Tensor fields invariant under subgroups of the conformal group of spacetime

J. Beckers, J. Harnad, M. Perroud, and P. Winternitz

Citation: Journal of Mathematical Physics 19, 2126 (1978); doi: 10.1063/1.523571

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 Tensor fields invariant under subgroups of the conformal
group of space-time8 )
J. Beckers,b) J. Hamad, M. Perroud,c) and P. Wintemitz
Centre de Recherches Mathematiques, Universite de Montreal, Montreal H3C 3J7, Canada

This work is concerned with the characterization of tensor fields in (compactified) Minkowski space which
are invariant under the action of subgroups of the conformal group. The general method for determining
all invariant fields under the smooth action of a Lie group G on a manifold M is given, both in global
and in local form. The maximal subgroups of the conformal group are divided into conjugacy classes under
the Poincare group and the most general fields of I-forms, 2-forms, symmetric (0,2) tensors and scalar
densities which are invariant under representatives of each class (as welJ as certain other subgroups) are
then determined. The results are then discussed from the viewpoint of physical interpretation (as, e.g.,
electromagnetic fields, metric tensors, etc.) and applicability; in particular, for studies of spontaneously
or otherwise broken conformal invariance.

I. INTRODUCTION tensor fields, except insofar as relating the fields de-

fined on different sheets of M, we shall make no use of
it.
Statement of the problem

In this paper we concern ourselves with the problem 1. Background

of determining the most general types of tensor fields
on the space-time manifold which are invariant under The conformal group as a local transformation group
subgroups of the conformal group. We shall obtain a in Minkowski space has long been known in physics as
complete solution of this problem for the following types the invariance group for Maxwell's equations, 4.5 and
of fields and groups: 1-forms, 2- forms, symmetric more generally, for a wide variety of other field equa-
(0,2) tensors and scalar densities invariant under the tions 6- 9 (particularly those describing massless parti-
maximal subgroups of the conformal group, as well as cles), Since it contains the invariance group of special
under certain other subgroups of interest in physics. relativity; that is, the Poincare or inhomogeneous
Such fields may be identified, for example, as electro- Lorentz group, it has often been suggested that the
magnetic potentials and fields, metric tensors, etc. conformal group might have an equally basic interpre-
The particular model for the space-time manifold that tation as fundamental symmetry group for the geometry
is used is the conformally compactified Minkowski of space-time and the equations defined in it. In par-
space £1, upon which the conformal group acts as a ticular, considerable study has been made of the im-
global transformation group. This is a compact, in- plications of conformal invariance in quantum field
finitely connected manifold which possesses the same theory, 10,11 primarily motivated by attempts to interpret
local structure as Minkowski space, That is, it admits the approximate scaling invariance found experimen-
a local (infinitesimal) causal orientation and therefore tally in high energy scattering of elementary particles, 12
a pseudo-Riemannian metric of signature (1,3), How- Apart from this, conformal invariance of field equa-
ever £1 does not possess a global causal structure since tions is a property shared by most models for the uni-
there exists within this space an infinity of closed, time- fied gauge field theories of weak, electromagnetic, and
like geodesics, and hence it is not quite an adequate strong interactions, 13.14 due to the fact that the fields
model for the space-time of relativistic physiCS. This involved (prior to renormalization) are all massless.
difficulty may be satisfactorally resolved l - 3 by replac- In this context, particle masses are introduced through
ing £1 by its noncompact, simply connected covering spontaneous breaking of the gauge symmetry which also
space M and, correspondingly, the conformal group leads to breaking of the conformal invariance at the
with its universal covering group. Since such a replace- level of solutions to the field equations, 15-18 Solutions
ment has no effect upon the invariance properties of which minimize the total energy represent the classical
analog of the quantum ground state, and the largest sub-
group of the invariance group of the field equations
which also leaves invariant these solutions will play the
role of the fundamental symmetry group for the physical
alSupported in part by the National Research Council of Canada, system. If for reasons of stability, either due to the
Le Minist~re de l'Education du Gouvernement du Qullbec, and type of interaction involved l9 or to the topological prop-
a NATO research grant. erties of space-time implied by the solutions, 20-22
blpermanent address: Institut de Physique au Sart Tilman,
this ground state possesses less symmetry than the
B-4000 Li~ge 1, Belgium.
dAfter Aug. I, 1977: Department de Mathematiques, Ecole dynamical equations, we necessarily have a spontan-
Polytechnique, Montreal. eously broken symmetry. At the level of space-time

2126 J. Math. Phys. 19(10), October 1978 0022-2488178/1910-2126$1.00 © 1978 American Institute of Physics 2126

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 symmetries it is therefore quite basic to ask which introduce all the definitions and notations needed with
solutions to conform ally invariant equations are them- regard to the conformal group, realized successively
selves invariant under large subgroups of the conformal as: a local transformation group in Minkowski space;
group. Certain particular cases of this question have a global transformation group in conformally compacti-
been studied in the literature. For example, the "one fied Minkowski space; the group 0(4, 2)/Z2' acting in a
instanton" solution for the SU(2) gauge theory is known six-dimensional real space; and the group SU(2, 2)/24
to be completely characterized by its property of in- acting in a four complex dimensional space. In
variance under the 0(5) subgroup of the conformal group Secs, IV 9-IV 11 the maximal subgroups of the con-
0(5, 1) for Euclidean space-time. 13 Similarly, for the formal group are divided into conjugacy classes under
A¢4 (scalar density) field theory, solutions invariant the Poincare group. The reason for dOing this is the
under 0(3,2) have been studied as a model for introduc- following. When obtaining the most general invariant
ing a fundamental scale of length, thereby spontaneously field under a given group, we may naturally generate
breaking the 0(4,2) invariance of the field equations. 17 an infinity of other fields by applying Poincare trans-
formations, and each of these fields will be invariant
under the group obtained from the original invariance
In general relativity, conformal transformations group through conjugation by the corresponding trans-
also playa central role, several of the known cosmo- formation, However, such fields will not be essentially
logical models having the property of conformal flat- different from one another in terms of interpretation
ness; that is, differing from a flat space by a conformal since, by virtue of the duality between point and co-
change of metric. The symmetry groups of some of ordinate transformations, they may be interpreted as
these models [e. g., 0(4,1) or 0(3,2) for the de Sitter the same field viewed in different Lorentz frames.
spaces] moreover, are maximal subgroups of the con- Therefore, it is sufficient to determine the invariant
formal group. Alternative models for explaining cosmo- fields corresponding to a single representative of each
logical data, such as the chronometric theory! of Segal, conjugacy class of groups under the Poincare group.
formulated in the space AI, have also put particular Such an argument holds, of course, for any invariant
emphasis on the underlying conformal invariance prop- quantity under a given group, making the conjugacy
erties of space-time. The conformal group has further- classes under the Poincare group the only relevant
more been studied as a gauge group for field theories characterization from the point of view of relativistic
formulated in a non- Riemannian geometry, in which it invariance. An essential distinction must be made,
plays a role analogous to that of the Lorentz group in however, between the restricted Poincare group, pre-
general relativity. 23 serving orientation and causal sequence, and the general
Poincare group, containing the transformation P (space
inversion), T (time inversion), and PT, Equivalence
The subalgebras of the conformal Lie algebra have of reference frames under the latter represents an
been studied recently as part of a general program for additional physical assumption, beyond that of relativis-
subalgebra structure analysis. 24 All the maximal sub- tic invariance, whose validity depends upon the nature
algebras have been identified, up to conjugacy under of the dynamical equations involved. With regard to the
the group,25 and their subalgebra structures, in turn, general problem of determining classes of subgroups,
have been determined completely. 26-21 These, and other conjugated under a particular subgroup, a formulation
subalgebra analyses have been applied in particular to is given in Sec, IV 11 in terms of double cosets which
the classification of symmetry breaking interactions in leads to two possible approaches to such an analysis,
the Scrodinger equation 28• 29 as well as to other prob- Both these methods are illustrated for the particular
lems of interest in physiCS. The electromagnetic fields cases of maximal subgroups of the conformal group and
invariant under certain subgroups of the Poincare the results are summarized in Table L In Secs, V 12-
group have been studied systematically with methods V 14, the methods of Secs. II 3-II 5 are applied with
similar to those developed in the present work, 30-33 respect to these maximal subgroups (as well as certain
nonmaximal ones) so as to obtain the most general in-
variant fields of 1- forms, 2- forms, (0,2) symmetric
tensors, and scalar densities, The cases for which
2. Outline of development nonzero invariant fields exist are also indicated in
Table L Finally, the results are discussed in Sec. V 15
from the viewpoint of physical interpretation and a sum-
The present work is divided into several distinct parts mary is given in which possible applications and exten-
and may be read in a variety of ways, depending upon sions are suggested.
the interests of the reader, Sections n 3-I15 deal with
the problem of characteriZing the most general tensor
field on a manifold, invariant under a transformation General references: For differential geometric
group. The discussion is presented first in coordinate- definitions and notations, as used in Secs, II 3-II 5,
free terms, both for global and local invariance we recommend the standard texts of Refs. 34-37 to
(Sec. II 3). The relevant equations are then given in the reader. For further background and references on
a coordinate representation [Eqs, (4,2), (4.3), and the conformal group, see Refs. 1, and 38-40, For a
(4.6»). Finally (Sec. II 5) a formulation in terms of summary regarding nonlinear group action on manifolds,
fibre bundles is given which makes precise certain see Ref. 41 and for more detailed mathematical back-
notions used in Sec. II 4, The following Secs. II16-II18 ground, Refs. 42-44.

2127 J. Math Phys., Vol. 19, No. 10, October 1978 Beckers et a/. 2127

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 TABLE I. Maximal Subalgebras of c(3, I).

Algebra
- tv ,,'8
=-
Co
JY'
(dim~)
[n,/Lno]
Normalizer
Nd.,Yr)
in 0(4,2)
Invariant
Subspace
V,
Transition
Element
g, : V\~Vi
Orbit in
C' under
Nd.,%,)
Basis
for
iil'i
C'
a a
A
C'

F
~.!:;!
...
G
l.p
sim(3,1) SIM(3, I) X Z, V, 1(O,O,O,O,-x,x) I [ '1"=0, ""+'1'=0 ILi,K"D,P~ I
---- --
(11) V,= (0,0,0,0,x,x) I (!!.-
exp 2 (P,-G,)
. ) ""+'1'*0 (L"K"D,C,,1

[001] V,= Ix,O,O, -x,O,O)! ex{f<po+Co) ) "'"*0, ""+'1'=0 (L"K"L,+K,,L, -K"

xexp(f (P,-C.) ) P"P"Po-P"C"C"Co-c.)

0(4,1) 0(4,I)XZ, V, ( (x,O,O,O,O,O) I [ ."~+.,,~+.,,j (D,L"P"C,I V V

+.,,!-.,,; =0
(10)

[100\ V,= 1(0,0,0,0, e AD 'I; +.,,~ +.,,~ (L"K"eAp" -e 'AC~ I V V

x sinM,x cosM) I xexp(f(po+Co) ) +.,,~_.,,;=eH

0(3,2) 0(3,2)XZ, V, = {(0,0,0,x,0,0) I 1 ."~-.,,;-.,,i (K"K"L,,D,Po, V V

-.,,~+.,,; =0 P"P"C"C"C, 1
(10)
[0 I 0] V, = { (0,0,0,0, e AD "'5-"'~-"'~-"'! (Li,K i,e AP ~ +e -AC~ I V V
X cosM,x sinM) I X exp(-f< P,-C,) ) +."l =e 2A

optO,l) OPT(3,I)XZ, V,= I(x,O,O, I .,,'+.,,'=.,,0+.,,' ID,L,,K,,L,+K,,

-x,-y,y)l ="1'='1'=0 L,-K"P~,C,-C,I

(1O)

(002] V,= l(x,O,O,-x,y,y)1 ex{f(p,-C,) ) 71'+71'*0 ID,L"K"L,+K"

Of 71°+.,,'*0 L, -K"C~,Po-P,1

S(U(2,I)XU(I» [
su(2,1) V, = 1(0,0,0,,,,) I C' IL"D-K"po+C"Po+P."
Z,
$ u(l) XZ, Co-C"K,+P,+C"
K,-P,-C"L,-P,+C"
ZL,-P,+C,1
(9) $ \2L,+Po-p,+Co-C,1

(010]

0(2) 0(2) X 0(4) (V, = 1(x,O,O,O,O,y) I) (1) ( 1I~+"';

=11 i+lI i+."i+lI ~= I)
$0(4)
(7)
iA=O
t
A=O

[200J V,(A) eAD "'~+7Il IL"eAP,-e -AC, I V V V

= 1(x,O,O,O,xsinM, =."i +7Ii +'1j +1I! =e ZA <!lleAPo+e-ACol
x cosM) \

0(2) 0(2) X 0(2,2) V, = 1(O,x,y,O,O,O) I 1 .,,;+7Il IK"D.P",C"p"C, I V V v'

$0(2,2) =115-11 ;-." h'7;=o <!liLd

(7) V,= 1(O,O,O,y, e AD exp ( '2L,

7r ) 7I;+7Il IK"K"L"eAPo+e' ACo, V V V
x cosM,xsinM)I X exp(- f(p, - C,») ="1 ~-7Il-." h71 ;=e lA e AP, + e -A C" e AP,+ e -A C,j
[020] <!leAP,-e .ACd

0(3) 0(3)XO(2,1) V, = I(O,x,y,z,O,O) I [ '7i+7ll+7li IL"L,,L,I <!lID,Po,C,1 V V V

$0(2,1) ='16-lIh.,,;=0
(6) V,= 1(O,x,y,O, e'D 71;+71 l+7l; iK" e APo+ e -A Co, e AP, V V V
ZA +e -AC,1 $ IL" eAp,_ e -AC"
z cosM,z sinM)1 xexp(- f<P'-C,») =1J6-'1!+1I;=e
[030J eAp,_e-AC,1

2128 J. Math Phys., Vol. 19, No.1 0, October 1978 Beckers et S/. 2128

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 TABLE 1. Maximal SUbalgebras of c(3, I).

