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REFERENCEIC/67/50
INTERNATIONAL ATOMIC ENERGY AGENCY
INTERNATIONAL CENTRE FOR THEORETICAL
PHYSICS
CLEBSCH - GORDAN COEFFICIENTSFOR THE LORENTZ GROUP - I
PRINCIPAL SERIES
R. L. ANDERSON
R. RACZKA
M. A. RASHIDAND
P. WINTERNITZ
1967PIAZZA OBERDAN
TRIESTE
10/67/50
CENTRE FOR THEORETICAL PHYSICS
TRIESTE
CLEBSCH-GOHDAET COEFFICIENTS FOR TEE LOHESTTZ GROUP
I - PRINCIPAL SERIES
R.L.R.
M.A
AndersonRaczka
, Rashid
and
P e Winternitz
ERRATA
Several lines were unfortunately left out when formulae were
written.
Please replaoe formula (28) by:
5« A,-rtA,ITU>
ddf
-1-
A *
(28)
Please replaoe formula (35)
r^,
The CleTjaoh-Oordan ooeffioient in (37) is
The 9-J symbol in (40) should reads
k - Tt 2. i
and the two subsequent formulae are;
- {
and
IC/67/5O
INTERNATIONAL ATOMIC ENERGX AGENCY
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
CLEBSCH-GOKDAKT COEFFICIENTS FOR THE LORENTZ GROUP
I - PRINCIPAL SERIES
R . L . Ander son *
R. Raczka **
M.A. Rashid ***
and
P. Winternitz *** #
TRIESTE
July 1967
* National Soience Foundation Post-doctoral Fellow,
** On leave of absence from the Insti tute of Buclear
Researoh, Warsaw, Poland.
*** On leave of ataenoe from the Pakistan Atonic Energy
Coniaission, Lahore, Pakistan.
**** On leave of absence from Joint Institute of Nuclear
Research, Dubna, U.S.S.R.jand from Institute of
Nuclear Research, Prague, Czechoslovakia.
ABSTRACT
Explicit expression for the Ciebach-Gordajra coefficient for
the -unitary representations of the principal series for the SL(2,C}
group is obtained. Their orthogonality and. symmetry properties
are disoussed. Special oases of physical interest are dealt with
in detail.
-1-
QLEBSCH-GOEDAI COEFFICIENTS FOS THE LOBEHTZ GROUP
I -PRINCIPAL SERI3S
IHTEODUGOJIOir
The knowledge of the SL(2,C) Clebsch-Gor&an coefficients is of
fundamental importance in such areas of elementary particle physics
as "the analysis of angular distributions in relativistic scatteringl) 2)
theory , the theory of relativistic wave equations J and the problem
of the construction of invariant vertex functions for finite or in-
finite partiole multiplets .
The problem of reducing the 'tensor products of two irreducible
unitary representations has been completely solved by ETADIARK ' in a
series of papers.' However, the calculation of the explicit form
of the Clebsoh-Gordan coefficients has "been carried out only in
oertain special oases by DOLGI2TOV et al.^\ BISIACCHI and FRONSDAL6)
and BA1OERG7'.
We present here the solution to the problem of the C.G.
coefficients for arbitrary unitary representations of the principal
series. The oontents of the paper are given in the table of
contents.
CONTENTS
INTRODUCTION
1 . INTEGRAL REPRESENTATION FOR THE CLE23CH-G0RDAN COEFFICIENT
IN TERMS OF SL(2,C) TRANSFORMATION MATRICES.'
Si Derivation.
Orthogonality and completeness relations.
Explioit expression for-the SL(2,C) transformation matrices.
2 . THE CLEBSCH-GORDAN COEFFICIENT.
General express ion .
£ 2 S impl i f ica t ion when V , V-i » % s a t i s f y a t r i a n g u l a r r e l a t i o n ,
3 . CLEBSCH-GORDAtf COEFFICIENT FOR THE COUPLING OF DEGENERATE
REPRESENTATIONS.
