Table Of ContentGroup Theory in Non-Linear Problems
NATO ADVANCED STUDY INSTITUTES SERIES
Proceedings of the Advanced Study Institute Programme, which aims
at the dissemination of advanced knowledge and
the formation of contacts among scientists from different countries
The series is published by an international board of publishers in conjunction
with NATO Scientific Affairs Division
A Life Sciences Plenum Publishing Corporation
B Physics London and New York
C Mathematical and D. Reidel Publishing Company
Physical Sciences Dordrecht and Boston
D Behavioral and Sijthoff International Publishing Company
Social Sciences Leiden
E Applied Sciences Noordhoff International Publishing
Leiden
Series C - Mathematical and Physical Sciences
Volume 7 - Group Theory in Non-Linear Problems
Group Theory in
Non-Linear Problems
Lectures Presented at the NATO Advanced Study Institute
on Mathematical Physics, held in Istanbul, Turkey,
August 7-18, 1972
edited by
A.O.HARUT
International Centre for Theoretical Physics, Trieste, Italy and
University of Colorado, Boulder, Colo., U.S.A.
D. Reidel Publishing Company
Dordrecht-Holland / Boston-U.S.A.
Published in cooperation with NATO Scientific Affairs Division
Library of Congress Catalog Card Number 73-91202
ISBN-13: 978-94-010-2146-3 e-ISBN-13: 978-94-010-2144-9
001: 10.1007/978-94-010-2144-9
Published by D. Reidel Publishing Company
P.O. Box 17, Dordrecht, Holland
Sold and distributed in the U.S.A., Canada, and Mexico
by D. Reidel Publishing Company, Inc.
306 Dartmouth Street, Boston, Mass. 02116, U.S.A.
All Rights Reserved
Copyright © 1974 by D. Reidel Publishing Company, Dordrecht
Softcover reprint of the hardcover 1st edition 1974
No part of this book may be reproduced in any form, by print, photoprint, microfilm,
or any other means, without written permission from the publisher
TABLE OF CONTENTS
Introduction VII
ROGER PENROSE / Relativistic Symmetry Groups 1
1. Orthogonal and Conformal Groups 1
2. Asymptotically Simple Space-Times 10
3. The B.M.S. Group 21
4. Twistor Theory 43
M. CARMEL I / SL(2,C) Symmetry of the Gravitational Field 59
1. Spinor Representation of the Group SL(2,C) 59
2. Connection between Spinors and Tensors 61
3. Maxwell, Wey~and Riemann Spinors 63
4. Classification of Maxwell Spinor 66
5. Classification of Weyl Spinor 73
6. Isotopic Spin and Gauge Fields 82
7. Lorentz Invariance and the Gravitational Field 88
8. SL(2,C) Invariance and the Gravitational Field 92
9. Gravitational Field Equations 100
Problems 105
M. HALPERN and S. MALIN / Coordinate Systems in
Riemannian Space-Time: Classifications and Transformations;
Generalization of the Poincare Group 111
1. Introduction 111
2. Riemannian and Normal Coordinates 114
3. Transformations between Normal Coordinate Systems 118
4. Generalization of Inertial Frames to Curved Space-Time 121
5. Analytic Characterization of Geodesic Fermi Frames 124
6. Classifications of Coordinate Systems 128
7. The Principle of the Pre-Assigned Measurements 134
8. The Degree of Invariance of the Laws of Nature 135
9. Generalization of the Poincare Group to Curved
Space-Time; and Concluding Remarks 138
HANS TILGNER / Symmetric Spaces in Relativity and
Quantum Theories 143
1. Introduction 143
2. Lie Transformation Groups, Lie Algebras, Covering and
Pseudo-Orthogonal Groups 144
PART I / Symmetric Spaces and Lie Triple Systems 147
3. Symmetric Spaces 147
4. Symmetric Spaces as Homogeneous Spaces of Groups 151
5. Lie Triple Systems as the Local Algebraic Structures
of Symmetric Spaces 153
6. On Symmetric Spaces of Pseudo-Orthogonal Groups 154
7. Conformal Groups of Pseudo-Orthogonal Vector Spaces 158
8. Light Cones as Homogeneous but not Symmetric Spaces
of the Pseudo-Orthogonal Groups 160
VI TABLE OF CONTENTS
9. Applications in General Relativity 161
PART II/Domains of Positivity and Formal Real Jordan
Algebras 164
10. Domains of Positivity or Self-Dual Convex Cones 165
11. Jordan Algebras 167
12. The Relation between Domains of Positivity and
Symmetric Spaces 169
13. The Jordan Algebra of Minkowski Space 171
14. The Jordan Algebra of Non-Relativistic Spin Observables 173
PART III / Halfspaces of Jordan Algebras and Bounded
Symmetric Domains 175
15. The Siegel Half Space 176