Algebra
,W Normalizer Invariant Transition Orbit in Basis
(dim$) Nd/lf) Subspace Element C' under for
[n.n_n o] in 0(4,2) Vi g i : V,--->V i NdY(",) Yi"i

0(2,1) 0(2, I)XO(2,l) XZ; V, = I(x,Y,Z,O,O,O) I I TJO-TJI-TJ,-O, TJ,oj=O

alo(2,1) orTJ,=TJ,=TJ.=O, TJooj=O
(6) (see text)
[120] V,=! (OJ',Z,O, e
AD
7J~+1J~-7J;= feu, ! 2K,-P,-C,,2K,-2D+Po-Co v V Y
xsinM,xcosM)1 xexp(f(po+Co») TJ6-TJ;-TJ~= ±e
u -P,-C,,2L ,-P,+C,1
alI2K I +P ,+C,,2K,-2D-Po
+Co+P,+C,2L,-P, +C, I

V'={( V2'Y'O, ex p( - f(p,-c,) ) 1) i+TJ ~-TJ ;=0, TJ,oj=O; IK"e'Po-e-'Co, :V,Y Y

z z X)} xex{fL,) 1)6 -TJ; -TJ~ =0, TJooj=O e'P, - e-'C, I
\1'2 \ / 2 \ / 2
xex p( - ~Po+co) ) al/L"e'P,+e 'c"
e"p,+e-"C,I

3. Coordinate independent formulation (3.2)

We address ourselves now to the following problem.
Given a C" manifold M of dimension n and a transforma- which represents a set of linear algebraic equations for
tion group G acting on M by C" diffeomorphisms (i. e. , each go. NOW, given a tensor I/J(po) at Po which satisfies
we have a homomorphismj: G- Diff(M) into the group (3.2), we may generate an invariant tensor field on Mo
of C" diffeomorphisms of M), determine the most gen- by defining
eral ¢rE.n(r,s)(M) (element of the module of s-covariant
r-contravariant tensor fields on M) such thatjg*(¢) = I/J (3.3)
for all g rE. G [where jg* is the mapping n(r •• )(M)
-n(r")(M) induced by the differential ofjg]. In fact,
it is sufficient to consider tensor fields of definite Because of (3.2), this mapping depends only on the left
symmetry under permutation of indices, since the sub- Go coset to which g belongs and hence I/J(p), PE Mo
modules of a given permutation symmetry are invariant really only depends on the point p and not the particular
under Diff(M). We may also extend the class of in- element g in the coset which maps Po to p. Conversely,
variant fields considered to include tensor densities. any invariant tensor field on Mo is uniquely determined
The requirement of invariance may be weakened to by its value at Po through Eq. (3.3). The particular
that of invariance under groups of local diffeomorphisms point Po chosen is evidently immaterial due to the
defined only in a neighborhood of each point. If the transitivity of the group action on the orbit, the isotropy
group G is a Lie group, this becomes equivalent to the group of any other point on a given orbit being conjugate
condition that the Lie derivative of ¢ with respect to to Go and the corresponding linear isotropy representa-
the vector fields X induced by the one-parameter sub- tion equivalent. The above remarks may be summarized
groups of G should vanish, as follows:

There is a one-to-one correspondence between the

(L ¢)p=lim ([¢-j'~tl (¢)]) =0, (3.1) irreducible tensor fields ¢ of type (r, s) on M o, in-
x t.\l t p
variant under G, and the GL(n)-irreducible tensors ifio
of the same permutation symmetry type as ifi, invariant
where X(p) is the tangent vector at prE. M to the curve
under the linear isotropy tensor representation at a
jg(t)(p) generated by the one-parameter subgroup get). fixed PoE Mo realized in the n(r,s)(po) tensor space;
the correspondence being given by Eqs. (3.2), (3.3)
Now. let Mo be the orbit of a point Po under the group with ¢(Po) identified as ¢o.
action and let Go be the isotropy subgroup of G at Po
[io e., Go ={g Ij,(po) =Po}]. Then for go '=- Go, j,* maps Suppose now that G is a Lie group and that the
the tensor space n(r··)(po) into itself linearly, ~nd mapping GXM-M defined by (g,p) -jlf(p) is C~. We
defines a representation of Go on this space which is a shall assume, moreover that M has an everywhere
(reduced) tensor product of the linear isotropy repre- dense submanifold M', which is the un_ 'n of a finite
sentation r times with itself and s times with its con- number of submanifolds {Mt } each of which is an open
jugate representation. Clearly, for the field to be stratum 41 (that is, each Mi is the union of all orbits
invariant under G, it must satisfy in particular, at with conjugate isotropy subgroups). We shall refer to

2129 J. Math Phys., Vol. 19, No.1 0, October 1978 Beckers et al. 2129

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 such fll;' s as the regular strata (and the orbits they
contain as regular orbits) and all others as singular
strata (and singular orbits). It will be assumed that where {K A } is an open covering of K. This covering
either Mi consists of a finite number of orbits or that may be used to defined a covering for GUo C M; by open
the space of regular orbits G \i'vl j is itself a differen- neighborhoods U A,
tiable manifold under the differentiable structure in-
herited from M; and Go In the former case, the identi-
fication of invariant fields is complete when they are
UA =? KA{pos)·
Os
known on each orbit, while in the latter we may use the
manifold structure of G\Mi to characterize the invariant Then a natural coordinate system for AI; in GUo is given
fields throughout each Mi' A local coordinate system by the mapping y: GUo -JR" defined as
for G \ 1\11i is equivalent to a functionally complete set of
scalar fields defined on the orbit of an open set UOCM j ,
invariant under the group G. To see this, let k be the
dimension of G \Mi and let
where k E: K A C K. Thus, the first n - h coordinates
[3:U-V (3.4) identify the point in the coset corresponding to the given
~oint on the orbit Gpos while the last k coordinates iden-
tlfy the orbit upon which POs lies. If the invariant scalar
be such a local coordinate system (Uo, V, and U being
fields {j3h} are defined and functionally independent
open neighborhoods). Then
throughout M i , this may be used to defined a coordi-
nate system throughout AI;.
fj(p) =[3(Gp) (3.5)
In general, let {xi(p)} denote the coordinates of a
defines a k-tuple {eo} (a = 1, •.. ,k) of functionally in-
pOi.nt PEe M j in any coordinate system and let x;(p)
dependent G-invariant scalar fields. Furthermore, if
=x'(tg(p» denote the coordinates of the image of this
under the mappingjg for a given gEe G. The Jacobian
(3.6) of this map, referred to these coordinates, is repre-
sented by the nXn matrix gJ(p) with element
is a G-invariant scalar field on GUo (considered now as
an open set in M i ), define the map:
(4.1)
1):U-JR (3.7)

l/J
Let the components of the tensor relative to the co-
(considering U as on open set in G \Mi) by ordinate frame be denoted l/J~::::~:. Then the conditions
for invariance (3.2) and (3. 3) become
1)(Gp) =l1(p). (3.8)

Then 11 = 1)0 0 (3, where 110 =1) 0 {3-1 is a mapping from V

into JR, and hence 11 is functionally dependent on the
{i3 o }.
and

4. Coordinate systems

Given a k- tuple {is"} of G-invariant functionally in- (Vf{EG) (4.3)

dependent scalar fields defined on GUo C M i , a coordi-
nate system for Mi in this open set may be defined as
follows. Let S be a submanifold of Mi which locally (where gJ represents the inverse of the Jacobian
intersects each orbit in GUo exactly once and more- matrix). Thus, the most general invariant field 1/) is
over such that the isotropy subgroup Go for each point obtained by solving (4. 2) for the independent compo-
POs E: S n GUo is locally the same. (For further discus- nents of </J(Po), allowing these to be arbitrary functions
sion concerning the existence of such an S, see the next of the scalars j3 O(Po) characterizing the orbits and then
section.) Now the space of cosets K = GIGo may be applying (4.3) to generate the field throughout the orbito
identified with each orbit in GUo in a unique way by the The above expressions become considerably simpler
correspondence within the {aa, j3b} coordinate system, since the Jacobian
matrix then takes on the block form

A coordinate system for K thus provides a coordinate

system for the orbit of any point POs. The orbit is neces-
J= [~~~~~--~-J
0: I.~
. (4.4)

sarily of dimension n - k and therefore such a coordi-

nate system is a mapping If invariant tensor densities are being considered,

2130 J. Math. Phys., Vol. 19, No.1 0, October 1978 Beckers et 81. 2130

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 rather than tensors, Eqs. (4.2) and (4.3) are modified isotropy condition (3.2)] which lies over x. A coordi-
by multiplying the right-hand side by a suitable power nate system for G \ Mi is a Co, 1-1 onto mapping be-
of Idet(J) I, depending upon the density weight. In tween open sets {VAt covering G\Mj and {U A} in JR".
particular, for a scalar density ¢ of weight 0, we have This may also be interpreted as a set of local cross
sections over {VA} of the bundle G\M(xJRk , each defin-
rjJ(po) = j detgoJjo rjJ(po), go~ Go (4.2') ing a set of k functionally independent, local G-invariant
and scalar fields in M j • That they are functionally indepen-
dent follows from the fact that the Jacobians of the C~
(4.3')
maps VA -JR" are nondegenerate; that k is moreover
Finally, let get) be a one-parameter Lie group which the maximum number is evident from the fact that the
induces the vector field X, denoted in the coordinate Jacobian's rank cannot exceed the dimension of G\M;.
system {Xi} by
X_ti
-s
a
W' (4.5)
III. THE CONFORMAL GROUP OF MINKOWSKI
SPACE
The requirement (3.1) for local invariance of the field
If! is expressed in terms of components as 6. The local transformation group

Let M(p, q) be a pseudo-Euclidean space with metric

gM of signature (p, q). That is, within any rectilinear
coordinate system Y, :M(p, q) -JRP+., gM is identified
with a nondegenerate bilinear form
g: JRP+. x JRPH -JR.
=0. (4.6)
Unless stated otherwise, we shall always identify ![M
Agail1, f9r tensor densities, a suitable multiple of with the diagonal matrix
~k,,,wn:::;: must be added to the expression for the Lie
diag(l, ... , 1, - 1, ... , - 1),
derivative. ,,~/~

q p
5. Fibre bundle formulation
using the same symbol for both. By O(p, q), we shall
In terms of fibre bundles, the preceding structures mean the orthogonal group corresponding to this particu-
may be described as follows. Let M j denote the ith lar ![Mo A C~ transformation T: UT - M(p, q) rUT being
stratum. Then the action of G on iVI; defines an equiva- an open set in M(p, q)] is called a local conformal trans-
lence relation turning i'vl; into a fibre bundle over the formation if, for any x E UT , the Jacobian matrix J of
base manifold G \ /VIi with group G, the fibres being the the transformation within the coordinate system Yr is
orbits, and the fibre-type, the coset space K = GIGo. of the form
An open subset of the sub manifold S intersecting each
J(x) = f(x) R(x), (6.1)
orbit once represents a local cross section of this
bundle which is invariant under Go. Such cross sections where R(x) ~ O(p, q) and f(x) E JR- (multiplicative group
may be shown to exist, in particular, if Go is com- of positive real numbers). This is equivalent to the
pact, 41-43 and under weaker assumptions as well, such requirement that
as the existence of a Go- invariant local metric. For
(6.2)
the cases treated in Sec. V either G\lvI; is discrete or
a smooth local section exists. A coordinate system for where ![M in this equation simply denotes the metric
G\lH; defined on any covering by open sets {VA} to- tensor, and not any particular coordinate representation
gether with a Go invariant local section SA over each of it. If the open set UT is all of M(p, q), then T is a
VA and a coordinate system in K gives rise to a coordi- global conformal transformation. The definition (6.2)
nate system for M; through the identification of the applies, moreover, both in local and in global form
identity coset with the intersection of SA with each fibre if M(p, q) is replaced by an arbitrary (pseudo)-
over VA (the open covering being {KBSA (VA)}' where Riemannian manifold. It is a well-known result that
{KB } is the covering by open sets of K). for p + q? 3, the conformal transformations of M(p, q)
An invariant tensor field If! is a cross section of the are generated by translations, pseudo-orthogonal
bundle T(r,s)(MI)IG over G\Mj as base manifold, transformations, dilatations and the inversion
T (r,s)(M i ) being the bundle of (r, s)-tensors on M i , and x- XlgM(X, x). These generate a local Lie group C(p, q)
the G-action on T(r,s)(M j ) defined by <p:. Each point of transformations on M(p, q) [local, because the open
in T(r,s)(Mi)IG over a point x E G \M j corresponds to a sets UT for certain T ~ C (p, q) do not cover the whole
space].
G-invariant tensor field defined on the orbit x, and
these are completely characterized by the Eqs. (3.2) For (p, q) = (3,1), M(p, q) becomes Minkowski space
and (3.3). Thus, the problem of finding the most gen- (denoted hereafter as M), for which Klein45 proved that
eral tensor field If! of type (r, s), invariant under G is C(3, 1) is isomorphic to the projective orthogonal group
reduced to that of characterizing the sections of the PO(4, 2) -0(4, 2)/Z z, where Z2 denotes the centre {t, - I}
bundle T(r,s)(Mj)IG. Locally this may be done by speci- of 0(4, 2). Klein's proof is based on the sphere geom-
fying the coordinates of a point x ~ G\M j and the com- etry of Lie which allows the local action of C (3, 1) on
ponents of the (r, s)-tensor [satisfying the linear M to be extended to a global action on another manifold,

2131 J. Math. Phys., Vol. 19, No.1 0, October 1978 Beckers et al. 2131

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 having the same local properties as M; namely, the
conformally compactified Minkowski space M, which
Mo;:C 5 / - ,
we shall define in the next section.
where C 5 is the cone of nonzero null vectors for the
The one parameter subgroups of C(3, 1) induce a set quadratic form associated with the matrix (gab); that is
of vector fields on M which close under commutation the set of points 17 = frio, 71\ 71 2, 71 3,71 3,714, 71 5) E mB for
to form a 15-dimensional Lie algebra (4,1). A basis for which
c(3, 1) which is convenient for distinguishing the Lorentz
subalgebra consists of the infinitesimal homogeneous (7.1)
Lorentz transformations M"v, translations P", dilata- [where (710,711> 712> 713, 714, 1)5) 0;: (1)0, - 1)\ - 71 2, _71 3, -1)\ 71 5) 1.
tions D and special conformal transformation C" The equivalence relation - is defined by
(/.L = 0, 1,2,3). Within a Cartesian coordinate system
these are of the form: 7) -7)' <=> 7)' = A7), *
A O. (7.2)
This gives an identification between points in M and
rays on the cone C 5 •
(6.3)
The action of 0(4,2) on M is the natural one induced
c" =x2a" - 2x,,~a" by t1"-c linear action of 0(4,2) on m6 ,
[summation convention used throughout, and raising or {[ :[17] - [g7J),
lowering of indices done with the Minkowski metric
~M =diag(1, - 1, - 1, - 1)]. Now, denoting an arbitrary where [7)] is the class of points on C 5 equivalent to 7).
vector field X E: C(3, 1) with components given by This action is transitive but not effective, since the
center Z2 = {I, - D.} acts as the identity at all points of
M. The projective orthogonal group PO(4, 2) =0(4, 2)/2 2
corresponding to the infinitesimal transformation acts effectively and the isotropy group at any point in
M is easily seen to be an ll-dimensional Lie group
x" - x" - w~ XV + a" - d" x 2 + 2c' xx" - AX" (6.5) which is isomorphic to the similitude group SIM(3, 1);
(w"V, p, a" ,c" being real constants), we have that is, the semidirect product of the inhomogeneous
Lorentz group P(3, 1) (Poincare group) with the group
(6.6) m+ of dilatationso Choosing the point [(0,0,0,0, -1, 1)1,
showing explicitly that these vector fields represent the isotropy group may be characterized as the set of
infinitesimal conformal transformations. The nonzero all matrices which can be written as a product
commutators within this basis are: {[(a, A, L)o;: exp(a" P".) exp(AD)gL, (7.3)
[NI ,,", Mrfr 1= ~"'alvlvr + ~vr M".a - g,,'T Mva - gvu M "r' where

--h-~~;';---l~;--J '
[M"v, P a] ={[" aP v - {[va P",
[D,P"]=-P,,, exp(a'" P,,) = [ ; ; ; : - (704a)
(6.7)
[D, C" ]=C", Ti, 2 1 I 2
- a gM I - 2a - CZa
I
[M llv , Ca] ={["a C v - {[va C",

[
[p", Cv1 = 2g"v D + 2M"v' ~4__ ~ ___ ~ _____ ~__ ]
exp ( AD ) = . (7.4b)
At the algebraic level, the 0(4,2) structure may be seen o I
I
coshA smhA
by defining a new basis Jab == - J ba (a, b = 0, 1,2, 3,4, 5) o I
I
sinhA COShA
with and

in terms of which the commutation relations become

(6 8)
0

gL= [~-H-~J
o I
I
0 1
(7.4c)

[Jab' Jed] =gac J bd + gbd J ac - gad Jbc - gbc J ad , (6.9)

where LE 0(3,1), aE m4 with components a" and
where a • b saTg M b for any a, bE m4 considered as column
gab =diag(+ 1, - 1, - 1, - 1, - 1, + 1). (6.10) vectors. This may readily be verified to define a six-
dimensional representation of the similitude group.
Another basis frequently used is obtained by separating We may thus identify M with the homogeneous space
M"v into rotations L; and boosts K; (i = 1, 2,3) PO(4, 2)/SIM(3, 1).
(6.11) From (7. 3) and (7.4) we see that P(3, 1) acts on the
cone C 5 as follows:
7. Compactified Minkowski space
In order to realize C(3, 1) as a global transformation
group, we proceed in a standard way, 1 replacing M by
another manifold M defined to be the projective cone of

