Revista Brasileira de Flsica, Vol. 17, n? 2, 1987
A Gauge Approach for the S0(3,1) o, U(1) Group
Departamento de Fhica, Universidade Federal do Paraná, Caixa Postal 19081, Curitiba, 80000, PR,
Brasil
Recebido em 9 de junho de 1986
A sourceless gauge model f o r g r a v i t a t i o n , coupled t o Maxwel l
theory i s examined. By c o n s i d e r i n g t h e Lorentz group SO(3,l) i n semi- .
d i r e c t product w i t h t h e U ( l ) group i n a bundle o f 1 inear frames,Maxwell
equations and Yangls g r a v i t a t i o n a l equation can be derived.
This approach has a gauge- like Lagrangian which leads t o the f i e l d equations.
Abstract
1. INTRODUCTION
I n s p i t e o f the analogies e x i s t i n g between Yang-Mi I 1 (YM) theo r y , a t t h e c l a s s i c a l l e v e l , and General R e l a t i v i t y (GR), t h e f a c t r e mains t h a t the E i n s t e i n - H i l b e r t Lagrangian i s n o t o f the YM
type,
and
t h e dynamical aspects o f b o t h t h e o r i e s a r e q u a l i t a t i v e l y d i f f e r e n t .
Gauge t h e o r i e s , besides t h e i r phenomenological successes, present a l s o renormal i z a b i l i t y as a b a s i c formal p r o p e r t y ( i s s p i t e o f t h e
f a c t t h a t n o t a11 gauge t h e o r i e s a r e a u t o m a t i c a l l y renormalized).
This
does n o t happen w i t h E i n s t e i n ' s g r a v i t a t i o n a l model.
Severa1 attempts t o renormal i z e the Einstein-Maxwell model s have
been made, w i t h o u t much success.
I t s e e m s t h a t any e l e c t r o - g r a v i t a t i o n a l
approach, combining b o t h Maxwell and E i n s t e i n t h e o r i e s w i 1 l n o t g i v e
any new r e s u l t s , t h i s being due t o t h e i m p o s s i b i 1 i t y o f e1 i m i n a t i n g
ghosts a f t e r q u a n t i z a t i o n .
A gauge orthodox method f o r pure g r a v i t a t i o n has been developed
before',
c o n s i d e r i ng
l e a d i n g t o a unique tagrangian densi t y , by
the
Poincaré group ~ 0 ( 3 , 1 ) 0 T3,1. This approach i s implemented by
t h e use
of weak c o n s t r a i n t s and r e q u i r e s viewing t h e ~ o i n c a r é g r o u p
as
Wigner-lnonu
2
the
c o n t r a c t i o n o f t h e de S i t t e r group. Yang's and E i n s t e i n ' s
g r a v i t a t i o n a l equations a r e e v e n t u a l l y obtained.
I n such a case,
- M i l l s equations a r e d e r i v e d from Bianchi i d e n t i t i e s , by
.
Work supported by CNPq ( B r a z i 1 i a n Governrnent ~ g e n c y )
duality
Yangsyrn-
Revista Brasileira de Física, Vol. 1'7, no 2, 1987
metry, f o r the sourceless case. The absence o f a nondegenerate K i l l i n g -Cartan m e t r i c on t h e group i s n o t by i t s e l f an impediment: t h e o r i e s f o r
the non semisimple groups a r e q u i t e f e a s i b l e through the use o f general
invariants o f the a d j o i n t representation.
We question here why n o t consider another g r a v i t a t i o n a l theory,
GR, coupled t o Maxwell's t h e o r y . The reason f o r such an i n -
instead o f
have
q u i r y i s t h a t i t i s p o s s i b l e , as demonstrated below, t o
gauge- like e l e c t r o - g r a v i t a t i o n a l model i n which,
another
instead o f GR,
Yang's
g r a v i t a t i o n a l theory i s obtained.
The gauge model approach, developed i n a f i b r e bundl e P o f l i n near frames
3
has f o r base m a n i f o l d Minkowski
space-time and
f o r gauge
group G t h e s e m i d i r e c t product between the Lorentz group and
0
modular group of r o t a t i o n s i n the c i r c l e : G = SO(3,l)
the
uni-
~ ( 1 ) . Starting
from YM equations, two independent s e t s o f equations a r e obtained: Yang's
equation f o r t h e Lorentz s e c t o r and Maxwell equations f o r t h e U(1) sect o r . F i n a l l y , a Lagrangian i s proposed which couples the above two theo r i e s . The formalism o f d i f f e r e n t i a l forms w i l l be used throughout.
