation theory is discrete: representations are direct sums of a collection of basic irreducible representations (governed by the Peter–Weyl theorem). The second means that there will be irreducible representations in dimensions greater than 1. (en) |
| rdfs:label | Darstellungstheorie der sl(2,C) (de) Teoría de la representación de SU(2) (es) Representation theory of SU(2) (en) |
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