4. CALCULATING NUMBERS.
5. RECURSION FORMULAE AND ANALYTIC CONTINUATION.
6. SYMMETRY PROPERTIES.
7. CONCLUSIONS.
-3-
1. INTEGRAL REPRESENTATION FOR TEE CLEBSCH-GORDM
COEFFICIENT Iff TEEKS OP SL(2,C) TRANSFORMATION MATRICES
Derivation
In the case of a finite or compaot group G, knowing the
matrix elements D (g) of an irreduoible unitary representation D"\
we oan construct the operator• a \
(n fixed)
T. (1)
which has the property
ZI (2)
Above M" = d/V\ where d = dimension of D and
h m "volmce" of Gt A denotes the set of invariant numbers which
characterize the given irreducible representation and m(n) represents
•the set of non-invariant numbers which determine the basis vectors
j A ; m]]> , dy.(g) is an invariant measure on the group space and
T->, is a -unitary, generally reducible, representation in some
Hilbert space H. Owing to the property (2) the set of unnormalized
veotors
.x ,AX 6 H n fixed (3)
spans the carrier space H of the irreducible unitary represent-
ation D A .
In the derivation of eq. (2) we have only used the
invarianoe of the measure dju-(g) on Q and the group properties
of the matrices JD (g) . The above construction of the basis
veotorfj (3) can therefore be extended also for every locally
oompaot group for which the invariant measure dul g) and the
matrices J 2)(g)i are known.
In the case of the Lorentz group, the explicit form of
the matrices {l>V^(g)] has been calculated recently "by various
authors"'. Thus we can construct explicitly the operators (l) and
the basis vectors (3) and utilize them for obtaining the Clebsch-
Oordan coefficients.
In fact, let us oonsider the tensor product of two
irreducible representations D ' ' 1 and D *•**- (V. integers or half-
integers - »<&£ n. - oo, i = 1»2) which are realized in Hilbert
spaoes H <1"i , H^^1 , and spanned Taj the seti of oanonical basis
veotors^Vj ^ ^ Mi yl and jjv2 j?2 J2 M ? / ^ respectively.
Due to ITAIMAHK's r e s u l t s we a l ready know tha t the decomposition
of the t enso r product B " ! J I ( ^ ) D i J t has the fol lowing form:
CO
if
where the summation extends over all V s suoh that
v + v- + Vp » integer (5)
In the t enso r product spaoe H » H ^ ' ' © H 1 2 ve can
oonstruot two s e t s of orthogonal b a s i s v e c t o r s . The f i r s t one
oons i s t s of the Kroneoker product of the o r i g i n a l b a s i s veotors
(6)
while the second one contains the basis veotors
v y J M J • v 1 f ^ j>4 > ( 7 )
- 5 -
which span an irreducible space H J5 contained in the direct integral
of the Hilbert spaces corresponding to the decomposition (4).
In what follows we shall omit indices v± P^^fz. in "fc3ie
ket vector (7).
The normalization property of the basis vectors (6) and
(7) are
Mlf
Wand
O ? JM 1 V'?'JV> - -^—4 ^ ^ Wv (9)'W r r
In ITaimark• s notations, the veotor v p J M^> corresponds
to the function fN^( E ) "belonging to the Eiltert space H(C) of
functions f(2 ) with the domain C being the complex plane (Ref. 8)
p. 162).
The connection between the basis vectors (6) and (7) oan be
aohieved with the help of the operators (l). We have
JM> -
(10)
In this formula for the state %f^ \^-^I \ vr &.J2_
we may take any veotor (6) from the tensor produot space
H 1"1 @> H r ' z . IT i s some normalization coefficient . The operator
TQ aots on the basis veotor (6) as
T ft
(11)
Using invariance of the measure dju(g) and group properties of thematrices D " (g) we can easily check that the basis vectors (10)have oorreotttitsnsforiaatiQn proper t ies .
