A relativistic extension of SU(6)-symmetry

.\NNSLS

OF

PHYSICS:

37, 303-314 (1966)

A Relativistic

Extension W.

C’ourunf

Inxfitufe

of Mathematical

of SU(6)-Symmetry*i’

ZIMMERIVIANR

Sciences,

Kew

Pork

Cniversily,

Xew

York,

~\‘ew

l’r~rk

A 11ew formulation of relativistic SU(6)-symmetry is proposed which al,plies t I) the full Lagrangian (including kinetic energy terms) and is consistent with the principles of quantum field theory. This symmetry requires the classic ficat ion of particles according to Szi (6). multiplets and yields the characteristic relations among bare coupling constants which are known from the phenomenological theories, While the bare masses within an SC(G)-sul)rrnl~~ltil)lel must be equal, the physical masses of different SC’@-multiplets are expectrd t,o split for interacting syst,ems. In the formulation proposed the S-matrix is lmitary. However, relativistic S/I(6)-symmetry of the Lagraugian does IIIII imply corresponding symmetry properties of the S-matrix. As examples the free sextet field and the sextet-sextet interactions are tiiscussed. A Lagrange formulation for free spin $$ fields is given and gcncralized 10 an iuvariant description of the 56.component baryou field.

I. INTROD1:(‘T101i

Recently several attempts have been made to cxt,end SLT((i j-syl~lrdl~~ ( I -.i J to n relativistically invariant theory (4-l 5). In these formulations higher synmctry groups are studied which contain SU(6) and t,he Lorentz group as \vell. Beg and Pais (5) and Salam, Delbourgo, and Strathdee (I,$ ) haw proposctl ~.‘henomenologit~alrules for effective IS-nlatrix element,swhich so far sew1 to be in good agreement with experiments. Unfortunately, these rules violate thca unitarity of the S-matrix (5, 16, 17). This difficulty is not present in other formallations (4, 6, 12, 13, 15) which use local Lagrangians recluiring higher svtlwwtr> for the interaction Lagrangian only. In such a theory the S-matrix is unitary, but the fret part of the Lagrangian violates higher symm(+y, an un~;:~tisf:rc+or~ situation. The purpose of this paper is to show that :L conventional low1 fielcl theor? (,intrluding the kinetic energy terms) can well display illvurianc*e under :I groul) which is ;t relativistic generalization of sr.‘(t;). The symmetry group IG’/,((;) which we use was recently proposed by Riihl ( IZ), it is closely rclatr~tl to t,htl inhomogeneous extension of 8L( 6) suggested by Sakit:~ ( J, 18 j :~II(I it tlcliocl * The research reported fi~ound:~tic~r~. t A more detailed version

in this of this

paper paper

Teas sllppc~rtetl is availnblr 303

ilr p:trl 011 wqrlest

by

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304

ZIMMERMANN

independently by Fulton and Wess (8, 19, 20). Let (x,6) and (x”“) denote hermitian 6 X 6 matrices. Then 1GL(6) is defined as the group of all transformations’ b’

XL = C:‘C, X,,Q + Aas

X

‘da

=

-(c’);,(c-‘>b&‘b’

+

A”b

a, b = 1, ... , 6

(l-1)

in the coordinate space (~~6 , 2”“) of 72 real dimensions. C is a nonsingular cornplex 6 X 6 matrix, (Aad) and (Akb) are hermitian 6 X 6 matrices. IGL(6) is an inhomogeneous extension of GL(6), the full linear group in six dimensions. By ISL (6) we denote as usual the subgroup with det C = 1, For the coordinates we use also the notation %I

=

X(,i)(+43:)

,

X

db =

x(d

MA

assigning a pair (CY, i) to every index a a = 1, a.. , 6, a = 1, 2, i = 1, 2, 3.

a ++ (a, i)

