- Email: [email protected]
On the reordering of biquadratic scalars
ANNALS
OF
PHYSICS:
On
40,
The
159-178
(1966)
Reordering
of Biquadratic
CHaRLES
Georgia
Institute
1’.
Scalars*
hAHI
of Technology,
Llflcmfn,
Georgitr
A t hrorem analogous to the Fierz reordering theorem for Lorentz scalars c*onstrwted from 4 Dirac spinors is proven for all semisimple compact grollps. The proof of the theorem depends on only -I factors, i he most important of which are the reality of the generalized Clehsch-Gordan coefficients for semsimple grollps and the uniqueness of the constrllction of a scalar from a rcpw sentatioll of a compact group and its contragredient. The reordering matrix asso&ted with the scalars formed from four “octets” of Sf’(3) is also givctl along with an orthogonal matrix that diagonalizes the octet-reordering matris. I. INTRO1)UCTIOiY
Since the formulation of the Dirac equation in 1928 several authors havc~ COIIsidered the behavior of the five Lorentx scalars formed from four A-spinors urld~ permutation of either the “barred” or “unbarred” spinors (1). The result of tJhis so-called PI, ( or Pz4) operation on the Lorentz scalars is contained in the “l~icrz reordering theorem” (2)) d’lscussions of which are available in several plain. (See ref. 3.) The invariants of the PI3 (or Pz4) operator have heer~ considrred by Feveral authors both in the theory of beta-decay and in the sexrcsh for a universal Fermi interaction (4). In fact, the accepted V-A theory of beta decay is such :UI invariant if the renormalization effects of tjhe strong int,eractions are neglected. This PI3 invariance of the bare V-A interaction is to a large extent ignored IIOW adays; holr-ever, one cannot help but wonder if it has some underlying sigllificxnce. C’onsidering this significance to lie in a self-consistent analogy of particalcs Ahrens (5 i has investigated the possible struct)ures of the weak intern&ion :IX determined by permut,atiori symmetry in isospace. A natural extension of llr:it work in light of the success of the S&‘(3)-oc*tet’ model for hadrons (6;) is IO illvestigate the permutation properties of biquadratjic S17( :?,) sAars. This p:1p~r is intended to lay t#he ground work for such investigat,iorls. Since the Lorentz group and the special unitary group in three dimc&ons, SU(3 j, are structurally quite different, the existence of a reordering t.heorrnl of the E’icrz type is questionable. It will be shown in Section I\: that not, only tlocts such a theorem exist for XU(3) but it exists at Icast for all scmisimple mnpwt groups. * Work
supported
in part
by
the National
Hcience 159
Fulultlat
iota.
160
FRAH.M
I<. M. Case (7) has shown previously that a Fierz type reordering theorem exists for complex orthogonal groups. His proof requires the existence of a set of quantities I’(i) having commutation properties of the form
r(i)r(j)
+ r(j)r(;)
= taij
Since such commutation rules do not occur for compact groups in general (sU(2) is an exception) Case’s proof is not applicable to such groups. It is worth noting that Case’s theorem and the reordering theorem presented here are essentially complementary and together they cover most groups of physical interest. Two examples of the PI3 operation will be presented in Section II. These examples concern the reordering of the simplest nontrivial xU(2) and SU(3) biquadratic scalars. As a preliminary to the statement and proof of the reordering theorem in Section IV the question of the reality of the Clebsch-Gordan (C-G) coefficients for semisimple groups will be considered in Section III. The general properties of the reordering matrix will be discussed in Section V. The next section will consider the definition and simplification of Racah coefficients, the results of which will be used in Section VII to obtain a relation between the reordering matrices associated with SU(3) and certain XU(3) Racah coefficients. Section VII will also give the simplest nontrivial reordering matrix, (Y*, occurring in the octet model of hadrons and an orthogonal matrix U which diagonalizes CX’. II.
EXAMPLES
Two elementary examples of possible physical importance will be discussed in this section in order to become familiar with the P13 operation. Consider first the isoscalars formed from two nucleons and two antinucleons. There are two such scalars : S = m,N&,N, and V = N~TNZ . fl,~Na where the N’s are isodoublets, matrix vector. gives
Expanding
e.g., N =
and assuming
P n , and T is the usual 2 X 2 isospin 0 the various field operators to commute
and
In this case and in the following
SU(3)
example the P24 and PI3 operations
are
REORDERING
equivalent
OF
BIQUADRATIC
l(il
SCBLARS
so that, P24 v = g$3 - xv,
In general Pz4 and PI3 are equivalent If t,he matrix (Y is defined by
etc.
up to a phase.
then clearly
Since the square of PI, leaves the scalars unchanged cy must, be t*he unit- matrix as is indeed the case. The PI3 invariants are now easily seen to bc V + 3S
and
1’ -
S
with eigenvalues +1 and - 1 respectively (6). Next consider the XU(3) scalars formed from two sakat’ons and two antisakatons (8). Again t,here are two scalars. This can be seen as follows. The sakatons belong to the irreducible representation (IR) 3 of SL’(3) while the antisakatons belong to the contragredient IR 3”. The IR’s of XU(3) are labeled by their dimensions and an asterisk is used to denote the corresponding cant ragredient IR. The Kronecker product 3* X 3 decomposes into the dirwt ~111 1 + 8. Hence (3* X 3) X (3* X 3) = (1 + 8) X (1 + 8) = (1 X 1) + (1 X 8:) + (8 X 1) + (8 X 8) Kow in the decomposition of a direct, product of t,wo IR’s of SI -(X ) :L w:~l:u will appear once and only once if and only if the product involves an IR and its contragredient (9). Furthermore, for StT(3) an IR is equivalent. to its c:ontr:rgredient if and only if its dimension is the cube of an integer. Thus it is clear I hat in the above expression one scalar appears in 1 X 1 and another in 8 X 8. Thwr scalars are respectively x = 6&26&l4 and I’ = c 6Ab&Xib4 z where h = the “bars”
P n is t’he sakaton and the X’s are Gelt-JIann’s ma.triws 0A simply denote t,he complex-conjugate transpose.
(10). Hcrca
162
FRAHM
Again expanding
and assuming the various - -
factors to commute gives
and
This can be written
in matrix
where the reordering
form as
matrix p is given by
In this case the P13 invariants
are
S + ?dV with
eigenvalues III.
and
S - p;V
+ 1 and - 1 respectively.
REALITY
OF
THE
CLEBSCH-GORDAN
COEFFICIENTS
After considerable effort the commut’ation relations for the generators of a semisimple Lie group can be cast into the following canonical form (11, Id). [Hi ) Hf] = 0
(1)
[Hi) E,] = criE,
(2) (3)
[E, , Eal = Nm&e+~,
if
a! + P # 0
(4)
The number of mutually commuting generators, Hi , is called the rank, I, of the Lie algebra of the group while the total number of independent generators is called the order, r. Equation (2) is conveniently thought of as an “eigenvector problem” where the E, are simultaneous eigenvectors of the 1 commuting generators, Hi, with eigenvalues, pi . The eigenvalues are said to form a root vector (or simply a root) a! = (011, a, These root vectors are notation Ea+p in Ey. eigenvector associated @ is not a root Nap is
.** ffl)
completely determined by the structure of the group. The (4) thus means that if a~ + /3 is a root then Ea+p is an with that root and Nao # 0. On the other hand if a! + zero.
REORDERING
OF
BIQUADRATIC
Ifi:
SCALARS
By suitably normalizing the generators (which has already been assumed ~II Eqs. (l)-(3)) one obtains ( *-Ii
T (Y@j = &j mcl
Nap
=
-N-a,-0
=
zt
Z/(k
+
l)(j
+
1)
/ B 1
i(j)
The sum in Eq. (,.5) is over all root vect,ors while the numbcrsj and k: in 1.21.( ti 1 are the smallest positive integers such that
[E&p
) E-)9]
=
0
respectively. The numbers j and k are completely tletc~rmilled by the roots :111tl can be easily evaluated by inspection of the root diagram ( 11). The N’s also s:tt,isfy (11) Nap = ND, -a-/j = AL-,
, cL
171
and t,he trivial relation Nao = -NBC
(XI
III pract,ice one useseq. (6) to select a suitable set of N's :md t$eu procectls to find the matrices of the generators in a given representation with t’he aid of l’(is. (3) and (4). When following this procedure the sign in Eq. (A) rnus~ he chos;rn so as t,o be consistent with Eqs. (7) and (8) but is otherwise arbit’rary. One apparent motive for casting the commutation relations into the c*anollic~;ll form, alt,hough it has not been explicitly indicated in the literature, is th:tt it allows one to choose the (matrix representatives of the) I?‘, real. Since by &;(I. (3) the E’, are simply raising and lowering operat,ors, the reality of thy /5,Z determines the reality of the C-G coefficients associated with the group. In order to show that the l~‘~ can be chosen real csonsidrra t~hcortm tlucb to Cartan and discussedby Racah (12) which proves that the eigenvetstors of l*:(l. (2) are nondegenerate if the root vectors are nonzero. Thus by t:~king t,he tr:~~s;pose (T ) and also the hermitian adjoint (t ) of the c:moni(*al commutation relations and making use of Eq. (6j one sees that 13,” and sat are both proport)ional to IL, . Thus either EaT or E’k cwl be chosen equal to K-, (‘hoositlg &J = Jj’_, Nld
i !I t
164
FRAHM
it only remains to be shown that the remaining can be set equal to one for all CYso that
constant
of proportionality,
Eh = EaT = Es,
A,
(11)
The equality in Eq. (9) is the conventional choice. However, no explicit statement concerning the equality in Eq. (10) nor of t,he freedom to establish Eq. (11) seems to appear in the literature. The Hi are only of secondary concern here since they can be taken to be hermitian and diagonal and hence real in all representations of the group. This in turn implies that the roots (Y are real (11). However, it is important to keep in mind that the Hi matrices along with the root vectors determine the representation of the group involved. Equations (9) and (10) establish that E, differs from a real matrix only by a phase which is the same for all the matrix elements of E, . If one sets E a = +E where E,' is real then becomes
E,'
automatically
’ c-4
satisfies Eqs. (2) and (3) ; and Eq. (4)
The exponential is real and thus equal to fl since the other factors are real. This is essentially the arbitrary sign in Eq. (6). Hence the exponential can be absorbed into Nap . In fact if one defines j&
= Naaei(+~+F--+8)
it is easily seen that the N& satisfy Eqs. (6)-(8) if the Nap do. This then establishes that the generators can be chosen real without disrupting the canonical commutation relations in any representation which diagonalizes the Hi . Now consider the C-G coefficients (jlcLl~l;
jZP2VZ
I j(Y)W)
defined by I (M~(~)PV)
= g (a‘1 PlV1 c2yz
;j2/.42v2
I j(Y)W)
/ jlWl)
( j2P2V2)
(12)
where 1. j, ,j2 and j label IR’s of the group. 2. The y’s are multiplicity labels which must be used if an IR can appear more than once in the decomposition of the Kronecker product of two IR’s of the group (i.e., the group is not simply reducible (12, 13)). 3. Vl ) vz ) and Y are vectors whose components are the eigenvalues of Hi which partially label the states of the IR, i.e., the v’s are weight vectors (11, I%‘), and,
REOR.DERING
OF
BIQUADRATIC
I 65
SCALARS
4. The P’S are other labels which are required to distinguish bet,ween stales with the same weight (12, 14). Here as in the remainder of this paper (j,~) denotes the st#ate labeled by p and v in the IR labeled byj. If the state has been obtained by coupling two other sets of states as in Eq. (12) these states will be placed in pztrentheses before the j while the multiplicity label, if there is one, will be placed in parent,heses xftcr the j. The states ( (j,j,)j(,),v) form a complete orthonormal set as do thy 1j,,). Thus
and (j,v I kw)
i 131))
= ~jk&&
If 1j1~1v1) and 1~Z~ZVZ)are assumed to be in different lowering operators for ) (j&)j(,),v) can be writ.ten E, = E&ii where
E,C i 1 operates E,(i)
only on 1j;pivi).
