P I I Y S I C 4 L R E V I E W D V O L U M E 1 8 , N U M B E R 8 1 5 O C T O B E R 1 9 7 8

General internal gauge symmetry

Kevin Cahill

Physics Departnzeizt, frldiana L'niverritq', Bloon~ington, hdiana 4 7401 * and Deparrmenrs o f Mathernutics und o f Physics and Astrorromy, Louisiana State L1niversit?j, Baton Rouge, Louisiana 70803

(Received 14 March 1978)

A theory isoutlined in which ti scalar field interact in a way that is invariant under all real, nonsingular local linear transfonnations of the mesons among themselves. The energy of the system is positive. The symmetry spontaneously breaks down to a compact subgroup of G L ( n , R ) and the gauge mesons of the hroken symmetry become massive. Their longitudinal components are supplied by the derivatives of an ~nternai metric tensor with which no physical particles are associated.

This paper i s about a theory that possesses max- = @(x),,, - AU(x )$ (x ) , (5) inial internal symmetry . I t i s a gauge theory of

where the subscript comma mu means a/a.ru and the n rea l fields @, whose Lagrangian i s invariant un- nxrz matr ix of Yang-Mills f i e lds A u ( x ) t r ans forms

d e r the local action a s

. - of the general l inear group of a l l r e a l , nonsingular This trallsformation for the gauge Au rzxn m a t r i c e s , G L ( n , R ) . Such a theory is m o r e e n s u r e s that the covariant der ivat ive cb;u trans- symmetr ica l and l e s s a r b i t r a r y than one whose in- f o r m s like +, ternal-symmetry group i s a compact subgroup of GL(n , R ) , a s i s usually t+(x);,]' = a(x)@(x),.. (7)

In what follows a suitable Lagrangian will be pro- posed and the equations of motion and conserved The curvature tensor F u Y ( x ) is

cur ren ts that follow f r o m i t will be derived. It Fuv(x) =Au(x ) ,Y - AV(xisu f [Au(x),A,(x)l (8)

will be shown that the Hamiltonian i s non-negative. The theory exhibits an interesting kind of Higgs

mechanism. The vacuum cannot be symmetr ic and the symmetry group GL(rz, R ) breaks down spon- taneously to a subgroup that i s s i m i l a r to SO(n). The gauge mesons associated with the noncompact p a r t of GL(IL ,RJ become mass ive . A symmetr ic internal m e t r i c t ensor supplies the needed longi- tudinal components.

In o r d e r to make objects that a r e invariant under the general l inear transformation ( I ) , i t i s nec- e s s a r y to introduce a m e t r i c t ensor g,,(x) that is symmetr ic and positive and that t rans forms as

and t rans forms as

F:,(x) = a(x)Pu,(x)a- '(x) . (9)

A suitable covariant derivative of the met r ic ten- s o r g is

g.(x),. =g(x ) ,& + g ( x ) A U ( x ) +A:(x)g(x) , (10)

which implies that g;u t r ans forms like g ,

[ g ( x ) , u ] '=a- 'T(x)g(x); ,a- ' (x) . (11)

Similar ly the covariant der ivat ives of Q;", F m , and g i V a r e

In mat r ix notation Eqs. ( I ) and (2) become' F z =Fyu t [ F m , A u ] , (13)

and The Lagrange density

where the T means t ranspose. Evidently the fo rm + (2f)-' t r ( g ; u g - l g ; u g - ' ) + 8+rug9;e qbT(x)g(x)@(r) i s invariant. The tensor g i j plays a somewhat s i m i l a r role to that of the met r ic ten- - i v ( @ T g $)

s o r in general relativity and contributes to the field is invariant under the gauge t ransformation ( 1 ) . equations t e r m s not p resen t when the gauge group The numbers e and f a r e independent coupling is compact. constants. The variational equations of i t s inte-

A suitable covariant derivative f o r Q i s g r a l over space-t ime a r e

18 - 3930 O 1978 The American Physical Society

18 - G E N E R A L I X T E R X A L G A U G E S Y M M E T R Y

+ t ~ + g ‘ l g ; , , @ ; ~ + r fi V r $ = O , (16)

F$ = [ F u " l g - l g ; y ] +e2@;uOTg- (e2/f2)g-'g;",

; U =g;,,g-lg;u + (f 2 e Z ) g [ g - l F ~ , g , FuY I (17) 4 ; u

+fZg4;u+T,,g-f ZgbV'$Tg, (18)

where V' 1s the derlvatlve of V The p=0 com- ponent of the equatlon f o r FuY isa constraint, which may be called Gauss ' s law.