2132 J. Math. Phys., Vol. 19, No.1 0, October 1978 Beckerset al. 2132

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 (where 17 = (17iJ. ) E 1R4]. conjugate to P(3, 1) and therefore we have the
5 identification
The orbit of the point (0,0,0,0,1, 1) in C under
P(3, 1) is thus the four-dimensional submanifold of C 5 -0(4, 2)/P(3, 1). (7.13)

pOint[s ,w::. rdina,es Finally, we should like to distinguish between the

different connected components of the conformal group
in a manner analogous to that done for the Lorentz
group46; namely, according to which, among certain
1- a
2
J discrete transformations, each component contains.
It is known that 0(4,2) as a manifold has four connected
This allows us to identify the orbit with Minkowski
components (as does each O(p, q) group for which
space M through the injection mapping
p, q > O}, and that these may be distinguished according

[~(l: x';].
to the values of two multiplicative discrete characters, 26
namely the determinant d (=± 1) and the spinor norm
j : x I- (7.6) n (=± 1). The latter may be defined as

[~:: ~:]
2"(1- x )
The image j(M] is then the set of all (17,17 4,175) for which sgn de'
114 +11 5 = 1. The injection of M into !Vi is given by
for any 0(4,2) element (Gab)' We thus have
J: x r-- [j(x»
E4 =0(4, 2)/SOo(4, 2), (7. 14)
and the image consists of the sub manifold of rays (11]
for which 114 + 11 5"* 0. The local inverse J.I (11] - M defines where the four cosets in Z4 may be identified with the
a coordinate system corresponding to the Cartesian following four elements of C(3, 1)-0(4,2)/E 2 ,
coordinates {~} of M,
1= e diag(l, 1, 1, 1, 1, 1) (n=+I,d=+I),
(7.7) P=ediag(l, -1, -1, -1, 1,1) (n=+I,d=-l),
(7.15)
The entire manifold M may be seen to be diffeomorphic T=€diag(-l, 1, 1, 1, 1, 1,) (n=-I,d=-l),
to the projective space [Sl x S3J/Z2 by introducing the PT=ediag(-l, -1, -1, -1, 1, 1) (n=-I,d=± 1)
projective coordinates
(e=±l).
(7.8) (Note that the center Jl 2 of 0(4,2) is not contained in
E" but is contained in SOo(4, 2).] Using notation which
which maps !Vi diffeomorphic ally into SI x S3/Z 2 since is standard for the Lorentz group, we may identify the
components of. C(3, 1) according to which components
u~+u3=u~+u~+u~+u~=1 (7.9)
of the Lorentz group'S they contain:
and the pOints {u } and {- u } must be identified to make
G G

C!(3, 1) - SOo(4, 2)/Z 2 ::J L:::J I,

the correspondence one-one. The set of points of !Vi
which are not the image of any point in M are just those C!(3,1)::JL!::JPT,
for which u4 + u 5 = 0, so that (7.16)
C:(3,1)::JL:::JP,
u~ - uf - u~ - u~ = 0. (7. 10)
C:(3, 1) ::JL:::J T.
This may be identified as a light- cone "at infinity" Correspondingly, we have the following five types of
(i. e., xiJ. _ ""). conformal groups:
The action of SIM(3, 1) defined by the mapping j is (i) C(3, 1) - 0(4, 2)/Jl 2 (general conformal group)
exactly that of the similitude group on M; namely
(ii) C:(3, 1) - SOo(4, 2)/Z2 (restricted conformal group)
g(a, X, L) : x I- exp(- X) Lx + a. (7.11)
(iii) C.(3, 1)-SO(4,2)/Z2-C!u C~ (proper conformal
The Abelian subgroup of PO(4, 2) consisting of elements group)
of the form
(iv) C'(3, l)-C!U C: (orthochronous conformal group)

g(fJ);: ( ~-:;;:-+-1-~~~~--~-~2-J (7. 12)

(v) C o(3, 1) - C: UC~ (orthochorous conformal group)

- fJ Tg M : tfJ 2 1 _ tfJ 2 8. The conformal group and SU (2,2)

I

moreover, defines the "special conformal In the following, we shall need an explicit construc-
transformations" tion41 of the isomorphism

x-x2b SU(2, 2)!E2 - SOo(4, 2), (a. 1)

g(fJ):xt-I_2box+bZxZ, (7.12')
where.f2 ={± I}, or, equivalently
These latter are clearly not defined globally on M. SU(2, 2)!E, - SOo(4, 2)/1: 2 - C~(3, 1), (a. 2)
The isotropy subgroups of 0(4,2) acting on C5 are all where.l4 ={± I, ± i I} is the discrete center of SU(2, 2).

2133 J. Math. Phys., Vol. 19, No. 10, October 1978 Beckers fit sl. 2133

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 Let h = ht denote a Hermitian form of signature (2,2) (where T denotes the complex conjugate of T). The fact
on <r: 4 • Then SU{2, 2) is the group of complex 4 x 4 that Dis involutive (D2 =1) follows from (8.11) and
matrices U such that (8. 12) and the relations
Ut/lU = II, deW = 1. (8.3) (8.14)

f"
h can always be chosen such that h 2 == 1 [and consequent- The set of matrices invariant under D,

::r,:)j_~d ~~
T D={T:=: T IDT == T} (8.15)

ly, (8.4) is a real vector space of dimension six which is in-

variant under p(U) since
(8016)
Let p(U) be the six complex dimensional, linear repre- Within T D, we have
sentation of SU(2, 2) on the space 1\ 2<1: 4 of bivectors
* T=hTh, It Tcc Tv (8.17)
(i. e., antisymmetric tensor product of <1: 4 with itself),
defined by and therefore the restriction of the form (8.6) to Tv
becomes
(8.5)
(T, T / )=- tr(hThT). (8. 18)
Now let us define the following symmetric form on
/\ 2<r:4 , By direct computation, one verifies that TD consists of
all matrices of the form
(Vi/\ v2, V3/\ V4) == det(I't. 1'2, V3, 1'4) (8.6)
(which extends, through bilinearity, to any pair of bi-
vectors). Since for each A EO: GL(4, <r:) we have
det(A vI, AV2,A 1)3, AV4) = detA • det(vh 112, 1'3, V4)
and since deW = 1 for U EO: SU(2, 2), the symmetric form
is invariant under p(U),

(P(U) : 1'1/\ 1'2, p(U): r3 f\ 1'4) == (1'1/\ 112, /'31' 1'4) (8.7)
and this invariance property defines a homomorphism
of SU(2, 2) into 800(6, <r:) (which incidently can be trivial-
ly extended to a homomorphism of 8L(4,<r:) into 80(6, <r:)1.
Moreover, the representation p is virtually real; that where
is, the complex vector space /\2<r:4, considered as a 11= (1]0,1]1, 1)\ 1)2, 1)3, 1]4, 1]5),,=, lRB.
twelve-dimensional real space, contains a six-dimen-
sionallinear subspace invariant under p(U) and such We then have
that the restriction of (8.6) is a real bilinear form of (r (1)), r (11) = 11~ - rJt - rJ~ - 1J~ - 7)1 + 7)€ :0 Q(l1) (8.21)
signature (4,2). This gives rise to the homomorphism
6
of SU(2, 2) onto SOo(4, 2). which thus defines a quadratic form in lR of signature
(4, 2). The homomorphism
To exhibit this decomposition explicity, it is conven-
ient to represent /\ 2<1: 4 by the space T of all antisym- <b : SU(2, 2) - 80 0 (4,2) is now defined by
metric 4x4 matrices T=- TT. The representation p (8.22)
acts on T as
and it is easily verified that ker <b == {± I} establishing
p(U) T=UTU T (808) the isomorphism (8. 1). The particular choice of co-
and the symmetric form becomes ordinatization in (8.20) is arbitrary up to an 0(4,2)
transformation and has been chosen so as to give a
(T, T') == tEijkl Tij T~l == - tr(* TT'), (8.9) simple form to the translations, homogeneous Lorentz
where EiJkl is the Levi-Civita symbol and *T denotes transformations, dilatations, and special conformal
the (affine) dual of T, transformations in the SU(2, 2) representation with
Hermitian form
(*T)iJ = tEiJkl T kl (8.10)
We note that
•
- [0 -il]
h= i il. 0
==ShS t
'
(8.23)

(8.11) where
and
(8.24)
(8.12)
Consider now the involutive, antilinear transformation Transl ations:
D on L defined by
(8.13)
- [II.0 kJII.
f{T= '
(8.25)

2134 J. Math Phys., Vol. 19, No.1 0, October 1978 Beckers at al. 2134

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 where A=Ha+d+i(b-c)],
3 i 2
k = [ aO1 + a 2 a o- ia3 ] E: lH (2) (Hermitian 2 x 2 matrix). B=Hd-a+i(b+c)],
a + ia a - a (8.36)
C = t(d - a - i(b + c)],
SPecial conformal transformations:
D=Hd +a - i(b - c)].
(8.26) The (t'4 nonsingular matrices defined by

- [A: BJ =t S gSt
---1---
g= C,D t t- (8.37)
I

(8.27) with

Homogeneous Lorentz transformations:

,. ~t-~\(;~-;)J
define a representation of SU(2, 2) which preserves the
(8.38)

g- L = [ao to
a-
IJ, aE: SL(2, a:). (8.28) diagonal form h whenever g preserves h. Moreover,
these two conditions define constraints for the matrices
Dilatations: {A,B,C,D} and {a,b,c,d} which guarantee that (8.35)
and (8.31) actually define transformations on lH(2) and
-
go=
[ex p (- A/2) II.
0
0
exp(+A/2)II.
J' AE: IR
.
(8.29) U(2) respectively.

As a final remark concerning the isomorphism (8.1),

note that with the choice (8.4) for the Hermitian form
The corresponding matrices in the basis with diagonal h, the matrix
invariant Hermitian form h is given, of course, by
conjugation with the matrix S. Another useful realiza-
tion of the action of C;(3, 1) directly on Minkowski space
J= UII. ~], (8.39)

may be defined using the representation (8.23)- (8.29). although not an element of SU(2, 2), nevertheless defines
Identifying the point of !vi with Cartesian coordinates an automorphism by conjugation since
{x"}by the Hermitian matrix
(8.40)
X=
Xu + x 3 xl - iX2] E:lH(2) (8.30) Moreover, the homomorphism (8.22) can be extended
[ xl + ix 2 x O_ x 3 to the two-component group obtained by multiplying
SU(2,2) elements by {I, J} (within this matrix represen-
we obtain a realization of SU(2, 2)/Z4 - C;(3. 1) as a local
tation). Applying the homomorphism to J, we find
transformation group acting effectively on M-lH(2)
b y 8: cfJ (J) : 1/ 1- (1]0,1]1,1]2,1]3, _1]4, _1]5). (8.41)
g : x r- x' '" (ax + b) (cx + d)-I, (8.31) That is,
where ± r/J(J) =PT (8.42)

If [~-i-~]
c , d
E: SU(2 2)
'
(8.32)
and the two component group is homomorphic to
SO(4, 2).
,
:gthg=h, a, b,c,d,E: GL(2,a::). (8. 33)
IV. MAXIMAL SUBGROUPS OF THE CONFORMAL
Moreover, the action may be extended to a global one, 1
GROUP
by replacing lH(2) by a space diffeomorphic to fl, name-
ly the group U(2). The diffeomorphic injection 1 (ef. 9. Remark on maximal subgroups, subalgebras, and
Eq. (7.6)] of lH(2) into U(2) is given by the Cayley normalizers
transform In the following, the notion of a "maximal" Lie sub-
1 +ix group of C(3, 1) is defined purely at the level of the Lie
l:xI-U=I--' EU(2), xElH(2) (8.34) algebra. Thus a subgroup of C(3, 1) will be referred
- LX
to as "algebraically maximal" if its Lie algebra is a
with local inverse maximal subalgebra of c(3, 1). Evidently, this will

=i G~ g] E lH(2),
allow the identification of several algebraically maximal
l-1 : U 1- x U E U(2), (8.34') subgroups corresponding to the same algebra. As will
become apparent from the discussion of Sec. IV 11, the
The "light-cone at infinity" is given precisely by the relevant group, from the pOint of view of conjugacy
subspace of U(2) for which the denominator in (8.34') classes, is the normalizer [in C (3, 1)] of any given group
is singular, that is; where det(1 + U) vanishes. The or algebra. Thus, two groups with equal (or conjugate)
global action of SU(2, 2) on U(2) corresponding to the normalizers will have a bijective correspondence be-
local action (8.32) on lH(2) is : tween their conjugacy classes under any given subgroup.
g: U 1- (Au + B)(CU + D)-I, (8.35) Therefore, we shall define a "normally maximal" sub-
group as any algebraically maximal subgroup whose
where normalizer equals the normalizer of its algebra. (Note

2135 J. Math Phys., Vol. 19, No.1 0, October 1978 Beckers et al. 2135

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 that the normalizer of any algebraically maximal sub- classes with signatures [n. n_ no] and [2 - n. - no,
group is always contained in the normalizer of its Lie 4 - n_ - no, no] are identical, we limit the range to
algebra and hence also that of any normally maximal n_ + n. + no -'S 3, n_ < 3. ) It follows from general theorems 26
subgroup with the same algebra. ) It is easy to verify that the following classes consist of maximal subalge-
in particular that the identity component of any alge- bras: [100], [OlD], [001], [200], [020], [002], [120],
braically maximal subgroup is also normally maximal. and [210]. All other classes of reducibly embedded sub-
Furthermore, the normalizer of any maximal sub- algebras within this representation are nonmaximal
algebra is either a maximal subgroup or the entire (e. g., any subalgebra HE [101] is properly contained
group, the latter being impossible for simple groups in a subalgebra H E [001] because the invariance of a
such as C(3, 1). subspace of signature [101] implies the invariance of
its isotropic subspace of signature [001)).
10. The maximal subalgebras of the conformal algebra
(iii) An irreducible subalgebra of a semisimple Lie
A complete classification of the subalgebras of c(3, 1) algebra can either be semisimple or the direct sum of
into conjugacy classes under C(3, 1) is in progress 25 - 27 ; a semisimple algebra with a one-dimensional compact
in particular, all the maximal subalgebras of c(3, 1) are Lie algebra. 26 The algebra c(3, 1) has precisely one such
known. We describe here briefly how the classification class [for the 0(4,2) representation], consisting of sub-
of the maximal subalgebras was made in order to char- algebras isomorphic to su(2, 1) (flu(l). When conjugacy
acterize these for use in the following sections. The under C!(3, 1) or C'(3, 1) is considered however, these
method given is applicable to any semisimple Lie split into two inequivalent classes. Furthermore, these
algebra. are reducible in the defining representation of su(2, 2)
on the space «;4 with the invariant Hermitian form h of
(i) We choose a specific representation of c(3, 1);
Sec. 1118. The remarks of paragraphs (0 and (ii) are
namely, the one provided by the 0(4,2) Lie algebra
equally applicable in this representation, with the cor-
[Eqs. (6.7), (6.9)], defined on lR6 with respect to the
responding signatures for subspaces V c ([4 relating
symmetric, bilinear form associated with the matrix
to the induced form obtained by the restriction of h to
r;ab [Eq. (6.10)]. In this representation, a subalgebra
V. The two different SU(2,2) [or C:(3, 1)] conjugacy
/i of c(3, 1) - 0(4, 2) is called reducible if it leaves some classes of su (2, 1)(fl u (1) subalgebras can be character-
proper linear subspace V C lR6 invariant, and irreduci-
ized by the signatures (1001 and [010] of the subspaces
ble otherwise. left invariant by the representatives of the two classes.
(ii) Due to the isomorphism of Eq. (7.17), the C(3, 1) If H belongs to one class, the subalgebra J-W J [see
conjugacy classes Eqs. (8.39)-(8.42)] belongs to the other one. We have
seen that this transformation induces a transformation
(10.1)
of SO(4, 2); consequently, depending upon whether we
are identical to the 0(4,2) classes. For reducible sub- classify under C.(3,1) or under C:(3, 1) we have,
algebras, these may be associated with 0(4, 2) equiva- respectively, either one or two classes of su(2)(flu(l)
lence classes of vector spaces, subalgebras.

