Group theory in a nutshell for physicists. (Q2800872)

The book under review is a textbook in group theory for graduate students in physics. It is written by a well-known physicist [\textit{A. Zee}, Quantum field theory in a nutshell. 2nd ed. Princeton: Princeton University Press (2010; Zbl 1277.81001)]. The general idea of the book is to pass to examples (simple, but non-trivial) as soon as possible. Mostly these examples are coming from quantum mechanics, but the author also touches applications in quantum field theory and particle physics (in particular the grand unification theories).NEWLINENEWLINE The content of the book is the following. Part 1 gives a definition of a group and considers examples of finite groups and rotation groups in dimension \(2\) and shortly in higher dimensions. In Part 2 the notion of a representation is introduced, the characters of representations are discussed, as examples the groups \(A_3\), \(A_4\), \(A_5\) are considered in details. Part 3 deals with basic facts about applications of group theory in quantum mechanics. This part begins with a non-standard introduction to Bloch's theorem and the Brillouin zone. In Part 4 the groups \(\mathrm{SO}(3)\), \(\mathrm{SO}(4)\) and their applications in quantum mechanics are discussed. In Part 5 the group \(\mathrm{SU}(3)\) and its applications in quantum mechanics is discussed. In Part 6 Cartan's classification of simple Lie algebras is presented. In Part 7 the spinor representations are defined. In Parts 8 and 9 the grand unification theory is discussed.NEWLINENEWLINE Every chapter contains a lot of non-trivial examples which are usually not included in textbooks devoted to applications of group theory in physics. The author tries to make the explanation as simple as possible. Instead of building general theory the author considers in details low-dimensional cases that have applications in quantum mechanics. Then the general theorems are shortly presented. This makes the book much simpler. It is especially suitable for students studying physics. Due to the great number of non-trivial examples the book is very suitable for mathematicians, too.