The Clebsch-Gor&an coefficients are the matrix elements
of the t rans i t ion matrix between the basis vectors (6) and (7)#
Ut i l iz ing (10) and ( l l ) we get
M
(12)
In order to find the normalization coefficient
we oaloulate the square of a basis veotor
(10). From the group properties of the matrioes Dvf* (g) and the
normalization (9.) we find that
-7-
(13)
Note that the normalization (S) i s chosen so as to canoel the faotor
AV/> - • — — T (U)
4f
•which appears in the integration defining normalization of the
V^ (g) functions.JM , J ' M1
The group element g oan "be represented in the form
g- u , ( > t ^ ; O ) g(a) Wt (?, ,6,fa ) (15)
where C1 and U^ "belong to SU(2) and g(a) is an element of the one-
parameter non-oompaot subgroup of SL(2,C). The decomposition (15)
induces the following for the D ^(g) matrices:
v ? , t min£J,J')
(16)
Referred to the parametrization (15) the invariant measure on
SL(2,C) has the form
-8-
- d(oos if/ ) dY[ d j , d ( o o s G ) d £ s inh 2 a d a ( l ? )
Using formulas (12), (13), (16) and (17), carrying out the elementary
integrations over the variables V» "'7 , V' , ^ , 9_,s we arrive at
\ V 1?1 J 1 M 1 J K J>
32TT
(2J + 1)
x f L- Vf (a) ]>V^ (a) L^ft. (a) sinh2 a d
(18)
The last integration will be carried out in Seo. 2
Orthogonality and completeness relations
The Clebsch-Gordan coefficients as defined hy (12) figure
in the decompoeition
cO
(19)
•where the oanonical "bases are normalized according to (8) and (9).
Calculating the norms of "both sides in (19) we obtain
the orthgonality relation
(20)
Using (20) it is easy to oheck that the formula inverse
to (21) is
>*
(21)
Combining (19) and (21), we obtain
JM
' M' >
which is completeness.
-10-
A3) Explicit expression for the SL(2,C) transformation matrioes
In the following we shall need the explicit form of the
transformation matrioes D ?(a). For our purpose the most convenient9d")
one is essentially that of DUC and HIBU ' except that we haveo \
adopted the canonical basis as defined "by 1TAIMAEK ^
(23)
where
O
and
v f _ _
The ohoice of the phase CX T is such that the veotors
7 j> J H y form a oanonical basis in the sense of EAJMAEK ' i.e.,
the operation of all generators on jv o J M > is completely prescribed,
-11-
2 . THE CLEBSCH-GOEDM COEFFICIENT
§1) General expression^
In th i s section we carry out the integrat ion appearing
in the formula (18) for the C.G. coeff ic ients . Substi tuting (23)—2a
in (18) and introducing a new variable x = 1 - e we reduce the
integral in (l8) to a sum of terms of the type
,_4 d
j X) x-
To perform this integration, we develop each of the
hypergeonetrio functions into a power series. These series converge
uniformly for x < "1 and we can perform the integration term by
term from 0 to 1 - £ and take the limit € 0 . Sinoe this limit
exists, Abel's theorem on the value of a function on the boundary
of its region of convergence ensures that the obtained value is the
correct answer. Performing the integration (see e.g. Eef, 13), p. 284)
we obtain
, M,, TJ.M,. I JM >< a/*/ , Tj K I r W
I. OIV^
r'- »fc),^
(25)
'The sums with respeot to A's, 4fs and d'a are finite,
their limits being
t a n
min(J± f J^) i - 1,2
(0, - X - v) '<$ d « min(J -A , J - v) e t c
The sums over n , n. , n2 are from 0 to <=o and are convergent.
We have used the notation
( a ) -v 'nP(a+n)
Let us remark that the triple infinite sumn n, rip
figuring in (25) and all subsequent expressions can be re- .
arranged in such a way.that two of the sums can be expressed as
generalized hypergeometric functions of unit argument. The
Clebsoh-Gordan coefficient oan thus be expressed using a single
infinite and many finite sums over a product of P functions,
a terminating F, function and an J&V, function. Sinoe this does
not simplify oaloulations, we shall not give the explicit
-13-
expression here.