lj;/.l;v;)
= c
Iri’
spaces t,he raising and
+ E,(2)
( 11)
Kow from F:q. ( 12 )
L&.hLjl,jipivi)
Ij;&;
+ a); i = 1, 2
(1.51
where the X, are real since they are just the matrix elements of E,( i‘). Operat.ing on Eq. (12 ) witch E’, t’hus gives F X,(,b4’,jc(vJ I (.hh)j(7)Lv
+ a> = ~~(jt~ln ll2VP
; jmv2 Ij(y)wj
In the first term on the right hand side replace the dummy index vl by v1 - u and interchange the dummy indices ~1 and ~1’. Similarly in the second tc>rm replace v2by V?- cx and interchange ~2 and p2’, Equakion i ltii then bec*otncls 5 Lb’,
jw) I (j&>j(7),‘v
+ a>
= C [CXrr(~l,jl~ulIvl-~)(jlcll’v~ PlVl 111’ P?V2
- ~Y;.~~P~v~I~(Y)cLV)
+ c X&2, j2dv2 - a)(jl~c11VI; J?c12’v2 - a lj(~jpv)] c-2’ x lj1,1v1)Ij2,2v*)
(17)
166
FRAHM
Solution of Eq. (17) for the 1 (j&)j(y)p’v for the C-G coefficients
in terms of the A, which
+ a> determines
a recursion
relation
are real and the coefficients
Clearly the C-G coefficients for the weight v + ar will be real if t.hose for v are. This will be the case if the C-G coefficient for the state in j with highest weight (11) (i.e., the state in the IR whose weight has the largest positive first component) is chosen real. The phase of this “highest state” can always be adjusted so that this is the case. IV.
THE
REORDERING
THEOREM
Let J/ and I#Jbe column matrices whose elements are the orthonormal basis states of the IR j of a semisimple compact group G. Similarly let the elements of q and CPbe the orthonormal basis states of the IR j* of G. Thus, for example, the (pv) component of 1c,is given by J/(pv) = 1j~v). Now assume that the IR K appears in the decomposition of the Iironecker product j* X j. The IR K” must also appear in the decomposition since j X j* is equivalent to j* X j. Thus one may, for example, construct from q and 4 quantities which transform according to the IR K” and from Q, and 4 quantities which transform according to the IR K. These can be expressed with the aid of the C-G coefficients as gl
(j*/m
; jw2
I K*(~Pv)
WPI~I)~(P~~~)
my2
and
respectively. by
From these quantities
it is possible to construct
&rskw, which transforms as a scalar under G. The Pz4 operation (which is equivalent interchanges # and 4. That is
P&,,a,w, The reordering
is then expressed
P&w
(W, w>
denoted
W) to P13 up to a phase)
WI = &a,m
theorem
a quantity
WI
in this case (18)
by the equation
= c ~~(rs)J(Br)S~(Br)(\k~, JBr
w>
(19)
KEORDERIPiG
OF
BIQUADRATIC
SCALilRS
167
where the sum is over all IR’s occurring in the dec~omposition of ,i* X j. liic~u:~t~ior~ (19) is convcnient~ly written in matrix form as P2&
= oljsj
! “0 I
where X’ is :L c~olunmmatrix whose elements are t,he sc~xl:~ S:coa, and LV’is 1hc “reordering matrix” whose elements are t,he numbers 0l~(,g,,,,,3~, . The proof of the reordering theorem depends on only four conditions: ( 11 the ‘i(,oilll~leteIless” of the C-G coefficients, ( ‘1) the reality of t)he C-G coefficients, ( 3 ) the symmeky of the C-G coefficients under interc~h:rngcof the two f:tc*tot states /jlFlvlj and ( J’~pzvz) in Eq. (12), and (4) the uniqueness of the construction of :I sc*xlnr from a11 IR and its c&or11 IXgredient . Condition (2 ) is snt,isfied by semisimple groups as was shown in the previous section, (4 i is satisfied by compact groups (see ref. 9, pl). 147 and 317j :tntl c*onditions ( 1 ) and (3) are satisfied in general (9 ) iat least whenever it is m(‘:uingful to define C-G csoefficients). Since the C-G coefficient,s are real and transform an orthonormal basis into XI orthonormal basis, they form a real orthogonal matrix. Definition ( 12) md its inverse then yield what will be referred to in t)his paper as the (‘4~ orthogonnlit y and completeness relations. CMhogonalit,y :
(‘ompl&eness:
where the 111’s represent all of the appropriate state labels. The symmetry property mentioned in condition ( 3) arisesfrom the equiv:ll(~rlc(~ of j, X jz andj2 X j, for arbitrary IR’s j1 andj3 . This rcquircs that (j2m2 ; jl~ml / J(-y)M)
= &(j,jlJ(y))(jlm
; j2ln2 1J( y‘) :\I)
t 2:i 1
where [I is a real phase factor which is independent of the st:tt,e labels 1~1, M:! , and ill. Since only those C-G coefficients will occur in the following which oo~plr~ :m IR and its cont,ragredient it is convenient to int)roduc*c tJhc matrices (j./(r ).I/ ) whose elements are defined by
where a and b are all of the appropriate state labels. The orthogonnlity, complA(~-
168
FRAHM
ness and symmetry
relations
then become respectively
and
(~*Jwcwn, Substituting
= c;l(ii*J(Y) )(jJ(rwbrwn,
(27)
Eq. (27) into Eq. (26) gives
NOW let 0 and e1 be two arbitrary nj X ni matrices where ni is the dimension of the IR j and multiply Eq. (28) by @cTe)m2&,$~Q(e’#),, . Summing on the m’s gives
Equation (29) is the analog of Eq. (81) in Case’s paper (7). Choosing 0 = (jK*(e) - N) and 0’ = (jK(s)N), multiplying by the “l-j symbol” (KlO)-NN and summing on N gives
(30) @TW*(d
- N)(~*J(~>M)(~K(~)N)~~(~J(~)M)~
The 1-j symbol @lo)- NN is simply the appropriate set of C-G coefficients for combining states which transform according to the IR’s K* and K to give a state which transforms as a scalar (which is labeled by 1 and whose single component is labeled by 0). The symbol -N labels that state which must be combined with the state labeled by N in forming a scalar quantity. Now the left side of Eq. (30) is a scalar under the group. Hence the right side is also a scalar. Since the quantity aT(jJ(y)M)4 transforms according to the IR J, the scalar character of Eq. (30) requires the remainder of the expression under the sum on M to be the appropriate 1-j symbol times a quantity which transforms as the -M component of the IR J*. This is a direct consequenceof condition (4). Hence
t,(j.;J(,))
F WO)-NNW*(~
- W(~*J(r)MX~~(GV = T &~J&~ (JlO)--M&J*(P)
- M)
(31)
REORDERING
Equat,ion
OF
BIQUADRATIC
169
SCALARS
(30) thus becomes ( 32 1
where
Equation
(32 ) is simply the reordering V. PROPERTIES
OF
theorem of Eqs. (18 1 and ( 19).
THE
REORDERING
MATRIX
Since the square of P,, leaves the scalars unaltered, (J)’ Thus (Y,’ is it,s own inverse. Now consider the following ~~~~ces,iw, W), = -g
“%Br~(W, az (jk’*(eJ
Eq. (20) implies that
= 1
inner product
( :3-l I
of the scalars in Eq. (33 )
W)) - N),b(K10)-NN(jhT(8)N)cri(j.l*(P)-ill)E~
ef !lh
( :i.-, I
X (JlO)-,,(jJ(y)n~),, x
(*(a), (4(b),
J/(e)
ddf))iw.c),
1) +(y))iticn),
1C/OI).r
This i, the usual inner product, for composite systems encountered in ctuantum mechanics. Sow by virtue of Eq. (13) (recalling that the state labels have been compressed t,o a single index) (*(a),
q(e))
= 6,, ,
etc.