[V]c={gV, gEO(4,2)} (10.2)

11. Conjugacy classes as orbits in homogeneous spaces
by the surjective mapping
fa) General method
gV f- g/ig-l • (10.3)
Before treating the maximal subgroups of C(3, 1) in
In general, the correspondence is not one-one; indeed, particular, we shall formulate the general problem of
the kernel of the mapping is identifiable with N(H)/H , determining conjugacy classes for subgroups of a given
where He 0(4,2) is the subgroup consisting of all ele- group G. Let K and H be two subgroups of G and let
ments leaving V invariant and NUn is the normalizer
[H]e ={g-lHg, gE G} (11. 1)
of /i [in 0(4,2)]. If /i is a maximal subalgebra of 0(4,2)
however, then N(/i) is a maximal subgroup of 0(4,2) denote the set of subgroups of G conjugate to H. We
and H a normally maximal one, both having H as Lie should like to divide [H]e into conjugacy classes under
algebra. For these cases N d/i)/H is always a finite K,
group and easy to determine by inspection.
The classification of the subspaces V is done by [g-lHr;]K=k 1g- 1Hgl?, kEK}. (11. 2)
direct application of a theorem of Witt,48 which for the If we define the action of K in [H]e by
present case states that any two subspaces V and V'
of lR6 are related by an 0(4,2) transformation if and k :g-tHg t- k- 1gHgk, (11.3)
only if they have the same signature under restriction then each class [g-lHg]K is a K-orbit, and [H]e is the
of the quadratic form Q [Eq. (8.21)] to V and V'. Since (disjoint) union of all the orbits. Let Ne(H) denote the
the induced quadratic forms on the subspaces are no normalizer of H in G. The mapping </>: [H]e - Ne(H)\G
longer necessarily nondegenerate, they must be charac-
terized by three integers [n. n_ no] denoting, respective- </>:g-lHgI-Ne (H)g (11.4)
ly, the number of positive, negative, and null eigen- is bijective and, under this mapping, the natural action
vectors. The conjugacy classes of reducible subalgebras
I/JG of G on [HJc,
may now be completely identified by the Signature of the
space they leave invariant. (Of course, since the (11.5)

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 and ec on Nc(H)\G, method may be applied with regard to the corresponding
invariant subspaces V c (1;4. The procedure for identify-
ei: Nc(H)g t- Nc(H) g,li, .iiE G (11. 6)
ing the classes [V]P(3,1) is as follows:
are equivariant.
(a) Choose a particular space V of given signature
Through the mapping cp, the problem of determining which is invariant under the algebra H and identify it
the classes [K- 1Hg]K becomes equivalent to that of by the 0(4,2) [or SU(2, 2)] signature.
identifying the double co sets N c(H) KK. The latter can
(3) Identify a basis for V in terms of m6 (or (1;4) co-
be seen either as the set of K-orbits in Nc(H)\G or as
ordinates and apply the most general P(3, 1) transfor-
Nc(H) orbits in GIK, and this dual interpretation gives
mation to it, as given by Eqs. (7.4) [or (8.25), (8.28)].
two equivalent methods of solution. Of course, the right
and left cosets are interchangeable; however, the (y) By suitable choice of the P(3, 1) transformation,
nature of the orbits under K and H in the respective reduce the basis elements to a set of standard forms
homogeneous spaces Nc (H)\G and GIK may be entire- which may no longer be related to each other by a
ly different, the one being more readily identifiable P(3, 1) transformation.
than the other. We should like furthermore to remark
(6) Identify for each orbit a representative set of basis
that if in the above, the group G is a Lie group and Ii
elements; one for each standard form.
is the Lie algebra for some subgroup H, the entire
analysis carries through mutatis mutandis if we make (E) Identify a set of elements KE 0(4, 2), one for each
the replacements H ~Ii, Nc(H) ~ Nc(Ii). standard basis, which maps the basis for V in its
standard form onto the other standard bases represent-
For the reasons outlined in the Introduction, we
ing different conjugacy classes.
should now like to characterize the maximal subgroups
of the conformal group by their conjugacy classes We now treat the maximal algebras of c(3, 1) by this
under the Poincare group P(3, 1); that is, for our case, procedure. In each case, rather than referring to the
G =0(4,2), J( =P(3, 1) and H is any normally maximal algebras, we shall identify the normally maximal group
subgroup of C(3, 1). [The central elements of 0(4, 2) H, defined as the largest subgroup (mod ~2) of 0(4,2)
do not, of course, affect the classification. 1We thus [or SU(2, 2)] to leave invariant the same space V which
have is invariant under its Lie algebra H. For the cases
(i)- (vi) the group H equals its normalizer (mod Z2)'
(11. 7)
For case (vii), there are two elements in N(H)IH and
and the particular choice of H for a given H is imma- this will be discussed separately.
terial since the conjugacy classes [H]P(3,1) and [H]P(3,1>
will be isomorphic under the correspondence (i) The Kroup 0(4,1): This group is identified by an
invariant subspace of signature [100], i. e., it is
K- 1HK-K- 1H'g-g-llig
spanned by one positive vector TE mG, which we nor-
for any two normally maximal groups H,H' with the malize to length Q(T) = L Applying translations, we
same algebra Ii. can immediately reduce T to the form T(A)
= (0, 0, 0, 0, sinhA, coshA) if T4 + T 5 "* 0. If T4 + T 5 = 0,
The two methods provided by the two interpretations then by translation and homogeneous Lorentz trans-
of the double cosets are both worthwhile considering, formations we reduce T to T(1) = (1, 0, 0, 0, 0, 0). Thus,
since the first [P(3, 1) orbits in 0(4, 2)/NdH)] is direct- we obtain two types of P(3, 1) orbits of [100] signature
ly related to the contents of the preceding section (i. e. , spaces: a one parameter family, represented by the
identification of maximal subalgebras) while the second spaces with basis elements exp(AD) T(2), with T(2)
[Nc(H) orbits in 0(4, 2)/P(3, 1)-C 5 ] is directly related = (0, 0, 0, 0, 0, 1), and an isolated one, represented by
to the contents of Sec. V. The computations are the basis element TO). Taking the latter as the standard
straightforward, and the results are summarized in basis for the defining representation of 0(4, 1), an
Table I. Readers mainly interested in the applications 0(4,2) element which maps this onto the corresponding
to invariant fields are invited to omit the details below. basis in the second family with parameter A, is K<A)
=exp[W] exp[1T/4(Po+C o)]. A basis for the 0(4, 1)
(b) P(3, 1) orbits in O(4,2)/NG (H) algebras leaving T(I) and exp(AD) T(2) invariant is given
by
For each reducible maximal sub algebra Ii, we have
{D,L;,P;,C;} and {L;,K it exp(X)P,,-exp(-X)C,,},
seen in Sec. IV 10 that the space 0(4, 2)INc (H)- [Ii]c
can be realized as the set of all vector subspaces V of
m6 which are related by 0(4,2) to a given H-invariant respectively.
subspace. Then each P(3, I)-orbit consists of an (ii) The Kroup 0(3,2): This corresponds to an in-
equivalence class of vector spaces [V]P(3 1> related variant subspace of signature [010], spanned by one
by P(3, 1) transformations, ' negative length vector 8, which analogously to the
[V]P(3,1> ={pV,PE P(3, I)} (11. 8) above, may be reduced by P(3, 1) transformations to
one of the two forms 8(1) = (0, 0, 0, 1,0,0) or exp(AD) 8(2),
which, in turn, corresponds to a conjugacy class where 8(2) = (0, 0, 0,0,1,0). The corresponding Lie
[fi]P(3,1> of algebras leaving the spaces in [V]P(3,t> in- algebras and the transition elements in 0(4,2) leading
variant, For subalgebras which are reducible within from one conjugacy class to the others are given in
the su(2, 2) realization on (1;\ precisely the same Table I (as they are for all the cases below).

2137 J. Math Phys., Vol. 19, No.1 0, October 1978 Beckers et al. 2137

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 (iii) The group SIM(3, 1): The invariant subspace is of orbit represented by the standardized set of basis ele-
signature [001] and is spanned by an isotropic vector ments {to, 1,0,0,0,0), (0,0,1,0,0,0), (0,0,0,1,0, OJ).
Lc:=: IR6 li. e., satisfying Q(L) = 0]. Applying P(3, 1)
(vii) The group 0(2, l)xO(2, 1): For this single case,
transformations this may be reduced to the form L(!)
the normalizer N(j/) is not equal to the group Hand
=(0,0,0,0,1,1) if L4 +L 5 4oO or to one of the forms
hence, according the discussion of Sec. IV 10, the
L(2) = (1, 0, 0, - 1,0,0) or L(3) = (0, 0, 0, 0, 1, - 1) if
mapping (10.3) does not define a one-one corre-
L 4 +L 5 =0.
spondence between the set of equivalence classes
(iv) The group OPT(3, 1): The invariant subspace is {[V]P(3 lJ of vector spaces invariant under the algebra
of signature [002], i. e., it is spanned by two ortho- 0(2,1)810(2, 1)=/t and the set of conjugacy classes
gonal vectors X and Y [g(X,X)=g(Y, Y)=g(X,X)=O]. {(H]P(3.1l}' However, N(H)/H has only two elements
Vector X can be transformed to one of the three and hence there will be at most two classes [V]P(3.1)
standard forms L(j), L(2), L(3) obtained above in the corresponding to the same class [/tjp(3,1)'
SIM(3,1) case and vector Y may then be standardized
To see this, we note first that the 0(4,2) conjugacy
by transformations in the subgroups of P(3, 1) which
class of groups 0(2, 1) x 0(2, 1) (and their Lie algebras)
leave invariant the space spanned by X. Doing this, and
is defined by the 0(4,2) class of subspaces V eIR6 with
taking suitable linear combinations among the pairs of
signature [120]. We may take, as the defining represen-
basis vectors to simplify their form, we obtain two
tation, the group (or algebra) leaving invariant the sub-
orbits of spaces, inequivalent under P(3, 1), repre-
space Vj spanned by the vectors {to, 1, 0, 0, 0, 0,),
sented by the following pair of standardized basis
(0,0,1,0,0,0), (1,0,0,0,0, OJ). Then the normalizer
vectors:
of the group H (or algebra If) leaving this space in-
{L, L}(I) ={1, 0, 0, - 1,0,0,), (0,0,0,0, - 1, In variant is
and N(H) = H x sZ;"
{L, L}m = {(I, 0, 0, - 1,0,0,), (0,0,0,0,1, 1, j). whereZz={I1,Il'}with

(v) The gro1lps 0(4)xO(2) and 0(2, 2)xO(2): These

two cases, corresponding, respectively to invariant
°° °° °° °° °1 °1
subspaces of signature [200] and [020] are treated 11'=
° ° °1 1 ° °
° °° °
°°1 °1 °° °° °° °°
analogously to the above; that is, one of the basis
vectors is put in a previously established standard form
of type S or type T, and the other is standardized us-
ing the stabilizer of the corresponding one-dimensional
The transformation 11' takes the space VI into its
space in P(3, 1). The results are summarized in
Table 1.
orthogonal complement vtand consequently two equiva-
lence classes [Vnp(3,1) and [VtlP(3,1) correspond to the
(vi) The group 0(3)XO(2, 1): The signature of the in- same conjugacy class [H]P(3 I)' In general, any other
variant subspace is [030], i. e., it is spanned by three subspace Vi of signature [120] will be mapped either
orthogonal negative length vectors S(I), S(Z), and S(3). into itself or its orthogonal compliment Vi~ by elements
Two of them may be put into one of the two standard of the normalizer of the group leaving Vi invariant, and
forms identifiable from Table 1 for the group 0(2, 2) hence will correspond to the same conjugacy class of
x 0(2). Considering each separately, let us first choose groups (or algebras). Turning to the detailed classifica-
{S(I>, S(Z)}={(O, 1,0,0,0,0), (0,0,1,0,0, OJ). Then S(3) tion of these, the [0201 subspace of any such space can
= (5°, 0, 0, 53, sl, 55), with 5% - 5~ - 5~ + 5~ = - 1. If be chosen in one of the forms given in Table I for the
*
54 + 55 0, we can translate 5(3) into invariant spaces under 0(2,2)XO(2). Consider first the
(0,0,0,0, coshA, sinhA) without altering {S(I), S(Z)}. If case when {s(1) , S(Z)} = {to, 1, 0, 0, 0, 0), (0,0,1,0,0, On.
sl + ~{, = 0, we can transform S(3) by a homogeneous Then the third basis vector, orthogonal to these first
Lorentz transformation and a translation into two, has the form T = (TO, 0, 0, T3, T\ T 5 ) with normali-
(0,0,0,1,0,0) without affecting {S(I), a(2)}. Now, take zation T5 - T~ - T~ + T~ = 1. If 1-.4 + T 5 = 0, we can trans-
the other case {SW, S(2)} = {to, 0, 0, 1,0,0), form the basis, using translations and homogeneous
(0,0,0,0, coshA, sinhA)}. Then S(3)= (5°,5 1,52, 0, sl sinhA, Lorentz transformation which leave a(1) and a(2) fixed
sl coshA). By an 0(2, 1) transformation 8(3) can be taken to precisely the form given above defining the space
into (0,0, coshll, 0, sinhll sinhX, sinhll coshA). The triplet *
Vi' If r + T 5 0, we can apply a translation to trans-
{S(l), 8(Z" S(3)} now determines a hyperplane in the 2345 form the basis to the form {S(1), S(2), T}= {to, 1, 0, 0, 0, 0),
subspace. A vector orthogonal to this hyperplane within (0,0,1,0,0,0), (0,0,0,0, SinhA, coshA)} defining a one-
the 2345 subspace determines the hyperplane completely parameter family of spaces V 2(A). Consider now the
and can be written as T~ = (0, 0, sinhll, 0, coshll sinhA, other possibility; namely, {a(l), S(2)} = {(o, 0, 0, 1,0,0),
coshll coshA). By a translation (a, 1) with (0,0,0,0, coshA, sinhX)}. We then have T
a = (0, - tanhll exp(- A), 0, 0) we can take T L into = (TO, T i , T2, 0, T4 sinhA, r coshA), with TTh - T~ - T~
(0,0,0,0, sinhA, coshA). The orthogonal hyperplane °
= 1 - T~ =K. If K < or K> 0, we can transform the
° °°
then goes into {S(I) "8(2) a(3)}={(0 , , 1, , , 0) , basis, in a manner analogous to case (vi) into one of
the standard bases defining Vb V 2(X) or their ortho-
(0,0,0,0, coshA, sinhA), (0,0,0,1,0, OJ). Thus the
classes for this caSe are characterized by this one
parameter family of orbits together with the single
gonal complements vt, °
V{(X). If K = with TO = Tl
= T2 = 0, we already have the space Vl. However, the