The number of fini-fce sums in formula (25) can be
reduced to at most four by a suitable choice of the superfluous
parameters J! , Mf. Using the symmetry relations (discussed in
Sec. 6) we oan always arrange the coefficient to be such that
Choosing
(26)
Jl " V l ' J2 " V2
we eliminate the sums over d' and d' and can perform the d- , doL t d x c.
summations with the help of
Thus ve obtain
-14-
0+:/
J
. fi.,:. ,
where
(T-V)l (J^M!(v+3v)J(y-^» (29)
pand B -was defined in eq. (24).
JJ» X dd»
In order to fix the C.G. coefficient completely it is
still necessary to ohoose the value of J' and the phase^for.
instance lay taking (see eq.(26))
J' - va - v 2 (26-0
and defining the coefficient
1 P i V l V l •• V 2 f 2 V2 ^ V2)| V f (Vl -
to be real and pos i t ive .
- 1 5 -
2) Simplification when v , v^ , "tf 2 satisfy a triangular
relation
If the invariant parameters "v , -v. , Vg satisfy the
relation
Vl "
we oan put (of. 26'•)
J' - v (26'»)
in eq,, (28). Thus achieving a farther simplification:
2.
(30)
The d , d' summations have completely disappeared
-16-
3. CLEBSCE-GOBDAN COEFFICIENT FOR THE COUPLING OF
REPRESENTATIONS
If we oonsider the case V -
reduces to
0, eq., (30)
J,, M1 , 0 p z J^ M 0 J3 J M
*
\ I?-
f ( H '
p , 0 j ^ 00 0 j) 0 0 )
re-f ( j - iPi/fc+ WQ v!., ;
In this case the "normalization Clehsoh-Gordan
coefficient" is specially simple and can "be oaloulated explicitly.
In fact, from eq. (31) "by manipulating the triple sum
f, oo 9 ofj,oo j o f
(32)
Inserting (32) into (3l) we obtain
- 1 7 -
In the evaluation (32) we have assumed positive phase
for the ooefficient.
From the presenoe of <(^ 0 , • Jg 0 j J o)> in the above
formula (33) i t is olear that, the only non-zero Clebsch-Gordan
ooeffioients are the ones with
J- + J« + J •» even integer
in this special oase.
So far we have made only a partial comparison of (33)
with the results of DOLGIUOV^'and DOMOKOS K This comparison
shows that our expressions (31) - (33) agree with their results.
-18-
4. CALCULATING LUMBERS
For ooncrete calculations of specifio Clebsoh-Cordan
coefficients it may prove advantageous to use a slightly different
approaoh.
So far we have made use of D ? matrices for the SL(2,C)
group expressed in terms of hypergeoir.etric functions. However,
they can also be expressed in terms of•elementary functions. Indeed,
using an integral representation for the hypergeometric function,
we have
X
** dd'
\ e •* /
^ . . — — — — — ,
Putting x » w ^ — and expanding the positive integer powers
of (l - e x ) and ( x - ea) into (finite) binomial series, we
obtain (this is a slight modification of the results in Ref. 9c))
)
(34)
where
- (-0 '
-19-
Using this D-matrix to calculate the Clebsch-Gor&an
coefficient, we get immediately
M2 I V f J M > < Vi ?1 *1 ' ^2 ?z J2 H |V f ™2 I V f J M > < Vi ?1 *1 ' ^2 ?z J2 H2
j J M >
Z <3>
(35)
The integrals in (35) are, in general, divergent and in
ezplioit calculations, they must be combined in such a way that the
divergences cancel.
Using (35) it is possible to calculate several simplest
Clebsch-Gordan coefficients for eaoh set of invariant parameters
v and p . Afterwards, using the obtained values and the recursion
relations of Sec. 5 it will be possible to calculate arbitrary
Clebsch-Gordan coefficients.
Let us demonstrate the application by two examples;
a) < Of, oo, *U°<>I Of06}
(36)
agreeing vi"ch (32) obtained previously by a different method.
b) A less trivial ClebGoh-Gordan coefficient which, can be computed
performing tedious ,though elementary, calculations is
4 * ,
- i joo >
or
R 4 4 t 4 R, 4 - 4 | 4 j > o o>
. f {H-p,l3f i+tf
(37)
In both cases we have ohosen the CW3 coefficients to be real
and positive.
