Symmetries and group theory in particle physics. An introduction to space-time and internal symmetries.
The aim of the monograph is two-fold, to give the basic notions and facts from group and group representation theory for particle physics and to give a survey of their applications in relativistic quantum field theory and particle physics. Therefore it should be considered to be a good handbook. Chapter 1 introduces the basic notions of Lie groups and algebras including the semi-simple Lie algebras and their representations together with Dynkin diagrams and classification of the classical semi-simple Lie algebras and Lie groups. Chapter 2 gives a short account of the main properties of the three-dimensional rotation group \(\mathrm{SO}(3)\) and its universal covering group \(\mathrm{SU}(2)\) as illustration of the contents of Chapter 1. It contains also the irreducible representations of \(\mathrm{SO}(3)\) and \(\mathrm{SU}(2)\) and introduces the Clebsch-Gordan coefficients. In Chapter 3 general properties of the homogeneous Lorentz transformations are considered paying basic attention to their group-theoretical aspects, which are important for special relativity in the description of elementary particle physics. The restricted Lorentz group \(L_{+}^{\uparrow}\) is analyzed through derivation of Lie algebra from its infinitesimal transformations. Irreducible representations of the group \(L_{+}^{\uparrow}\) and of the complete Lorentz group are discussed. In Chapter 4 the general properties of the Poincaré group (i.e. the group of inhomogeneous Lorentz transformations) are given as the space-time coordinate transformations between any two inertial frames of reference. Then, unitary irreducible representations of this group are considered, which are needed for the description of the quantum-mechanical states of one or more particles in Chapter 5. Chapter 6 analyzes the properties of the discrete operations, space inversion \(I_{s}\) and time reversal \(I_{t}\), belonging to the complete Lorentz group \(L\) and the Poincaré group \(P\). The transformation properties of the one- and two-particle states under these operations are analyzed with discussion of their physical implications. In Chapter 7 the relativistic wave-functions are introduced from one-particle states considered in Chapter 5. The lower spin cases \(0,1\) and \(\frac{1}{2}\) are examined and the relevant relativistic equations both for massive and massless particles are derived. In particular, the group-theoretical derivations of the Dirac and Maxwell equations are given. The second part of the monograph studies the various kinds of internal symmetries. In Chapter 8 the wide use of unitary groups and their representations is presented, which are very important to the investigation of the particle interactions. They allow to obtain the hidden symmetries and new invariance principles. The main part of Chapter 8 is devoted to the analysis of hadrons participating in strong interactions. They have the same spin, parity and strangeness and differ only in electric charges, suggesting to classify their different families according to the lower representations of \(\mathrm{SU}(2)\) using the concept of isospin. The further analysis of the structure of the hadron multiplets leads to the introduction of three quarks as the fundamental constituents of matter. Each quark is identified by a specific quantum number, the flavor \(\mathrm{SU}(3)\), and an additional property of more fundamental symmetry, the color \(\mathrm{SU}(3)\). These notions and classifications are introduced in Chapter 8 on the basis of group representation theory. In Chapter 9 the gauge groups are studied, in which the parameters of the group transformations are continuous functions of the space-time coordinates. The gauge groups are widely used in quantum field theory and especially in the field theory of elementary particles. Among the non-abelian gauge field theories in this chapter the case of quantum chromodynamics (QCD) is considered, which is based on the gauged version of the group color \(\mathrm{SU}(3)_{c}\). QCD describes the strong interactions of quarks, mediated by eight vector bosons called gluons, which correspond to the eight generators of \(\mathrm{SU}(3)\). Here the electroweak standard model is considered, the field theory based on the gauge \(\mathrm{SU}(2)\otimes \mathrm{U}(1)\) group, which reproduces with great accuracy the properties of weak interactions of leptons and quarks, combined with the electromagnetic ones. Another kind of symmetry is based on spontaneous symmetry breaking when the Lagrangian of the system is invariant under a specific gauge symmetry but the solutions of the equations of motion possess a lower symmetry. It is illustrated here in the frame of a couple of simple models. Also in Chapter 9 the higher gauge symmetries are considered, on the basis of which several grand unified theories are constructed combining strong and electroweak interactions. The monograph is finished by three appendices: A. Rotation matrices and Clebsch-Gordan coefficients; B. Symmetric group and identical particles; C. Young tableaux and irreducible representations of the unitary groups.
scientific article; zbMATH DE number 5800327