VOLUME 14, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 12 APRIL 1965

*Work supported by the U. S. Atomic Energy C o m m i s ­sion.

tAccepted without review under policy announced in Edi tor ia l of 20 July 1964 [Phys. Rev. L e t t e r s 13, 79 (1964)].

JDur ing the pe r fo rmance of th is exper iment , t hese authors held guest appointments at Brookhaven Nation­al Labora tory .

§Pa r t of th is work is the subject of a thes i s to be s u b ­mit ted by J . T. Reed in pa r t i a l fulfillment for the r e ­qu i rements of the degree of Doctor of Philosophy at The Universi ty of Roches te r .

1A. C. Mel iss inos , T. Yamanouchi, G. G. Fazio , S. J . Lindenbaum, and L. C. L. Yuan, Phys . Rev. 128, 2373 (1962).

2 E. J . Sachar id is , S. Ozaki, J . T. Reed, J . J . R u s ­se l l , S. J0 Lindenbaum, and L. C. L. Yuan, to be pub ­l ished.

3R. I. Louttit , T. W. M o r r i s , D. C. Rahm, R. R. Rau, A. M. Thorndike, and W. J . Wil l is , Phys . Rev. 123, 1465 (1961).

4M. M. Block, P h y s . Rev. 101, 796 (1956). 5If t hese two uncer ta in t ies a r e added in quadra tu re

we have Q = +4±6 MeV. 6See, for example , R. G. Ammar , R. Levi Setti,

W. E. Slater , S„ Limentani , P . E. Schlein, and P . H.

Steinberg, Nuovo Cimento lj>, 181(1960) . Also R. H. Dali tz, in Proceed ings of the Ninth Annual In ternat ion­al Conference on High-Energy Phys i c s , Kiev, 1959 (Academy of Sciences, U.S.S.R., Moscow, 1960), pB 517.

7 J . J . de Swart and C. Dullemond, Ann. Phys . (N.Y.) 19, 458 (1962).

See, for example , K. M. Watson and R. N. Stuart , Phys . Rev. 82, 738 (1951). We multiplied the t h r e e -body phase space by

47T

a ~fe 2 + l l / a - i ro fe 2 ] 2

with a r b i t r a r y normal iza t ion; h e r e k is the wave num­ber cor responding to the momentum of the A in the A-p s y s t e m which is uniquely de te rmined by the m o ­mentum of the K+ meson.

9 B. Sechi -Zorn , R. A. Burns te in , T. B. Day, B. Ke-hoe, and G. A. Snow, P h y s . Rev. L e t t e r s 13, 282 (1964).

10G. Alexander, U. Karshon, A. Shapira, G. Yekutieli , R. Engelman, H. Fil thuth, A. F r i d m a n , and A. Minguz-z i -Ranz i , P h y s . Rev. Le t t e r s JL3, 484 (1964).

" H . O. Conn, K. H. Bhatt , and W. M. Bugg, Phys . Rev. Le t t e r s 13, 668 (1964).

1 2 P. A. P i r o u e , P h y s , Le t t e r s 11 , 164 (1964).

INTERNAL SYMMETRY AND RELATIVISTIC WAVE FUNCTIONS*

C. L. Critchfield

Univers i ty of California, Los Alamos Scientific Labora tory , Los Alamos , New Mexico (Received 23 Feb rua ry 1965)

It is known that states of single particles hav­ing spin may be represented by symmetric poly­nomials of Dirac wave functions.1 For spins •|, 1, | , 2 , • • •, the number of components is 4, 10, 20, 35, • • •, respectively. When the spin is zero, a five-component wave function is r e -required in order to obtain first-order, cou­pled differential equations.

The generators for the spinor-index t rans­formations of the homogeneous Lorentz group form an antisymmetrical tensor (six-vector) and may be expressed as a commutator of the four-vector components y^ of Dirac's theory,

mn -*K>*n]- a)

We also have

where gn^ is the metric tensor of special rela­tivity. Equations (1) and (2) plus commutators between Smn's form the basis of an algebra

of order 10 and rank 2 in which the elements oz and y4 may be diagonal in their matrix form. The algebra is therefore C2 (or B2).

