Angel "Java" Lopez en Blog

Publicado el 11 de Abril, 2013, 14:32

Ya mencioné a Herman Weyl y su libro The Theory of Groups and Quantum Mechanics en Hermann Weyl, fisica y matematicas. Hacia el final de la introducción, encuentro:

Our generation is witness to a development of physical knowledge such as has not been seen since the days of Kepler, Galileo and , and mathematics has scarcely ever experienced such a stormy epoch. Mathematical thought removes the spirit from its worldly haunts to solitude and renounces the unveiling of the secrets of Nature. But as recompense, mathematics is less bound to the course of worldly events than physics. While the quantum theory can be traced back only as far as 1900, the origin of the theory of groups is lost in a past scarcely accessible to history; the earliest works of art show that the symmetry groups of plane figures were even then already known, although the theory of these was only given definite form in the latter part of the eighteenth and in the nineteenth centuries. F. Klein considered the group concept. as most characteristic of nineteenth century mathematics.

Es interesante ver cómo algo como la teoría de grupos tuvo una larga historia, aunque hay que reconocer que como teoría matemática aparece realmente en el siglo XIX. Igual, es de destacar que tenemos otro ejemplo de matemáticas desarrolladas ANTES de tener una aplicación física.

Until the present, its most important application to natural science lay in the description of the symmetry of crystals, but it has recently been recognized that group theory is of fundamental importance for quantum physics; it here reveals the essential features which are not contingent on a special form of the dynamical laws nor on special assumptions concerning the forces involved. We may well expect that it is just this part of quantum physics which is most certain of a lasting place.

Y ahora menciona dos grupos que van a tener un rol en su libro:

Two groups, the group of rotations in 3-dimensional space and the permutation group, play here the principal role, for the laws governing the possible electronic configurations grouped about the stationary nucleus of an atom or an ion are spherically symmetric with respect to the nucleus, and since the various electrons of which the atom or ion is composed are identical, these possible configurations are invariant under a permutation of the individual electrons.

Y otro tema que aparece, cuando los grupos se relacionan con la física: sus representaciones por transformaciones lineales:

The investigation of groups first becomes a connected and complete theory in the theory of the representaration of groups by linear transformations, and it is exactly this mathematically most important part which is necessary for an adequate description of the quantum mechanical relations. All quantum numbers, with the exception of the so-called prirtcipal quantum number, are indices characterizing representations of groups.

Nunca traté todavía el tema de representaciones de grupos por transformaciones lineales. Pero es interesante encontrar esta temprana referencia en Weyl, que luego llegaría a aplicarse más allá de la teoría cuántica que menciona. Ver

Group Representation
Character theory
Group Representations
Group Representation Theory ahí leo lo principal: Informally, a representation of a group is a way of writing it down as a group of matrices

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Angel "Java" Lopez
http://www.ajlopez.com
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Por ajlopez, en: Ciencia