It is easy to write down IGL(6)-’ invariant differential equations for field quantities transforming linearly under IGL(6). (Various examples of free field equations have been discussed by Riihl.) In addition, Riihl assumesunitary representations of ISL(6) in a Hilbert spaceof (physical and unphysical) state vectors. It seems, however, that this assumption is responsible for the inconsistencies discussedin refs. 5, 16 and 1Y. In order to avoid these difficulties we start from a conventional quantum field theory in Minkowski space. We will formulate a criterion under which we consider a conventional theory as IGL( 6) -invariant. Let

A’ $I?),

B’ ‘(x”),

*f*

(1.3)

denote the field operators of a local and Lorentz invariant quantum field theory, which is invariant under the unitary spin group of Gell-Mann ($1) and Ne’eman (22). ( ) indicates the various spin and unitary spin indices. We form the expression %9k)kx) + U”d

X(&C)~ak) k=l

(summed over (11,fl = 1, 2) where the a”d’S(J‘,b) are the three Pauli matrices k uk ( -u ) extended by the unit matrix -&y

1 It is more appropriate composition law corresponding

.) = 4(,“49) - fJIca@

to define IGL(6) to (1.1).

by

the

=

k

k = 1,2,3

(1.5) elements

(C/,

Aab , Aid)

with

the

RELATIVISTIC

:m

,S’U(6)-SYM3IETKY

Under the subgroup XL(% j X SU(3) of GL(6 j the expressions ( 1.1) tr:ur&~rm as the components of a Minkowski vector. We extend the field opernt’ors j I .:<1 to functions of 72 variables x,6 , .A? by subst,ituting ( 1.4) for the arguments .Y. Thus we have defined field operators il’

)(zr& ) P)

3 A’ ‘(:l.yr,,i; ) Pi

), . . .

I I.(il

in the coordinate space (r,d , xcd). With t’his definition we st,ate the followitlg crit’crion for IGL(6)-invariance. The theory giwn by the lie/d operators (I ..I) is said to be invariant under IGL(6) if ihe operators (1.6) (suitably ar,rarl.ywl US tht components of GL( 6) -tensors) are solutions of jiekl equations invariant ,undrl, ( 1. I I This invariance requirement imposes restrictive conditions on the Lagrangiat~ of a conventional field theory regarding the free parts and the coupling terms :ts well. In order to find such t,heorics one starts from an ~GL((j)-ill\.:1ri:tllt fic4(l theory in the coordinate space (~~6 , .xtd), r&ricts the manifold of solut ioIl t)y (1.6), and reduces fir&y to SL(2) X SsC’(3). Applied to the free pnrt oi’ tl~~ 1,:~. grangian the criterion requires the classification of particles accaording to SI ‘t,(i) supermultiplets. For the interaction term the c~ontlition of ~Gl,((i)-irlv:lri:Ill(,(. yields the characteristic relat’ions among coupling constants whic~h ar( ktlowtl from the phenomenological t#heories. Since the criterion proposed refers t,o a local quantum field theory of the (‘otiventional type there is no difficulty with the physical interpretation of the thfaory. Norrover, the criterion requires that t,he field equations can be written in :I forn1 whic*h is fully invariant under IGL(6). It should he not,cd, howevclr, that t tl(b I’(‘st.ricat,ion (I .6) clefirles a subset of solutions which is irlv:tri:intuncirr ,V,(‘>) X .s( ‘(:il only.

Unitary representations of ISL(6 j in Hilhert space are of no cotlcertl h(:ro. Indeed, if a GL(G)-tensor is of the form ( 1.C no unitary rcprescatatiotl rS( (I! of ISL( 6 ) es&s such that x’ = U(g

j-lxF( g j

g t t,sI,((;

