IL NUOV0 CIMENTO VOL. XXXVI, N. 2 16 Marzo 1965

57~.~ and B r o k e n S U~ S y m m e t r y .

n . DELBOLTRGO, ABDUS SALAlk[ and J. STRATH]:)EE

International Atomic Energy Agency International Centre ]or Theoretical Physics - Trieste

(ricevuto il 18 Gennaio 1965)

1. - Starting with a spin-�89 quark model the most general algebraic structure is the U(12) ring of matrices y ~ T i (li:. We wish to point out that this ~(12) structure can be used to give a direct covariant formulation of the S U 6 symmetry of Gt3]~s~u I~ADICATI and SAKITA (~), provided that for the physie]lly re~lised multiplets one writes not only the composite field operators but also their conjugate-momentum operator as , independent ~) components within the same multiplet. The mo~i- ra t ion of our remark is as tollows: a number of authors (3) have recently suggested that the S U 6 symmetry of ref. (*) may be looked upon as a noncovariant approxi- mation to a symmetry W(6) = U~(6) • Uz(6) which itself is a straightforward genera- lization of the UL(2)• UR(2) symmetry associated with the homogeneous Lorentz group (4). Starting with this, a number of examples of interaction Lagrangians invariant for W(6) h~ve been writ ten down.

Now there are serious difficulties in elaboration el these ideas. First, the right and left split of the basic quark implies that mq= 0 and therefore W(6) must be b~dly broken. Second, physical particles correspond to representations of the inhomo- geneous Lorentz group, and since kinetic-energy terms are not invariant for U~(6)• (in contrast to the Lorentz UL(2)• Us(2) case) it has so far been possible to develop theories of physical states at zero momenta (5) only. A third difficulty is related to the second; so long as there is no analogue of the inhomogencous- Lorentz-grou 2 structure, it is impossible to assign physical particles unambiguously

(1) The nine T i are t he 3 • g e r m i t i a n genera tors of U(3) while the s ix teen Dirac y's genera te U(4).

(~) F. GURSEY a nd L. A. RADIOATI: P h y s . Rev . Lef t . 13, 173 (1964); ]3. SAKIT~k: P h y s . Rev. (to be published).

(a) A. SAI,A~: P h y s . Left . , (to be published); 1R. DELBOURGO, A. ShL~M an d J. STRATtIDEE: s ubm i t t ed to Phys . Rev.; R. P. FEYN~lV~AI~ r, hi. GELL-~CIANI~ ~ an d G. ZWEIG: Phys . Rev . Left , . 13, 678 (196~); K. BAEDAKCI, 5. M" CORNWALL, :P. G. O. FRI~UND a n d B. ~V. LEE: PhYs . Rev. Lett . , 13, 698 (196~); T. FULTON &tld J . ~r p repr in t (Vienna)

(4) Adopt ing ~Veyl's (, u n i t a r y t r ick ,,, one considers a pseudo-Eucl id ian met r ic for space- t ime. For a Dirac par t ic le the homogeneous Loren tz group is t hen g e n e r a t e d by UL(2) • U~(2) ~(l:t=iys)a #v.

(~) }L BEG and A PAts: s u b m t t e d to P h y s . Rev . We are g ra te fu l to the au thors fdr s end ing us a prepr in t .

690 R. D~L~OU~GO~ A~D~8 SALA~ and J. ST~ATHDEE

to t h e mnl t ip le ts of W(6); thus ba ryon ec te t and deeimet can belong equal ly to (56, 1 )+(1 , 56) or to (6, 21)+(21 , 6).

F o r t he 4-component Dirac equat ion , which includes t he mass t e rm, t he pas- sage to the inhomogeneous group is m a d e in t he wel l -known fashion by ex tend ing the subalgobras Uz(2) x Ua(2) (with six generators a~) to t he full Dirao algebra U(4). This takes place essent ial ly because U(4) contains in aeldition to t he a ~v, the four (transla~iomlike) ma*rioos y~'s wi th commuta t ion rules

(1)

al lowing one to wri te equat ions invar ian* for t he inhomogenoous group

( y~p t * - m) ~p = 0 , wt*p a ~p = 0 ,

where

i (2) w" = ~ 7 ~ [7", ~ P ~ ] .

No te in passing t h a t iyt 'ySw ~ = ~ 7 ~ p ~, so t h a t t he first equa t ion m a y be wr i t t en in t he form

(3) [ y ' ( ~ ' q- iy~ w t~) -b �89 : 0 .