2. THE GAUGE FIELD
We s t a r t by c o n s i d e r i n g a l i n e a r connection
r
on the
o f l inear frames, represented by a m a t r i x o f 1-forms,
{hU)
o f Minkowskils
a1 gebra so ( 3 , l ) :
basis
Here,
l a t i n i n d i c e s a,b,
and greek i n d i c e s p , v ,
...
wr i t t e n
space-time, and w i t h values
r
.. .
=
P-bundle
in
~~~r~hu
(2.1)
blJ
.. . $ 4 correspond
= 1,
t o t h e ~ 0 ( 3 , 1 )algebra,
= I,...,'+
r e f e r t o space-time. The Ja
4
generators o f the Lorentz algebra
,
a
i n the Lorentz
P
and the components
n e c t i o n form w i l l be i n t e r p r e t e d as a gauge p o t e n t i a l
i
.
b
arethe
o f the con-
blJ
The
r
connec-
t i o n d e f i n e s c o v a r i a n t d e r i v a t i v e s o f tensors belonging toany represent a t i o n o f the Lorentz group. For simpl i c i t y , a l l forms w i l l be considered
as p r o j e c t e d on t h e base m a n i f o l d .
Next, we replace the Lorentz g r o u p b y
G = SO(3,l)
0
the extended group
U ( I ) . This a l l o w s the i n t r o d u c t i o n o f another l i n e a r con-
Revista Brasileira de Física, Vol. 17, n? 2, 1987
n e c t i o n A , w r i t t e n i n the same b a s i s {&'}
above, and
wi t h values
in
the ~ ( 1 )algebra
where
I
i s the s i n g l e generator o f U(1). Our bundle P has now Minkowski
space-time f o r base m a n i f o l d and SO(3,l)
0
~ ( 1 )f o r symmetry group.
Since P i s a bundle o f l i n e a r frames, i t i s p o s s i b l e t o ~ t a k ea
l i n e a r connection 5
(2.3)
W = T + A
w i t h valueç i n t h e L i e algebra o f G. Now, the components o f
be
W will
i n t e r p r e t e d as the t o t a l e l e c t r o - g r a v i t a t i o n a l p o t e n t i a l , w h i l e the components
Pby and
A
wi 1 l be i n t e r p r e t e d as p a r t ia1
1-i
gauge
p o t e n t i a l S,
corresponding t o each one o f the two algebra sectors.
The L i e algebra o f G has the commutation r u l e s
where
n
=
(+,+,+,-)
6
i s t h e Minkowski m e t r i c .
The c u r v a t u r e
F o f t h e W connection
and can be decomposed i n t o
where (on account of t h e commutation r u l e s eq. ( 2 . 4 ) ) ,
the curvature
F,
corresponding t o t h e Lorentz s e c t o r i s
F=dr+rAr
and the c u r v a t u r e
f,
corresponding t o the ~ ( 1 )s e c t o r ,
f =dA
(2.7)
is
Revista Brasileira de Física, Vol. 17, n? 2, 1987
r
The i n t e r p r e t a t i o n o f t h e l i n e a r connections W ,
gauge p o t e n t i a l s makes o f t h e corresponding c u r v a t u r e s
F,
and
A
as
f the
F and
gauge f i e l d s o f the model. Thus, the components
represent the gauge f i e l d o f the Lorentz algebra s e c t o r ,
while
the
componen t s
represent the gauge f i e l d o f the ~ ( 1 algebra
)
sector.
From eq. (2.8)
we have
df =
O
(2.11)
7
whi ch corresponds t o t h e homogeneous p a i r o f Maxwell equations.
3. FIELD EQUATIONS
D i f f e r e n t i a t i o n o f eq. (2.5)
furnishes Bianchils i d e n t i t y
which guarantees t h a t t h e c o v a r i a n t d e r i v a t i v e o f the t o t a l gauge f i e l d
F
3
i s a u t o m a t i c a l l y zero
.