In the usual way, the ant lsymmetry of F N Y lm- pl ies that the mat r ix of n2 c u r r e n t s

i s conserved, Jru = 0 . By using Gauss ' s law, one may wri te the com-

ponent T"O of the canonical s t ress -energy tensor a s the sum of

and the total divergence

D =e-2 t r ( g " ~ ~ ~ g A ~ ) * ~

Thus a p a r t f rom sur face t e r m s , the Hamil tmian may be taken a s the integral of the density H. Now H is non-negative a s long a s g i s symmetr ic and of non-negative s ignature. h he la t t e r con- s t ra in t may be enforced by the device of writing g = h T h . The field equations f o r h , which may be chosen to be symmetr ic , a r e those that resu l t f rom the substitution of hTh f o r g in (18).] The energy i s therefore non-negative.

The met r ic t ensor gi, part ic ipates in an inter- esting variation of the Higgs m e ~ h a n i s m . ~ If the potential V a s s u m e s i t s minimum value a t $ J ~ ~ $ I

= O , then in the lowest approximation the (super-) vacuum expectation values of the fields 4 and A,, vanish, while that of the met r ic g i s a positive symmetr ic mat r ix go. By expanding the fields

about these vacuum values with

one may identify the quadratic par t of the Lagran- gian (15) a s

where E,,, i s the c u r l U7u,,- U',,u of the symmetr ic combination

while G,,, is the c u r l C,,,, - C,,,, of the antisym- met r ic combination

The m a s s spec t rum of physical par t i c les is c l e a r f rom the s t ruc ture of L,. There a r e )z(tz+1),:2 vec- to r mesons Ur,, of m a s s .V f ) . The longitud- inal components of W , a r e contributed by the n(n + l ) :2 components of the symmetr ic met r ic tensor g-. There a r e n(n - 1 ) / 2 m a s s l e s s vector mes- ons CLL . There a r e n s c a l a r mesons of m a s s p= [ ~ ' ( 0 ) ] ' / ~ . There a r e no physical par t i c les corresponding to the met r ic g .

If the potential V a s s u m e s i t s minimum value a t $ 1 ~ ~ 6 >O, then the usual Higgs mechanism comes into play a s well. The part ic le spec t rum becomes &l(n + 3) - 1 mass ive vector mesons , &1(12 - 3) + 1 m a s s l e s s vector mesons , and 1 mass ive s c a l a r meson. Thus f o r n = 2 there a r e no n lass less par- t i c les , while f o r n = 3 only one gauge meson is m a s s l e s s .

It i s perhaps worth emphasizing that the gauge mesons Wp a r e mass ive even in the absence of the s c a l a r mesons $. They generate the noncompact par t of GL(n ,R) , while the gauge mesons C, g.:n- e r a t e the compact subgroup SO(n).

*Present address. and B. Zumino, ibid. fi, 81 (1975); and T. T. \ITu and ' ~ o r n ~ l e x groups such a s =(n) here a r e thought of a s C. N. Yang, Phys. Rev. D 2, 3233 (1976).

subgroups of GL(272, R) . 3A somewhat s imi lar mechanism has been observed by %oncompact internal-symmetry groups have been con- R. Arnowitt and P. Nath, Phys. Rev. Lett. s, 1526

sidered by various authors, e.g., P. Nath and R. Arno- (1976). witt, Phys. Lett.=, 177 (1975); R. Arnowitt, P. Nath,