Lie Groups, Physics, and Geometry

Prof. Robert Gilmore
Physics Department

This book has been published by Cambridge University Press during Janaury, 2008 (ISBN13 978-0-521-88400-6). Several chapter from this book are presented below to whet the appetite.

[PDF] Table of Contents

[PDF] Chapter 1: Introduction

Lie groups were initially introduced as a tool to solve or simplify ordinary and partial differential equations. The model for this application was Galois' use of finite groups to solve algebraic equations of degree two, three, and four, and to show that the general polynomial equation of degree greater than four could not be solved by radicals. In this chapter we show how the structure of the finite group that leaves a quadratic, cubic, or quartic equation invariant can be used to develop an algorithm to solve that equation.

[PDF] Chapter 2: Lie Groups

Lie groups are beautiful, important, and useful because they have one foot in each of the two great divisions of mathematics --- algebra and geometry. Their algebraic properties derive from the group axioms. Their geometric properties derive from the identification of group operations with points in a topological space. The rigidity of their structure comes from the continuity requirements of the group inversion map. In this chapter we present the axioms that define a Lie group.

[PDF] Chapter 3: Matrix Groups

Almost all Lie groups encountered in the physical sciences are matrix groups. In this chapter we describe most of the matrix groups that are typically encountered. These include the general linear groups $GL(n;F)$ of nonsingular $n \times n$ matrices over the fields $F$ of real numbers, complex numbers, and quaternions, and various of their subgroups obtained by imposing linear, bilinear and quadratic, and $n$-linear constraints on these matrix groups.

[PDF] Chapter 4: Lie Algebras

The study of Lie groups can be greatly facilitated by linearizing the group in the neighborhood of its identity. This results in a structure called a Lie algebra. The Lie algebra retains most, but not quite all, of the properties of the original Lie group. Moreover, most of the Lie group properties can be recovered by the inverse of the linearization operation, carried out by the EXPonential mapping. Since the Lie algebra is a linear vector space, it can be studied using all the standard tools available for linear vector spaces. In particular, we can define convenient inner products and make standard choices of basis vectors. The properties of a Lie algebra are identified with the properties of the original Lie group in the neighborhood of the origin. These structures, such as inner product and volume element, are then extended over the entire group manifold using the group multiplication operation.

[PDF] Chapter 5: Matrix Algebras

The Lie algebras of the matrix Lie groups described in Chapter 3 are constructed. This is done by linearizing the constraints defining these matrix groups in the neighborhood of the identity operation.

[PDF] Chapter 6: Operator Algebras

Lie algebras of matrices can be mapped onto Lie algebras of operators in a number of different ways. Three useful matrix algebra to operator algebra mappings are described in this chapter.

[PDF] Chapter 7: EXPonentiation

Linearization of a Lie group to form a Lie algebra introduces an enormous simplification in the study of Lie groups. The inverse process, reconstructing the Lie group from the Lie algebra, is carried out by the EXPonential map. We return to a more thorough study of the exponential map in this chapter. In particular, we address the three problems raised in Chapter 4: Does the EXPonential operation map the Lie algebra back onto the Lie group? Are Lie groups with isomorphic Lie algebras themselves isomorphic? Are there natural ways to parameterize Lie groups? We close this chapter with a spectrum of applications of the EXPonential mapping in Physics. Applications include computing the dynamic evolution of quantum systems and their\ thermal expectation values.

[PDF] Chapter 8: Structure Theory for Lie Algebras

In this chapter we discuss the structure of Lie algebras. A typical Lie algebra is a semidirect sum of a semisimple Lie algebra and a solvable subalgebra that is invariant. By inspection of the regular representation `in suitable form,' we are able to determine the maximal nilpotent and solvable invariant subalgebras of the Lie algebra and its semisimple part. We show how to use the Cartan-Killing inner product to determine which subalgebras in the Lie algebra are nilpotent, solvable, semisimple, and compact.

[PDF] Chapter 9: Structure Theory for Simple Lie Algebras

In this chapter we continue the development begun in the previous chapter. These two chapters focus on determining the structure of a Lie algebra and putting it into some canonical form. In the previous chapter we determined the types of subalgebras that every Lie algebra is constructed from. In this chapter we put the commutation relations into a standard form. This can be done for any Lie algebra. For semisimple Lie algebras this standard form has a very rigid structure whose usefulness is surpassed only by its beauty.

[PDF] Chapter 10: Root Spaces and Dynkin Diagrams

In the previous chapter the canonical commutation relations for semisimple Lie algebras were elegantly expressed in terms of roots. Although roots were introduced to simplify the expression of commutation relations, they can be used to classify Lie algebras and to provide a complete list of simple Lie algebras. We achieve both aims in this chapter. However, we use two different methods to accomplish this. We classify Lie algebras by specifying their root space diagrams. This is a relatively simple job using a `building up' approach, adding roots to rank $l$ root space diagrams to construct rank $l+1$ root space diagrams. However, it is not easy to prove the completeness of root space diagrams by this method. Completeness is obtained by introducing Dynkin diagrams. These specify the inner products among a fundamental set of basis roots in the root space diagram. In this approach completeness is relatively simple to prove, while enumeration of the remaining roots within a root space diagram is less so.