2 The or­thogonal group of five dimensions, of which B2 is the algebra, has been investigated exten­sively in its relationship to relativistic wave functions by Bhabha.3

We may then expect that supermultiplets, in representations of SU(4) for nuclear states4

and SU(6) for particle states,5 have similar relationships to larger groups which include Lorentz transforms. For wave functions in nuclei one would require a rank of four since spin is generalized to three quantum numbers which may be designated as spin, isospin, and seniority. The most likely algebras are then B4 and C4. Irreducible representations may be expressed as collections of sets of integers with either sign, four to a set. Each distinct set is a weight, the multiplicity of which can be calculated from Freudenthal's recursion formula.6 In such a representation of C4 the

607

VOLUME 14, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 12 APRIL 1965

sum of the four numbers i s e i ther even or odd for al l s e t s , whereas for B4 both occur . Rep­resen ta t ions of the exceptional a lgebra F4 in­volve ha l f - in tegers as well a s even and odd s u m s . In application to nucleons, the re fore , C4 i s indicated. Similar express ions for the r ep resen ta t ions of A39 which is the a lgebra of SU(4), a r e most conveniently expressed in t e r m s of four number s , a l so . They contain only posit ive in tegers , and their sum is a con­stant for a given i r reduc ib le r ep resen ta t ion . For a common highest weight, {N1,N2,N3,N4}, all weights of A3 will occur in C4 , and it r e ­mains only to consider the mul t ip l ic i t ies .

Let D(X) be the dimension of a pa r t i cu la r r ep resen ta t ion of the a lgebra X. Each r e p r e ­sentation of SU(4) may be expressed a s a sum of t e r m s of specified dimensions in spin and isospin space: (D(Al),D(Al)) = (2T + l,2S + l). In going over to re la t iv i s t i c waves we have the cor respondence between the dimensions of the ord inary spin pa r t and that in C2 which is shown in the following a r r a y . If

I>(Ai) = l , 2 , 3 , 4 , 5 , - - - , (3a)

then

D(C2) = 5 , 4 , 1 0 , 2 0 , 3 5 , • • • . (3b)

Equations (3) provide the bas i s for c o m p a r i ­son of a given expansion for SU(4) with that for the s a m e highest weight in Sp(8) which will be in t e r m s of (D(A1),D(C2)). As is to be ex ­pected, the multipl ici ty of these la t te r t e r m s may be g r e a t e r than nece s sa ry to include those of SU(4), and a lso r ep resen ta t ions of C2 occur which do not appear in E q s . (3), e.g., of d imen­sions 1, 14, 16, another 35, and so on. Ca l ­culat ions of a l l r ep resen ta t ions having a sum of iV's equal to four and l e s s show that those of A3 a r e indeed contained in those of C4 in the sense of the cor respondence in E q s . (3).

In a given subs ta te {nxn2n3n4) of an i r reduc ib le r ep resen ta t ion of C4 , the sum nx + n2+n3+n4

gives the number of la rge components minus the number of smal l components p resen t , and l(n1 + n2-n3-n4) i s the z component of ord inary spin. All these e igens ta tes of y4 and spin a r e operated upon in the genera l ized Di rac equa­tion for free pa r t i c l e s

k YkP ^ = mip (4)

a s in r e fe rence 1. However, ip and y^ h e r e belong to a represen ta t ion of C4 in which, in­

cidentally, the m a s s values have not been spl i t . Interact ion t e r m s would be subject to Lorentz invar iance as a minimum requ i r emen t .

We may now genera l ize the p rocedure sug­gested for SU(4) to the i n t e r n a l - s y m m e t r y group of baryons and mesons , including spin, v iz . SU(6). Instead of nucleons we have "quarks , " 7

and we look for the decomposit ion of the r e p r e ­sentat ion of C6 with highest weight { 3 0 0 0 0 0 } in t e r m s of the d imensions in SU(3) and Sp(4), (D(A2),D(C2)):

{3 0 0 0 0 0} = (10, 20 )+ (8 ,16 )+ (8 , 4)+ (1 ,4) . (5)

This is of 364 dimensions and is exactly the r e su l t of Roman and Aghass i , 8 although they do not cons ider the genera l symplect ic group of dimension 12 (of which C6 is the a lgebra) .

In Eq. (5) the t e r m (10, 20) co r r e sponds to the | decuplet and e i ther (8,16) or (8, 4) to the \ octet of ba ryons . According to the identifica­tion in E q s . (3), (8, 4) s tands for a s imple mu l -tiplet of spin \y whereas (8,16) has al l the sma l l components that a r e r equ i red for t h ree Di rac pa r t i c l e s to have a total spin one -half.