Incidentally, the coupling of representation V = •§- , 0 = 0
is of special interest, since this and the representation \> = 0 ,
y = i/2 belonging to the supplementary series, are the only unitary
irreduoible repreaentations of the SL(2,C) group, the basis functions8
of which satisfy invariant wave equations J (these are the Majoranarepresentations),
-21-
5. RECURSION FORMULAE A M ANALYTIC COHTIMJATIOtf
Recursion relations for the SL(2,C) Clebsch-C-ordan
coaffioients can "be obtained from the knowledge of the action of
the generators H. and F. ^ on an arbitrary canonical basis
funotion. The procedure is to apply H. and F. separately to
eq. (21), expand the l.h.s. of the result by using ,(2l) again, and
then equate the coefficients of the orthonormal basis functions
| v £>„ J, M J v 2 p Jo p / * There its an important well-known
simplification which occurs, namely that the application of the
generators H. of the oompact subgroup SU(2) leads to recursion
relation*whose solution up to a multiplicative factor is the SU(2)
Clebeoh-Gordan coefficient. With the usual choice of phases for
the SU(2) Clebsch-Gordan coefficients we can therefore • faotorize
the SL(2,C) coefficients as follows:
X ^PiJ1 '^2 Pz J2
(33)
Applying the above procedure, we arrive at the following
recursion relations for X (v J-, f -tfp p J£ t V P J ) :
-22-
&o<!CD
.o r>-'r
•3fi v - ^
/v?
r.
r<
*4
M
TT -H
7*
H
i s .
A.1
r-
J4
— >H
•f
»•
V
V
t* •
CJ.
M
c
-r
zl
A .
oo
4M
oHj
oH
'H(aCD
" • *
-4-
73"
M
*zf-
V
- 2 3 -
73-
MJX
V
The above are a system of three recursion relations
satisfied "by the X functions. In order to use them as adjuncts
to the analytio approach, we note that we can eliminate the X*
functions with arguments containing J ' + 1 and CP ' + 1. This
results in a recursion relation which allows us to increase the
value of the argument J. Similarly we can obtain two other
recursion relations whioh enable increases in J ' or J*1 '. Every
time the relation involves at most 5 terms. Suppressing the
arguments v. p . , one of these looks like
J+l) + 0( 2X*(J 1J 2J) + 0(3 X ^ - l , J2, J)
+ K 4 X*(j, , J2-l , J) + tX5 X*(j^ , J2 , j-l) -.0
For the oases where v , v^ , v 2 form a triangle (this includes the
degenerate case), we evidently require only one starting value
namely the C.G. coefficient with J, => v,, J« = v 2, J • V,to generate
all the other C.G. coefficients. Thus the recursion relations do^
in these caseSj define the problem up to a "normalization coefficient"
which represents a special C.G. coefficient. This starting C.G.
coefficient can be chosen to be real and positive.
However, we can verify that, as expected, in other
cases also we require just one "normalization coefficient" to fix
all for fixed v , "V,, v 2 . Of course there is some arbitrariness
here in its choice and no canonical choice seems to be evident
exoept for two of the J values selected as the minimum. This
cannot be done in an arbitrary fashion. Indeed if
then ve can take the normalization coefficient to be the one
corresponding to
-24-
' 1 . 7 1
o r Vl » J
The above difficulty i-ras reflected in our inability
to do two of the d summations in the formula (28) for the C.G.
coefficient whenever v , v.. , v ? did not form a triangle.
H"ow "we discuss the important question of whether the
solution to our problem can be obtained "by "analytically continuing"
the corresponding C.G. coefficient for the finite-dimensional
(non—unitary) representations. Indeed it is well known that for
the finite-dimensional cases v,
fc + «
( 2 J , + l ) ( f
(-1) Z ((2J1
J M
n
(40)
•where | r is a 9-J symbol. Here the conventions in the book15)
by De-SEALIT ' have been employed. The connection between v , Pand ^ , n is
fe-n
ioj » ( + n) +1
-25-
The recursion relations given above are indeed "analytic
continuation" of the ones for the finita-dimensional case. They can
be obtained by replacing the invariant numbers P< , n characterizing
the finite-dimonsional irreducible representations by
z 12 -2-- ~z i® i
respectively. Thus we can define a new 9J-syffibol with complex
parameters whioh will rigorously satisfy the recursion relations.