W h a t we wish to emphasize is t h a t the re is a dose analogy be tween the g roup complet ion U~(2) X U~(2) r ~(4) and Ur(6) • U~(6) ~-~ U(12). The g e n e r a t o r s for U~(6) • U~(6) are t he 72 matr ices a ~ T / , T ~, 7 ~ ~. In addi t ion to *hose, U(12) contains ano ther set of 72 matr ices

1 ~ : ya T i , ff_~5 : i y~ 5 Tt , A = (#i) , i = 0, ..., 8 , # = 0, ..., 3

wi th t he typ ica l commuta t ion rules (similar to (1))

�9 i j~ al~ v 2v t~ T ~ ]~Jke2,uvoyoy5 [ y X T i , ~ v T ~ ] = * d (g Y - - O Y ) + , [YaY 5 T ~ ' T s ] = i f j ~ y ~ y S T ~

Defining ~ 72-component vec to r (pa, W a) (~) once again one m a y wr i te ~n ~< (inhomo- genoous) W(6)~ iuva r i an t equa t ion

(4) [F~(P ~ + i T ~ W a) + M ] ~ = 0 .

Note t h a t 8 [ P a P A - - W ' ~ W a ] = 0 and Mso 8 [ p z p ~ - - w ~ w ~ ] = 0 where ~oU=P a~ w ~ = W ~~ I% is wor th stressing r t h a t t he Lagrang ian mass t e r m remains invar ian*

as well (~).

(6) A 36-componente vector of this type was first considered by FULTON and WESS (ref. (3)). For the case we are considering, i t is necessary however that *he'vector have 72 components

(v) In fact the mass term is invariant also for a full ~(12) transformation

8 W = i( et + eJ5 + e#t yt t +ieJ#5 yl~ y5 + �89 e~tw c~l~v) ~.

if all the s are reaa.

U(12) ~ND B~OKEN S U s SYMMETnY 691

I t is p e r f e c t l y p o s s i b l e n o w to w r i t e a cova r i an~ U~(6) • ~ ( 6 ) S - m a t r i x f o r m a l i s m , u s i n g t h e U(12) a l g e b r a in eomp le r a n a l o g y w i t h a L o r e n t z e o v a r i a n t f o r m a l i s m f o r spin- �89 p a r t i c l e s w h i c h u t i l i ze s t h e U(4) a l g e b r a . T h e ch i e f p r o b l e m is t h e p a s s a g e r t h e p h y s i c a l l i m i t o f s u c h S - m a t r i x e l e m e n t s , %he p h y s i c a l l i m i t b e i n g d e f i n e d (s) as P a -~ p~', a l l o t h e r c o m p o n e n t s of P a n d W v a n i s h i n g . T h e l a s t s t e p wi l l n ~ t a r a l l y b r e a k t h e U J 6 ) • Ua(6) s y m m e t r y in a w e l l - d e f i n e d a n d d e t e r m i n a t e m a n n e r l e a v i n g a f o r m a l i s m w h i c h is f l f l ly L o r e n t z cov~rian%. T h e s y m m e t r y b r e a k i n g is we l l d e f i n e d in t h e s ense t h a t w e k n o w p r e c i s e l y t h e t r a n s f o r m a t i o n p r o p e r t i e s of t h e b r o k e n v e c t o r (p~, 0).

2. - C o n s i d e r n o w t h e p r o b l e m of h i g h e r r e p r e s e n t a t i o n s . S t a r t i n g i t e m u s i n g l e D i r a e f ie ld y~ a n d a 4 - c o m p o n e n t s p i n o r o n e g e n e r a t e s succes s ive ly h i g h e r m u l t i p l e t s a n d t h e i r a l g e b r a s b y $~king o u t e r p r o d u c t s (~)

?~ l X l X ... X y ~ X l X . . . x 1 .

T h e f i r s t c o n c r e t e e x a m p l e of t h i s is t h e 4 • r e p r e s e n t a t i o n of DUFFIN a n d K n ~ M E ~ (10) w i t h t h e a s s o c i a t e d a lgeb r~ fl~ ~ �89 (7~ • l -~ 1 • 7~'). Th i s g ives r ice i n t h e w e l l - k n o w n m a n n e r $o p a r t i c l e s of s p i n o n e (10 c o m p o n e n t s ) a n d s p i n z e r o (5 c o m p o n e n t s ) b o t h of t h e s ~ m e p a r i t y w i t h i n one m u l t i p l e t . T h e c ruc i a l po in~ is n o t t h a t t h i s is o b v i o u s l y t h e (~ n a t u r a l ,) f o r m a l i s m fo r e x t e n s i o n of U(6) i d e a s i n the% i t c o m b i n e s ze ro s p i n a n d s p i n o n e ; i t is m o r e , for b y i m p o s i n g t h e r e q u i r e m e n t t h a t t h e f ie ld q u a n t i t y sa t i s f ies a l i n e a r e q u a t i o n , K ~ n c o u l d s h o w t h a t t h e s p i n - o n e f ield is c o m p o s e d of t h e p o t e n t i a l A ~ a s w e l l a s t h e f ie ld t e n s o r (11.~2) F :~ . L i k e w i s e t h e s p i n - z e r o p a r t cons i s t s o f 9 as we l l as i t s c o n j u g a t e m o m e n t u m ~ 9 . A l t o g e t h e r t h e s p i n d e c o m p o s i t i o n is 1 6 ~ 1 0 O 5 O 1. (The ~ 1 ~ does n o t cor - r e s p o n d %o a n y d y n a m i c a l s i t u a t i o n a n d is c a l l ed t h e t r i v i a l r e p r e s e n t a t i o n of t h e a lgeb ra . )