Thus Eq. (,35) becomes &,dw,
W,
&dw,
w> j
where the last equality follows after several applications of Eq. ( 2.5) _ Equation ( ;36j demonstrates t,hat the reordering matrix transforms one orthonormal set of quantities into another orthonormal set. Hence the LY’ must he unitary. l’urt,hermore the cuj are real by virtue of Eq. (31) and the fact &at the C-G coefficients are real. Combining these results with Eq. ( 34) implies that the reordering matrices are real, symmetric, and orthogonal. Thus Lyj z!T (g)* whew
* means complex conjugate.
= (&T
= $-’
(37 j
170
FRAHM
An additional consequence of the orthogonality of the reordering matrix is the implication that there are the same number of P,, (or PI,) invariants as there are scalars in the decomposition of the Kronecker product (j* X j) X (j* X j). VI.
SU(3)
RACAH
COEFFICIENTS
In this section the definition and structure of the Racah W-coefficients for the group G will be discussed. The definition of the Racah W-coefficients to be used here is analogous to that given by Rose (15) for the corresponding SU(2) quantities. Consider the coupling according to Eq. (12) of three orthonormal sets of quantities ( jlnzl), ( j,m,> and 1j,m,) which transform according to the IR’s j1 , j, and j, respectively of G. One could couple first ]j,ml) and 1j,nz,) to give 1 (j,j~)j’(r’)m’) and th en couple this to 1j,m,) to give 1 (( j,j,)j’(,‘)j,)j(,),,) or as one alternative couple / jgm,) and / j,m,) to give 1 (j&)j”($‘)m”) and then couple this to (j,mJ to give 1 (j,(j,j,)j”(y”))j(p)m). The two sets of orthonormal quantities I ( (j,j,)j’(,‘)j,)j(,)n2) and ( (j,(j,j,)j” (y”) )j(p)nl) span the same space and hence must be related by a unitary transformation.
I ((j,j,)j’(,‘)j,)j(,>m) This relation defines the W-coefficients for G. It is important to note that since the quantities in Eq. (38) belong to the same m the W-coefficients are independent of m. (In this connection see ref. 16, pp. 115 and 298.) Using Eq. (12) to express the composite quantities in terms of ( J&I), 1j2m2), and I j,m,) gives
= & X (Jim;
%;? ~~w(j,j,j(YP)j3;j’(y’)j,r(yN)) TlQ?lZ”
jnmrr I j(Ph)(j
(39)
~m~;j~m~Ij”(~“)mn)jj~m~)Ij~m~)Ij3m3)
The orthonormality of the products 1jImI) 1j,m,) / j,m,) requires their coefficients on the two sides of Eq. (39) to be equal. Thus Zt$ (j’m’; jm
Ij(y>m)(jm = jg,,
;j2m Ij’(-f’)m’)
~~jp~jrW(jlj!ij(~~); i(T')j"(T") 1 X (jImI;
Using the orthogonality
jrrmlr Ij(P)m)(j,
of the C-G coefficients
rn~;j3m3(j”(~“)rn”)
twice then gives
(40)
REORDERING
W(j,
j2 j(yPjj,;
j'(y')j"(y"))
OF
=
BIQUADRATIC
(12jnnj')-l"
171
SCALARS
C
j3m3
(.ym';
j j(y)m)
m,m2m3 WL'7fS"
X (j, ml; j2m2 Ij’(y’)m’)
(j2m2 ;j3m3 Ij”(y”)m”)(j~
ill)
ml;j”mN
/.j(p>m)
for all values of HLin the IR j. This result is, of course, the same as in t,he 1st ’ ( 2 ) wse with two generalizat,ions ( 1)) the multiplicity labels have been int.roduwcl, and (2 j, the state labels and corresponding coupling rules may now be mow complicated than in the XIY(2) case. Specializing G to the group SC(3) one can split the #C-(3) C-C; coefficients in t#he customary way into an sL’(2) C-CT coefficient and an “isoswlar fwtor” (j*??l~ ;j2 ?)121j(y)nL)
= (j,I,Y,
; jJ,I’,
1j(y)IIr)((I*I,r
; IrIp, j IZz))
(41’ I
where the double “bra-kets,” ((e . . 1 . . e)) denote the SI ‘(I)) (‘-G coefi?c.ients and the quantities (. . . I . . . ) are the isoscalar factors (17 j. The st.att: lab& have been chosen to be t,he “isospin, ” “z-component of isospin,” and “hyprrc*harge.” That is, w = (I, I, , Y), etc. Making t,he splitting for S17(8) in l’q. (41) and recombining t#he X1’(2) C-G coefficient H according to the same ~qu:~tion gives
x Cj’l’Y’;
j, 13 Ys I j(y)IY)(j,
x (j, I2 Yz;j313
II Yl ;j, I, z’s / j’(y’)l’Y)
I’3 ~j”(YN)I~YN)ijlllY1;j”l”I”’
lj(/3)IY)
where WZ denot’es the XI-(2) Racah coefficients. The additivity of hypercharge can be used to evaluate the sums on I’, , Y2 , and I73 . I;urthermore t.here is a unique state in every IR of Sf-(3) for which I = 0. (See ref. 18, pp. 26-28.) This state can be used to simplify Eq. (43) since the SC(3) W-coefficients are independent of the choice of I and I’ (provided, of course, that a state with the labels I and I’ exists in the IR j). This gives ~l’(jljzj(rP)j3;j’(r’)j”(r”))
=
(TLj"njT)-"2
x (j’I3 I”; j, 13Y x (j, II Y -
Y’ 1j(y)OY)
YN; j, 1,Y + Y” -
x (j,I, Y’ + I”’ - Y;j313Y x (j, II I7 -
-
y /j’(y’)& Y’ ljN(-fn)ll
I-“; j”1, Y” 1j(p)or-)
I-‘) I-“j
(44)
172
FRAHM
In obtaining this expression the sums on I’ and I” were evaluated by noting that in order to obtain I = 0 it is necessary that I’ = I3 and I” = Il. The remaining Wp coefficient is well known in the literature (16, 19). It is given by
W2U1I2OI3; I3Il) =
6(L) I,, I2) 4211
(45)
+ l>(21, + 1)
where 6(11,13,12)
= 1 ifII+Iz+13isaninteger and fl + I3 >= 1, >= / I1 - I, 1 = 0
otherwise
Thus W(j&j(rp)j,
;j’(y’)j’(,fl))
= (?7++,)-1’2
x (j’I,Y’;
j&Y
-
x (jlIlY
-
x (j2IzY’
+ Y” -
C
C
It1213
Y’YN
fi(Ii,
Is,
12)
Y’ 1j(y)OY)
Y’f;j,I,Y’
+ Y” Y; jJ3Y x
-
(jlI,Y
Y Jj’(q)IsY’)
(46)
Y’ Ij”(yN)IIY”) -
YN;j”IIY”
1j(P)OY
This expression can now be used to evaluate the simpler XU( 3) Racah coefficients using tables of isoscalar factors (see, e.g., ref. 17). VII.
APPLICATION
TO
THE
/W(3)-OCTET
MODEL
A relationship between the SU(3) reordering matrices associated with the octet model of hadrons and certain #U(3) W-coefficients will be established in this section. Such a relation would follow immediately in the SU(2) case where the extensive Racah algebra of 6-j and 9-j symbols is available (15, 19). By multiplying Eq. (31) by [&(jj*J*(p))(JIO)-MM]-‘(j*J*(P) - M) and taking the trace the following expression for the matrix elements of cr’ is obtained. 1 ah(ea)J(py) = El(jj*J*(p>)sl(jj*J(r)) X
Td(S*(r)
c (KlO)-NN N (JlO)-MM
- N)(j*J(r)2M)(jli(G)N)(j”J*(P)
(47)
- WI
Now in the octet model the hadrons are assigned to the various IR’s of SU(3) which occur in the decompositions of the Kronecker products of the 8-dimensional IR of XU(3) with itself. For all of these IR’s the hypercharge is integral. Because of this one can introduce a particularly convenient phase convention due
REORDERING
to de Swart
(17) relating
OF
BIQUADRATIC
173
SCALARS
the states of the IR’s j and j*: / j*,t,g = (j --rn 1 ( - 1)”
(48)
where (j - ~1 1 = ( j -no)*
(*denotes
complex conjugate)
and
m = I: + Y/2 Using this phase convention de Swart ties of the SVi3) C-G coefficients. (j1m;j*m
lj3(,h3)
=
E2(jl.M3(Y))(--
0
1)"'
x
(jl ml ; j2 rn? 1j3(7)m3)
derived the following
(jl
symmetry
proper-
l/2
E
ml ; j,*
-
( 49 ) m3
= (3(j1 j2j3C-i) > (.A* - ml ; j2* -
1j2*(7')
-
m3)
m2 I j3*(-f> - m3)
CM)
The t’s are real phase factors which are independent of the state labels 1)~~, 111:!) and ‘w. The validity of Eq. (49) is questionable whenever there is more than one value of y. However de Swart, has established explicitly that it is satisfied for all casesof present interest. These symmetry relations can now be used to rearrange Eq. (47). Several phase factors will be introduced by the symmetry relations during this rearrangement process. These phase factors will be denoted collectively by x. Of course, t’he value of x will change from step to step but only the phase factor for the final expression will be recorded. Indicating explicitly the sums on t’he matrix indices, changing the dummy index N to -N and applying Eq. (50) to the last two factors in the sum Eq. ( -I-i’) becomes
x (j*u;jb 1K*(E)N)(jb; j*c 1J(yjM)(j
(51)
- c;j* - cl j k’*mq x (j* - d;j - a I J(B)Jo