16. Halfspaces and Bounded Symmetric Domains 179
17. The Halfspace of Minkowski Space 181
W. RUHL / Boundary Values of Holomorphic Functions that
Belong to Hilbert Spaces Carrying Analytic Representations
of Semi simple Lie Groups 185
O. Preliminaries 185
1. The Discrete Series of SU(l,l) 190
2. The Discrete Series of SU(2,2) 212
Appendix 228
BRUNO GRUBER / The Semi simple Subalgebras of the Algebra
B3(SO(7» and Their Inclusion Relations 231
1. Introduction 231
2. Classification Scheme 232
3. Actual Classification 233
4. Index of Embedding; Defining Matrix 234
5. Classification of Semisimple Subalgebras of B3 238
A.O. BARUT / External (Kinematical) and Internal (Dynamical)
Conformal Symmetry and Discrete Mass Spectrum 249
1. Introduction 249
2. Conformal Transformations on External Co-ordinates 249
3. Conformal Transformations on Internal Co-ordinates 254
4. The Connection between the External and Internal
Conformal Algebras. Discrete Mass Spectrum 256
PAUL ZWEIFEL / Non-Linear Problems in Transport Theory 261
Introduction 261
1. A Non-Linear Transport Equation 261
2. General Properties of the Solution 265
3. Solution of the Milne Problem 272
4. Explicit Evaluation of the Milne Problem Solution 277
INTRODUCTION
This is the second volume of a series of books in various aspects
of Mathematical Physics. Mathematical Physics has made great
strides in recent years, and is rapidly becoming an important dis
cipline in its own right. The fact that physical ideas can help
create new mathematical theories, and rigorous mathematical theo
rems can help to push the limits of physical theories and solve
problems is generally acknowledged. We believe that continuous con
tacts between mathematicians and physicists and the resulting
dialogue and the cross fertilization of ideas is a good thing.
This series of studies is published with this goal in mind.
The present volume contains contributions which were original
ly presented at the Second NATO Advanced Study Institute on Mathe
matical Physics held in Istanbul in the Summer of 1972. The main
theme was the application of group theoretical methods in general
relativity and in particle physics. Modern group theory, in par
ticular, the theory of unitary irreducibl~ infinite-dimensional
representations of Lie groups is being increasingly important in
the formulation and solution of dynamical problems in various bran
ches of physics. There is moreover a general trend of approchement
of the methods of general relativity and elementary particle
physics. We hope it will be useful to present these investigations
to a larger audience.
A.O. BARUT
RELATIVISTIC SYMMETRY GROUPS
Roger Penrose
Department of Mathematics
Birkbeck College, London
1. ORTHOGONAL AND CONFORMAL GROUPS
A mathematical fact of very great significance for relativity
theory is the existence of the familiar homomorphism* between the
group SL(2,C) of complex unimodular (2 x 2) matrices and the
Lorentz group 0(1,3). This homomorphism
SL(2,C) ~ 0(1,3) (1. 1)
is a local isomorphism and maps SL(2,C) onto the identity-connected
component of 0(1,3), in an essentially (2-1) fashion. The term
"essentially" here refers to the fact that SL(2,C) is connected.
If A and B are the two elements of SL(2,C) which map to some given
element Q of 0(1,3) (actually B = -A), then A may be connected to
B by some curve in SL(2,C). The image of this curve in 0(1,3) is a
closed curve K through Q (topologically equivalent to a continuous
rotation through 2n). Neither A nor B can be preferred as the
SL(2,C) image of the Lorentz transformation Q. For as we pass from
Q back to Q along the curve K in 0(1,3), the inverse image in
* A homomorphism between continuous groups is simply a continuous
mapping from the first to the second which preserves the group
operations. A local isomorphism is such a mapping which is 1-1
in the neighbourhood of the identity elements; then there is in
duced an isomorphism between the corresponding Lie algebras of
infinitesimal group elements. The identity-connected component
of a continuous group is its maximal connected subgroup (i.e.
consisting of elements continuously connected with the identity
element). For a discussion of the classical groups SL(2,C),
0(1,3), etc. see references [1,2].