(~) We realize tha t the precise meaning of the Hiibert-space operators ~A and yyA (with eigen- values pA and l~ re) is, to Say the least, obscure. I r i s interesting none the less to observe tha t the W ~ occurring in eq. (4) could be identified with the generalized Pauli-Lubanski-t~arg]nann spin operators if one relates (in complete analogy with the ease of the normal Dirac eq. (3)) W A and pA by the implicit equation

(5) I, FA(P) = const • s [ i ,4 F ~ ( p B + i t5 W$)].

For the physical limit p A __>p~, eq. (5) solves $o give

W~i(p) oc rs[v~ T ~, r~ p ~] .

Clearly, p f f w # i ( p ) ~ 0 for all i, and when i = 0, W~~ is just the normal spin pseudoveetor. More generally the 36 operators w # i ( p ) and T i generate the little group ~6 (or Uw(6) of ref. U)). We have thus demonstrated that the sohltions of the free equation [ I ~ ( P A § i r 5 WA)§ ~ = 0, in what we have called the physical limit PA -+pff, can be labelled by the eigenvalues of p~, wPi(~) and T i. Note that (p~m p~0 _w#o W#0) = constant now implies with this interpretation of W pc, the relation pZp~ = const + J(i + 1).

(~ F. J. ]~ELINFANTE, I~I. A. I~RAI~IERS and J. ~ . LunAr-SKI: Physicao 8, 597 (1941). (L,) R~ J. DUFFIN: Phys . Rev . , 54, 1114 (1938); N. !~EI~II~IER: Prec. Roy. ~'oe., A:[73, 91 (1939), (zz) For a Lie-grouP gauge theory F/~ would correspond to the ~generalized ,~ conjugate

momentum. For a u type of theory for example F ~ = 0 ,a A ~ ' - 8 ~ A ~ + 2,4~ • A ~. (~) N. KEI~I)IER: Hev. Phys . .Acre , 23, 829 (1960), has stressed that both types of quantities A/~

as well as F #v are completely on par SO far as dynamics is concerned and neither of the quantities is in any sense more (r fundamental ,, than the other.

6 9 2 ~ . DELBOURGO, ABDUS SALA~ a n d J . STRATHDEE

The next algebra is generated by the matrices (lS)

~ : y~X 1 -4- lXf l s ,

~he reducible representations describing particles of spins �89 and ~. In a future paper ~his decomposition will be exhibited in detail; like for the case of the fl-~lgebra, both the field operators ~s well as their conjugate momenta occur together in the description of a physical system.

The extension of the above to include uni ta ry spin (passage from U(4)to U(12)) presents no essential complications though the iormalism gets tedious as is well known from pas~ experience o~ caleulatlons involving for example fl-formalism for mesons. Bu~ the compensations are two-fold; first the ambiguities of UL(6)• Un(6) assignments for the same physical multiplet are avoided (~4) ; the formalism incor- porates them all in a specific manner. Second, using the methods of Sect. i , a broken but covariant S U 6 formalism can readily be eonstrueted. In practice, since one is hardly ever going to work out the dynamics of particles of spins > ], we hope one can set up the necessary formal machinery once and for a l l This will be treated elsewhere.

Our thanks are due to Dr. M. A. I~ASHID for numerous helpful suggestions.

(is) A deta i led worked-out example of a s l ight ly modified ve r s ion of the above is the a lgebra gene ra t ed by the ma t r i ces a~ = ?I~ xT~ + 7 a x fl g See HARISH-CItANDRA: Proc. R o y . *%~oc., 192, 195 (1947). This theory describes also spin-(~ a n d ~) part icles. The descr ipt ion of the spin-�89 par t ic le is g iven essent ial ly by the de r iva t ive of a spinor ~#W r a t h e r t h a n b y a f u n d a m e n t a l spinor y'.

(14) Note 12 x 12 ~ 143(~1 = 9 x l 0 + 8 x 5 + 5 (ba r r ing the 9 t r iv ia l components) . Also, 12 • 12 x 12 = 220(~2 (572)(~364, where the U ~ ( 6 ) x U~(6) con ten t of the mul t ip le t is as follows:

220 = (20, 1) + (15,6) + (6, 15) + (1, 20) ,

572 = (70, 1) + (21, 6) + (15, 6) + (6, 15) + (6, 21) + (1, 70),

364 = (50, 1) + (21, 6) + (6, 21) + (1, 56) .

I n t e rm s of U(3)x U(4) decomposi t ion

364 ~ 10 • 2 0 + 8 • 2 0 + 1 • 4 .