The pedagogical role and epistemological significance of group theory in quantum mechanics

The action of an arbitrary (but finite or compact) group on an arbitrary Hilbert space is studied. The application of group theory to physical calculations is often based on the Wigner-Eckart theorem, and one of the aims is to lead up to a general proof of this theorem. The group’s action gives irreducible ket-vector representation spaces, products of which lead to a definition of coupling (Wigner, or Clebsch-Gordan) coefficients and jm and j symbols. The properties of these objects are studied in detail, beginning with properties that are independent of the basis chosen for the representation spaces. We then explore some of the consequences of choosing bases by using the action of a subgroup. This leads to the Racah factorization lemma and the definition of jm factors, also a general statement of Racah’s reciprocity. In the third part, we add to these ideas, some properties of the space of all linear operators taking the Hilbert space to itself. This leads to a proof of the Wigner—Eckart theorem which is both succinct and in the language of quantum mechanics.

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Equivariant Localization on Simply Connected Phase Spaces: Applications to Quantum Mechanics, Group Theory and Spin Systems

Lecture Notes in Physics Monographs - Equivariant Cohomology and Localization of Path Integrals ◽

10.1007/3-540-46550-2_5 ◽

2000 ◽

pp. 127-201

Keyword(s):

Quantum Mechanics ◽

Group Theory ◽

Spin Systems ◽

Simply Connected

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Quantum statistics: the indistinguishability principle and the permutation group theory

Revista Brasileira de Ensino de Física ◽

10.1590/s1806-11172007000300013 ◽

2007 ◽

Vol 29 (3) ◽

pp. 405-414

Author(s):

M. Cattani

Keyword(s):

Quantum Mechanics ◽

Group Theory ◽

Graduate Students ◽

Permutation Group ◽

Quantum Statistics ◽

Basic Knowledge ◽

Clear Understanding ◽

Theoretical Predictions

About two decades ago we have shown mathematically that besides bosons and fermions, it could exist a third kind of particles in nature that was named gentileons. Our results have been obtained rigorously within the framework of quantum mechanics and the permutation group theory. However, these papers are somewhat intricate for physicists not familiarized with the permutation group theory. In the present paper we show in details, step by step, how to obtain our theoretical predictions. This was done in order to permit a clear understanding of our approach by graduate students with a basic knowledge of group theory.

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