Applying Eq. (49) to the second and fourth factors in the sum gives ah)-%%) = (J” _ fif; J&, 110) z 0
F zd (-
x (j*a; K - N Ij*(X> - b) (jb; j*c / J(y)M)(j
l)“-‘:(K*N;K
- N I 10)
- c; K - N ) j(X’)
d>
x (j* - cl; j - a ) J@)M)
“2‘)
174
FRAHM
where
X and X’ are such that
(j*u;jb
( K*(c)N)
= C;2(j*jK*(~))(-
1)”
2
0 3 (j - c; j* - d ( K*(6)N)
= &(jj*K*(S))(-
l)-’
“* (j*u; K - N 1j*(X)
- b)
z 1’2 0 1
x (j - c; K - N 1j(h’) Now using Eq. (23) on the last two factors &~).%%)
=
(J”
_
$
J&f
z
( lo>
d)
gives
T azd (-
1)“~‘(K*N;
K - N ( 10)
0
X (j*q
K - N 1j*(X)
- b) (jb; j*c 1J(r)M)(K
- N; j - c 1j(x’)
d)
(53)
X (j - a; j* - d I J(P)M) Finally gives
using Eq. (50) on the second and fourth
&).Qr)
=
factors in the sum of Eq. (53)
K - N 110) j 10) 0 2 T & (- l)“-‘(K*N; (54) Jj(X)b) (jb; j*c ] J(y)M)(K*N; j*c Jj*(h’) - d)
(J* - M;JM
X (j - a; K*N
X (j - a;j* ( - 1 )“-’ (K*N;
T\Tow consider the factor According
to de Swart
- d( J(p)M)
K - N 1 lO)/(J*
-
M; JM 110)
(17) (J*
-
M;JMj
lo)=
'-;y+ nJ
where M, * is the highest weight in J*. Thus
a E (K*N; (-
‘)
-
(J*
_
K - N 110) M;
JM
j 10)
(55) =
(-
‘)
(NH* is the highest weight in the IR K*). Clearly the summand of Eq. (54) gives a nonzero contribution only when 6+c=iL?and-d+iV=6 or equivalently when m --B + F = i@. Thus recalling t,hat the barred quantities are integers in the octet model it follows that - (-1 )a--i+N+M = ( -l)‘N-“+“‘+H-- = (47 = 1. Hence
&s,rcpr)
= xdny
3
5 agd (j - a; K*N
I j(X)b)(jh
j*c I J(r)M)
X (K*N; j*c I j*(X’>
- d)(j - a; j* - d I J(@)M)
(56)
REORDERING
OF
BIQUADRATIC
SCALARS
In Eq. (56) the value of x is given by
Comparing
Eq. (56) and Eq. (41
j it is clear that
= x d?LJ7Lg Tl’(jK*J(rp)j*;
&,P)J(P,)
j(x)j*(x’,
( .5x i‘
\
By using Eqs. (16 ), (57), and (58) one can now evaluate the elements of the reordering matrices associat’ed with the octet model whenever Eq. (49) is vali(l. Of course, all the reordering matrices associated with Xrr(3) may be evnluat,ed using Ey. (47) even if instances are encountered where Eq. (39 ) is not valid. In this circumstance Eq. (581 would have to be abandoned but, one c+ould si ill simplify Eq. (47) for computational purposes in the same manrler as the It-coefficients were simplified in Section VI. One would, of course, have to be careful to use a cxonsist#ent phase convention. The result of this simplification nsillg de Swart’s phase convention for t,he octet model is
x (j*r, I- - I”; jr, I” / J(y)OY) (j*r, I’ - Y’;j& - Y” [ K(B)12 1’ - I”’ - 1.’ ! (j*r, I-” - F;jl, - IT” / .l*(fgo - 1-1 Equat’ion ( 59) was actually used to evaluate the matrix elements of the S I ‘( 3 ! reordering matrix a8. Recall that’ the IR’s of SC’(3) arc labeled by their tlimensions. Since Csee refs. 10 and 11) 8 X 8 = 1 + 8(l)
+ 8(2l
+ 10 + lO* + 27
((i0)
t,here are eight scalars occurring in the decomposition of (8 X 8) X (8 X X ) ; one occurring in the decomposition of each of the following terms 1 x 1,
8(l)
X 8(l), 8(2)
B(l) X 8(Z),
X S(Z),
8(2)
10* x 10,
X 8(l), 10 X lO*,
and
27 X
27
This is a consequence of condition (4) of Section IV and t’he fact that for Sc’( 3 ) an IR is equivalent t#o its contragredient if and only if its dimension is the cube of
10 lo* 27
81 (1, 1) 8 (1, 2) 8 (271) 8 (2, a
k’(cS)/J@r)
1 8 (1,1) 8 (1, 2) 8 (2, 1) 8 (2, 2) 10 10* 27
li’@)lJ@r)
- 3 d/55/44
d&3 - dE/56 - di/2a 0 d&,11
1
- -\/ii98 3 d/3/8
d10/8
l/8 - da4 0 $a/4 -
1
0 0
0
5 l/ii/22 3 &a/110
d&o
w,
1)
2/w - VW5 - 3 ~/6/20
l/2
- VW4 -3/10 0 0
8U,1)
2) 0l/2 :
0 0
&/‘3
d/3
80,
0
l/2 w
0 0 -l/2 -l/2 0
so, 2)
1)
u
0 0
VW2
-l/20 0 8
80,
TABLE
0 0 --1/2 -l/2 0 -l/2 -l/2 0
80,
TABLE a*
II
1)
I
- m/22
0 2&o/7 x6/14 0 d&22
w, 2)
0 0 VW4
l/2
0 0
l/2
- VW4
w, 2)
d/6/6 ~59’5 - d!w22 2/si?i/44
-
1/i/4 l/2 -SF?
10
0 l/4 -l/4 - &G/40
-l/2
l/2
10
l/2
&z/22 - vz?/44
~‘36
- d/3/4 l/2 d/35/28 12;;
lo*
0 -l/4 1/4 da/40
-l/2
10*
d/S 5 2/z/56 -3 m/140 0 0
27
3 d/3/8 - 3 2/i/20 0 0 v%/4 - diw40 d/30/40 7/40
27
E
2
REORDERING
OF BIQUADRATIC
SCALARS
177
an integer ( 11) . These scalars will be denoted respect8ively by
The corresponding reordering matrix a8 with the rows and c~olumnslabeled in the sameorder as in the set (61) is given by Table I. It is worth noting that Tr CZ’= 0 so t,hat in this case there are four Pz4 iI]variants with eigenvalue +l and four with eigenvalue - 1. The real ort,hogonal matrix t’ given in Table II diagonalizes LY’.Thnt is
where 1 is the 4 X 1 unit matrix. Thus I’ can be used to construct, the P,., variank VIII.
irl-
CONCLUSION
4 proof of a reordering t,heorem of t.he Icierz t.ype has been given which is wrlitl for all semisimple compact groups. It, is, of course, equally valid for finit’e groups. The restriction to semisimplecompact groups, although sufhrient, for the proof of the t.heorem present,ed here, is not. in general necessary as is exemplified by ihe Fierz theorem itself. ,411auxiliary proof of t’he freedom to chooset)he C-G coefficients real in the (we of semisimple groups was also provided since it is important to the proof of t.he reordering theorem and apparently does not appear elsewhere in the literaklre. The reordering matrix (Y*for SU(3) and the matrix /T which diagonalizes (Y’ were given as nontrivial examples of the reordering theorem.
The author wishes to thank Dr. T. Ahrens for many discussions from which the basic ideas of this paper evolved and for several suggestions during its completion. The antthor wishes also to thank Dr. C. H. Braden for several discllssions and recommendations COIIcerning the use of Racah coefficient.s and Dr. H. R. Brewer for some enlightening commeut5 OII raising and lowering operators and their relation to Clebsch-Gordun coefficients.
RECEIVED:
March 1, 1966 REFERENCES
I. W. P.\~ILI,
.-inn. In.st. Henri PoincaG 6, 109 (19%). 2. M. FIERZ, %. Physik 104, 553 (1937). 3. J. I). JACKSON, “1962 Brandeis Summer Institute Lectures in Theoretical Vol. 1, p. 279. Benjamin, New York, 1963; G. KALLEX, “Elementary Part,icle p. 376. Addison-Wesley, Reading, Mass., 1964; R. H. (:OOD, Jlt.. Rer. Mod. 187 (1955).
Physics,” Physics,” Phf1.s. 27.
123 4. 5. 6. 7. 8. 9. 10. lf. 11. 13.
f4. 15. 16. 17. 18. 19.
FRAHAM
L. MICHEL, Proc. Phys. Sot. 63A, 514 (1950); E. R. CAIANELLO, Nuovo Cimento 10, 43 (1953); D. C. PEASELEE, Phys. Rev. 91, 1447 (1953); C. L. CRITCHFIELD AND E. P. WIGNER, Phys. Rev. 60,412 (1941); C. L. CRITCHFIELD, Phys. Rev. 63,417 (1943). T. AHRENS, Progr. Theoret. Phys. (Ryoto) 34, 867 (1965); see also, T. AHRENS, C. P. FRAHM, AND B. D. QUANG, Nucl. Phys. 78, 641 (1966). R. E. CUTKOSKY, Ann. Rev. Nucl. Sci. 14, 175 (1964). K. M. CASE, Phys. Rev. 97, 810 (1955). H. J. LIPKIN, “Lie Groups for Pedestrians.” North-Holland, Amsterdam, 1965. M. HAMERMESH, “Group Theory.” Addison-Wesley, Reading, Mass., 1964. M. GELL-MANN, “The Eightfold Way, A Theory of Strong Interaction Symmetry,” CTSL-20 (1961); M. GELL-MANN, Phys. Rev. 136, 1067 (1962). C. FRONSDAL, “1962 Brandeis Summer Inst,itute Lectures in Theoretical Physics,” p. 427. Benjamin, New York, 1963; R. E. BEHRENDS, et al., Rev. Mod. Phys. 34, 1 (1962). G. Rnca~, “Princeron Lectures on Group Theory and Spectroscopy,” reprinted in CERN 61-8. G. E. BAIRD AND L.C. BIEDENHARN,J.Math. Phys.6,1730 (1964). L. C. BIEDENHARN,J. Math. Phys. 4,436 (1963). M. E. ROSE, “Elementary Theory of Angular Momentum.” Wiley, New York, 1957. E. P. WIGNER, “Group Theory. ” Academic Press, New York, 1958. J. J. DE SWART, Rev. Mod. Phys. 36, 916 (1963). S. GA~IOROWICZ, “A Simple Graphical Method in the Analysis of SU(3),” ANL-6729 (June 1963). U. FANO AND G. Racl~, “Irreducible Tensorial Sets.” Academic Press, New York, 1959.