[PDF] Chapter 11: Real Forms

Root space diagrams classify all the simple Lie algebras and summarize their commutation relations. The Lie algebras so classified exist over the field of complex numbers. Each simple Lie algebra over $C$ of complex dimension $n$ has a number of inequivalent real subalgebras over $R$ of real dimension $n$. These are obtained by putting reality restrictions on the coordinates in the complex Lie algebra. The different real forms of a complex simple Lie algebra are obtained systematically by a simple eigenvalue decomposition. For the classical (matrix) Lie algebras, three different procedures suffice to construct all real forms. These are: block submatrix decomposition; subfield restriction; and field embedding.

[PDF] Chapter 12: Riemannian Symmetric Spaces

In the classification of the real forms of the simple Lie algebras we encountered subspaces {\frak p}, $i${\frak p} on which the Cartan-Killing inner product was negative-definite (on {\frak p}) or positive-definite (on $i${\frak p}). In either case these subspaces exponentiate onto algebraic manifolds on which the invariant metric $g_{ij}$ is definite, either negative or positive. Manifolds with a definite metric are Riemannian spaces. These spaces are also globally symmetric in the sense that every point looks like every other point --- because each point in the space $EXP({\frak p})$ or $EXP(i{\frak p})$ is the image of the origin under some group operation. We briefly discuss the properties of these Riemannian globally symmetric spaces in this chapter.

[PDF] Chapter 13: Contraction

New Lie groups can be constructed from old by a process called group contraction. Contraction involves reparameterization of the Lie group's parameter space in such a way that the group multiplication properties, or commutation relations in the Lie algebra, remain well defined even in a singular limit. In general, the properties of the original Lie group have well-defined limits in the contracted Lie group. For example, the parameter space for the contracted group is well-defined and noncompact. Other properties with well-defined limits include: Casimir operators; basis states of representations; matrix elements of operators; and Baker--Campbell--Hausdorff formulas. Contraction provides limiting relations among the special functions of mathematical physics. We describe a particularly simple class of contractions --- the Inonu--Wigner contractions --- and treat one example of a contraction not in this class.

[PDF] Chapter 14: Hydrogenic Atoms

Many physical systems exhibit symmetry. When a symmetry exists it is possible to use Group theory to simplify both the treatment and the understanding of the problem. Central two-body forces, such as the gravitational and Coulomb interactions, give rise to systems exhibiting spherical symmetry (two particles) or broken spherical symmetry (planetary systems). In this Chapter we see how spherical symmetry has been used to probe the details of the hydrogen atom. We find a heirarchy of symmetries and symmetry groups. At the most obvious level is the geometric symmetry group, $SO(3)$, which describes invariance under rotations. At a less obvious level is the dynamical symmetry group, $SO(4)$, which accounts for the degeneracy of the levels in the hydrogen atom with the same principal quantum number. At an even higher level are the spectrum generating groups, $SO(4,1)$ and $SO(4,2)$, which do not maintain energy degeneracy at all, but rather map any bound (scattering) state of the hydrogen atom into linear combinations of all bound (scattering) states. We begin with a description of the fundamental principles underlying the application of group theory to the study of physical systems. These are the Principle of Relativity (Galileo) and the Principle of Equivalence (Einstein).
[PDF] Chapter 15: Maxwell's Equations

The electromagnetic field ${\bf E(x},t)$, ${\bf B(x},t)$ is determined by Maxwell's equations.These equations are linear in the space and time derivatives. In the momentum representation, obtained by taking a Fourier transform of the electric and magnetic fields, Maxwell's equations impose a set of four linear constraints on the six amplitudes ${\bf E}(k), {\bf B}(k)$. Why? At a more fundamental level, the electromagnetic field is described by photons. For each photon momentum state there are only two degrees of freedom, the helicity (polarization) states, corresponding to an angular momentum 1 aligned either in or opposite to the direction of propagation. Thus, the classical description of the electromagnetic field is profligate, introducing six amplitudes for each $k$ when in fact only two are independent. The remaining four degrees must be absent in any description of a physically allowed field. The equations that annihilate these four nonphysical linear combinations are the equations of Maxwell. We derive these equations, in the absence of sources, by comparing the transformation properties of the helicity and classical field states for each four-momentum.

[PDF] Chapter 16: Lie Groups and Differential Equations

Lie group theory was initially developed to facilitate the solution of differential equations. In this guise its many powerful tools and results are not extensively known in the physics community. This Appendix is designed as an antidote to this anemia. Lie's methods are an extension of Galois methods for algebraic equations to the study of differential equations. The extension is in the spirit of Galois' work: the technical details are not similar. The principle observation --- Lie's great insight --- is that the simple constant that can by added to any indefinite integral of $dy/dx=g(x)$ is in fact an element of a continuous symmetry group --- the group that maps solutions of the differential equation into other solutions. This observation was used --- exploited --- by Lie to develop an algorithm for determining when a differential equation had an invariance group. If such a group exists, then a first order ODE can be integrated by quadratures, or the order of a higher order ODE can be reduced.

[PDF] References

[PDF] Index

[PDF] Index

A number of readers have pointed to additional useful applications to problems in the natural sciences. Before a survey of such applications is provided.

Prof. Steven Fox has derived the Lie group that describes transformations among noninertial frames in Hamilton's mechanics:

Prof. H. Katsura and H. Aoki have described a supersymmetry group for the relativistic hydrogen atom:

Last update: March 11, 2008