Lie Groups, Physics, and Geometry
Prof.
Robert Gilmore
Physics Department
Email: robert.gilmore@drexel.edu
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.
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Table of Contents
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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).
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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.
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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.
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References
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Index
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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