OF
PHYSICS:
On
40,
The
159-178
(1966)
Reordering
of Biquadratic
CHaRLES
Georgia
Institute
1’.
Scalars*
hAHI
of Technology,
Llflcmfn,
Georgitr
A t hrorem analogous to the Fierz reordering theorem for Lorentz scalars c*onstrwted from 4 Dirac spinors is proven for all semisimple compact grollps. The proof of the theorem depends on only -I factors, i he most important of which are the reality of the generalized Clehsch-Gordan coefficients for semsimple grollps and the uniqueness of the constrllction of a scalar from a rcpw sentatioll of a compact group and its contragredient. The reordering matrix asso&ted with the scalars formed from four “octets” of Sf’(3) is also givctl along with an orthogonal matrix that diagonalizes the octet-reordering matris. I. INTRO1)UCTIOiY
Since the formulation of the Dirac equation in 1928 several authors havc~ COIIsidered the behavior of the five Lorentx scalars formed from four A-spinors urld~ permutation of either the “barred” or “unbarred” spinors (1). The result of tJhis so-called PI, ( or Pz4) operation on the Lorentz scalars is contained in the “l~icrz reordering theorem” (2)) d’lscussions of which are available in several plain. (See ref. 3.) The invariants of the PI3 (or Pz4) operator have heer~ considrred by Feveral authors both in the theory of beta-decay and in the sexrcsh for a universal Fermi interaction (4). In fact, the accepted V-A theory of beta decay is such :UI invariant if the renormalization effects of tjhe strong int,eractions are neglected. This PI3 invariance of the bare V-A interaction is to a large extent ignored IIOW adays; holr-ever, one cannot help but wonder if it has some underlying sigllificxnce. C’onsidering this significance to lie in a self-consistent analogy of particalcs Ahrens (5 i has investigated the possible struct)ures of the weak intern&ion :IX determined by permut,atiori symmetry in isospace. A natural extension of llr:it work in light of the success of the S&‘(3)-oc*tet’ model for hadrons (6;) is IO illvestigate the permutation properties of biquadratjic S17( :?,) sAars. This p:1p~r is intended to lay t#he ground work for such investigat,iorls. Since the Lorentz group and the special unitary group in three dimc&ons, SU(3 j, are structurally quite different, the existence of a reordering t.heorrnl of the E’icrz type is questionable. It will be shown in Section I\: that not, only tlocts such a theorem exist for XU(3) but it exists at Icast for all scmisimple mnpwt groups. * Work
supported
in part
by
the National
Hcience 159
Fulultlat
iota.
160
FRAH.M
I<. M. Case (7) has shown previously that a Fierz type reordering theorem exists for complex orthogonal groups. His proof requires the existence of a set of quantities I’(i) having commutation properties of the form
r(i)r(j)
+ r(j)r(;)
= taij
Since such commutation rules do not occur for compact groups in general (sU(2) is an exception) Case’s proof is not applicable to such groups. It is worth noting that Case’s theorem and the reordering theorem presented here are essentially complementary and together they cover most groups of physical interest. Two examples of the PI3 operation will be presented in Section II. These examples concern the reordering of the simplest nontrivial xU(2) and SU(3) biquadratic scalars. As a preliminary to the statement and proof of the reordering theorem in Section IV the question of the reality of the Clebsch-Gordan (C-G) coefficients for semisimple groups will be considered in Section III. The general properties of the reordering matrix will be discussed in Section V. The next section will consider the definition and simplification of Racah coefficients, the results of which will be used in Section VII to obtain a relation between the reordering matrices associated with SU(3) and certain XU(3) Racah coefficients. Section VII will also give the simplest nontrivial reordering matrix, (Y*, occurring in the octet model of hadrons and an orthogonal matrix U which diagonalizes CX’. II.
EXAMPLES
Two elementary examples of possible physical importance will be discussed in this section in order to become familiar with the P13 operation. Consider first the isoscalars formed from two nucleons and two antinucleons. There are two such scalars : S = m,N&,N, and V = N~TNZ . fl,~Na where the N’s are isodoublets, matrix vector. gives
Expanding
e.g., N =
and assuming
P n , and T is the usual 2 X 2 isospin 0 the various field operators to commute
and
In this case and in the following
SU(3)
example the P24 and PI3 operations
are
REORDERING
equivalent
OF
BIQUADRATIC
l(il
SCBLARS
so that, P24 v = g$3 - xv,
In general Pz4 and PI3 are equivalent If t,he matrix (Y is defined by
etc.
up to a phase.
then clearly
Since the square of PI, leaves the scalars unchanged cy must, be t*he unit- matrix as is indeed the case. The PI3 invariants are now easily seen to bc V + 3S
and
1’ -
S
with eigenvalues +1 and - 1 respectively (6). Next consider the XU(3) scalars formed from two sakat’ons and two antisakatons (8). Again t,here are two scalars. This can be seen as follows. The sakatons belong to the irreducible representation (IR) 3 of SL’(3) while the antisakatons belong to the contragredient IR 3”. The IR’s of XU(3) are labeled by their dimensions and an asterisk is used to denote the corresponding cant ragredient IR. The Kronecker product 3* X 3 decomposes into the dirwt ~111 1 + 8. Hence (3* X 3) X (3* X 3) = (1 + 8) X (1 + 8) = (1 X 1) + (1 X 8:) + (8 X 1) + (8 X 8) Kow in the decomposition of a direct, product of t,wo IR’s of SI -(X ) :L w:~l:u will appear once and only once if and only if the product involves an IR and its contragredient (9). Furthermore, for StT(3) an IR is equivalent. to its c:ontr:rgredient if and only if its dimension is the cube of an integer. Thus it is clear I hat in the above expression one scalar appears in 1 X 1 and another in 8 X 8. Thwr scalars are respectively x = 6&26&l4 and I’ = c 6Ab&Xib4 z where h = the “bars”
P n is t’he sakaton and the X’s are Gelt-JIann’s ma.triws 0A simply denote t,he complex-conjugate transpose.
(10). Hcrca
162
FRAHM
Again expanding
and assuming the various - -
factors to commute gives
and
This can be written
in matrix
where the reordering
form as
matrix p is given by
In this case the P13 invariants
are
S + ?dV with
eigenvalues III.
and
S - p;V
+ 1 and - 1 respectively.
REALITY
OF
THE
CLEBSCH-GORDAN
COEFFICIENTS
After considerable effort the commut’ation relations for the generators of a semisimple Lie group can be cast into the following canonical form (11, Id). [Hi ) Hf] = 0
(1)
[Hi) E,] = criE,
(2) (3)
[E, , Eal = Nm&e+~,
if
a! + P # 0
(4)
The number of mutually commuting generators, Hi , is called the rank, I, of the Lie algebra of the group while the total number of independent generators is called the order, r. Equation (2) is conveniently thought of as an “eigenvector problem” where the E, are simultaneous eigenvectors of the 1 commuting generators, Hi, with eigenvalues, pi . The eigenvalues are said to form a root vector (or simply a root) a! = (011, a, These root vectors are notation Ea+p in Ey. eigenvector associated @ is not a root Nap is
.** ffl)
completely determined by the structure of the group. The (4) thus means that if a~ + /3 is a root then Ea+p is an with that root and Nao # 0. On the other hand if a! + zero.
REORDERING
OF
BIQUADRATIC
Ifi:
SCALARS
By suitably normalizing the generators (which has already been assumed ~II Eqs. (l)-(3)) one obtains ( *-Ii
T (Y@j = &j mcl
Nap
=
-N-a,-0
=
zt
Z/(k
+
l)(j
+
1)
/ B 1
i(j)
The sum in Eq. (,.5) is over all root vect,ors while the numbcrsj and k: in 1.21.( ti 1 are the smallest positive integers such that
[E&p
) E-)9]
=
0
respectively. The numbers j and k are completely tletc~rmilled by the roots :111tl can be easily evaluated by inspection of the root diagram ( 11). The N’s also s:tt,isfy (11) Nap = ND, -a-/j = AL-,
, cL
171
and t,he trivial relation Nao = -NBC
(XI
III pract,ice one useseq. (6) to select a suitable set of N's :md t$eu procectls to find the matrices of the generators in a given representation with t’he aid of l’(is. (3) and (4). When following this procedure the sign in Eq. (A) rnus~ he chos;rn so as t,o be consistent with Eqs. (7) and (8) but is otherwise arbit’rary. One apparent motive for casting the commutation relations into the c*anollic~;ll form, alt,hough it has not been explicitly indicated in the literature, is th:tt it allows one to choose the (matrix representatives of the) I?‘, real. Since by &;(I. (3) the E’, are simply raising and lowering operat,ors, the reality of thy /5,Z determines the reality of the C-G coefficients associated with the group. In order to show that the l~‘~ can be chosen real csonsidrra t~hcortm tlucb to Cartan and discussedby Racah (12) which proves that the eigenvetstors of l*:(l. (2) are nondegenerate if the root vectors are nonzero. Thus by t:~king t,he tr:~~s;pose (T ) and also the hermitian adjoint (t ) of the c:moni(*al commutation relations and making use of Eq. (6j one sees that 13,” and sat are both proport)ional to IL, . Thus either EaT or E’k cwl be chosen equal to K-, (‘hoositlg &J = Jj’_, Nld
i !I t
164
FRAHM
it only remains to be shown that the remaining can be set equal to one for all CYso that
constant
of proportionality,
Eh = EaT = Es,
A,
(11)
The equality in Eq. (9) is the conventional choice. However, no explicit statement concerning the equality in Eq. (10) nor of t,he freedom to establish Eq. (11) seems to appear in the literature. The Hi are only of secondary concern here since they can be taken to be hermitian and diagonal and hence real in all representations of the group. This in turn implies that the roots (Y are real (11). However, it is important to keep in mind that the Hi matrices along with the root vectors determine the representation of the group involved. Equations (9) and (10) establish that E, differs from a real matrix only by a phase which is the same for all the matrix elements of E, . If one sets E a = +E where E,' is real then becomes
E,'
automatically
’ c-4
satisfies Eqs. (2) and (3) ; and Eq. (4)
The exponential is real and thus equal to fl since the other factors are real. This is essentially the arbitrary sign in Eq. (6). Hence the exponential can be absorbed into Nap . In fact if one defines j&
= Naaei(+~+F--+8)
it is easily seen that the N& satisfy Eqs. (6)-(8) if the Nap do. This then establishes that the generators can be chosen real without disrupting the canonical commutation relations in any representation which diagonalizes the Hi . Now consider the C-G coefficients (jlcLl~l;
jZP2VZ
I j(Y)W)
defined by I (M~(~)PV)
= g (a‘1 PlV1 c2yz
;j2/.42v2
I j(Y)W)
/ jlWl)
( j2P2V2)
(12)
where 1. j, ,j2 and j label IR’s of the group. 2. The y’s are multiplicity labels which must be used if an IR can appear more than once in the decomposition of the Kronecker product of two IR’s of the group (i.e., the group is not simply reducible (12, 13)). 3. Vl ) vz ) and Y are vectors whose components are the eigenvalues of Hi which partially label the states of the IR, i.e., the v’s are weight vectors (11, I%‘), and,
REOR.DERING
OF
BIQUADRATIC
I 65
SCALARS
4. The P’S are other labels which are required to distinguish bet,ween stales with the same weight (12, 14). Here as in the remainder of this paper (j,~) denotes the st#ate labeled by p and v in the IR labeled byj. If the state has been obtained by coupling two other sets of states as in Eq. (12) these states will be placed in pztrentheses before the j while the multiplicity label, if there is one, will be placed in parent,heses xftcr the j. The states ( (j,j,)j(,),v) form a complete orthonormal set as do thy 1j,,). Thus
and (j,v I kw)
i 131))
= ~jk&&
If 1j1~1v1) and 1~Z~ZVZ)are assumed to be in different lowering operators for ) (j&)j(,),v) can be writ.ten E, = E&ii where
E,C i 1 operates E,(i)
only on 1j;pivi).
