Volume 14, number 4
E OUIVA LENT
P HYSI CS L E T T E R S
FORMULATIONS
OF
THE
15 February '1965
SU 6 G R O U P
OUARKS
FOR
F. G'IYRSE Y
Middle East Technical University, Ankara
Received 19 January 1965
A r e l a t i v i s t i c f o r m u l a t i o n of the SU 6 group
based on the little group of Wigner was sketched
p r e v i o u s l y [1] and spelled out r e c e n t l y [2] for an
a r b i t r a r y SU 6 multiplet. In view of the d i v e r g e n t
s t a t e m e n t s [3] about the r e l a t i v i s t i c i n v a r i a n c e
of SU6, a p p l i c a t i o n s of the g e n e r a l t h e o r y to the
s p e c i a l c a s e of the lowest ( 6 - d i m e n s i o n a l ) r e p r e s e n t a t i o n of SU 6 c o r r e s p o n d i n g to q u a r k s [4] *
will be given in this note. F o r this p a r t i c u l a r
m u l t i p l e t t h e r e a r e m a n y s i m p l i f i c a t i o n s and all
the o p e r a t o r s of the group can be w r i t t e n explicitly in equivalent f o r m s which a r e a l r e a d y fam i l i a r to p h y s i c i s t s . The following r e s u l t s which
a r e e s s e n t i a l l y contained or implied in ref. 2,
will be d e r i v e d d i r e c t l y in this c a s e ;
1. The o p e r a t o r s X i of the little group, when
applied on q u a r k p a r t i c l e s t a t e s , a r e equivalent
to the H e r m i t i a n m e a n spin o p e r a t o r s of Foldy
and Wouthuysen [5].
2. In the r e p r e s e n t a t i o n of Foldy and Wouthuysen for p a r t i c l e q u a r k s the g e n e r a t o r s of SU 6 a r e
-~mzz, F~, ~il F~, where the ~i a r e the u s u a l P a u l i
m a t r i c e s o p e r a t i n g on the 3 q u a r k s s i m u l t a n e o u s ly and the FX a r e the SU 3 m a t r i c e s .
3. The f r e e q u a r k H a m i l t o n i a n i s i n v a r i a n t
u n d e r SU 6.
4. The Newton-Wigner [6] position o p e r a t o r
R f o r f r e e q u a r k s and the o r b i t a l a n g u l a r m o m e n t u m R × P c o m m u t e with the SU6 g e n e r a t o r s .
5. I f u ( p , n ) is the c - n u m b e r D i r a c spinor a s sociated with a q u a r k p a r t i c l e state with m o m e n t u m p and p o l a r i z a t i o n d i r e c t i o n n, then the
SU 2 t r a n s p o r t a t i o n s of the little group induce
the following P a u l i t r a n s f o r m a t i o n [7] ** in
m o m e n t u m space
• The quarks need not have fractional charge in the
atomic model proposed in ref. 1, where they repel
each other and are held together by a boson core
that is a SU6 singlet and has suitable charge and
baryon number.
• * While Pauli's transformation was originally defined
for a zero mass 4-component Dirae field inx space,
we have here invarianee under a formally similar
SU2 group in momentum space where zero-mass is
not implied by this invarianee.
330
u ( p , n ) ~ u ( p , n ' ) = au(p,n) + b~5uC(p,n),
(]a[2+ ]b]2=l).
(1)
We s t a r t with the expansion of the q u a r k o p e r a tor ~V(x),
~(x) = ~ h ( p ) ~ ( p )
(2)
P
where
(3)
A ( p ) = ~---~n--fl7 4 7 =m+pO+75a'p~ ,
and
(PO = ~ )
al)
a2
-b
c0(p) = e i v 4 p x x x
(4)
\ bl¢/
in the s t a n d a r d D i r a c r e p r e s e n t a t i o n with 74 d i a gonal, a i,b i are the u s u a l a n n i h i l a t i o n o p e r a t o r s
for p a r t i c l e s and a n t i p a r t i c l e s , each with spin up
o r down. w(+) and w(-), the p o s i t i v e and negative
f r e q u e n c y p a r t s of w, a r e p r o j e c t e d by ½(1 • 74)The non local o p e r a t o r s X i of the little group
w e r e shown to be [2]
Xi('il~V ) = ½A(-i~V) ~ A ' I ( ' i ~ V ) ,
(5)
where A is the boost o p e r a t o r that r e d u c e s to
eq. (3) for a D i r a c p a r t i c l e ¢. O p e r a t i n g with X i
on @(+), the positive f r e q u e n c y p a r t of ~ , we obtain
X i ~(+)(x) = ½~ X i (p) A(p)(l+74) co(p).
(6)
P
On the other hand, u s i n g eq. (3), we have
A(p)
1+~4
i/~uP0
= [ ~ - vO)
1+74
2 '
(7)
¢ In the s = ½ ease the operators X i are identical with
the spin operators introduced by Chakrabarti [8],
who has also shown their connection with the relativistic classical mechanics of a system of particles.
Our operator wip) of eq. 2 is essentially the momentum space field operator in the Chakrabarti representation.
Volu~ne 14, number 4
PHYSICS
LETTERS
where
u' = 75uC ,
(m+iv.p~½ m+Po+iT"P
V(p)=\
PO / - ~Po(Po +m)
(8)
i s the u n i t a r y F o l d y - W o u t h u y s e n o p e r a t o r [5].