The many-pa r t i c l e aspec t of the theory is of pa r t i cu la r impor tance to r ep resen ta t ions for the meson because the ant iquark is now a "hole ." The s imples t quark-an t iquark s igna­tu re in C6 is {2 1 1 1 1 0}, which has 6006 d imen­s ions . A s ix -pa r t i c l e s inglet in C2 is of 30 d imens ions , and the t r ip le t has 81, and

{ 2 1 1 1 1 0 }

= (9, 81) +(8, 30) +(15, 35) +(64, 35')

+ (48,14) +(110,10) +(90, 5) + (50,1) . (6)

The f i r s t two t e r m s in Eq. (6) a r e to be in t e r ­pre ted a s the (pseudo)vector nonet and sca l a r octet of the m e s o n s .

Thus the noncompact symplect ic group of rank six, Sp(12), p e r m i t s the s a m e quantum numbers a s SU(6) plus that of y4 which, when cons idered to be an infini tesimal opera to r , is a form of pa r t i c l e gauge opera to r s ince the sign of i ts eigenvalues is that of the energy in hole theory .

•Work performed under the auspices of the U. S. Atomic Energy Commission.

!V. Bargmann and E. P. Wigner, Proc. Natl. Acad. Sci. U. S. 34, 211 (1948).

2E. B. Dynkin, American Mathematical Society Translations (American Mathematical Society, Provi-

608

VOLUME 14, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 12 APRIL 1965

dence, Rhode Island, 1957), Ser. H, Vol. 6, p. 319. 3H. J. Bhabha, Rev. Mod. Phys. 1/7, 200 (1945). 4E. P. Wigner, Phys. Rev. 51, 106 (1937). 5F. Glirsey and L. Radicati, Phys. Rev. Letters JJ3,

173 (1964); A. Pais, Phys. Rev. Letters 13., 175 (1964).

6M. Konuma, K. Shima, and M. Wada, Progr. Theo-ret. Phys. (Kyoto), Suppl. 28., 44 (1963).

7M. Gell-Mann, Phys. Letters 8., 214 (1964). 8P. Roman and J. J. Aghassi, Phys. Letters 14, 68

(1965).

U(12) PREDICTIONS FOR pp ANNIHILATION INTO MESONS

R. Delbourgo, Y. C. Leung, M. A. Rashid, and J. Strathdee

International Centre for Theoretical Physics, Trieste, Italy (Received 15 March 1965)

The noncompact symmetry group U(12) which, with appropriate prescriptions,1 can be regard­ed as a relativistic generalization of SU(6), preserves the successful aspects of that group. At the same time, through its incorporation of momentum effects, it is applicable to a wider class of processes. Already the U(12)-invari-ant three-point functions have been studied ex­tensively,1 '2 and attention is shifting to the four-point functions. Blankenbecler, Goldberger, Johnson, and Treiman,3 in their treatment of baryon-meson elastic scattering have shown that U(12), in addition to preserving those con­sequences of SU(6) which could be drawn for the case of forward scattering,4 both good and bad, leads to a number of new relationships and, in particular, to predictions about parti­cle polarizations which seem to be at variance with experiment. A preliminary examination of the associated nucleon-antinucleon annihila­tion problem has been carried out by Harari and Lipkin,5 again leading to predictions that are not fulfilled, in this case the relative mag­nitudes for the production of TI^H", K+K~, and K°K°.

Here we shall consider the annihilation prob­

lem. The first point to notice is that the ampli­tude for pp annihilation into two mesons,6

B A B C ( - r^ADE^f*!**^*

vanishes at threshold p'=p. This is a direct consequence of the Bargmann-Wigner equations which are necessary for the physical interpre­tation of U(12) states.1 It is , no doubt, here that our prescriptions7 differ from those of Harari and Lipkin. Having seen that the two-meson decays8 are suppressed by U(12), we can proceed to the consideration of three-me­son decays for which the amplitude survives at threshold. It is given by

E/"C. (-P)BDEF(p)MAD(kl)MB

E(k2)McF{k3).

Remarkably, the nucleon-antinucleon part of this structure contains only the nonstrange me­sons 77, 77, p, and a>. The amplitudes AJJ, where J and / denote, respectively, the spin and iso-spin of the nucleon-antinucleon system, are easily extracted. They are

^ 0 - ^ p e r m ^ 1 ^ ^ ^

A 01 ^ p e r m

-f€ Q € . , f w ( l )p . . (2)p ,(3)}ryAT, Q?j3y jkl a fy yk J I ' 0 <Zj3y jkl a" '^Pf yAT 'J I '

11 perm *{Wl)r,(2)-yi)y2^

-'w^V--* i(2)^.(3)-Pa.(l)p0.(2)p^(3)-paZ(l)p^(2)p /3.(3)]}aarziV, (1)

609