Moreover, s i n c e gauge t h e o r i e s
exh i b i t
du-
a1 i t y s y m t r y , the dynamical YM f i e l d equations ( f o r t h e s o u r c e l e s s
case) a r e eq.(3.1),
where
*
b u t w r i t t e n i n terms o f the dual o f
F'
i s Hodgels d i f f e r e n t i a l s t a r o p e r a t o r ?. Here, we s h a l l deal with
t h e sourceless case o n l y . A f t e r s e p a r a t i n g eq. (3.2)
for
each
algebra
s e c t o r we a r e l e d t o
Sf
=
o
(3.3)
f o r the U(1) sector, and
o~ +
*-i
[r, *F]
= O
Revisía Brasileira de Física, Vol. 17, n? 2, 1987
f o r t h e Lorentz s e c t w . Here, 6 stands f o r t h e c o d e r i v a t i v e o p e r a t o r 7 .
As
f
i s a 2-form, we have i n t h e c a r t e s i a n b a s i s
f=
7 fpv
&V
(&'I
A 'hv
(3.5)
and
So, eq.(3,3)
is,
i n terms o f i t s components
S i m i l a r l y , i n o r d e r t o w r i t e eq.(3.4)
i n terms o f
its
com-
nents, we consider
Taking i n t o account t h e f i r s t commutation r u l e from eq.
(2.4)
we o b t a i n
and
a
By r e a r r a n g i n g t h e terms o f eqs. (3.8) and (3.91, we
the components o f eq.(3.4)
(3.9)
bvp
have
for
Revista Brasileira d e Fisica, Vol. 17, no 2, 1987
I n t e r p r e t ing f as t h e electromagnet i c f i e l d , eq. (3.6)
Maxwell dynamical equations i n t h e sourceless case.
In
eq.
becomes
(3.10)
the
componentsFa
areskew- symmetrical i n p , v , a n d t h i s e q u a t i o n e s t a b h v
1 ishes t h a t t h e c o v a r i a n t d e r i v a t i v e o f F~
i s nu1 1 , f o r t h e algebra
buv
indices.
a
which
By means o f t h e v i e r b e i n f i e l d s h and t h e i r i n v e r s e ,h:
a a
a
r a t i s f y the conditions h
: h; = 6$, h h
= ,6;
we s t a b l i s h an isomor-
a B
phisrn between t h e bundle space and t h e base space o f t h e f i b e r r n a n i f o l d ,
so t h a t we may take t h e 1 inear connection
r
on the b u n d l e
of
1inear
frames as a L e v i - C i v i t a connection on the base space
which we now take as t h e components o f t h e Riemann c u r v a t u r e t e n s o r
Thus, i n s p i t e o f the f a c t t h a t we s t a r t e d w i t h a f l a t
m a n i f o l d (Minkoswski space-time),
sector,
base
space-time
the gauge f i e l d o f the L o r e n t z a l g e b r a
f , generates a c u r v a t u r e i n t h a t base space through eq.(3.11).
a
By choosing a b a s i s such t h a t h =
a
equations f o r the Lorentz s e c t o r , eq. (3.10)
due t o t h e skew-symmetry
wr i t t e n as 8
Ra
we can w r i t e t h e f i e l d
as
.
This l a s t e q u a t i o n c a n
Buv= - ~ ~ B v p
which leads t o Yang's g r a v i t a t i o n a l e q uation
be
g
These equations, which have been proposed by Popov and ~ a i k h i n "
on t h e b a s i s o f a h e u r i s t i c argument, have, as a very p a r t i c u l a r s o l u t i o n , E i n s t e i n ' s sourceless equation
R
aB
= O .
(3.14)
Revista ~rasilèirade Física, Vol. 17,
n? 2, 1987
4. LAGRANGIAN FORMALISM
I n o r d e r t o w r i t e t h e t o t a l Lagrangian, we have t o keep i n mind
t h a t G i s n o t a semisimple group, so i t does n o t admi t a K i l l ing- Cartan
m e t r i c 4.
I n t h i s case, we can deal w i t h a new technique t o
obtain
in-
v a r i a n t s and such i n v a r i a n t s may be taken as Lagrangians.