lj;/.l;v;)
= c
Iri’
spaces t,he raising and
+ E,(2)
( 11)
Kow from F:q. ( 12 )
L&.hLjl,jipivi)
Ij;&;
+ a); i = 1, 2
(1.51
where the X, are real since they are just the matrix elements of E,( i‘). Operat.ing on Eq. (12 ) witch E’, t’hus gives F X,(,b4’,jc(vJ I (.hh)j(7)Lv
+ a> = ~~(jt~ln ll2VP
; jmv2 Ij(y)wj
In the first term on the right hand side replace the dummy index vl by v1 - u and interchange the dummy indices ~1 and ~1’. Similarly in the second tc>rm replace v2by V?- cx and interchange ~2 and p2’, Equakion i ltii then bec*otncls 5 Lb’,
jw) I (j&>j(7),‘v
+ a>
= C [CXrr(~l,jl~ulIvl-~)(jlcll’v~ PlVl 111’ P?V2
- ~Y;.~~P~v~I~(Y)cLV)
+ c X&2, j2dv2 - a)(jl~c11VI; J?c12’v2 - a lj(~jpv)] c-2’ x lj1,1v1)Ij2,2v*)
(17)
166
FRAHM
Solution of Eq. (17) for the 1 (j&)j(y)p’v for the C-G coefficients
in terms of the A, which
+ a> determines
a recursion
relation
are real and the coefficients
Clearly the C-G coefficients for the weight v + ar will be real if t.hose for v are. This will be the case if the C-G coefficient for the state in j with highest weight (11) (i.e., the state in the IR whose weight has the largest positive first component) is chosen real. The phase of this “highest state” can always be adjusted so that this is the case. IV.
THE
REORDERING
THEOREM
Let J/ and I#Jbe column matrices whose elements are the orthonormal basis states of the IR j of a semisimple compact group G. Similarly let the elements of q and CPbe the orthonormal basis states of the IR j* of G. Thus, for example, the (pv) component of 1c,is given by J/(pv) = 1j~v). Now assume that the IR K appears in the decomposition of the Iironecker product j* X j. The IR K” must also appear in the decomposition since j X j* is equivalent to j* X j. Thus one may, for example, construct from q and 4 quantities which transform according to the IR K” and from Q, and 4 quantities which transform according to the IR K. These can be expressed with the aid of the C-G coefficients as gl
(j*/m
; jw2
I K*(~Pv)
WPI~I)~(P~~~)
my2
and
respectively. by
From these quantities
it is possible to construct
&rskw, which transforms as a scalar under G. The Pz4 operation (which is equivalent interchanges # and 4. That is
P&,,a,w, The reordering
is then expressed
P&w
(W, w>
denoted
W) to P13 up to a phase)
WI = &a,m
theorem
a quantity
WI
in this case (18)
by the equation
= c ~~(rs)J(Br)S~(Br)(\k~, JBr
w>
(19)
KEORDERIPiG
OF
BIQUADRATIC
SCALilRS
167
where the sum is over all IR’s occurring in the dec~omposition of ,i* X j. liic~u:~t~ior~ (19) is convcnient~ly written in matrix form as P2&
= oljsj
! “0 I
where X’ is :L c~olunmmatrix whose elements are t,he sc~xl:~ S:coa, and LV’is 1hc “reordering matrix” whose elements are t,he numbers 0l~(,g,,,,,3~, . The proof of the reordering theorem depends on only four conditions: ( 11 the ‘i(,oilll~leteIless” of the C-G coefficients, ( ‘1) the reality of t)he C-G coefficients, ( 3 ) the symmeky of the C-G coefficients under interc~h:rngcof the two f:tc*tot states /jlFlvlj and ( J’~pzvz) in Eq. (12), and (4) the uniqueness of the construction of :I sc*xlnr from a11 IR and its c&or11 IXgredient . Condition (2 ) is snt,isfied by semisimple groups as was shown in the previous section, (4 i is satisfied by compact groups (see ref. 9, pl). 147 and 317j :tntl c*onditions ( 1 ) and (3) are satisfied in general (9 ) iat least whenever it is m(‘:uingful to define C-G csoefficients). Since the C-G coefficient,s are real and transform an orthonormal basis into XI orthonormal basis, they form a real orthogonal matrix. Definition ( 12) md its inverse then yield what will be referred to in t)his paper as the (‘4~ orthogonnlit y and completeness relations. CMhogonalit,y :
(‘ompl&eness:
where the 111’s represent all of the appropriate state labels. The symmetry property mentioned in condition ( 3) arisesfrom the equiv:ll(~rlc(~ of j, X jz andj2 X j, for arbitrary IR’s j1 andj3 . This rcquircs that (j2m2 ; jl~ml / J(-y)M)
= &(j,jlJ(y))(jlm
; j2ln2 1J( y‘) :\I)
t 2:i 1
where [I is a real phase factor which is independent of the st:tt,e labels 1~1, M:! , and ill. Since only those C-G coefficients will occur in the following which oo~plr~ :m IR and its cont,ragredient it is convenient to int)roduc*c tJhc matrices (j./(r ).I/ ) whose elements are defined by
where a and b are all of the appropriate state labels. The orthogonnlity, complA(~-
168
FRAHM
ness and symmetry
relations
then become respectively
and
(~*Jwcwn, Substituting
= c;l(ii*J(Y) )(jJ(rwbrwn,
(27)
Eq. (27) into Eq. (26) gives
NOW let 0 and e1 be two arbitrary nj X ni matrices where ni is the dimension of the IR j and multiply Eq. (28) by @cTe)m2&,$~Q(e’#),, . Summing on the m’s gives
Equation (29) is the analog of Eq. (81) in Case’s paper (7). Choosing 0 = (jK*(e) - N) and 0’ = (jK(s)N), multiplying by the “l-j symbol” (KlO)-NN and summing on N gives
(30) @TW*(d
- N)(~*J(~>M)(~K(~)N)~~(~J(~)M)~
The 1-j symbol @lo)- NN is simply the appropriate set of C-G coefficients for combining states which transform according to the IR’s K* and K to give a state which transforms as a scalar (which is labeled by 1 and whose single component is labeled by 0). The symbol -N labels that state which must be combined with the state labeled by N in forming a scalar quantity. Now the left side of Eq. (30) is a scalar under the group. Hence the right side is also a scalar. Since the quantity aT(jJ(y)M)4 transforms according to the IR J, the scalar character of Eq. (30) requires the remainder of the expression under the sum on M to be the appropriate 1-j symbol times a quantity which transforms as the -M component of the IR J*. This is a direct consequenceof condition (4). Hence
t,(j.;J(,))
F WO)-NNW*(~
- W(~*J(r)MX~~(GV = T &~J&~ (JlO)--M&J*(P)
- M)
(31)
REORDERING
Equat,ion
OF
BIQUADRATIC
169
SCALARS
(30) thus becomes ( 32 1
where
Equation
(32 ) is simply the reordering V. PROPERTIES
OF
theorem of Eqs. (18 1 and ( 19).
THE
REORDERING
MATRIX
Since the square of P,, leaves the scalars unaltered, (J)’ Thus (Y,’ is it,s own inverse. Now consider the following ~~~~ces,iw, W), = -g
“%Br~(W, az (jk’*(eJ
Eq. (20) implies that
= 1
inner product
( :3-l I
of the scalars in Eq. (33 )
W)) - N),b(K10)-NN(jhT(8)N)cri(j.l*(P)-ill)E~
ef !lh
( :i.-, I
X (JlO)-,,(jJ(y)n~),, x
(*(a), (4(b),
J/(e)
ddf))iw.c),
1) +(y))iticn),
1C/OI).r
This i, the usual inner product, for composite systems encountered in ctuantum mechanics. Sow by virtue of Eq. (13) (recalling that the state labels have been compressed t,o a single index) (*(a),
q(e))
= 6,, ,
etc.