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 case K = 0, TO *° is different. We then have, after a (c) NG (H)·Orbits in O(4,2)/P(3, 1)
suitable rotation T = (TO, - TO, 0, 0, sinhA, coshA). Apply-
ing a translation with a= (aO,a 1 ,0,0) we obtain: S(I) The homogeneous space 0(4, 2)/P(3, 1) has been iden-
= (0, 0, 0,1,0,0), S(2):= (aOe\ a 1 e\ 0, 0, coshA tified in Sec. III 7 with the five-dimensional cone C5 in
+[(a5-ai}/2]e\ sinhX-[(a~-aV2]eA), andT lR6 defined by Eq. (7. 1), with the identity coset identi-
= (TO + aOe\ _ TO + aiel., 0, 0, (ao + al)T 1 + sinhA fied with the point (0,0,0,0, - 1, 1). The conjugacy
+[a% - aV21 eA. - (aO + a1 ) To + coshA - [(a~ - ai)/2] e A). classes therefore, under P(3, 1), for any subalgebra
We now replace S(2) and T by if = coshbT + sinhbS(2) g-lf/g of 0{4,2), (or subgroupg- 1Hg) are identified with
and S:=sinhhT+coshbS(2). Choosing eb =e-·V12. the orbits in C 5 under the group NG(f/) [or NG(H)], the
m=e->'/2, oO_a 1 =(,-2>./2, aO+«=I, we obtain a space orbit of the identity coset being identified with the class
spanned by{(1,0,0,0,0,l), (0,1,0,0,1,0), (0,0,0,1, of fI (or H) itself. Since the action of any subgroup of
O,O)}. (Note that TO may be chosen to have any nonzero 0{4,2) on C 5 is just given by the linear transformation
value by applying a Lorentz boost in the 1- direction on lR6 , these orbits are easily identified. We therefore
to tile original expression for T.). To identify the simply list below the equations defining the different
conjugacy class, we apply another rotation in the 1-3 types of orbits on C 5 corresponding to each maximal
plane to yield a space V3 spanned by basis vectors subalgebra, together with a geometric characterization
{O, 0, 0, 0, 0,1), (0,1,0,0,0,0), (0,0,0,1,1, OJ}. We note of the type of space they describe. In order to recover
that, unlike the space VI and V(A), the space V3 may a representative of the class to which these orbits cor-
be mapped into its orthogonal complement V~ by an respond, we merely conjugate the original algebra f/,
element of the (general) Poincare group, consisting which is on the identity orbit, with any group element
of a product of spatial rotations and a time inversion. gE 0{4, 2) which maps the point (0,0,0,0, - 1, 1) into
Hence, we have five types of equivalence classes of the orbit considered. But such a group element is given
vector spaces; [V 1 Jp (3,l)' [Vilp(sd)' [V2 (A)]p (3,ll' precisely by the (inverse of the) mapping which takes
[V~ (A) ]p (S.l)' [V s]p (3,1)' but only three conj ugacy the representative V left invariant under fI, into another
classes of groups (or algebras) leaving invariant V', invariant under the conjugate algebrafl'=g-lf/g.
spaces in the 1'1 (and Vi), 1'2 (and V~), and V3 - V~ This allows a direct translation of the results as ob-
classes. tained in the present section into those obtained in the
previous one. Since these are essentially identical, we
(viii) The group S[U(2, l)xU(l)]: As discussed in only treat three representative examples below in order
Sec. IV 10, there is only one C(3, 1) [or C.(3, 1)] con- to illustrate the method.
jugacy class of subalgebras of SU(2, 2) leaving, within
{i} The group 0(4, 1): Taking, as the original defini-
the defining representation on a;4, a one-complex dimen-
tion of H, the subgroup of 0(4,2) leaving invariant the
Sional, nonisotropic subspace invariant, whereas under
space spanned by (1,0,0,0,0,0), we obtain the following
SU(2,2), C;(3,1), or C'(3, 1), the conjugacy classes
two types of orbits on C 5 :
split into two, characterized by SU(2, 2) signature [100]
and [OlD]. Similarly there is just one conjugacy class (0) 7)I + 7)~ + 7)§ + 7)~ - 7)~ = 0,
of this algebra under the general Poincare group P(3, 1), (11. 10)
({3) 7)I +7)~ +7)~ +7)~ -1)~ =e 2A •
but two, characterized by the Signature, under the
restricted [P:(3, 1)] or orthochronous [P'(3, 1)] Poincare The first, defining a four-dimensional cone c4, is the
groups, To see this, we may proceed as follows. Work- orbit of the identity, while the second, corresponding to
ing in the basis with off-diagonal form h [Eqs. (a. 25)- a one-parameter family of single-sheeted hyperboloids
(a. 29)] we may transform an arbitrary, nonisotropic H4(A) is obtained from the identity by applying the
vector in a;4 into one of the forms: (0, (a + i) q, 0, q) or mapping exp[-(1T/4)(P o + Co)] exp(- AD).
(0, (a - i) q, 0, q), with q E a;, a E lR, through a suitably
(ii) The group SIM(3, 1): We define H as the subgroup
chosen homogeneous Lorentz transformation. Next, by
leaving invariant the space of null vectors
multiplication with a normalization factor and applica-
tion of a translation in the (0,3) plane, we may trans-
CO, 0, 0, 0, - x, x). This gives three orbits in C 5 :
form this to the form (0, i, 0, 1) or (0, 1,0, i) depending (a) 7)", = 0, 7)4 + 7)5 = 0,
upon whether the signature is + or -. No further trans-
formation in the identity component of P(3, 1) can relate ({3) 7)4+7)5*0, (11. 11)
the two spaces which these vectors span. However the
transformation J (i. e., PT) [Sec. IlIa, Eq. (8.39)],
(Y) 7)iJ-7)'" = 0, 7)4 + 7)5 = 0, 7)0 * 0.
which in the off-diagonal basis takes the form The identity orbit (a) defines a line L 1. Orbit ({3) de-
fines a five-dimensional subspace of C 5 , which geome-
J ~
= [- U. iO[ ] ' (11.9) trically may be identified as a product of a four-
dimensional paraboloid p4 with a line L' minus a point.
maps these two spaces into each other. Thus, we have A group element mapping the identity coset into this
either one or two conjugacy classes of the group orbit is given by exp[- (1T/2)(P 3 - C 3 )]. The third orbit
S(u(2, l)XU(l» under the groups P(3, 1) [P.(3, 1)] and (y) defines the product C 3 XL I of a cone with a line and
P!(3, 1) [or P'(3, 1)], respectively. Within the diagonal may be related to the identity by the mapping
representation with Hermitian form h [Eq. (a.4)], the exp[- (1T/4)(P 3 -C 3 )]exp[- (1T/4)(P o +C o)]' The orbits
bases for the two subspaces of positive and negative for the remaining cases of reducible subgroups of
signature have the form (0,1,0,0) and (0,0,0,1), 0(4,2) are equally simple to identify, and the corre-
respectively. sponding results are listed in Table I.

2139 J. Math Phys., Vol. 19, No.1 0, October 1978 Beckers sf a/. 2139

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 (iii) The group H=S(u(2, l)XU(l»: This case is a little more involved, because the subgroup is simply defined
only within the SU(2,2) realization. Therefore, to determine its action on C 5 , we must make use of the homo-
morphism et> given by Eq. (S.22). We first identify three subgroups of S(o(2, 1) x U(l» that are useful for the orbital
analysis. Working in the basis with diagonal form h, let Go be the isotropy subgroup of the origin in M-U(2). Using
the realization of Sec. III S, Eqs. (S. 31) or (S. 35) for the action on this space, we find Go to consist of all elements
go of the form
go =uogj, (11.12)

'" ~ r
where

exp(- ;_1
± exp(i,p)
exp(- i,p)
± exp(icp)
1 (11.13)

and
Iz 12 Iz 12
cosh;\ - exp(;\) -2- - iQ - iz Q+i sinh;\ - exp(;\) 2 0

- i exp(;\) z 1 exp(;\) z 0
(11.14)
Iz 12
Q-i sinh;\ + exp(y) ~ Z cosh;\ + exp(;\) -2- + iQ 0
2
0 0 0 1
0"'et>--::27T, ;\, QE JR, Z=X+iYE<C.
Now, we identify the maximal compact subgroup K=S(U(2)xU(1)XU(1» whose elements we parametrize in the form
(11.15)

where

J; _ [exp (- iV2) v : 0 ]
\-- -----------~---------- ,0"'~<7T, vESU(2). (11.16)
o I exp(i~/2)

Finally, let Vo be the U(l)x Z2 subgroup consisting of elements of the form Uo and note that this is precisely the
intersection of K with Go. The set K of elements of the form k is not a group (due to the limits on the range of the
angle 0, however, it may be seen as locally isomorphic to the group U(2) under the mapping
Ii -- exp(- iO vE U(2). (11.17)
This group U(2) acts on the space M transitively and freely [Eq. (S.35), Sec. IllS]. It therefore follows that each
element gE S(U(2, l)xU(l» may be decomposed uniquely into a product
g=kuogb kER, uoEVo, g1EGb (ILlS)
where G1 is the noncompact group conSisting of elements of the form (11.14). Thus the orbit in AI under S(U(2, 1)
XU(l» covers the entire space and the space R may be identified with the coset S(U(2, l)xU(l»/G o• To determine
the orbits in C5 , we make use of the homomorphism et> given by Eq. (S.22). Parametrizing the SU(2) element v as

(11.19)

we obtain
cos~ 0 0 0 0 - sin~
0 a b -c d 0
0 -b a d c 0 (11.20)
0 c -d a b 0
0 -d -c -b a 0
sin~ 0 0 0 0 cos~

1 0 0 0 0 0
0 coset> - sinet> 0 0 0
0 sinet> coset> 0 0 0 (11.21)
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1

2140 J. Math Phys., Vol. 19, NO.1 0, October 1978 Beckers et al. 2140

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 Iz 12 Iz 12
coshX + exp(X) -2- -y -x sinhX + exp(X) -2- Q -Q

yexp(X) 1 yexp(X) - x exp(><) x exp(X)

- x exp(:\.)
°1 - x exp(X) yexp(X) - y exp(X)
Izl2
°y Iz 12
cosh:\. - exp(X) -2- -Q Q
<fJ(go) = sinhX - exp(:\.) -2- x

( cosh:\. - exp(:\.) -2-

Iz 12 ) ( Iz 12
Q X -y - sinh:\, - exp(X) -2-

- ( sinhX + exp(:\.) -2-

Iz 12) ( sinh:\. + exp(X) -2-
Iz 12)
Q X -y

(11. 22)
Applying <fJ(go) to the point (0,0,0,0, - 1, 1), we obtain (0,0,0,0, - exp(X), exp(X». Next applying the transformations
2 3 5
<fJ (7i) E K, we obtain all points (17°,171, 17 ,17 ,17\ 17 ) with

1}~ + 17~ = exp{2X), 171 + 1}~ + 1)~ + 1)1 = exp(2 X) (11.23)

5
(that is, submanifolds diffeomorphic to S1 >< S3). Since X may take any value, this covers the entire cone C • There-
fore, there is only one conjugacy class of S(u(2, 1) x U(I» c SU(2, 2) under the group P(3, 1).

V. INVARIANT FIELDS was studied. We thus define the three- dimensional

vectors E, B, ¢, and 9 with components
12. Local invariance and Lie derivatives
EI =FO I , BI =tEiJkFJk' (12.4)
Although the invariant fields to be discussed in Sec.
V 13 have been obtained by use of the global method out- <fJI =wO I
, 0 1
= tEIJkW Jk • (12.5)
lined in Sec. II, we give here the differential equations ro simplify the expressions below, we also define
[ef. Eq. (4. 6)J describing local invariance of fields
under infinitesimal conformal transformations. Using C;:;"";: W"" + 2(c"x V
- e"x"), (12.6a)
the notations of Sec. III 6 Eqs. (6.3), (6.4) to express (12.6b)
X=:\.-2e"'x""
the most general field X generating local conformal
transformations the condition, in Cartesian coordi- and correspondingly
nates, for invariance of a (symmetric or antisymmetric) ¢ =¢ + 2(cOx_xOc], (12.7a)
tensor of type (0,2) with components F "" is ...
8=9 +4cXx, (12.7b)
(X" w,,8 08 + a'" 0",) F"" + (W(:FOlV + w~ F ""')
and the differential operator
- X(2 + x"o",)F,," - (x 2c'" 0", - 2c"x",[2 + x80,s]) F ""
15 =a"'o", + w",,sx,,,08 - Xx"'o", + e'" [2x",x 8 - o!x2 ] 0,6 (12.8)
- 2 (c '" x" - c"x"') F ",,, - 2{e'" Xv - ell x"')F"'" = 0.
[which is just the negative of the vector field X of
(12. 1)
Eq. (6.4) defining the infinitesimal transformation]. In
For a covariant vector (I-form) with components A", this notation Eq. (12. 1) decomposes into the two vector
the condition for invariance becomes equations:
(x",W"'8 08 + a"'o",)A" + w~A", - X(1 +x"'o",)A" DE + '§xE+ "¢XB - 2X"E=0, (12.9a)
- (x 2c"'0", - 2c"'x",[1 +x808J)A" - 2x"c"'A" +2c"x"'A", =0. DB + ~xB - ~xE - 2XB=0, (12.9b)
(12.2) while Eq. (12.2) decomposes into the scalar and vector
Finally, for a scalar density <fJ with scaling dimension equations
d (density weight - d/4), the invariance condition is DAo - ~·1 - MO =0 (12.10a)
(X'" w",.6 0,6 + a"'a",) <p - X(x'" 0", - d) <p
- (x 2e"'0", - 2c'''X,,,[X808- dJ) <p =0. (12.3)
and
...
DA + 8x1 -A°.p- Xl =0,
- (12. lOb)
For the case of antisymmetric F"v (2-forms), we where AI' ={A o? - A} may, for instance, be interpreted
should like to make a decomposition of Eq. (12.1) into as the electromagnetic 4-potential.
space and time parts, with a separation of what would
be the electric and magnetic components of FUll, if this Now, depending on the dimensionality of the particu-
were interpreted as an electromagnetic field tensor. lar subalgebra of c(3, 1) that is considered, the resolu-
The notations we use are chosen to correspond with tion of these differential equations may be relatively
those of Refs. 30-32, where invariance of the electro- simple or rather complicated. In any case, for the ex-
magnetic field under subgroups of the Poincar~ group amples treated in the following section we have found

2141 J. Math Phys., Vol. 19, No.1 0, October 1978 Beckers sf at 2141

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 them useful in corroborating the results obtained by the results will be re-established with the global method in
global method and in some cases complementing it. For the following section, where the form of the invariant
example, it is evident from these equations that no symmetric tensor and scalar density fields will also be
tensor fields exist (other than the trivial constant given"
scalar) which are invariant under the groups SIM(3, 1)
or OPT(3, 1), since the algebra of each contains all the Before proceeding to the detailed examples treated
translation and dilatation generators, Local invariance there, we should like to make a remark that bears upon
under the translations implies that the fields must be the results obtainedo In the case of a closed, invariant
constant, and the dilatation invariance implies further- 2-form F (eo g., the electromagnetic field), one might
more that this constant is zero (except for a scalar ask whether, at least locally, the problem may not be
field). This observation leads rather simply to the re- reduced to that of finding the most general invariant
sult, whereas trying to solve the linear isotropy condi- local I-form field A (e, g., the vector potential) and
tions at a point on a regular orbit is somewhat more taking its exterior derivative. (Since we are referring
involved. here to local quantities only, the distinction between
closed and exact forms arising from a nontrivial co-
An example for which Eqs. (12,9) and (12.10) lead homology group will be disregarded. ) Suppose that F
to nonvanishing invariant fields relatively simply is the is such a closed form and that it is invariant under the
group SO(3) x SO(2, 1) which leaves invariant the space local transformation group generated by the set of
Y! = {(O, x, y, 2,,0,0)} (see Table I), The Lie algebra is vector fields {Xi};.!" .• , r ' We thus have
defined by the basis (Lt> L 2 , L 3 , 1), Po, Co), which corre-
sponds to setting all parameters in Eqs. (12,9) and dF=O, (12.15)
(12.10) equal to zero except for e,_a°l.. r~.. and:l. This (12.16)
yields the following equations for E, B, A, and AU.
(8) ({jo;X~)E_exE=O, (B';X~)A0'-"oO, Since F is closed, there exists a local I-form A, such
that
(e'xx~)B-exB=O, (B.;x~)A-exA=O,
F=dA. (12.17)
(12. 11 a)
Since Lie derivatives commute with exterior deriva-
tives, Eqs. (12.15)-(12.17) imply
(12.11b)
:l(Lx.A)=O
, (12.18)
(Je) (; .~) E+2E =0, (X. V)A +A =0,
(12. 11c) and hence the LXiA are closed forms, but not neces-
(X .~) 13 + 2B =0, (x '~)AO +Ao =00,
sarily zero. Again, this implies that there exists a set
° - -
(c) 2x oC¥'Y')E+2)x '-1 2 coE+2xxB+4xoE=0,
-~ - - - of local scalar fields 0i such that

2.10(;' v) B+ 2 Ix 12noB + 2xx E+ 4x/3 =0,

(120 19)

2.10(; 'V)A + 2\;I'iJoi1- 2xXA + 2xev1 °= 0,

U Adding a closed I-form riA to A (i. e., making a local
change of gauge) does not alter the relation (12.17),
2xo(' 'V)A + 21; 1001\ - 2Ao; + 2xoA =0.
2
therefore we may inquire whether it is possible to do
Equations (12. llb) imply that the fields are independent so in such a way that the resulting I-form has vanishing
of time and (112. 11a) (rotational invariance) then imply Lie derivatives with respect to the Xi' We define
they are of the form: if::oA +dA;
E=oc(lxl) x, B=IJ(lxl)x, A will satisfy
(12. 12)
A=a(X)x, AO=/(\;i), Lx.~'L,O
,
for some scalar functions c(!X,), Hi:\-!), (/(I~'i), and
/(1:\-',). Solving Eqs. (12.l1c) shows that these func- provided that
tions have the form:
d(XiA-a;)=O, (12020)

1) (I-I)
that is, provided
e ('_I)
Ix = N
1;!3, Jj
x = :;13,
(12.21)
(12. 13)
F }"
a(;)=~, , f(lxl)=~I"
IX,' IX
where the C i are arbitrary constants. Thus, the
existence of a locally invariant I-form A, whose ex-
where K, K', :\I, and N are arbitrary constants. Finally, terior derivative is F reduces to the existence of a
Eqs. (12. lld) are automatically satisfied by the fields solution for Eqso (12.21L This is a set of r linear,
E and B given by Eqs. (12.12) and (12.13) but cannot be first-order partial differential equations in n (=4)
satisfied for nonzero A, AO. Thu~we filld the following dimension, If y = 1, a solution will always exist, pro-
solutions for the invariant fields E and B, vided the fields considered satisfy sufficient smooth-
-E =N l,yJ3'
x B-. =}\1 1)'1-; 3 , (120 14)
ness requirements. In general, the number of indepen-
dent equations in (12.21) may be fewer than r, due to
the Lie algebra commutation relations between the
while no invariant fields of the type A or A ° exist. These fields Xi' but they need not necessarily be compatible.