For non semisimple groups we can b u i l d up i n v a r i a n t s
u s i n g a m e t r i c f o r t h e group manifoldl.
wi thout
We can take a m a t r i x X = ~ ~ $ i n
t h e a d j o i n t r e p r e s e n t a t i o n o f t h e l inear group GL(~,R) and develop t h e
express i o n
d e t ( ~ +X)
where
I
(4.1)
X i s a parameter. By expanding (4.1)
i s the i d e n t i t y m a t r i x and
i n t h e p o l i n o m i a l form i n A we have
where
In i s the n - t h i n v a r i a n t . The f i r s t i n v a r i a n t i n eq.íb.2)
= t r X a n d the n - t h invariant i s I
n
= d e t X.
i s I, =
I n c l a s s i c a l electromagnetic
theory t h e Lagrangian i s e q u i v a l e n t t o the i n v a r i a n t tr(f A
*f),
where
f i s t h e f i e l d strengh, because i n t h i s case t r f = O . The same happens
t o Yang-Mills theory, when we w r i t e the f i e l d i n t h e a d j o i n t r e p r e s e n t a t i o n o f t h e SU(2) algebra.
I n t h e case we a r e deal i n g w i t h ,
take as i n v a r i a n t the q u a n t i t y t r ( F A X F ) and t h e a c t i o n
chosen t o be
s=
-i
1 d4&
t r ( F A *F)
equat ions
pbpv
da,,
-
6fdo
a
ís
(4.3)
= F(r,dr,dA)
we o b t a i n Eu1 e r
"aw
can
integral
.
By c o n s i d e r i n g t h e f u n c t i o n a l dependences F
by s e t t i n g the extrema1 c o n d i t i o n 6.5' = O ,
we
and
- Lagrange
J
&(aPr Cd -o
L]
*Ao
6 (aXA,)
(4.4)
=
O
Revista Brasileira de F isica, Vol. 17,n? 2,1987
for each group algebra separately. The terms in F lead to eq. (3.10) and
the terms in f to eq.(3.6).
5. CONCLUSION
We have proposed a geometrical setting for a gauge niodel of
grav i tation coupled to Maxwel l 's theory. The model presented here leads
to Yang's gravitat ional theory, instead of GR, suggest ing that a unif ied
approach to an electro-gravitational gauge model can be obtained if we
are willing to replace GR by Yang's theory. We were able also to write
a n leads to the f ield equations of each
down a gauge-l ike ~ a ~ r a n ~ iwhich
algebra sector separately. Further developments to be considered would
be to introduce sources in eq. (3.2) and also to anal ise the possibi 1 i ty
of a quantízatíon procedure.
The author i s very grateful to the referee of this paper for
many helpful suggestions.
REFERENCES
1. R.Aldrovandi and E.Stédi le, Intern.Journ.of Theor.Phys. v01
.
23 (4)
(1984).
2. E. l nonu, i n Group TheoreticaZ Concepts and Methods in Elementary
ParticZe Physics, F.Gursey, ed. p.391, Gordon and Breach, New York,
(1964).
3. S.Kobayashi and K.Nomizu, Foundations of Diff erential Geometry,
vols. I , I I. Interscience, New York (1969).
4. B. G .Wybourne, CZassicaZ Groups for Physicists, J .Wi 1 ey (1 964).
5. ~.~tédile,
Rev.Bras.Fis. 13(1) (1983).
6. B.S.de Witt, D y m i c a Z Theory of Groups and FieZds, Blackie & Sons
(1965).
7. C. von Westenholtz, DifferentiaZ Forms in MathernaticaZ Physics,North
-H01 l and (1 978)
8. C.W.Misner, K.S.Thorne and J.A.Whealer, Gravitation, W. H. Freeman
and Co., San Francisco (1973).
9. C.N.Yang, Phys.Rev.Lett. 33, 445 (1974).
10. D.A.Popov and L. I .Daikhin, Sov.Phys.Dokl. 20, 12 (1975).
.
Revista Brasileira de F lsica, Vol. 17, no 2, 1987
p r o p õ e - s e um formal ismo de gauge para a g r a v i tação, acoplada
sobre o espato-tempo de Minkowski e com o grupo de gauge SO(3,l) a U(1). As equaçoes
de Maxwel l e a equação g r a v i t a c i o n a l de Yang surgem de maneira n a t u r a l .
O presente modelo apresenta uma Lagrangeana de gauge t í p i c a , que conduz
às equações de campo correspondentes de Maxwell e de Yang.
à t e o r i a de Maxwel 1, desenvolvi da num espaço f ibrado