One can, for example, express it in terms of analytically continued
Racah coefficients with complex parameters
Y
(41)
In the above equation, a's and b's are complex while'J's
are non-negative integers or half-integers. Thus ~Y summation is
indeed finite ', In terms of this 9-J symbol we oan easily express
the C.G. coefficient up to a normalization factor. This factor,
as emphasized before,, is a special C.G. coefficient and oan be
computed from the formulas in (28) or (35).
-26-
6. SYMMETRY PROPERTIES
Formula (18) determines the product of two Clebsoh-Gordan
coefficients. An arbitrary Clebsch-Gordan coefficient can "be obxained
"by specifying the values of the redundant parameters X'l-I etc. in a
convenient manner, calculating the square modulus of this "normalization
coefficient" from (18) and defining the overall phase for each set of
invariant parameters "V p.. , v ? P ? , v p . Dividing (18) by this
chosen coefficient we obtain a formula for the general coefficient
< Y. f, J1 M-, , Y2 pz J2 Mg | V p J M y , So far, except for the
cases like v.= v. =» v 2 a 0 » V, =• v^ = %- , v = 0 we have not been
able to reduce, in general, the normalization coefficient to a single
term. Thus we cannot establish the phase factors in the symmetry
relations and we shall only consider the relations between the moduli
of various Clebsch-Gordan coefficients.
The simplest symmetry relations of these coefficients
can be obtained directly from the properties of the D ' matrioes
(eq.3. (2 24))» Up to phase factors connected with the quantities
Ql Jwhioh are not important for the present purposes, these relations
are given in Ref.
It is easy to establish the following:
v f J M>
J-B ,
2 ?2 J^'l-V-fJ
2)^f J K>
(42)
-27-
Thase are of oourse "the dost o~bvious independent
relations. Applying some of than .repeatedly, we can also obtain
relations of the type
V2 ?2
-V -
It is clear that we have only found the simplest symmetry
relations and that the symmetry properties of 8L(2,C) Clebsch-Gordan
coefficients will ba muoh richer. Symmetries similar to those dis-
oovered by HEGGE ; for the SU"(2) group should also exist here, but
so far no attempt to disoover them has been made.
7. CONCLUSIONS
We have oonsidered the problem of the Clebsoh-Gordan
coefficients for the SL(2tC) group and have obtained general
expression for these coefficients for the principal series re-
presentations. The merit of our derivation is its rigour. We have
been able to achieve it from the knowledge of the transformation
matrices. Previous attempts by Dolginov et_al. for calculating
speoial oases were based on the use of recursion relations and the
observation that Pano functions satisfied the same recursion relations.
Effectively this was an analytic continuation approach. We have also
arrived at ooncrete results on analytio continuation from the finite-
dimensional case .
Our methods can "be applied for the calculation ofClebsch-
Gordan coefficients in all cases where multiplicity does not appear.
In particular, one might apply them for the couplings between principal
and supplementary, supplementary and supplementary, or unitary and
finite-dimensional non-unitary representations. This programme will
be the subjeot of future publications in which we shall also return
in detail to the problems of analytic continuation.
-28-
ACKNOWLEDGMENTS
We are indebted to Professors Abdus Salam and P. Budini
and to the IAEA for hospitality extended to us at the International
Centre for Theoretical Physios, Trieste. We are grateful to Professor
Abdus Salain also for suggesting the problem and for continued
encouragement. We are grateful /or very profitable discussions with
Dr. John Strathdee.