1 f 1.7 i

(x’ denotes the tensor transformed under g, see (2.1) ). This is not surlxisillg sinrse (1.1) is no transformation of observables in the sense of cluantunl ihclory. The int,roduction of elementary meson fields in this contrxtP leads to $onle peculiar problems. It is easy to find various invariant Lagranginns tlcscribitlg free meson fields involving 3.5 or 70 independent components. 1lost of thcth(> formulations fail, however, when an intjeract~iotl with a sextet or baryon field is intJroduced. These problems will be studied in a forthcoming paper. In this ~):~pcr we restrict ourselves to the discussion of the free sextet, and bnryon fic& :itl(l tllc sextet-sextet inberactions. Some concluding remarks concern the problem of renormalization. I*:sc~pt for trivial PRSCSthe IGL(B)-invariant, interactions :n-~ tronrcnorn~nliz:~~~l~~. l.‘or

306

ZIMMERMANN

some interactions renormalization may be possible by nonperturbative methods (25-25). Steinmann (26) has shown that even in nonrenormalizable theories consistent results can be obtained to any order of perturbation theory. However, infinitely many parameters must be introduced corresponding to an infinite number of counter terms in the Lagrangian. Another problem is the mass renormalization. IGL(B)-invariance implies that that the members of a GL( 6) -multiplet have the same bare mass. This need not be the case for the true masses of the particles defined as discrete eigenvalues’ of 4E. Although the Lagrangian is invariant under the larger group, one expects in general that the masses of different SCi(3)-multiplets split when the interaction is turned on. Indeed, Beg and Pais (5) have shown for a mesonsextet interaction that the pseudo-scalar and vector meson self-energies are different in second order of perturbation theory. Alternat,ively, one can renormalize the theory such that the true masses of a SupermultipIet are equa1 and the renormalized coupling constants have the ratios characteristic for GL( 6) -invariant couplings. This description should come closest to the SU(6)-invariance of the phenomenological theories. But the IGL(6)invariance is then broken by the renormalization terms. In Section II the properties of IGL(6)-tensor fields are discussed. The free sextet field and the sextet-sextet interactions are studied in Section III. A Lagrangian formulation of free spin-g fields is given (Section IV) and generalized to an IGL(G)-invariant description of Baryon fields (Section V). II.

IGL(6)-TENSOR

FIELDS

We introduce field quantities transforming denote an N-component quantity depending

linearly under IGL(6). Let xU(z) on the coordinate vector

x = (zag, xEd), We consider a linear transformation

a,b,c,d=

1;..,6

law

XU’W = Cl suu~(c>xu~(x), qJ= 1, . . . ) N

(2.1)

where x and x’ are related by ( 1.l) . S,,f (C) denotes an N-dimensional representation of GL(6). A few examples are 1. Scalar field x’(z’) = x(x). 2. Sextet field x.(z) with S = GL(6), N = 6, a = 1, .a. , 6. 3. Symmetric tensor field xalaraa (x) of rank 3. In this case S is the representation with Young tableaux (3, 0, 0, 0, 0, 0), N = 56. * As usual, P,, denote the group in Minkowski space.

generators

of the

unitary

representation

of the

translation

RELtlTIVISTIC

4. Covarimt

(a(

307

N(6)-SYMMETRY

vector field xa6(x) with s = GA(G) x GL(6),

6) denotes the complex conjugate 5,. f’ontrnvarimt vector field Xdb(.22 j, s = (f(6)

N = 36

representation.)

x C(6),

N = 36

(zL(6.j denotes the contragredient, represeutat~ion. 1) The simplest exa~nplcs of vector fields are the coordinate vectors .r:,~ , x ‘ib. WC wtmt to introduw uow tlwi\-:t tives whkh trmsform c*ontragrediently t,o the c*oordin:ite vectors. It slm~ltl I)(, noted th:lt, only the diagonal e1ement.s .I’,~,~, x”” at-cbreal v:viables, while .r,,,; , J.“” are conlples for a # 6. We will make tlifferentiabilityy assurnptiotls only with respect to the real and imaginary part of th(l cwortlin:11 w. In that case it is ITHIveriient lo use complex diffcrentiitl opcmtors :LStlefili~cl, for indt~nw, in I~OC~~IIC~I~ :m(l YIartill (27). I,et,.fC 2) tw :L furic~tion of 3, cwlnplcs v:iri;tblc 2 = .c -

z = cc + i/J, theu dcfinc the differential

i//

operut’ors

(22)