Thus Eq. (,35) becomes &,dw,
W,
&dw,
w> j
where the last equality follows after several applications of Eq. ( 2.5) _ Equation ( ;36j demonstrates t,hat the reordering matrix transforms one orthonormal set of quantities into another orthonormal set. Hence the LY’ must he unitary. l’urt,hermore the cuj are real by virtue of Eq. (31) and the fact &at the C-G coefficients are real. Combining these results with Eq. ( 34) implies that the reordering matrices are real, symmetric, and orthogonal. Thus Lyj z!T (g)* whew
* means complex conjugate.
= (&T
= $-’
(37 j
170
FRAHM
An additional consequence of the orthogonality of the reordering matrix is the implication that there are the same number of P,, (or PI,) invariants as there are scalars in the decomposition of the Kronecker product (j* X j) X (j* X j). VI.
SU(3)
RACAH
COEFFICIENTS
In this section the definition and structure of the Racah W-coefficients for the group G will be discussed. The definition of the Racah W-coefficients to be used here is analogous to that given by Rose (15) for the corresponding SU(2) quantities. Consider the coupling according to Eq. (12) of three orthonormal sets of quantities ( jlnzl), ( j,m,> and 1j,m,) which transform according to the IR’s j1 , j, and j, respectively of G. One could couple first ]j,ml) and 1j,nz,) to give 1 (j,j~)j’(r’)m’) and th en couple this to 1j,m,) to give 1 (( j,j,)j’(,‘)j,)j(,),,) or as one alternative couple / jgm,) and / j,m,) to give 1 (j&)j”($‘)m”) and then couple this to (j,mJ to give 1 (j,(j,j,)j”(y”))j(p)m). The two sets of orthonormal quantities I ( (j,j,)j’(,‘)j,)j(,)n2) and ( (j,(j,j,)j” (y”) )j(p)nl) span the same space and hence must be related by a unitary transformation.
I ((j,j,)j’(,‘)j,)j(,>m) This relation defines the W-coefficients for G. It is important to note that since the quantities in Eq. (38) belong to the same m the W-coefficients are independent of m. (In this connection see ref. 16, pp. 115 and 298.) Using Eq. (12) to express the composite quantities in terms of ( J&I), 1j2m2), and I j,m,) gives
= & X (Jim;
%;? ~~w(j,j,j(YP)j3;j’(y’)j,r(yN)) TlQ?lZ”
jnmrr I j(Ph)(j
(39)
~m~;j~m~Ij”(~“)mn)jj~m~)Ij~m~)Ij3m3)
The orthonormality of the products 1jImI) 1j,m,) / j,m,) requires their coefficients on the two sides of Eq. (39) to be equal. Thus Zt$ (j’m’; jm
Ij(y>m)(jm = jg,,
;j2m Ij’(-f’)m’)
~~jp~jrW(jlj!ij(~~); i(T')j"(T") 1 X (jImI;
Using the orthogonality
jrrmlr Ij(P)m)(j,
of the C-G coefficients
rn~;j3m3(j”(~“)rn”)
twice then gives
(40)
REORDERING
W(j,
j2 j(yPjj,;
j'(y')j"(y"))
OF
=
BIQUADRATIC
(12jnnj')-l"
171
SCALARS
C
j3m3
(.ym';
j j(y)m)
m,m2m3 WL'7fS"
X (j, ml; j2m2 Ij’(y’)m’)
(j2m2 ;j3m3 Ij”(y”)m”)(j~
ill)
ml;j”mN
/.j(p>m)
for all values of HLin the IR j. This result is, of course, the same as in t,he 1st ’ ( 2 ) wse with two generalizat,ions ( 1)) the multiplicity labels have been int.roduwcl, and (2 j, the state labels and corresponding coupling rules may now be mow complicated than in the XIY(2) case. Specializing G to the group SC(3) one can split the #C-(3) C-C; coefficients in t#he customary way into an sL’(2) C-CT coefficient and an “isoswlar fwtor” (j*??l~ ;j2 ?)121j(y)nL)
= (j,I,Y,
; jJ,I’,
1j(y)IIr)((I*I,r
; IrIp, j IZz))
(41’ I
where the double “bra-kets,” ((e . . 1 . . e)) denote the SI ‘(I)) (‘-G coefi?c.ients and the quantities (. . . I . . . ) are the isoscalar factors (17 j. The st.att: lab& have been chosen to be t,he “isospin, ” “z-component of isospin,” and “hyprrc*harge.” That is, w = (I, I, , Y), etc. Making t,he splitting for S17(8) in l’q. (41) and recombining t#he X1’(2) C-G coefficient H according to the same ~qu:~tion gives
x Cj’l’Y’;
j, 13 Ys I j(y)IY)(j,
x (j, I2 Yz;j313
II Yl ;j, I, z’s / j’(y’)l’Y)
I’3 ~j”(YN)I~YN)ijlllY1;j”l”I”’
lj(/3)IY)
where WZ denot’es the XI-(2) Racah coefficients. The additivity of hypercharge can be used to evaluate the sums on I’, , Y2 , and I73 . I;urthermore t.here is a unique state in every IR of Sf-(3) for which I = 0. (See ref. 18, pp. 26-28.) This state can be used to simplify Eq. (43) since the SC(3) W-coefficients are independent of the choice of I and I’ (provided, of course, that a state with the labels I and I’ exists in the IR j). This gives ~l’(jljzj(rP)j3;j’(r’)j”(r”))
=
(TLj"njT)-"2
x (j’I3 I”; j, 13Y x (j, II Y -
Y’ 1j(y)OY)
YN; j, 1,Y + Y” -
x (j,I, Y’ + I”’ - Y;j313Y x (j, II I7 -
-
y /j’(y’)& Y’ ljN(-fn)ll
I-“; j”1, Y” 1j(p)or-)
I-‘) I-“j
(44)
172
FRAHM
In obtaining this expression the sums on I’ and I” were evaluated by noting that in order to obtain I = 0 it is necessary that I’ = I3 and I” = Il. The remaining Wp coefficient is well known in the literature (16, 19). It is given by
W2U1I2OI3; I3Il) =
6(L) I,, I2) 4211
(45)
+ l>(21, + 1)
where 6(11,13,12)
= 1 ifII+Iz+13isaninteger and fl + I3 >= 1, >= / I1 - I, 1 = 0
otherwise
Thus W(j&j(rp)j,
;j’(y’)j’(,fl))
= (?7++,)-1’2
x (j’I,Y’;
j&Y
-
x (jlIlY
-
x (j2IzY’
+ Y” -
C
C
It1213
Y’YN
fi(Ii,
Is,
12)
Y’ 1j(y)OY)
Y’f;j,I,Y’
+ Y” Y; jJ3Y x
-
(jlI,Y
Y Jj’(q)IsY’)
(46)
Y’ Ij”(yN)IIY”) -
YN;j”IIY”
1j(P)OY
This expression can now be used to evaluate the simpler XU( 3) Racah coefficients using tables of isoscalar factors (see, e.g., ref. 17). VII.
APPLICATION
TO
THE
/W(3)-OCTET
MODEL
A relationship between the SU(3) reordering matrices associated with the octet model of hadrons and certain #U(3) W-coefficients will be established in this section. Such a relation would follow immediately in the SU(2) case where the extensive Racah algebra of 6-j and 9-j symbols is available (15, 19). By multiplying Eq. (31) by [&(jj*J*(p))(JIO)-MM]-‘(j*J*(P) - M) and taking the trace the following expression for the matrix elements of cr’ is obtained. 1 ah(ea)J(py) = El(jj*J*(p>)sl(jj*J(r)) X
Td(S*(r)
c (KlO)-NN N (JlO)-MM
- N)(j*J(r)2M)(jli(G)N)(j”J*(P)
(47)
- WI
Now in the octet model the hadrons are assigned to the various IR’s of SU(3) which occur in the decompositions of the Kronecker products of the 8-dimensional IR of XU(3) with itself. For all of these IR’s the hypercharge is integral. Because of this one can introduce a particularly convenient phase convention due
REORDERING
to de Swart
(17) relating
OF
BIQUADRATIC
173
SCALARS
the states of the IR’s j and j*: / j*,t,g = (j --rn 1 ( - 1)”
(48)
where (j - ~1 1 = ( j -no)*
(*denotes
complex conjugate)
and
m = I: + Y/2 Using this phase convention de Swart ties of the SVi3) C-G coefficients. (j1m;j*m
lj3(,h3)
=
E2(jl.M3(Y))(--
0
1)"'
x
(jl ml ; j2 rn? 1j3(7)m3)
derived the following
(jl
symmetry
proper-
l/2
E
ml ; j,*
-
( 49 ) m3
= (3(j1 j2j3C-i) > (.A* - ml ; j2* -
1j2*(7')
-
m3)
m2 I j3*(-f> - m3)
CM)
The t’s are real phase factors which are independent of the state labels 1)~~, 111:!) and ‘w. The validity of Eq. (49) is questionable whenever there is more than one value of y. However de Swart, has established explicitly that it is satisfied for all casesof present interest. These symmetry relations can now be used to rearrange Eq. (47). Several phase factors will be introduced by the symmetry relations during this rearrangement process. These phase factors will be denoted collectively by x. Of course, t’he value of x will change from step to step but only the phase factor for the final expression will be recorded. Indicating explicitly the sums on t’he matrix indices, changing the dummy index N to -N and applying Eq. (50) to the last two factors in the sum Eq. ( -I-i’) becomes
x (j*u;jb 1K*(E)N)(jb; j*c 1J(yjM)(j
(51)
- c;j* - cl j k’*mq x (j* - d;j - a I J(B)Jo