-29-
H3EEHE1TCES
1 ) a) M. TOLLER, C3RN preprint Th-750 (1967). This
oontains an extensive l i s t of references and
summarizes author's "work on the subject.
b) J . F . BOYCE, R. DELBOURGO, AHDUS SALAM and J . STRATHBEE,
ICTP, Tr ies te , preprint IC/61/9 (1967).
o) 1ST. Ya. VILEflKnr and Ta. A, SMOROXCTSKT, Zb. Sksperim.
i Teor. F i z . 35,, 794 (1958) (Soviet Phys., - JBTP
1£, 1209 (1964)).
2) See e . g . , C . FROffSDAL, UCLA, prepr in t -Oct . 1966.
3) See e . g . , C. FEOHSDAL and R. WHITS, Phys. Sev. 1^1, 1287 (1966).
4) M.A. HAIMAEK, Trudy Moakov. Mat. Obsc. _8, 121 (1959);
- % 237 (I960)} JLO, 181 (1961)5 English t r ans l a t i on :
Amer. Math. Soo. Trans l , Series 2, Vol. 36. (1964)
pages 101-229.
5) A.Z. DOLGIJOV and IJST. TOPTYGIIT, Zh. Eksperim. i Teor. ? i a .
35., 794 (1958); Soviet Phys. - JETP Bt 550 (1959);
Ibid, Zh. Ekeperim. i Teor. F i z . _3_7 1441 (l959)j Soviet Phys.
- CG3TF 10, 1022 (l96O)j
A.Z. DOLGINOV and AJJ. MOSKALEV, Zh. Eksperim. i Theor. F i z .
3X, 1697 (1959); Soviet Phys. - JEPT 10, 1202 ( I960) .
6) G. BISIACCEI and C. HL02JSDAL, Uuovo Cimento 41, 35 (1965).
7) P.G. BA1BEEG, Clarendon Lab ., Oxford, preprint 196 (1966).
8) M.A. HAIMAEK, "Linear Representations of the Lorentz group",
Pergamon Press , London (1964).
9) a) LJD. E-SKIU, I zv . Vysshikh Uohebnykh Zavedeni, Matematika
§j (25) , 179-184 (1961).
b) ' S. STROM, Arkiv fur Fysik 2£, 467 (1965).
o) A. SCIARRIKO and M. TOLLER, Hota In t ema no . 108
Universi ta d i Roma (1966).
d) DAO WAFG DUC and UGTHW VAU HIEU, Lubna preprint - JIITR
P-2777.
-30-
Ibid: Annals Ins t i tu t Henri Poincare S_, IT (19°7).
e) I.A. VEPJ3IEV, L.A. DADASHEV, Ins t i tu te cf Theoretical
and Experimental Physics, preprint no. 476 (i960).
10) See ;e.g. ;K. KJTOPP, "Theory and application of infini te
series(t2nd English edition, Blackio, London (1564).
11) I . S . GRADSHTEYU and I J i . HTZIIIK, "Tables of in-ce.graio,
series and products", Academic Press, JTew York.
12) G. DOKOKOS, University of California, Berkeley, preprint l /o? .
13) H. and P. are the generators of the SL(2,C) group. Theirmatrix elements are expl ici t ly knovn. v;o are u t i l i z ing
A. and C. as given "oy 1TABUHX (Eef. S) above).J <3
14) R. DSLBOUHGO, J . STRATHD33 and AHDUS SALAE, ICTP, TTieste
prepr int IQ/SI/21, sea also Ref. .7).
This expression i s equivalent to the one given in
JJSI. GSL'PASD, E.A.KI1IL0S and Z.Ya. SHAPIRO, "Bapresanta-Sio-.-.s
of the rotation and Lorenxg ;_rroupg and their ar;plic:,'c.icns"T
Pergamon Press, Oxford (1963).
15) A. de-SEALIT and I . TALKI, "Iluclear Shell Theory", Academic
Press, Hew York (1963).
16) 9—J symbol with complex parameters i s knotm as ?ar.o functions.
Their explici t expression can "be seen in H. KATSUiTOBU and
H. TAKE3E, Progr. Theoret. Phys ..(Kyoxo) 1A, 1^39 (1955).
17) T. EEGGE, JTuovo Cimento _1£, 545 (1956).
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