Lie Groups, Physics, and Geometry: An Introduction for Physicists, Engineers and Chemists

Description

This book is a historical tour from Galois theory to solve polynomial equations by radicals using the permutation group of the roots (first chapter), to Lie theory to solve differential equations using symmetries (last chapter). In between, the author describes many of the important aspects of Lie group theory in fourteen chapters, focusing on applications, rather than on demonstrating theorems. This contrasts with the other book of the author [Lie groups, Lie algebras, and some of their applications, Reprint of the 1974 original, Krieger, Malabar, FL, 1994; MR1275599], which is a longer book and more difficult to read, although the spirit is the same.
The book contains many worked examples, helping the reader to understand the general theory, and a lot of proposed problems, the majority of them on applications in many areas of physics, with detailed hints (although solutions would be welcome).
The content of the book is the following:
In the Introduction a few notions on groups are given, and a description of the Galois method to solve by radicals polynomial equations up to fourth degree is given, using the permutation group of the roots.
In Chapter 2 on Lie groups the basics of Lie groups are explained, with a concentration on examples, mainly SL(2,R) and other matrix groups.
Chapter 3 is devoted to matrix groups, since the majority of Lie groups are matrix groups, subgroups of matrix groups obtained by imposing constraints, or coverings of them (apart from the exceptional Lie groups). Particularly interesting are the problems about 1D scattering by means of transfer, SU(1,1), matrices or scattering, U(2), S-matrices.
Chapter 4 concentrates on Lie algebras, as the linearization of a Lie group. A brief introduction to the exponential map is given (it will be studied in more detail in Chapter 7). Structure constants, the regular (adjoint) representation and the Cartan-Killing inner product are introduced.
Chapter 5 deals with matrix algebras, and parallels Chapter 3. In the problems the Cartan decomposition of a Lie algebra is addressed.
In Chapter 6 operator algebras are introduced as realizations of Lie algebras in terms of either boson or fermion operators and in terms of differential operators. Many examples of applications in quantum mechanics are given in the problems section.
Chapter 7 elaborates on the exponential map, the covering problem and the universal covering groups, focusing on the examples of SU(2) and SL(2,R). The BCH formulas are illustrated by a few examples. Important applications of the exponential map in physics are given, like the evolution operator for the Schrödinger equation, computing it in the standard way as a time-ordered product (although a Magnus series expansion would have been more adequate since it is directly related with the BCH formula; see, for instance, [S. Blanes et al., Phys. Rep. 470 (2009), no. 5-6, 151–238; MR2494199]), and the computation of expectation values in equilibrium thermodynamics.
Chapter 8 on structure theory for Lie algebras begins with the classification problem of Lie algebras accomplished through Chapters 8, 9 and 10. In this chapter a decomposition of a Lie algebra in its semisimple and maximal solvable (also known as radical) parts by means of the Cartan-Killing inner product is considered. Some examples of this decomposition are considered, like the six-dimensional algebra of the two photon operators nˆ=a†a+12,a†2,a2,a†,a,I. However, after decomposing the algebra into its semisimple, nˆ,a†2,a2, and solvable (here nilpotent) part a†,a,I, the conclusion is reached that the operator nˆ is non-compact. There must be some problem with the conventions (or some imaginary constant is lacking) since this is, up to a constant, the energy operator for the harmonic oscillator, which is clearly a compact operator with discrete spectrum.
Chapter 9, on the structure theory for simple Lie algebras, continues with the classification problem. Here the commutation relations for simple Lie algebras are written in standard form, diagonalizing the regular representation. Roots and their properties are studied. The rank and the invariant operators are introduced.
Chapter 10, on root spaces and Dynkin diagrams, finishes the classification program. All root diagrams are given, and the proof that they exhaust all possible cases is given in terms of Dynkin diagrams.
Chapter 11 goes on to study real forms of complex simple Lie algebras. Cartan's procedure is used for this purpose, i.e. starting with the real compact form and finding all involutive automorphisms preserving the inner product, with three ways of obtaining them: block matrix decomposition, subfield restriction and field embedding. Tables with the real forms obtained with each procedure for the classical simple Lie algebras are given. Some examples are given in the problems, but some inconsistencies appear. For instance, the algebra of problem 1 seems to be U(2) (it is related to the algebras appearing in problems 2 and 3 of Chapter 3); however, the classification of the generators into compact and non-compact ones seems to be inverted. For instance, the generator a†1a1−a†2a2, which coincides with the generator J3 of SU(2), is claimed to be non-compact. It seems as if the Cartan-Killing inner product should be multiplied by −1 to obtain the correct result. Also, in the construction of the unitary irreducible representations of SU(1,1) by analytical continuation from those of SU(2), finite-dimensional, non-unitary, irreducible representations could also have been discussed, by simply dropping the Hermiticity condition (non-unitary infinite-dimensional representations also appear). These non-unitary, finite-dimensional representations are very common. They appear, for instance, as the defining or the regular (adjoint) representation of non-compact groups.
