03054 - Lie Groups, Lie Algebras, and Some of Their Applications [Gilmore].pdf

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Lie Groups, Lie Algebras,
and Some of Their Applications
ROBERT GILMORE
University of South Florida
A WILEY-INTERSCIENCE PUBLICATION
JOHN WILEY
&
SONS, New York . London . Sydney . Toronto
Copyright
©
1974 by John Wiley
&
Sons, Inc.
All rights reserved. Published simultaneously in Canada.
No part of this book may be reproduced by any means, nor
transmitted, nor translated into a machine language with-
out the written permission of the publisher.
Library of Congress Cataloging in Publication Data:
Gilmore, Robert, 1941-
Lie groups, Lie algebras, and some of their
applications.
i.
"A Wiley-Interscience publication."
Bibliography: p.
1.
Lie groups. 2. Lie algebras. I. Title.
QA387.G54
512'.55
73-10030
ISBN 0-471-30179-5
Printed in the United States of America
10 9 8 7 6 5 4 3
J
1
Preface
Only a century has elapsed since
1873,
when Marius Sophus Lie began his
research on what has evolved into one of the most fruitful and beautiful
branches of modern mathematics-the theory of Lie groups. These researches
culminated twenty years later with the publication of landmark treatises
by S. Lie and F. Engel
[1-3]
between
1888
and
1893,
and by W. Killing
[1-4]
from
1888
to
1890.
Matrices and matrix groups had been introduced
by A. Cayley, Sir W. R. Hamilton, and
J. J.
Sylvester
(1850-1859)
about
twenty years before the researches of Lie and Engel began. At that time
mathematicians felt that they had finally invented something of no possible
use to natural scientists. However, Lie groups have come to play an
increasingly important role in modern physical theories. In fact, Lie groups
enter physics primarily through their finite- and infinite-dimensional matrix
representa tions.
Certain natural questions arise. For example, just how does it happen
that Lie groups play such a fundamental role in physics? And how are
they used?
Lie groups found their way into physics even before the development
of the quantum theory. They were useful for the description of pseudo-
Riemannian (locally) homogeneous symmetric spaces, being used in
particular in geometric theories of gravitation. But Lie groups were virtually
forced into physics by the development of the modern quantum theory in
1925-1926.
In this theory, physical observables appear through their
hermitian matrix representatives, whereas processes producing transforma-
tions are described by their unitary or antiunitary matrix representations.
Operators that close under commutation belong to a finite-dimensional
Lie algebra; transformation processes described by a finite number of
continuous parameters belong to a Lie group.
The kinds of applications of Lie group theory in modern physics fall
into three distinct stages:
v
VI
PREFACE
1. As symmetry groups (1929-1960). Symmetry
i~plies
degeneracy. The
greater the symmetry, the greater the degeneracy. Assume that a Lie group
G with Lie algebra 9 commutes with a Hamiltonian
Yf:
GYfG-
1
=
Yf<=>[Yf,
g]
=
0
Then by Wigner's theorem the basis vectors spanning a fixed energy
eigenspace carry a representation of
G.
For example, the three-dimensional
isotropic harmonic oscillator whose Hamiltonian is
Yf
=
hw(alal
+
a1 a2
+
a1 a3
+
3/2)
where
[at
a.]
=
-b··
"
J
1J
[ai' a
j]
=
[a
J,
aI]
=
0
has spherical symmetry. Therefore,
Yf
commutes with the infinitesimal
generators
Li
of the rotation group
SO(3):
(i,
j,
k)
=
(1, 2,
3)
cyc!.
The oscillator eigenfunctions therefore carry representations of the rotation
group
SO(3).
However, the existence of an "accidental" degeneracy in this example
gives a larger degeneracy than is demanded by the obvious geometric
invariance group
SO(3).
This suggests that a larger group, containing
SO(3)
as a subgroup, may be a more useful symmetry group for this Hamiltonian.
The group is
U(3),
with Lie algebra
Uij
:
..
[ Yf
,
U
1J
]
=
0
Yf
=
hw
U··
=
ata·
1J
1 J
L
(alai
+
1/2);
[Uij' U
rs ]
=
Uis
bjr
-
Urj
bsi
In fact, it is useful and even desirable from a calculational standpoint to
label the oscillator eigenfunctions with
SU(3)
representation labels (J. M.
Jauch and E. L. Hill [1], J. P. Elliott [1 ]).
2. As nonsymmetry groups (1960-
). Around 1960 physicists were
gradually forced to realize that groups that do not commute with
Yf
can
be even more useful than symmetry groups from a computational viewpoint.
As an example, it is possible to find a 16-dimensional nonsymmetry group
.
WIt h generators
ai
t t
a
l
a
j '
ai'
j '
[Yf,
alaj]
=
0
[
Yf,
al]
=
+
hwaI
[Yf,
aj]
=
-hwaj
[Yf, I]
=
0
This nonsymmetry group is contracted from the noncompact group
U(3, 1).
Using this noncompact algebra, any eigenstate can be obtained from any

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