In this sense we define

by the complex derivative for a # b and t’he ordimry derivative for II = 0. It can then bc shown that the diffcrcntial operat,ors (?..
we caonsider two sextet fields

di Xa , ; transforming according is defined by (12)

to GL(6)

or E(6)

t .‘.3 -. resp. P’or parity

+l

the operator

P

308

ZIMMERMANN

(Px)&‘>= tw,

(Pf)“(x’) = --X6?(5), .r’ = Px

The extension of (2.6) to tensors of arbitrary III.

SEXTET

(2.6)

rank is obvious. FIELD

IGL(6)-invariant differential equations for a free sextet field have been given by Riihl (Id). We briefly sketch a derivation of these equations from a variational principle since a Lagrangian formulation will be needed for the extension to interacting problems. We introduce two spinors (2.5) transforming under IGL(6) and P as described in the preceding section. Starting from the expression

(3.1) we construct

the Lagrangian

L = A $ PA + h.c. L is hermitian and transforms of the Lagrangian is explicitly

(3.2)

as a scalar under IGL(6)

and P. The secondterm

PA = i With this Lagrange function

(3.3)

we form the variational 6

sY

principle

L&=0

(3.4)

The integration extends over a volume V in the 72-dimensional space of real and imaginary part of the coordinate vectors. Treating the 12 field components (2.5) as independent we obtain from (3.4)

Since we will eventually SU(3) werelabeltheindicesa

reduce the field equations = 1, ... ,6by a = (a, i),

Under XL(2) tion

X XU(3)

the field quantities Xa --

Xai

2

transform 4” = pi

with

respect to SL(2)

X

a! = 1, 2, i = 1, 2, 3. as indicat,ed by the nota-

(3.6)

We restrict now the manifold of solutions of (3.5) by imposing the condition that the field depend only on the four real SU( 3)-invariants ( 1.4). This property

is preserved under SL(2) x SCY(3) t ransformat~iotis wit 11 .rp I ransfornring :rs ;I Rlinliomski vector. Acting 011:I fwction @ of .I! alone the IGI,( ti )-differential ~perat~ors rcciuw to

Hence WC obtain the field equations

This is simply the Dirac equation of an SU(S)-triplet tation. The r&e&ion property ( 2.6) becomes (Px,,l,(,.e’

1 = fern,

(P[)“k(X’)

This is the conventional transformation terms of van der Waerden spinors. It is evident t’hat the field operators Xak(.f),

in v:tn tier VV:wr&~n IIO~.:~-

= -XaF(J), of a spin-! 2 triplet

&

i, .rp

.r’ = f’.r.

(;2.10 i

under I-’ writ,ten

in

)

satisfying c 3.91 solve the field equat,ions (3.5) if ext.endctl by (:!,A), ( 1.1 I to fun&ions of .r,i , .rih. Hence a free quantized SC!(Z)-triplet of spin-j y fields fulfills the criterion of IGL( 6) -’invariance state(l in the introduction. One may as well reduce the Lagrangian (3.2) clirectly by assuming t,hut the fields (3.6,) depend on 2’ only. The J,agrangian obtained is

(3.11) becomesthe conventional Lagrange function of a spin-,J2 triplet

310

ZIMMERMANN

It is not difficult to introduce Lagrange function we consider

sextet-sextet L = LO +

interactions

in this context. As the (3.14)

Lint

where Lo is the Lagrangian (3.2) of a free sextet field, Lint denotes a hermitian, quatric, and nonderivative coupling term which transforms as a scalar under IGL( 6) and I’. The most general Lint is (gl , g2 real) Lint

=

-

8glt”XaXd”

-

4g2(taXatbXb

A system of IGL(6)-invariant field equations form the coupling term becomes Lint = 39, + g2)P~&z

follows

+ a71 -

in agreement with results previously obtained and Lee (4) and Charap and Mathews (6). IV. SPIN-g

+

(3.15)

Xd&&)

from (3.14).