Chapter 12 is a turn to geometry. Given a Cartan decomposition g=h+p of a compact simple Lie algebra, the exponentials EXP(p) and EXP(ip) lead to globally symmetric Riemannian spaces, the Cartan-Killing metric being negative definite in the first case and positive definite in the second one. Globally symmetric pseudo-Riemannian spaces can also be defined if g admits two different metric preserving involutive automorphisms. The rank can be defined, and the metric and the measure can be computed, as well as the invariant operators (Laplace-Beltrami).
Chapter 13 is devoted to contraction, i.e. going from a semisimple group to a non-semisimple one by means of a one-parameter family of isomorphisms with a singular limit. Inönü-Wigner contractions are explained in detail in examples. An example of a contraction that is not of Inönü-Wigner type is also given (although emphasis should be given on why it is not Inönü-Wigner). Contraction of the representations, Casimir operators, matrix elements, etc. are also given in examples.
Chapter 14 considers one of the most important applications of Lie groups in accessible physics, the hydrogenic atoms. Different symmetry groups can be obtained. The geometric symmetry, SO(3), is due to the central character of the Coulomb potential. The dynamical symmetry SO(4) (or SO(3,1) for scattering states) accounts for the degeneracy of the energy levels, and has its origin in the existence of another conserved quantity, besides the angular momentum, the Runge-Lenz vector. The spectrum generating symmetry is SO(4,2), the conformal group, which transforms energy eigenstates among themselves. Three different realizations of this symmetry are given: the Schwinger (bosonic) realization, the Kustaanheimo-Stiefel (KS) construction in terms of four harmonic oscillators in 1-1-1-1 resonance and a constraint, and the Lie algebra of physical operators (angular momentum, Runge-Lenz, etc.). These groups are very useful in obtaining the energy levels of the hydrogen atom, and even help in classifying the different perturbations of the Coulomb potential.
Chapter 15 deals with Maxwell equations, invariant under the Poincaré group. The transformation properties of the electromagnetic field are studied in two different realizations, the manifestly covariant, which is redundant, and the unitary irreducible representation. To eliminate the superfluous states of the manifestly covariant realization, some restrictions must be imposed, and these turn out to be the Maxwell equations. Some interesting applications are considered in the problems, such as the different kinds of redshift observed in starlight (Doppler, gravitational, etc.).
The last chapter is the closure of this historical trip, returning to Lie theory to solve differential equations using symmetries. From the observation that the integration constant added in an indefinite integral is an element of a continuous symmetry group, a sophisticated theory is developed to solve ordinary and partial differential equations. The general method is explained and an example to illustrate the procedure is worked out.
One of the things one misses is that there are no references to the omitted proofs, mostly auto-references to the examples developed.
Summarizing, this is a great how-to book, where one can find detailed examples worked out completely, covering many and interesting aspects and applications of group theory. Important mathematical aspects of Lie group theory, like harmonic analysis on homogeneous spaces, Plancherel formulas, representation theory, and many more, are absent, but they are outside the scope of the book, which purports to be accessible to graduate and undergraduate students.

Reviewed by Julio Guerrero

Describing many of the most important aspects of Lie group theory, this book presents the subject in a 'hands on' way. Rather than concentrating on theorems and proofs, the book shows the applications of the material to physical sciences and applied mathematics. Many examples of Lie groups and Lie algebras are given throughout the text. The relation between Lie group theory and algorithms for solving ordinary differential equations is presented and shown to be analogous to the relation between Galois groups and algorithms for solving polynomial equations. Other chapters are devoted to differential geometry, relativity, electrodynamics, and the hydrogen atom. Problems are given at the end of each chapter so readers can monitor their understanding of the materials. This is a fascinating introduction to Lie groups for graduate and undergraduate students in physics, mathematics and electrical engineering, as well as researchers in these fields.

the original (unabridged) edition of this work was recommended in Kadeisvili's "An Introduction to The Lie–Santilli Isotopic Theory" as a physical treatment of current Lie theory