2142 J. Math Phys., Vol. 19, NO.1 0, October 1978 Beckers et al. 2142

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 Thus, it may easily occur that even though the number and J( is a constant. Since these orbits are all projec-
of independent equations is less than the dimension of tively equivalent they may be identified with one and the
the space, no solution exists. same orbit in 5J (which is, in fact, the entire space).
We therefore introduce coordinates for iff by the follow-
This implies that there may be invariant fields F
ing six-dimensional parametrization
which, may not be derived from an invariant field A,
but only one satisfying the weaker condition (12.19) of 1]" 1],4/5
11"= U4 / 5 - (13.2)
invariance up to a gauge transformation. Indeed, this - (11~ +115 2)112 , - (1]~ + 1]&2)1/ 2
is precisely what occurs in the example treated above
in terms of which, we have [Sec. III, Eq. (7.7) 1
(SO(3)xSO(2,1», where no invariant I-form exists and
will be seen to reoccur in several of the examples 1]" au'"
x " - ---,----,. - ---,-----,. a '" e). (13.3)
following. - 1]" + 1]" - 11" + 11" ,

13. Global method and results with

We shall now apply the method described in Sec. II 1l~ + 1I~ = 111 + u~ + u~ + u~ = 1. (13.4)
to determine the most general fields of I-forms, 2-
forms, symmetric (0,2) tensors and scalar densities
Equation (13.2) may be interpreted as defining a mapping
invariant under the subgroups of C(3, 1) whose algebras
from C 5 - :iR 6 (the latter space having coordinates {u 4 })
are listed in Table I. Since several groups may corre-
whose image is the submanifold 51 x5 3 defined by (13.4).
spond to the same Lie algebra, we shall select among
Each ray in C 5 maps onto a pair of antipodal points,
these the ones which lead to the fewest vanishing fields.
allowing the identification iJ - (51 X5 3)/Zz as discussed
For example in the case of the 0(3) EP 0(2,1) algebra,
in Sec. III. The mapping of each orbit (13.1) of C 5 thus
as we have seen, the group SO(3) x SO(2, 1) leaves in-
covers the space iJ twice. Indeed, inverting Eq. (13.3),
variant certain 2-form fields; but the group 0(3)xO(2, 1)
taking into account (13.4), we obtain
does not. For most cases, however, the particular
choice is immaterial, since the same types of fields x"
u"=~-, (13.5)
arise for each. T

To summarize the method, we proceed as follows.

First, a convenient choice of coordinate system is made
for the space lIJ. This will always be chosen as the
natural coordinates describing the regular orbits of the where
group on the cone C 5 , since these proj ect onto the regu-
(13.7)
lar orbits in :11. Secondly, a convenient point Po is
chosen on each orbit Mo and the isotropy subgroup at
and
this point is identified. Next, the linear isotropy condi-
tions [Sec. II, Eq. (4.2) 1are solved to determine the E =± 1
independent nonvanishing components at Po. Finally, the 2
(x denotes the square of the 4-vector x).
group transformation [Sec. II, Eq. (4.3)1 is applied to
obtain the field at any point on the orbit of each Po and We may furthermore introduce the natural curvilinear
the independent parameters are taken as functions of coordinates for SI XS3 which can also serve as local co-
the scalar invariants defining the orbits (if any such ordinates for the projective space if we limit the angu-
exist). To simplify the calculations differential forms lar ranges to cover half the space and identify the anti-
are used throughout, thereby permitting the use of six- podes, Thus, define:
dimensional coordinates, in which the group action is
always linear, by identification of the orbits as lower ito = COsU), It
l
= sin a sinO cosdJ, 1t
2
= sina sinO sin¢,
dimensional submanifolds of 1R6 with suitable projections 5
u = sini/J, 11
3
= sin a cosO, 1l~ = cosa, (13.8)
made onto the cotangent spaces of these submanifolds.

(a) The group 0(2) X 0(4) o 1': 27T,

c 0 < (]I <: 7T, 0 < e <: 7T, 0 < rjJ . 27T •

We shall treat this example in somewhat greater de-

tail than the others in order to illustrate the method. Since the group action is linear in terms of the coordi-
As we have seen in Sec. IV, a one-parameter family nates rna} and hence the Jacobian matrix of the trans-
of subgroups of this type exists and each member can formation is identical to the group element itself, it is
be characterized by the fact that it leaves a space of convenient to utilize the basis [du a } for the cotangent
the form {(x, 0, 0, 0, y sinhA, y coshA)} invariant. For the space in :iRs. We can then express I-forms, 2-forms
group corresponding to a given fixed value of the and symmetric (0,2) tensors as
parameter A, the regular orbits in C 5 , which in this
case cover it entirely, are given by the equations (I-form),

1]~ + 1]5 2 = 1]~ + 1]~ + 1]~ + 1],,2 = J(2 (J( > 0), (13. 1) (13.10)
where [symmetric (0,2) tensor J,
1] 14 = coshA1]4 + sinhA1]5,
The cotangent space for 51 x5 3 is then given by the
1]15 = sinhA1]4 + coshA1]5, conditions

2143 J. Math Phys., Vol. 19, No.1 0, October 1978 Beckers et at. 2143

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 uOdu O+u 5du 5 = 0, G = Cdl)!2 +D(da 2 + sin2a d8 2 + sin 2a sin28 d</>2]
(13.11)
u l du l +u 2du 2 +u 3du 3 +u 4du 4 =0. C
= [4
aT
t~][(a2 +X5 +lxI2)dxo- 2xox 'dx]2
We now choose the point Po E M corresponding to the
point in SI XS3 with coordinates
-..g. dx .. dx".
T
(13. 18c)
(UO, u l , u 2, u 3, u\ u 5 ) Po = (0, 0,0,0,1,1). (13.12)
The isotropy subgroup at Po is the group 0(3) consisting
of elements of the form Although the expression for the tensor G is quite cum-
bersome in Cartesian coordinates, we may note that if

l !-:-~-+~-J
the arbitrary constants satisfy C = - D, it simplifies
to a form which may be interpreted as a conformally
go= 0' R 0 , (13.13) flat metriC, namely
--1----'---
o I 0 I [2
=g dx .. dx" =.g gil'
I I
G (13.19)
where [2 is the 2 x 2 identity matrix and R is any 0(3) T T
matrix. The linear isotropy conditions (4.2) of Sec. II To obtain the most general invariant scalar density
imply that at this point, we have: of scaling dimension d (density weight o=-d/4), we
A(Po)=Kduo, apply formulas (4.2'), (4.3') of Sec. II. To satisfy the
linear isotropy condition (4.2'), we must have
F(po) =0, (13.14) det(oJ(po»=±l for all go in the isotropy group at Po.
G(po) =C(du~) +D(duI +du~ +du~), In order to treat the problem in a linear way we
could determine the invariant 4-forms in 1t6 and project
for any A, F, and G invariant under the group, Sinc e M onto the four-dimensional subspace, analogously to the
consists of just one orbit under 0(2)XO(4), there are no fields A, F, and G, thus defining invariant volume
nontrivial scalar invariants, and hence the parameters forms, which (up to a sign) are equivalent to densities.
K, C, and D must be constants. Instead, we shall indicate how the calculation is done
The invariant fields at an arbitrary point can now be directly in a four-dimensional coordinate system, for
obtained by applying formulas (4.3). To do this, we which the 0(2)xO(4) action is nonlinear. In terms of
parametrize a general element of 0(2) XO(4) in terms CarteSian coordinates for M, the action of an arbitrary
of the coordinates by group element, parametrized as in Eq. (13.15) is
uO OT 0 sinl)! o acosl)!xo+~(a2-x2)sin1jJ
....
u -
cosl)!
0 M 7 0
'fiT k
0
0
(13. 15a) ....
x
x - _ sin1jJ x o+h' +talk(a 2 +X2) + cos1jJ(a 2 _ x 2)]
aMx+~(a2+x2)i
u4 0 0 1 x- -sin1jJ x o+h.t+talk(a 2 +X2) +cos1jJ(a 2 -x2)] .
u5 - sin1jJ OT 0 cos1jJ 1 (13.20)
where At the isotropy pOint Po, we have x" '" 0, and hence
... - { I}
U= U 1=1.2.3. and [~ 7] k E 0(4). (13. 15b)
aX"sg)/ = (KJ:(PO»
ax xv.o
We thus have
uO=cos1jJ, u 5 =sin1jJ, u=7, u 4 =k.
Applying formula (4.3) for this transformation and mak-
(13.16) 2
= (k +cos1jJ)Z t-;i~~~!--'(;:-:~~~!;:1i')] .
ing use of the conditions (13.11), we obtain the following (13.21)
expression for the fields at pOint P with coordinates
(u 4 ): In view of (13.15'), we have
A =K(u 5duo _ u Odu 5 ) , 11 =- k-l !TM
1" =0, (13.17) and hence
G = C(du~ +du~) +D(dui +du~ +du~ +du~).

The simple form (13.17) for the invariant fields is a

consequence of the linear action of the group expressed 2
in these coordinates. In spherical coordinates for
SI XS3 or Cartesian coordinates for M, they take the
form:
A=Kd1jJ

= 2K 2 [(a 2 +x~ + Ix 12) dx o - 2xox .dX], (13. 18a)

aT

F=O, (13. 18b) (13.22)

2144 J. Math. Phys., Vol. 19, No. 10, October 1978 Beckers et al. 2144

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 We thus have [using the further relations implied by The isotropy subgroup Go for any pOint on the regular
(13.15)] orbit in M is 0(2,1). In fact the intersection of the

detIrJ(po) = ± (k +;os1j!) 4 = ± e r
) 4. (13.23)
regular orbit (of which there is only one) with Minkowski
space M is the entire space minus the singular surface
defined by
For elements of the form (13 013) this equals ± 1 and
xl =X2 = 0. (13.31)
hence the linear isotropy condition (4.2) is satisfied.
Letting the value of the scalar density </> at Po be L(a/2)d, Away from the Singular surface the most general
Eq. (4.2) gives it at any point p with coordinates (xl') 0(2) x 0(2,2) invariant fields now have the form:
as
A =K(vl dV2 - v 2dvt>
(13.24)
=Kdl/J
(130 32a)
We have used here the general procedure to derive
K
(13024). However, if a nondegenerate invariant tensor = (~)(XldX2-X2dxl)'
Xl +X2
such as G is known, its determinant may be calculated
and used to determine the form (13.23) for detlrJ, since F=O, (13. 32b)
(13.25) G =C(dv~ +dv~) +D(dv~ - dv~ - dv~ +dv~)

This gives a much simpler method for arriving at the =CdI/J2 +D(cosh2bda 2- sinh2adi3 2_ db 2)
expression (13.24) for the invariant scalar density.
=C (dxi +~X~ _ (Xl dx~ +X;~X2)2 )
The procedure for each of the other groups treated is Xl +X 2 (XI +X2)
identical to the above, and therefore we present the re-
sults in summary form only. +D (dX~-~_ (xldx~ +x~~x-Zf) , (13.32c)
XI +X2 (Xl +X2)
(b) The group O(2} X O(2,2} </>=L(x~ +X~)d/2, (13,32d)
(i) We first consider the conjugacy class representa- where K, C, D, and L are all cons tants (since there are
tive defined as leaving invariant a space of type no nontrivial invariant scalar fields), Note furthermore
{(a, x ,y, 0, 0, a)}. that if C = - D, we have
Orbit M 0: The regular orbits on the cone C 5 are
(13033)
diffeomorphic to the space SI XH3 [circle times three-
dimensional hyperboloid with signature (2,2)]. The
(ii) The representative leaving invariant a space of
orbit in M, therefore, is the projective space
the form: {(O,O,O,y,xcoshA,xsinhA)}:
(SI XH5)/Z20 The four-dimensional surface, embedded
in a six-dimensional space is defined by Orbit M 0: (SI XH3)/Z2
(v o, VI' v 2, v 3, V4, V5) with
(13.34)
(13.26)
Isotropy subgroup Go: 0(2,1),
The projection onto the cotangent space is thus given
by Coordinates:

I'
=--
ati"
X
fj4 + v5 (13.35)
(13.27)
Coordinates: The projective coordinates {xl'} are re-
lated to the six-dimensional ones {va} by
(13.36)
vI'
where
x"=~o (13.28)
v +v
a=[x~+:2(1+ ~rr/2
Inverting (13.28) using (13.26), we have
vO=coshbcosa, i?=cos1j!,
~XI'
vI' = (x¥ + xnm (€ = ± 1), iJ1 = sinhb cos{3, v 4 = sin1j!, (13.37)
(13.29)
4/5 (1 ±X2) v2 = sinhb sin{3, if' = coshb sinO',
v = ~ 2[Xl2 +X22]172' O,,:b<oo, 0.,,: {3< 21T,
We may furthermore introduce the following curvilinear 0.,,:0' <21T, O"':I/J<21T.
coordinates for SI XH5:
Singular surface:
Vi =cos1j!, v O= coshb cosO',
X3=O, x~ +x~=a2 +x~.
v 2 = sin1j!, v 5 = coshb sinO',
(13.30) (In 3-space, a circle in the xlx2 plane with radius in-
v 3 =sinhbcosi3, O,,:b<oo, 0"':1' <21T, creasing with the velocity of light. )
v 4 =sinhb sini3, 0": 0'< 21T, 0": 1j! < 21T. Invariant fields:

2145 J. Math. Phys., Vol. 19, No. 10, October 1978 2145
Beckers et a/.

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 A =K(v3dv4 - v 4 dV 3) =Kdlj! Again, M,N,L, C,D are all constants since there are
no nontrivial invariant scalar fields, The expression
K
="F"2
au [ 3(XO O- 2Xl dXl- 2x 2 2)
2X dx dx for F must be understood as applying to one half of the
space S2 x IP only [as expressed in the coordinates
_(a 2 +X2 +2xn dx 3L (13.38a) wa(e, cp, a, b)] since it is not invariant under the Zz
mapping {w a} - { - wa}. If we require invariance, more-
p=O, (13,38b)
over, under O(3)xSO(2,1) or SO(3)xO(2,O, this implies
G =C(dv~ +dVi) +D(dv~- dVi- d~ +clV]) the vanishing of the constant M or N, respectively.
(13.38c)
=Cd1j!2 +D(cosh 2bda 2 - sinh 2ad{32_db 2),
(ii) The representative leaving invariant a space of
cp =Lu d , (13. 38d) the type {O, X ,y, 0, Z coshA, z sinhA)}
Orbit M 0: (S2 XH2)/Z2 (in i11)
(13.39)
wr + w~ + w~ = w~ - w~ + w~ = 1, (13,47)
(c) The group SO(3) X 80(2,1) Isotropy subgroup Go: SO(2)xSO(1,1).
(i) The representative leaving invariant a space of the Coordinates:
type {(O,x,y,z,O,O)}
" aw" (13.48)
Orbit M 0: (52 XH 2)/Z2 (in Ni) x = [;4 +w5 '
wi + w~ + w~ = w~ - w~ + w~ = 1. (13.40) -",
W
EX"
=-, (13.49)
P
Isotropy subgroup Go: SO(2) x SO(1, 1).
where
Coordinates:
P = [ xi + x~ + "4
a2 (
1+
X2)2J1I2 .
G'l
(13.41)

WO= coshb sina, w 3 = sinhb,

(13.42)
WI =sine coscp, w =cose,
4 (13.50)
where: w = sine sin cp,
2 5
w = coshb cosa,
Ixl '" (xi +x~ + x 5)11 2
•

wO = coshb sina, w 3 = cose,

Singular surface:
WI =sine coscp, w" =sinhb, (13.43)
xl=X2=0, x~=x5+a2, (13.51)
w 2 = sine coscp, w 5 = coshb cos a ,
0~cp<27T, _oo<b<oo,
Invariant fields: A =° (13. 52a)

Singular surface:
Xl =x 2=x 3 =0' (13.44) =M sinede !\dcp +N coshbda Idb

Invariant fields: A = ° (13.45a)

= -apr
Al [
(xodxo - X3 dx 3) II (Xj dX2 - X2dx j )
lvl Id d N W ad d
F = 2" E/ik W
j
W 1\ w
k
+ 2' Eabc b
W 1\ W
C

- [xi +x~ +~(a2 + x2 )] dx j l\dx21

N
=M sine de I\dcp +N coshbda II db +-;:-:-s[{xjdxj +x2dx2)/\{X3dxo-xodx3)
ap
M .. N --
+ [xi + x~ + ~(a2 + x 2)] dXo!1 dx 3 ],
k
= 2IxI3Eiikx'dxJj\,dx - 1X-13 dxoxodx, (13. 45b)

{ijk} = 1 ,2,3, {a, b, c}=0, 4, 5, {ijk} = 1, 2, 4, {abc}=O, 3, 5 (13,52b)

G=C(dwi +dw~ +dw§j +D(dw5-dw~ +dw~)
G = C{dwi +dw~ +dwn +D(dw5 - dw~ +dw~)
= C (de 2 + sin 2 e dcp2) _ D (db 2 _ cosh 2bd( 2)
=C(de 2 + sin 2edcp2) _D{db 2 _ cosh2 bd( 2) (13,52c)
_ C (dX"2 _ (x 0 dX)2 ) ( dx~ (x 0 dX)2) cp=Lpd, (13,52d)
- 1X-1 2 ---r.;rr +D!.'f1 2 - ~ ,
If
(13.45c)
D
(13,45d) C=-D, G=-::2g\l' (13,53)
P .
D
If C=-D, G= Ixl 2g.\l' (13.46) The comments above for case (c) (i) apply equally here,

2146 J. Math. Phys., Vol. 19, No.1 0, October 1978 Beckers et al. 2146

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 (d) The group SO (2, 1) X SO(2, 1) in region B: (ijk)=(534), (abc)=(120), C=D, D=C,
(13,59d)
(i) The representative leaving invariant a space of
If
the type {(x, y, Z, 0, 0, O)}:
Orbits M o: Two regular orbits in M (one regular
stratum), each diffeomorphic to (H;XH 2)/Z2- H:xH 2
(where H; is the upper sheet of the two-sheeted The values of the constants C,D,K,M,N need not be
hyperboloid H;): the same in regions A and B, If 0(2,1) x 0(2,1) in-
variance is required, then M =N = O. The 2-form F,
P5-pr-p~=p~ +P~-P~=Eo=±I, (13.54)
expressed in the pd(b, c, y, (3) coordinates is only locally
)J. -c x)J. 4/5 (1 ±X2)
(13.55) defined on H; xH 2 for the half-space defined by E = + 1,
P - A' P ==E~, but this covers the entire projective space (H; XH2)/'~2
and hence the regular orbits,
where
A '" -
Ix~ XI - x~ 1112 . (13.56) (ii) The representative leaving invariant a space of
type {(O,y, z, 0, x sinhA, X coshAn
Region A: xr+x~-x~>O (E o=-I),
Orbits Mo: H; XH 2/Z 2 (as above; two regular orbits,
pO == sinhb, p3 = sinhc cOSY, one stratum):
pi =coshb cos{3, p4 = sinhc siny, (13. 57a)
(13.60)
p2 = coshb sin{3, p5 = coshc,
(Again, the orbits corresponding to EO = - 1 and Eo == + 1
RegionB: xl+x~-x~<O (E o=+I),
respectively will be referred to as region A and region
pO=coshc, p3 = coshb cos{3, B. )

pi = sinhc cosy, p4 = coshb sin{3, (13.57b)

Isotopy subgroup: SO(2)xSO(I,1).
p2=sinhcsiny, p5==sinhb,
Coordinates:
0~y<21T, 0<{3<21T,
(13.61)
Singular surface: XI +x~ =x~. (13.58)
x""
Invariant fields: A =0 (13. 59a) q""=E-,
TJ
(13.62)

M dk N • b
F = "2 I i
EWP dp 1\ P +"2 E.bcP dp II dp
c where
a2 ( x2 ) 2jl /2
= - NI coshbdb 1\ d{3 +N sinhc dc /\dy I
TJ '" XI + x~ -"4 1 - Q2 • (13,63)

XI [ElmnX I d X m/\ dx n]
="2;\1 Region A:

XI + x~ - T
a
2
(
1 - Q2
x2 )2 < 0, EO== -1,
(13.59b)
qO = sinhb, q3 = coshb cos{3,
where in region A:
ql = sinhc cosy, q4 = coshb sin{3, (13, 64a)
(ijk) = (012), (abc)=(345), M=M, N=N,
q2=sinhcsiny, q5=coshc.
and in region B:
Region B:
(ijk)= (534), (abc)= (120), Ai =N, N =M,
~ ) 2 > 0,
2
2 2 a
and in both regions (lrnn) = (012), XI +X2-"4 (1 - EO = 1,
G = C (TJ/j dpl dpi) + Df'ij.b dP· dpb)
qO =coshc, q3 = sinhc cosy,
= C(db 2 - cosh 2bd{32) + D(dc 2 + sinh 2c dy2)
ql = coshb cos{3, q4 = sinhc siny, (13,64b)
= fz 5
(-EO(dX -dX -dX i V + (XodXO-X~I-X2dX2)2) q2 = coshb sin{3, q5 = sinhb,
O~c<"", _oo<b<"",
D ( d
+>;1 -EO
2
X3 + (XQdXQ-XldXl-X2d-:.J.:)
A2 , 0"" y'-: 21T, Os {3 < 21T,

where (13,59c) Singular surfaces:

TJ/j := diag( + 1, - 1, - 1), 1J.b = diag( + 1, + 1 , - 1) (a ± y' xi +x~)2 +X~ =xij, (13,65)
in region A: (ijk)=(012), (abc)=(345), C=C, D=D, Invariant fields: A =0 (13.66a)

2147 J. Math. Phys., Vol. 19, No.1 0, October 1978 Beckers et a/. 2147

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 1
r2 = coshb sin/3, r 5 = 12 (sinhc - sinhb),
(13.66b)
=- M coshbdb I\dj3 +N sinhcdc I\dy, (13,71b)
G =C(TJudql dqi) +D(Tjabdq"dqb)
(13.66c)
= C[db 2 _ cosh 2bd/3 2] +D[dc 2 + sinh2c dy2],
!/>=LTJd, (13.66d) Singular surface:

where x~ - xi - x 2 (XO +x 3) + (xo - x 3 ) = 0. (13,72)

TJIJ = diag( + 1, - 1, - 1), Tjab = diag( + 1, + 1, - 1),

Invariant fields (for region A: {ijl?}={0+5,3 +4,1},
in region A: (ijk) =(034), (abc) = (125), {abc}={2,3-4,0-5}, for regionB: {ijl?}={0-5,2,3-4},
in region B: (ijl?) =(512), (abc)=(340).
{abc}={3 +4,1, + 5}): °
If
A=O, (13, 73a)
(13,67)
1\1 Ej Jk r I d
F' = 2 r i1\d r k +"2
N Eabc r adr b 1\ d r C
The above expressions, within curvilinear coordinates, (13,73b)
are identical to those for the previous case. They may = - M coshbdb 1\ d/3 + N sinhc dc 1\ da,
be straightforwardly expressed in Cartesian coordi- G = CTJI i dr l drJ + DTjab dr adrb
nates, using Eqs. (13.63) and (13.64), but the expres- (13,73c)
sions are cumbersome. All the comments made above = C[db 2 - cosh 2bd/3 2] +D[dc 2 + sinh 2c d}21,
for case (d) (i) apply equally here. ¢ =Lv d • (13. 73d)
(iii) The representative leaving invariant a space of If
type {(x,y,O,z,z,x)}:
C
Orbits: (H; XH2)/Z2 - H;xH2 (as above, two regular C=-D, G=V2 g.\!, (13.74)
orbits, one stratum):
Again, the comments above for cases (i) and (ii) apply
}(r o+r5F - ~(r3 +r4 )2 - ri =r~ - ;',(r o- r5)2 + ~(r3 - r4)2 =E O' equally here,
(13.68)
(e) The group 0(4,1}
(Again, values EO = - 1 and EO = + 1 identify regions A and
B, respectively.) (i) The representative leaving invariant a space of
type {(a, 0, 0, 0, X sinhA, x coshA)}:
IsotrojJy subgroup: SO(2)xSO(I, 1).
Orbit Mo: H4 [one-sheeted hyperboloid, signature
Coordinates: (4,1)]

(13,69) ti + t~ + t3 + t~ - t~ = 1, (13.75)

EX'" 4/5 E(1 ±X2)

Isotropy subgroup: 0(3,1)
r'"--
- v ' r =-2-v-' (13.70)
Coordinates:
where
at"
x" = 1 + l' (a =e~), (13.76)

2 2
t4/5_a +x (13.77)
Region A: x~_xI_X2(xO+X3)+(xO_x3)<0, Eo =-I,
- a2 _ x2 ,

rO = Jz (sinhb + coshc), r3 = Ji (coshb cos/3 + sinhc cOSY),

to = sinhb,
t 1 = coshb sinjJ sine cos!/>,
t 3 = coshb sinJ3 cos e,
t4 = coshb cosJ3, (13.78)

rl = coshb sin/3, r4= Jz (coshb cos/3 - sinhc COSy), t2 = coshb sinfl sine sin!/>,

r2 = sinhc siny, r 5 = Jz (sinhb - sinhc). 0~J3~1T, O~CP<21T.

(13. 71 a) Singular sUrface:

Region B: x~ - xi - x 2 3 3
(XO +x ) + (xo - x ) > 0, EO = + 1, 1-12 =xo-a.
X
2 2 (13.79)
In1)ariant fields:
rO = ~ (sinhb + coshc), r3 = ~ (sinhc cosy + coshb cos!3) ,
A=O, (13.80a)
rl = sinhc siny, r = 4
~ (sinhc cosy- coshb cos/3), F=O, (13.80b)

2148 J. Math. Phys., Vol. 19, No.1 0, October 1978 Beckers et al. 2148

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 G ::::C(d~ - dli - dt~ - dfa - dtl) Isotropy subgroup: 0(3,1)
:::: C[db 2 - cosh2b(d/32 + sin2j3de 2 + sin 2i3 sin2e d</J2) J Coordinates:
4a2C v as"
:::: (a 2 _ X2)2 TJ,w dx" dx , x"':::: ~ (a::::e A) (13.88)
l+s '
(13. 80c)
s" -
2ax" a2 _
s5 - --..---."
r (13089)
-a2 +x2 ' -a'+x'
(13.80d)
so:::: coshb sin/3, S3 = sinhb cose,
There are no nontrivial scalar fields, therefore C
and L are constants. Note that G may be interpreted sl :::: sinhb sine cos</J, S5 :::: coshb cos/3, (13 90)
0

as the metric for the de Sitter space with constant

s2 == sinhb sine sin</J,
positive curvature.
(ii) The representative leaving invariant a space of
the type {(x, 0, 0, 0, 0, O)}: 0-"'/3<27T, 0-"'</J<27To
Orbit Mo: H4 (one-sheeted hyperboloid) Singular surface:

'if +'il +'il +tl- t5 = 1. 2 (13 81)

0 x =x~ + a2 •
1 12 (13 91)
0

Isotropy subgroup: 0(3,1) Invariant fields:

Coordinates: A::::O, (13. 92a)
/ 'i/ 0 1
(13.82)
F=O, (13.92b)
x::::~, x=~,
G =C{ds~ + ds~ - ds~ - dsi - dsV
/12
t' =XU,
x 74/5= ±x
""27' (13.83) = - C(db 2 - cosh 2bd{:32 + sinh 2 bde 2 + sinh2b sin2 e d</J2)
4a2C
'i l =coshb sin/3 sine cos</J, 'i4:::: coshb cos/3, = (a2 + x2) 7]"v dx" dx
Y
,

P =coshb sini3 sine sin</J, f5 :::: sinhb, (13 84)

0
(13.92c)
P =coshb sin/3 cose,
(13. 92d)

Again, C and L are constants and G may be inter-

preted as the metric for the de Sitter space of constant
negative curvature.
Singular sUrface:
(ii) The representative leaving invariant the space
x O =0. (13.85)
{CO, 0, 0, x, 0, OJ}:
Invariant fields:
Orbit Mo: 114 [one-sheeted hyperboloid, signature
A::::O, (13. 86a) (2, 3) 1
F=O, (13. 86b) ~+S5-Si-S1-5.i=L (13.93)
-2 -2 -2 -2 -2]
G ==C[dt5 - dt! - dt 2 - dt3 - dt4 Isotropy group: 0(3,1)
2
:::: C(db - cosh2 b(d/32 + sin2/3 de 2 + sin2(3 sin2 e d</J2) 1 Coordinates:

.gXo TJ"y dx'" dr, ? 1

== x"=~ (jH'3), x3=~, (13.94)

(13. 86c) x'" :::-4/51±X2

?::::? (11=0,1,2), s =~ (13095)
</J::::Lxg. (13. 86d)
S1l == coshb sin{:3, S"4 = sinhb cose,
The group in this case may be interpreted as the con-
formal group for a three-dimensional Euclidean space, Si == sinhb sine cos</J, st = coshb cos{:3, (13.96)
embedded in a four-dimensional space by the addition
of a scaling parameter xo.
SI. == sinhb sine sin</J,

(f) The group 0(3,2)

0-"'{:3<27T, 0-"'</J<2rr.
(i) The representative leaving invariant a space of
Singular surface:
type {CO, 0, 0, 0, x coshy, x sinhy)}:
Orbit Mo: Jj4 [one-sheeted hyperboloid, signature x 3 =0. (13.97)
(2,3) ] Invariant fields:
s~ + s% - s~ - s~ - s; = 1. (13.87) A==O, (13. 98a)

2149 J. Math. Phys., Vol. 19, No.1 0, October 1978 Beckers et al. 2149

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 F=O, (13. 98b) (h) The group OPT(3J)
G =C(dsg + d~ - dSf - d~ - dS,f) For each of the two cases [leaving invariant spaces
= - C(db 2 cosh 2bd{32 + sinh2 b df}2 + sinh2 b sin2 f} dq}) of the type {(x, O,O,x, ±y,y)}], there is one regular
orbit in lW which is diffeomorphic to Minkowski space
=.g X3
1)uv d:? dx v , M plus a three-dimensional Euclidean space "at in-
finity. " The isotropy subgroup is the direct product of
(13.98c) a Euclidean group E(2) with a three-dimensional solva-
ble group. There are no invariant tensor fields other
¢=Lx1. (13. 98d) than the trivial constant scalar.