Applying Eq. (49) to the second and fourth factors in the sum gives ah)-%%) = (J” _ fif; J&, 110) z 0
F zd (-
x (j*a; K - N Ij*(X> - b) (jb; j*c / J(y)M)(j
l)“-‘:(K*N;K
- N I 10)
- c; K - N ) j(X’)
d>
x (j* - cl; j - a ) J@)M)
“2‘)
174
FRAHM
where
X and X’ are such that
(j*u;jb
( K*(c)N)
= C;2(j*jK*(~))(-
1)”
2
0 3 (j - c; j* - d ( K*(6)N)
= &(jj*K*(S))(-
l)-’
“* (j*u; K - N 1j*(X)
- b)
z 1’2 0 1
x (j - c; K - N 1j(h’) Now using Eq. (23) on the last two factors &~).%%)
=
(J”
_
$
J&f
z
( lo>
d)
gives
T azd (-
1)“~‘(K*N;
K - N ( 10)
0
X (j*q
K - N 1j*(X)
- b) (jb; j*c 1J(r)M)(K
- N; j - c 1j(x’)
d)
(53)
X (j - a; j* - d I J(P)M) Finally gives
using Eq. (50) on the second and fourth
&).Qr)
=
factors in the sum of Eq. (53)
K - N 110) j 10) 0 2 T & (- l)“-‘(K*N; (54) Jj(X)b) (jb; j*c ] J(y)M)(K*N; j*c Jj*(h’) - d)
(J* - M;JM
X (j - a; K*N
X (j - a;j* ( - 1 )“-’ (K*N;
T\Tow consider the factor According
to de Swart
- d( J(p)M)
K - N 1 lO)/(J*
-
M; JM 110)
(17) (J*
-
M;JMj
lo)=
'-;y+ nJ
where M, * is the highest weight in J*. Thus
a E (K*N; (-
‘)
-
(J*
_
K - N 110) M;
JM
j 10)
(55) =
(-
‘)
(NH* is the highest weight in the IR K*). Clearly the summand of Eq. (54) gives a nonzero contribution only when 6+c=iL?and-d+iV=6 or equivalently when m --B + F = i@. Thus recalling t,hat the barred quantities are integers in the octet model it follows that - (-1 )a--i+N+M = ( -l)‘N-“+“‘+H-- = (47 = 1. Hence
&s,rcpr)
= xdny
3
5 agd (j - a; K*N
I j(X)b)(jh
j*c I J(r)M)
X (K*N; j*c I j*(X’>
- d)(j - a; j* - d I J(@)M)
(56)
REORDERING
OF
BIQUADRATIC
SCALARS
In Eq. (56) the value of x is given by
Comparing
Eq. (56) and Eq. (41
j it is clear that
= x d?LJ7Lg Tl’(jK*J(rp)j*;
&,P)J(P,)
j(x)j*(x’,
( .5x i‘
\
By using Eqs. (16 ), (57), and (58) one can now evaluate the elements of the reordering matrices associat’ed with the octet model whenever Eq. (49) is vali(l. Of course, all the reordering matrices associated with Xrr(3) may be evnluat,ed using Ey. (47) even if instances are encountered where Eq. (39 ) is not valid. In this circumstance Eq. (581 would have to be abandoned but, one c+ould si ill simplify Eq. (47) for computational purposes in the same manrler as the It-coefficients were simplified in Section VI. One would, of course, have to be careful to use a cxonsist#ent phase convention. The result of this simplification nsillg de Swart’s phase convention for t,he octet model is
x (j*r, I- - I”; jr, I” / J(y)OY) (j*r, I’ - Y’;j& - Y” [ K(B)12 1’ - I”’ - 1.’ ! (j*r, I-” - F;jl, - IT” / .l*(fgo - 1-1 Equat’ion ( 59) was actually used to evaluate the matrix elements of the S I ‘( 3 ! reordering matrix a8. Recall that’ the IR’s of SC’(3) arc labeled by their tlimensions. Since Csee refs. 10 and 11) 8 X 8 = 1 + 8(l)
+ 8(2l
+ 10 + lO* + 27
((i0)
t,here are eight scalars occurring in the decomposition of (8 X 8) X (8 X X ) ; one occurring in the decomposition of each of the following terms 1 x 1,
8(l)
X 8(l), 8(2)
B(l) X 8(Z),
X S(Z),
8(2)
10* x 10,
X 8(l), 10 X lO*,
and
27 X
27
This is a consequence of condition (4) of Section IV and t’he fact that for Sc’( 3 ) an IR is equivalent t#o its contragredient if and only if its dimension is the cube of
10 lo* 27
81 (1, 1) 8 (1, 2) 8 (271) 8 (2, a
k’(cS)/J@r)
1 8 (1,1) 8 (1, 2) 8 (2, 1) 8 (2, 2) 10 10* 27
li’@)lJ@r)
- 3 d/55/44
d&3 - dE/56 - di/2a 0 d&,11
1
- -\/ii98 3 d/3/8
d10/8
l/8 - da4 0 $a/4 -
1
0 0
0
5 l/ii/22 3 &a/110
d&o
w,
1)
2/w - VW5 - 3 ~/6/20
l/2
- VW4 -3/10 0 0
8U,1)
2) 0l/2 :
0 0
&/‘3
d/3
80,
0
l/2 w
0 0 -l/2 -l/2 0
so, 2)
1)
u
0 0
VW2
-l/20 0 8
80,
TABLE
0 0 --1/2 -l/2 0 -l/2 -l/2 0
80,
TABLE a*
II
1)
I
- m/22
0 2&o/7 x6/14 0 d&22
w, 2)
0 0 VW4
l/2
0 0
l/2
- VW4
w, 2)
d/6/6 ~59’5 - d!w22 2/si?i/44
-
1/i/4 l/2 -SF?
10
0 l/4 -l/4 - &G/40
-l/2
l/2
10
l/2
&z/22 - vz?/44
~‘36
- d/3/4 l/2 d/35/28 12;;
lo*
0 -l/4 1/4 da/40
-l/2
10*
d/S 5 2/z/56 -3 m/140 0 0
27
3 d/3/8 - 3 2/i/20 0 0 v%/4 - diw40 d/30/40 7/40
27
E
2
REORDERING
OF BIQUADRATIC
SCALARS
177
an integer ( 11) . These scalars will be denoted respect8ively by
The corresponding reordering matrix a8 with the rows and c~olumnslabeled in the sameorder as in the set (61) is given by Table I. It is worth noting that Tr CZ’= 0 so t,hat in this case there are four Pz4 iI]variants with eigenvalue +l and four with eigenvalue - 1. The real ort,hogonal matrix t’ given in Table II diagonalizes LY’.Thnt is
where 1 is the 4 X 1 unit matrix. Thus I’ can be used to construct, the P,., variank VIII.
irl-
CONCLUSION
4 proof of a reordering t,heorem of t.he Icierz t.ype has been given which is wrlitl for all semisimple compact groups. It, is, of course, equally valid for finit’e groups. The restriction to semisimplecompact groups, although sufhrient, for the proof of the t.heorem present,ed here, is not. in general necessary as is exemplified by ihe Fierz theorem itself. ,411auxiliary proof of t’he freedom to chooset)he C-G coefficients real in the (we of semisimple groups was also provided since it is important to the proof of t.he reordering theorem and apparently does not appear elsewhere in the literaklre. The reordering matrix (Y*for SU(3) and the matrix /T which diagonalizes (Y’ were given as nontrivial examples of the reordering theorem.
The author wishes to thank Dr. T. Ahrens for many discussions from which the basic ideas of this paper evolved and for several suggestions during its completion. The antthor wishes also to thank Dr. C. H. Braden for several discllssions and recommendations COIIcerning the use of Racah coefficient.s and Dr. H. R. Brewer for some enlightening commeut5 OII raising and lowering operators and their relation to Clebsch-Gordun coefficients.
RECEIVED:
March 1, 1966 REFERENCES
I. W. P.\~ILI,
.-inn. In.st. Henri PoincaG 6, 109 (19%). 2. M. FIERZ, %. Physik 104, 553 (1937). 3. J. I). JACKSON, “1962 Brandeis Summer Institute Lectures in Theoretical Vol. 1, p. 279. Benjamin, New York, 1963; G. KALLEX, “Elementary Part,icle p. 376. Addison-Wesley, Reading, Mass., 1964; R. H. (:OOD, Jlt.. Rer. Mod. 187 (1955).
Physics,” Physics,” Phf1.s. 27.
123 4. 5. 6. 7. 8. 9. 10. lf. 11. 13.
f4. 15. 16. 17. 18. 19.
FRAHAM
L. MICHEL, Proc. Phys. Sot. 63A, 514 (1950); E. R. CAIANELLO, Nuovo Cimento 10, 43 (1953); D. C. PEASELEE, Phys. Rev. 91, 1447 (1953); C. L. CRITCHFIELD AND E. P. WIGNER, Phys. Rev. 60,412 (1941); C. L. CRITCHFIELD, Phys. Rev. 63,417 (1943). T. AHRENS, Progr. Theoret. Phys. (Ryoto) 34, 867 (1965); see also, T. AHRENS, C. P. FRAHM, AND B. D. QUANG, Nucl. Phys. 78, 641 (1966). R. E. CUTKOSKY, Ann. Rev. Nucl. Sci. 14, 175 (1964). K. M. CASE, Phys. Rev. 97, 810 (1955). H. J. LIPKIN, “Lie Groups for Pedestrians.” North-Holland, Amsterdam, 1965. M. HAMERMESH, “Group Theory.” Addison-Wesley, Reading, Mass., 1964. M. GELL-MANN, “The Eightfold Way, A Theory of Strong Interaction Symmetry,” CTSL-20 (1961); M. GELL-MANN, Phys. Rev. 136, 1067 (1962). C. FRONSDAL, “1962 Brandeis Summer Inst,itute Lectures in Theoretical Physics,” p. 427. Benjamin, New York, 1963; R. E. BEHRENDS, et al., Rev. Mod. Phys. 34, 1 (1962). G. Rnca~, “Princeron Lectures on Group Theory and Spectroscopy,” reprinted in CERN 61-8. G. E. BAIRD AND L.C. BIEDENHARN,J.Math. Phys.6,1730 (1964). L. C. BIEDENHARN,J. Math. Phys. 4,436 (1963). M. E. ROSE, “Elementary Theory of Angular Momentum.” Wiley, New York, 1957. E. P. WIGNER, “Group Theory. ” Academic Press, New York, 1958. J. J. DE SWART, Rev. Mod. Phys. 36, 916 (1963). S. GA~IOROWICZ, “A Simple Graphical Method in the Analysis of SU(3),” ANL-6729 (June 1963). U. FANO AND G. Racl~, “Irreducible Tensorial Sets.” Academic Press, New York, 1959.