(3.16)

g2P%~&4h

by Bardakci,

In reduced

Cornwall,

Freund,

FIELDS

A spin-pi field has some similarity with a 56-component baryon field since in both cases the field quantities are described by symmetric tensors of rank 3. We begin therefore with a Lagrangian formulation of spin-x particles. A spin-% field of mass ‘rn # 0 is described ’ by a pair of symmetric tensors xa1(IzoL8 and te1aza3satisfying the Klein-Gordon equation, the anticommutation relations b (4.1) (the pairs ~(~1, X(B)and [@, $” anticommute) and the conjugation condition4 (4.2) This is equivalent to the formulation of Rarita and Schwinger (SO). Under space inversion x and .$transform according to (parity +l) (x’) = p.yx) (PxL,...a,

2' = Px (4.3)

(Pp-&3(x’)

= -Xal...aa(x)

field equations and commutation relations can be derived from the Lagrange function L = AI + Az + h.c. (4.4) 3In order to have P-invariance the formulation in ref. $6 must be modifiedby doubling the number of field components. 4E(b) denotesthe hermitian conjugate of Eta).

(The SUI~IRcxtcnd over cyclic perniutntjions 7j ant1 X the tick1 equat,ions hedonic:

of a1 , 01~, wi. i 2Iftw

6 IIifIiculties arise for instance for interacting spill-S; tions are uot observed n-hen carrying ollt t,he variations. then inconsislellt with the field equatiorls.

particlt3 The

eliinin:~t ion (11

if t hc: symmetry symmetry propcrt

condiies are

312

ZIMMEHMANN

(4.8) is identical with the Klein-Gordon condition (4.2) is imposed as subsidiary

equation for x and-t. The conjugation condition.6

V. BARYON

FIELD

An SU(S)-octet of free quantized spin-s fields does not satisfy the criterion of IGL(G)-invariance. There is no representation of GL(6) which under SU(3) X SL(2) would be equivalent to the product of the octet and the spin->; representation. An SU(3)-octet of spin-55 fields must therefore necessarily be extended by other fields in order to meet the requirement of IGL(G)-invariance. Following Gursey and Radicati (1) we choose the 56-dimensional representation of GL(6) with Young tableau (3, 0, 0, 0, 0, 0) which combines the spin-pi octet with the spin-35 decuplet into one supermultiplet. Accordingly we describe the system by two symmetric tensors of rank 3 xalaZaZ, la1a2ag.In analogy to the case of spin-95 fields we introduce the auxiliary quantities v(iblb2 , Xib1b2,which are assumed to be symmetric in 61, bz . Under P we have (parity +l) the transformation properties

(PX)(&')

= tY4,

(f?#“‘(d)

In analogy to the spin-s

and form the invariant

=

Lagrangian

-x(a)(x)

we start

(a)

=

al&a3

,%’

Px

(5.1)

from the expression

Lagrangian

L = A + PA + h.c. The field equations

=

become after elimination

(5.3)

of q and X:

(5.4) The tensors

g and x may be linked by the conjugation

condition

in analogy to (4.2). 6 In the case of interacting fields the conjugation condition order to be consistent with the field equations.

(4.2)

must be modified in

( The last cquatiou follows hy using the Klein-(;ortloll equation. ) (5.10) i.G t htb equation of an SU(3)-octet. For the SU( S’I-tlwuplet WC oht,airl with ( .?.!I ) and the Klein-Gordon equation precisely the formulation of spilt-:‘2 I);trtit-Ies. Diruc

~CKE;~\~L~~)G~L~~:N.I.H

The reading Dyson,

tutthor uurdd like to thank Drs. >I. ilrurts, the manuscript and for valuable commrttts. and A. Pais for helpful discussions.

