Group theory and physics.

*(English)*Zbl 0816.53002
Cambridge: Cambridge University Press. xiii, 429 p. (1994).

This excellent book contains an introduction into the theory of abstract groups, Lie groups and their representations combined with applications of this theory in physics. The author succeeded in giving a cohesive presentation of physics and mathematics, using physical problems for introducing and developing mathematical concepts and, after that, applying the mathematical results to physics. Rather small prerequisites (linear algebra, calculus, elementary physics) are demanded from the reader.

The contents of the book is, in short, as follows. Ch. 1: Groups, actions and homomorphisms, applications to crystallography. Ch. 2: Representations of finite groups. Ch. 3: Vector bundles and induced representations (motivated by molecular vibrations), representations of semidirect products with applications to the Poincaré group, the Mackey theorems. Ch. 4: Compact groups, Lie groups and Lie algebras, the irreducible representations of \(SU(2)\) and \(SO(3)\), the periodic table and the shell model of the nucleus, relativistic wave equation. Ch. 5: The representations of \(GL(V)\) and \(SU(n)\), strangeness, \(SU(3)\) symmetry for elementary particles, quarks. There are 7 appendices devoted to some further topics.

The contents of the book is, in short, as follows. Ch. 1: Groups, actions and homomorphisms, applications to crystallography. Ch. 2: Representations of finite groups. Ch. 3: Vector bundles and induced representations (motivated by molecular vibrations), representations of semidirect products with applications to the Poincaré group, the Mackey theorems. Ch. 4: Compact groups, Lie groups and Lie algebras, the irreducible representations of \(SU(2)\) and \(SO(3)\), the periodic table and the shell model of the nucleus, relativistic wave equation. Ch. 5: The representations of \(GL(V)\) and \(SU(n)\), strangeness, \(SU(3)\) symmetry for elementary particles, quarks. There are 7 appendices devoted to some further topics.

Reviewer: A.L.Onishchik (Yaroslavl)

##### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

22-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups |

81-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory |

20H15 | Other geometric groups, including crystallographic groups |

22E70 | Applications of Lie groups to the sciences; explicit representations |

20C35 | Applications of group representations to physics and other areas of science |