Eq. (6) now t a k e s the f o r m
x i ,(+)(x)= ~
p
x(+)
z ~) A~) ~(+)(p),
l+)(p)
=
½ u(p),~u*(p)
~
= ~a -
i
O)
(15)
ei x ' O qJ(x) ='~ A ( p , . ) e ½ i a ' 0 co(p) , (t6)
where n is the polarization direction chosen as
theOz axis in the r e s t frame, The new polarization direction n' can be defined by
A(p,n') = A ( p , - ) R ( 0 ) ,
2Po(Po+m )
where
~ F W = U¢~.
v' =Y5u.
P
are the Hermitian and non local mean
(i0)
spin operators. They coincide with the little
group operators X i on the manifold of the positive
frequency states and are known to commute with
the Hamiltonian. Thus we have proved statement 1.
We can now go over to the Foldy-Wouthuysen
representation through
(11)
In t h i s r e p r e s e n t a t i o n the D i r a c H a m i l t o n i a n
HD and the little group o p e r a t o r s h a v e the f o r m
H FW = u¢HDu = 74P0 ,
(12)
(X(+)) FW : U*X(+)U : ½a .
(13)
T h i s p r o v e s the s t a t e m e n t s 2 and 3. H e n c e ,
1
to f o r m u l a t e SU 6 we can u s e ~ a as spin o p e r a t o r s
only in the F o l d y - W o u t h u y s e n r e p r e s e n t a t i o n . X i
should be u s e d in the D i r a c r e p r e s e n t a t i o n . T h e
confusion b et ween the two r e p r e s e n t a t i o n s m a y
h a v e led s o m e a u t h o r s to c l a i m the non i n v a r i a n c e
of the k i n e t i c p a r t of t h e f r e e H a m i l t o n i a n u n d e r
SU6.
Statement 4 follows from the fact that the Newton-Wigner [6] position operator R for particle
states is identical with the Foldy-Wouthuysen
mean position operator. Thus ~ = x in the FoldyWouthuysen r e p r e s e n t a t i o n , and R and R x p
c o m m u t e with the SU 6 g e n e r a t o r s . In the D i r a c
r e p r e s e n t a t i o n ~ and ~ x p a r e non l o c a l o p e r a tors,
To p r o v e s t a t e m e n t 5 we r e w r i t e eq. (2) in
the f o r m
~(x) = ~'~{[ual+u'a2] p e i p u @ + [vb~-v'b~]p eiPUxv},
o
=
axp - px (axe)-
7475 2p 0
v =u c = 72u* ,
Under a transformation of the little group with
parameters O, we have
~'(x)
where
x
15 February 1965
(14)
w h e r e u, u ' , v ' and v a r e the 4 c o l u m n s of the
L o r e n t z m a t r i x A(p) in the r e p r e s e n t a t i o n with V4
diagonal. Hence u , u ' a n d v , v ' a r e r e s p e c t i v e l y
the spin up and down w a v e f u n c t io n s in m o m e n tum s p a c e f o r p a r t i c l e s and a n t i p a r t i c l e s r e s p e c t i v e l y . B e c a u s e A(p) i s d e t e r m i n e d by i t s f i r s t
column, one finds
R(O)
(17)
is the unitary rotation matrix
R(O)=e½iq'O
=
-b*
<; a*)"
(t8)
Rewriting eq. (17) in terms of the first columns
of each side we find eq. (I).
We also note that the transformations
(I) or
(16) can be applied to each quark separately, corresponding to the subgroup G s = SU2× SU 2 × SU 2
of U 6 with generators
HiXn, II2Xn mid II3Xn,
where H I , II2 and II3 are the projection operators
for the three quarks ql, q2 and q3.
The author would like to thank L. A. Radicati
for stimulating discussions.
1. F.Gffrsey and L.A.Radieati, Phys.Rev. Letters 13
(1964) 173.
2. F.Gffrsey and L.A.Radieati, Lorentz covariant definition of the SU6 group, to be published. In this
paper it was shown that the littIe group operators
X i can be defined by means of a self dual tensor
X. u constructed from the Bargman-Wigner operatots WK that commute with the energy-momentum
operator s.
3. B.Sakita, Phys.Rev. 136 (1964)B1756;
B.Sakita, Phys.Rev,Letters 13 (1964) 643;
M.A.Beg and Pals, Lorentz invariance and the interpretation of SU(6)-theory, preprint;
R. P. Feynman, M. Gell-Marm and G. Zweig, Phys.
Rev. Letters i3 (1964) 678;
K. Bardak~i, J.M. Cornwall, P.G.O. Freund and
B.W. Lee 13 (1964) 698;
M. Gell-Marm, Angular momentum and the algebra
of current components, Phys.Rev. Letters, to be
published;
B. Sehr6er, SU6 in relativistic particle mechanics,
preprint;
T. Fulton and J. Wess, Symmetry group containing
Lorentz invariance and unitary spin, preprint;
A.Salam, Physics Letters 13 (1964) 354;
R. Delbourgo, A.Salam and J.Strathdee, The r e lativistic structure of SU(6), preprint.
4, M.Gelt-Mam% Physics Letters 8 (1964) 214;
G. Zweig, CERN preprint.
5. L.Foldy and S.A.Wouthuysen, Phys.Rev.78
(1950) 29.
6. T.D.Newton and E.P.Wig~er, Rev.Mod. Phys.21
(1949) 400.
7. W.Pauli, Nuovo Cimento 6 (1957) 204.
8. A. Chakrabarti, J.Math.Phys. 10 (1963) 1215,1223.
331