2 ROGER PENROSE
8L(2,C) must pass continuously from A to B, or else back from B to
A. The ambiguity between A and B is therefore essential.
There is a higher-dimensional analogue of (1.1) which also
has considerable importance for relativity theory, namely the
homomorphism
8U(2,2) ~ 0(2,4) (1.2)
which is also a local isomorphism, and which similarly maps
8U(2,2) onto the identity-connected component of 0(2,4) in an es
sentially (2-1) fashion. The group 8U(2,2) of unimodular pseudo
unitary (++--) (4 x 4)-matrices gives rise to the algebra of
twistors, analogously to the way that 8L(2,C) gives rise to the
algebra of 2-component spinors. Twistors will be discussed in
8ection 4. The significance of the pseudo-orthogonal group 0(2,4),
for relativity theory, lies in its relation to the ls-parameter
conformal group of Minkowski space-time. I shall denote this con
formal group by C(1,3) and give its precise definition shortly.
We have, in fact, a homomorphism
0(2,4) ~ C(1,3) (1. 3)
which is again a local isomorphism, mapping 0(2,4) onto C(1,3) in
an essentially (2-1) fashion. The homomorphism which is the com
posite of (1.2) with (1.3)
8U(2,2) ~ C(1,3) (1.4)
is thus a local isomorphism which maps 8U(2,2) onto the identity
connected component of C(1,3) in an essentially (4-1) fashion.
8imilar to (1.3) is a homomorphism
0(1,3) ~ C(2) (1.5)
where C(2) denotes a 6-parameter conformal group for the Euclidean
plane analogous to C(1,3). (I shall be more precise shortly.) The
homomorphism (1.5) is a local isomorphism which is (2-1), but it
is not essentially (2-1). The map from the identity-connected com
ponent of 0(1,3) onto the identity-connected component of C(2) is
actually (1-1). The homomorphisms (1.3), (1.5) are part of a more
general pattern of local isomorphisms:
O(p+l, q+l) ~ C(p,q) (1.6)
The local isomorphisms (1.1) and (1.2) are, on the other hand,
special features of the low dimensionalities involved. (We may
note, in passing, there is the "non-relativistic" essentially (2-1)
local isomorphism 8U(2) ~ 0(3), which is closely related to these,
to quaternions and to non-relativistic spinors.) For the remainder
of this section I shall be primarily concerned with (1.6) and its
RELATIVISTIC SYMMETRY GROUPS 3
particular instances (1.3), (1.5). The special isomorphism (1.4)
will play a basic role in Section 4.
First, consider (1.5). I have yet to define what I mean by
the conformal group C(2). The orientation-preserving local confor
mal maps of the plane to itself may be conveniently represented by
f (t;) , (1. 7)
where f is a holomorphic (i.e. complex-analytic) function and where
1;; = x + iy,
x and y being standard Cartesian coordinates for the plane. We
have, for the line-element,
(1.8)
so
illustrating the conformal nature of (1.7). Since f is arbitrary
holomorphic, the local conformal maps of the Euclidean plane to
itself constitute an infinite-parameter system. For a global map,
we would require that both f and its inverse map be non-singular
over the whole plane. This restricts f to be a linear function
f(t;) = at; + S
showing that the group of (orientation-preserving) conformal maps
of the plane to itself is described by four real parameters (real
and imaginary parts of a,S). These maps may be generated by the
Euclidean motions (10.1=1) and the dilations (a real, S=O).
This group is not what I mean by C(2), however. For that, we
require to compacti~the plane by the addition of a point at in
finity. This is a standard procedure in complex variable theory,
and is most graphically illustrated by means of a stereographic
projection of the unit sphere S2 to the plane (Figure 1). Let S2
be given by the equation
Fig. 1. The unit sphere S2 is projected stereographically from the
=
north pole (0,0,1) to the plane Z 0, this plane being regarded as
the Argand plane of the complex number t; = x+iy. The first formula
(1.9) is readily obtained from the geometry of the picture.