The group here may be interpreted as the conformal (i) The group SI M (3,1)
group for a three-dimensional Lorentzian space em-
The three representatives leave invariant the spaces
bedded in a four-dimensional space by the addition of a
scaling parameter X3'
{(O, 0, 0, 0, - x, x)}, {(o, 0, 0, 0, x, xl}, and {(x, 0, 0, - x, 0, OJ}.
In each case, the regular orbit is diffeomorphic to lvI,
the isotropy subgroup is 0(3, I)XO(1, 1) and there are
(g) The group S(U(2,1) X U(1))
no nontrivial tensor fields.
We consider the subgroup of SU(2, 2) defined, in the (j) Nonmaximal subgroups
diagonal representation with Hermitian form h (Secs.
IV 10 (iii) and IV 11 (viii)], as leaving invariant the sub- In all the above cases there was only one regular
space of (1;4 consisting of vectors of the form (0,0,0, w). stratum in M, containing at most two orbits. Conse-
From the decomposition of Eq. (11. 18) and the repre- quently, no nontrivial invariant scalar fields existed
sentation on C 5 defined by Eqs. (11.20)-(11. 22), we and the invariant fields obtained are distinguished only
obtain the following results: by the values taken for the constants K, L, lvI, N, C, and
D. For nonmaximal subgroups, the strata may contain
Orbit ]\11o: As observed in Sec. IV 11, the group acts an infinity of orbits, making G VvI i (AI i the ith regular
transitively on il,ILthat is, there is one orbit covering stratum) a manifold rather than a finite set of points,
the entire space M - (51 x5 3 )/Z2' We may thus coord i- and hence there exist nontrivial scalar fields. We con-
natize, as for example (a) [the group 0(2) x 0(4)] by sider three illustrative examples below.
Eqs. (13.5)-(13.9).
(Q) The group 0(4) L 0(4)XO(2) [contained in the
Isotropy subgroup: The group Go of 8ec. IV 11 (iii), maximal subgroup of example (a) above 1:
with elements of the form (11.12)-(11.14), or equiva-
lently, (11.21) and (11. 22) [where the isotropy point is The same coordinates may be used as for example
taken as the origin (0,0,0,0,1, 1) under the embedding (a), however the orbits are now diffeomorphic to a 3-
j of Eq. (7.6)1. sphere 53. These are distinguished by the value of the
single nontrivial scalar invariant, which may be taken
Invariant fields: The linear isotropy conditions at the as uO=xo/T=cos~ [see Eqs. (13.4)-(13,9) above for
origin imply that there is no nonvanishing 1-form, notation]. The isotropy group for this case is still 0(3).
2-form, symmetric (0,2) tensor or scalar density field Hence, the invariant fields are of the same form as
invariant under this group. those for the maximal group 0(2)xO(4) [Eqs. (13.18)
This is most directly seen by using the realization and (13.24) above], but with the constants [(,L,C, and
D replaced by arbitrary functions [(uo), L(uo), C(l/o),
given by Eq. (8.31) in a neighborhood of the origin
(for which the local transformation is sufficient) to and D(uo) of the scalar invariant.
evaluate the Jacobian of the isotropy subgroup trans- ((3) The homogeneous Lorentz group 0(3, 1) contained
formations. On the other hand, nonzero invariant in the maximal subgroup 81M(3, 1) which leaves in-
vector fields do exist, having the general form: variant the space {O, 0, 0, 0, - x, x}~
Here the orbital analysis is familiar and may be
v= K (u Z il
w _ ul
w0 _ u4 iii?a + u3 ax;<a + u5 wa _ u o w
a) described entirely in Minkowski space :H upon which
8IM(3, 1) acts as a global transformation group. The
regular orbits are two- sheeted (timelike) or one-
= K [(x~ - x 3XO + ~(1 - X2»a? + (XOx l - x 3X1 + x 2) a? sheeted (spacelike) hyperboloids: the corresponding
strata having 0(3) and 0(2,1), respectively, as iso-
+ (x°x z - x 3x 2 - xl) bax + (x°x 3 - x~ - ~(1 + X2» ~] tropy subgroups. The single scalar invariant is and X2
ax ' the invariant fields are of the form:

(13.99) A = [(X2) 1)"v x" dx v , (13.100a)

F=O, (13.100b)
where K is constant. The fact that an invariant vector
field can exist, while an invariant 1-form (covector) g = C (X2)x" Xv dx'" dx + D(X2) 1),," dx" dx", (13. 100c)
does not is due to the nonexistence of an invariant
metric field [(0,2) tensor] which could relate the two.
rp =L(X2). (13.100d)
The vector field (13.99) is just the one induced by the (y) The group 0(3, l)xO(l, 1) contained in the maxi-
invariant 1-parameter subgroup U(lL mal subgroup 81M(3, 1):

2150 J. Math. Phys" Vol. 19, No, 10, October 1978 Beckers et al. 2150

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 This is just the homogeneous Lorentz group F=dA, (15.1)
augmented by scaling transformations. The strata are
where
the same as those for 0(3, 1) given above, but each one
contains just one orbit. There are therefore no non- A "'lv1cose d¢ - NsinhbdO'.
trivial scalar invariants, and the form of the arbitrary
(ii) SO(2, I)XSO(2, 1): From Eq. (13.59), [or (13.66),
functions in Eq. (13.100) becomes uniquely determined,
(13.73)1
giving the following scalar fields:
F=dA, (15.2)
(13. lOla)
where
F=O, (13.101b) A", - M sinhb d{3 + N coshe dO'.
Thus, the first set of Maxwell's equations,
g=C 7x x dx/J. d:i' + xzD TJ/J. V dx/J. dx
V
, (13.101c)
(15.3)
dF=O,
(13.10Id)
apply everywhere on the regular strata, though not on
the singular ones. To verify the second set of Maxwell's
14. Invariance under P, T, and PT
equations, we need the dual forms *F. This is most
As discussed in the Introduction, the choice of one easily obtained by noting that the dual of a 2-form is
representative of each conjugacy class under the unchanged under a conformal change in metric. Since
Poincare group for the maximal subgroups of C(3, 1) the choice C = - D for the symmetric (0,2) tensors ob-
is motivated by considerations of relativistic invariance, tained above yields what can be identified as a confor-
insofar as proper Lorentz transformations are con- mally flat metric which has a very simple form in the
cerned. For the discrete transformations P, T, and curvilinear coordinates, the Hodge star duality can be
PT, however, the interpretation is not quite the same, applied with respect to this metric (indeed, it is quite
since the physical equivalence of two different fields simple to alternatively express the Minkowski metric
related only by such transformations implies further gAl in the curvilinear system). In any case, the result
dynamical assumptions depending, for instance, on the is surprisingly simple; the dual of each F is just ob-
invariance of the equations governing such fields. tained by interchanging the two constants AI - N;
Therefore, we are led to examine how these transfor-
(i) SO(3)xSO(2, 1):
mations act upon the fields obtained above. By inspec-
tion of the six-dimensional forms of these fields, we *F = N sine de /\ d <b + AI coshb dO' 1\ db,
can see that these transformations [vid. Eq. (7.15)1
(ii) 80(2, l)x80(2, 1):
map the fields A, G, and ¢ onto themselves in every
case, therefore yielding no new invariant fields For o * F =- N coshb db !, dp + AI sinhe de !, da
the case of the nonvanishing 2-forms F [i. eo for the
and hence
groups SO(3)xSO(2, 1) and SO(2, l)xSO(2, 1)], we see
that these discrete transformations have the effect at (15.4)
most of changing the signs of the constants .11 and N,
Therefore, the sources vanish everywhere, except on
which are arbitrary. Therefore, again, no new types
the singular strata. This is not surprising, since the
of fields are obtained for these cases, even though the
singular strata exactly correspond to the singularities
invariance groups involved are not members of the
in all the fields. We now discuss these in particular:
same conjugacy class under the proper Poincare group.
(i) SO(3) XSO(2, 1): {O,x, y, z, 0, Or
15. Discussion
The singular stratum defined by Eq. (13.44) corre-
With regard to phYSical interpretation of the fields sponds to a point Singularity at the origin. The field
obtained here, the following questions may be asked: F may be interpreted as that of a static point electric
(a) Which, if any of these fields may be identified charge, plus a point magnetic charge, located at the
as solutions of conformally invariant field equations, origin (i. e., Coulomb field plus magnetic monopole
and in particular, can the 2-form fields be identified field), the two charges being proportional to .11 and N,
as satisfying Maxwell's equations? respectively.

(b) If the singularities of these fields can be identified (ii) SO(3) x SO(2, 1): {O, x, .1',0, z coshX, z sinh"}
as defining distributions of localized sources, how The singular stratum is defined by Eq. (13.51) and
may these sources be characterized in terms of spatial corresponds to a point source (e. g., electric plus mag-
extent and time dependence? netic) which is moving along the X3 axis with a constant
acceleration (in ·the relativistic sense that the 4-
In fact, the reply to these questions is very easily
acceleration is Fermi transported along the trajectory
obtained. First, we note that everywhere on the regular
remaining, of course, orthogonal to the 4-velocity).
orbits, the nonvanishing 2-forms are locally exact,
and hence closed. This may be seen explicitly in terms (iii) SO(2, 1)xSO(2, 1): {x,y,z, 0, 0, O}
of the curvilinear coordinates introduced above:
The singular stratum is defined by Eq. (13.58) and
(i) SO(3)XSO(2, 1): From Eq. (13.45) [or (13.52)], we corresponds to a point source (e. g., electric plus mag-
have netic) which is moving along the X~ axis with a constant

2151 J. Math. Phys., Vol. 19, No. 10, October 1978 Beckers et sf. 2151

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 (iv) SO(2, 1) x SO(2, 1): {O, y, z, 0, x sinhA, x coshA} have involved an explicit examination of such solutions
concerned 0(3, 2) invariant scalar densities, 16 and 0(5)
The singular stratum is given by Eq. (13.65), which
invariant Euclidean SU(2) gauge fields. 18,22,53 The latter
defines a toroid, symmetrical about the X3 axis, with
falls somewhat outside the scope of the results obtained
major radius a, and minor radius increasing with the
velocity of light. here, since it involves the invariance of connections
under "simultaneous" gauge and space-time transfor-
(v) SO(2, l)xSO(2, 1): {-~ ,y, 0, ~, ~, ~} mations. However, it is quite simple to relate this
to results regarding invariant tensor fields, as will be
,"'2 v2 v2 v2
shown elsewhere. 49
For this case, the singular stratum is defined by the
cubic equation (13.72) with sections that are hyperbolas (ii) Conserved quantities and inteKrals of molion: If
and cardioids respectively in the planes parallel to and the fields studied here are regarded as influencing the
normal to the X1X2 plane. The other cases of singular motion of particles, either classically or quantum
surfaces will not be discussed further here. They all mechanically, through a suitable coupling, the symme-
correspond to simple configurations such as lines, try group for the equations of motion will be deter-
planes, spheres expanding with the velocity of light, etc:. mined by the symmetry of the fields. The invariants
However, since no invariant 2-form fields exist of this symmetry group will then be of importance in
for these cases, it might be more natural to give a defining the integrals of motion (conserved quantum
different interpretation to these singularities. Looking numbers) and its representations will be relevant to
for instance at the invariant fields G for the groups defining the quantum mechanical Hilbert space. 50
0(4,1) and 0(3,2), we obtain what may be interpreted
(iii) Separation of l'ariables: A knowledge of the sub-
as the metrics for de Sitter spaces of constant positive
group structure of the invariance group for a symmetric
or negative curvature, However that is not consistent,
(0, 2) tensor, interpreted as the group of isometries for
of course, with the interpretation of {xl'} as Cartesian
the Riemannian space with the given tensor as metric,
coordinates, Instead, we must interpret these as singu-
is central to the problem of separation of variables in
lar coordinates for a curved space with metric G, the
the Laplace-Beltrami, Klein-Gordon and other
singularities at x 2 = ± a2 being spuriously introduced
differential equations defined in such a space. 51
through these coordinates, Within this interpretation,
we obtain a list of metrics for spaces diffeomorphic (iv) Classi(icalion of G-s!rucf!wcs: Invariant tensor
to the regular orbits on C 5 whose isometry groups are fields are directly related to the automorphisms of G-
the groups discussed above. In particular, the choices structures. 52 Therefore, if the symmetric tensor fields
C = - D for the constants defining these fields all yield which have been obtained here are interpreted as
conformally flat metrics. Finally, we may remark metric tensors, their invariance group is precisely
that the scalar density Ib, with weight d = - 1 obtained the automorphism group for the Riemannian (or pseudo-
for the 0(3,2) de Sitter group is exactly that given in Riemannian) structures they define. The scalar densi-
Ref. 16 as a solution of the conformal invariant field ties similarly define volume structures, the I-forms
equation of the ArjJ4 field theory. If this is taken as locally causal structures (given a Riemannian metriC),
representing the classical ground state of the system, the 2-forms symplectic or almost symplectic struc-
we have a spontaneous breakdown of the C(3, 1) tures, and their corresponding invariance groups may
symmetry. likewise be regarded as automorphism groups of these
structures.
VI. SUMMARY (v) Finally, regarding generalizations of the methods
used in the present work, two interesting possibilities
The principle results of the present work are two-
fold, First we have obtained a characterization of the that suggest themselves are: (i) the extension to non-
maximal subgroups of the conformal group of space- tensorial fields (eo g., spinoI' fields, other induced field
time through their conjugacy classes under the Poincare representations of the conformal group, 3,8 or connec-
group. In the process, a fairly general method for tions) and (ii) the study of fields which are con(onllally
determination of such conjugacy classes has been uti- invariant (i. e., up to a multiplicative factor) under
lized which has applicability both to the continued study various space-time transformation groups.
of the conformal group and to other subgroup analyses.
Secondly, we have determined the most general tensor ACKNOWLEDGMENTS
fields, of the types frequently encountered in phySics, The authors are indebted to J. Patera, who partici-
that are invariant under these subgroups. Again, the pated in the early stages of this work, and to R. B.
differential geometric methods have been formulated in Pettitt and S. Shnider for discussions.
such a way as to be applicable to a wide range of simi-
lar studies. The particular results obtained may them-
selves be further utilized in a variety of problems, of
which we mention a few here: lr. E. Segal, A[(lti1clI/(ltic(I[ Cosmology (lild Extragal(lctic
(i) Spontaneous sYlllllletry IJrea/dnK: As discussed in Astronomy (AcademiC, New York, 1976).
2T. II. Go, H. A. Kastrup, and D. Mayer, Rep. l\lath. Phys.
the Introduction, a knowledge of the invariance proper- 6, :195 (1976).
ties of solutions to conformally invariant field equations 3T.H. Go, Commun. Math. Phys. 41, 1:,7 (197,,).
is basic to any study of the spontaneous breaking of con- 4£. Cunningham, Proe. London Math. Soc. 8, 77 (1910).
formal symmetry. At present, the only studies which 'H. Bateman, Prot:. London Math. Soc. 8, Z~:l W:no).

2152 J. Math. Phys., Vol. 19, No.1 0, October 1978 Beckers et al. 2152

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2153 J. Math. Phys., Vol. 19, No. 10, October 1978 Beckers ef a/. 2153

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