RECEIVED:

A. (;rossmatttt, He is gratefrtl

August 23, 1963 I:EFE:1:ErU’C&

1. F. G~RSEIAXD L. A. I~ADI~ATI, Phys. Rec. Letfe,.s 2. A. Pals. I’ht~s. Rec. Lellers 13, 17.5 (1964). 3. B. 6~KlT.4, P/g/s. Reo. 136, B 1756 (l!#ii).

13, 173 (I<&~).

and 0. Steittmann ftlt 10 I)rs. 11. Beg, 12.

314 4.

ZIMMERMANN

K. BARDAKCI, J. M. CORNWALL, P. G. 0. FREUND, AND B. W. LEE, Phys. Rev. Letters 13, 698 (1964); ibid. 14, 48 (1964). 5. M. A. B. BEG AND A. PAIS, Phys. Rev. Letters 14, 267 (1965); Phys. Rev. 137, B 1514 (1965); Ibid. 138, B 692 (1965). 6. J. M. CHARAP AND P. T. MATTHEWS, Proc. Roy. Sot. 286, 300 (1965). 7. R. P. FEYNMAN, M. GELL-MANN, AND G. ZWEIG, Phys. Rev. Letters 13, 678 (1964). 8. T. FULTON AND J. WESS, Phys. Letters 14, 57, 334 (1965). 9. R. OEHME, Proc. Seminar High Energy Phys. and Elementary Particles (International Atomic Energy Agency, Trieste, 1965). 10. S. OKUBO AND R. E. MARSHAK, Phys. Rev. Letters 13, 818 (1964). 11. K. T. MAHANTHAPPA AND E. C. G. SUDARSHAN, Phys. Rev. Letters 14, 163 (1965). 12. W. R~~HL, n’uouo Cimento 37, 301, 319 (1965). 13. B. SAKITA AND K. C. WALI, Phys. Rev. Letters 14, 404 (1965). 14. A. SALAM, It. DELBOURGO, AND J. STRATHDEE, Proc. Roy. Sot. (London) A 284, 146 (1965). 15. A. SALAM, R. DELBOUHGO, M. A. RASHID, AND J. STRATHDEE, Proc. Roy. Sot. A 286,312 (1965). 16. R. BLANCENBECLER, M. L. GOLDBERGER, K. JOHIUSON, AND S. B. TREIMAN, Phys. Rev. Letters 14, 518 (1965). 17. S. COLEMAN, Phys. Rev. 138, B 1262 (1965). 18. L. MICHEL AND B. SAKITA, Ann. Inst. Henri Poincarb, in press. 19. H. BACRY AND J. NUYTS, Nuovo Cimento 37, 1702 (1965). 20. S. K. BOSE AND Y. M. SHIROKOV, Phys. Rev. Letters 14, 398 (1965). 21. M. GELL-MANN, Phys. Rev. 123, 1067 (1962). 22. Y. NE’EMAN, Nucl. Phys. 26, 419 (1961). 23. G. FEINBERG AND A. PAIS, Phys. Rev. 131, 2724 (1963); Phys. Rev. 133, B 477 (1964). 24. A. SALAM, Phys. Rev. 130, 1287 (1963). 25. A. SALAM AND It. DELBOURGO, Phys. Rev. 136, B 1398 (1964). 26. 0. STEINMANN, Ann.. Phys. (N.Y.) 29, 76 (1964). 27. S. BOCHNER AND W. T. MARTIN, “Several Complex Variables,” Princeton Univ. Press, 1948. 28. M. FIERZ, Helv. Phys. Acta 12, 3 (1938). 29. J. SCHWARTZ, J. Math. Phys. 2, 271 (1961). SO. P. RARITA AND J. SCHWIKGER, Phys. Rev. 60, 61 (1941).